research article chess-like games may have no...

Hindawi Publishing CorporationGameTheoryVolume 2013 Article ID 534875 10 pageshttpdxdoiorg1011552013534875

Research ArticleChess-Like Games May Have No Uniform Nash EquilibriaEven in Mixed Strategies

Endre Boros Vladimir Gurvich and Emre Yamangil

RUTCOR Rutgers University 640 Bartholomew Road Piscataway NJ 08854-8003 USA

Correspondence should be addressed to Vladimir Gurvich gurvichrutcorrutgersedu

Received 2 February 2013 Accepted 22 April 2013

Academic Editor Walter Briec

Copyright copy 2013 Endre Boros et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Recently it was shown that Chess-like games may have no uniform (subgame perfect) Nash equilibria in pure positional strategiesMoreover Nash equilibria may fail to exist already in two-person games in which all infinite plays are equivalent and ranked asthe worst outcome by both players In this paper we extend this negative result further providing examples that are uniform Nashequilibria free even inmixed or independentlymixed strategies Additionally in case of independentlymixed strategies we considertwo different definitions for effective payoff the Markovian and the a priori realization

1 Introduction

11 Nash-Solvability in Pure and Mixed Strategies MainResults There are two very important classes of the so-called uniformlyNash-solvable positional games with perfectinformation for which a Nash equilibrium (NE) in purestationary strategies which are also independent of the initialposition exists for arbitrary payoffsThese two classes are thetwo-person zero-sum games and the 119899-person acyclic games

However when (directed) cycles are allowed and thegame is not zero sum then a positional game with perfectinformation may have no uniform NE in pure stationarystrategies This may occur already in the special case of twoplayers with all cycles equivalent and ranked as the worstoutcome by both players Such an example was recentlyconstructed in [1]

Here we strengthen this result and show that for thesame example no uniformNE exists even inmixed stationarystrategies not only in pure ones Moreover the same negativeresult holds for the so-called independently mixed strategiesIn the latter case we consider two different definitions forthe effective payoffs based on Markovian and a priorirealizations

In the rest of the introduction we give precise definitionsand explain the above result in more details

Remark 1 In contrast for the case of a fixed initial positionNash-solvability in pure positional strategies holds for thetwo-person case and remains an open problem for 119899 gt 2see [1] for more details see also [2ndash20] for different cases ofNash-solvability in pure strategies

Furthermore for a fixed initial position the solvability inmixed strategies becomes trivial due to the general result ofNash [21 22] Thus our main example shows that Nashrsquos the-orem cannot be extended for positional games to the case ofuniform equilibria It is shown for the following four types ofpositional strategies pure mixed and independently mixedwhere in the last case we consider two types of effectivepayoffs defined by Markovian and a priori realizations

12 Positional Game Structures Given a finite directed graph(digraph) 119866 = (119881 119864) in which loops and multiple arcs areallowed a vertex V isin 119881 is interpreted as a position and adirected edge (arc) 119890 = (V V1015840) isin 119864 as a move from V toV1015840 A position of outdegree 0 (one with no moves) is calledterminal Let 119881

119879= 1198861 119886

119898 be the set of all terminal

positions Let us also introduce a set of 119899 players 119868 = 1 119899and a partition 119863 119881 = 119881

1cup sdot sdot sdot cup 119881

119899cup 119881119879 assuming that

each player 119894 isin 119868 is in control of all positions in 119881119894

2 GameTheory

1

3

2

1

2

1

2

1

21198861

1198862

1198863

1199701

1199702

1198861

1198862

1198863

1198864

1198865

1198866

Figure 1 Two Chess-like game structuresG1andG

2 InG

1 there are 3 players controlling one position each (ieG

1is play-once) while in

G2there are only two players who alternate turns hence each of them controls 3 positions V

1 V3 V5are controlled by player 1 and V

2 V4 V6by

player 2 In each position Vℓ the corresponding player has only two options (119901) to proceed to V

ℓ+1and (119905) to terminate at 119886

ℓ where ℓ isin 1 2 3

(V4= V1) inG

1and ℓ isin 1 6 (V

7= V1) inG

2 To save space we show only symbols 119886

ℓ while the corresponding vertex names are omitted

An initial position V0isin 119881 may be fixed The triplet

(119866119863 V0) or pair G = (119866119863) is called a Chess-like positional

game structure (or just a game structure for short) initializedor noninitialized respectively By default we assume that it isnot initialized

Two examples of (noninitialized) game structuresG1and

G2are given in Figure 1

13 Plays Outcomes Preferences and Payoffs Given an ini-tialized positional game structure (119866119863 V

0) a play is defined

as a directed path that begins in V0and either ends in a

terminal position 119886 isin 119881119879or is infinite In this paper we

assume that all infinite plays form one outcome 119886infin

(or 119888)in addition to the standard Terminal outcomes of 119881

119879 (In [5]

this condition was referred to as AIPFOOT)A utility (or payoff) function is amapping 119906 119868times119860 rarr R

whose value 119906(119894 119886) is interpreted as a profit of the player 119894 isin119868 = 1 119899 in case of the outcome 119886 isin 119860 = 119886

1 119886

119898 119886infin

A payoff is called zero-sum ifsum119894isin119868119906(119894 119886) = 0 for every 119886 isin

119860 Two-person zero sum games are important For examplethe standard Chess and Backgammon are two-person zero-sum games in which every infinite play is a draw 119906(1 119886

infin) =

119906(2 119886infin) = 0 It is easy to realize that 119906(119894 119886

infin) = 0 can be

assumed for all players 119894 isin 119868 without any loss of generalityAnother important class of payoffs is defined by the

condition 119906(119894 119886infin) lt 119906(119894 119886) for all 119894 isin 119868 and 119886 isin 119881

119879 in other

words the infinite outcome 119886infin

is ranked as the worst one byall players Several possible motivations for this assumptionare discussed in [4 5]

A quadruple (119866119863 V0 119906) and triplet (119866119863 119906) will be

called a Chess-like game initialized and noninitializedrespectively

Remark 2 From the other side the Chess-like games can beviewed as the transition-free deterministic stochastic gameswith perfect information see for example [8ndash11]

In these games every nonterminal position is controlledby a player 119894 isin 119868 and the local reward 119903(119894 119890) is 0 for each player119894 isin 119868 and move 119890 unless 119890 = (V1015840 V) is a terminal move that isV isin 119881119879 Obviously in the considered case all infinite plays are

equivalent since the effective payoff is 0 for every such playFurthermore obviously 119886

infinis the worst outcome for a player

119894 isin 119868 if and only if 119903(119894 119890) gt 0 for every terminal move 119890If |119868| = 119899 = 119898 = |119860| = 2 then the zero-sum Chess-like

games turn into a subclass of the so-called simple stochasticgames which were introduced by Condon in [23]

14 Pure Positional Strategies Given game structure G =

(119866119863) a (pure positional) strategy 119909119894of a player 119894 isin 119868 is a

mapping 119909119894 119881119894rarr 119864119894that assigns to each position V isin 119881

119894a

move (V V1015840) from this positionThe concept of mixed strategies will be considered in

Section 110 till then only pure strategies are consideredMoreover in this paper we restrict the players to theirpositional (pure) strategies In other words the move (V V1015840)of a player 119894 isin 119868 in a position V isin 119881

119894depends only on the

position V itself not on the preceding positions or movesLet 119883

119894be the set of all strategies of a player 119894 isin 119868 and

119883 = prod119894isin119868119883119894be the direct product of these sets An element

119909 = 1199091 119909

119899 isin 119883 is called a strategy profile or situation

15 Normal Forms A positional game structure can berepresented in the normal (or strategic) form

Let us begin with the initialized case Given a gamestructure G = (119866119863 V

0) and a strategy profile 119909 isin 119883 a play

119901(119909) is uniquely defined by the following rules it begins inV0and in each position V isin 119881

119894proceeds with the arc (V V1015840)

determined by the strategy 119909119894 Obviously 119901(119909) either ends

in a terminal position 119886 isin 119881119879or 119901(119909) is infinite In the

latter case 119901(119909) is a lasso that is it consists of an initial partand a directed cycle (dicycle) repeated infinitely This holdsbecause all players are restricted to their positional strategiesIn either case an outcome 119886 = 119892(119909) isin 119860 = 119886

1 119886

119898 119886infin

GameTheory 3

1

2

3

1198921

119905

119901

119901119905

119905

119901

119901

119901

119905

1199051198861 1198862 1198863 1198861 1198863 1198863

1198862 1198862 1198863 1198863 1198863 1198863

1198861 1198862 1198861 1198861 1198861 1198861

1198862 1198862 1198862 119888 119888 119888

119906(1 1198862) gt 119906(1 1198861) gt 119906(1 1198863) gt 119906(1 119888)

119906(2 1198863) gt 119906(2 1198862) gt 119906(2 1198861) gt 119906(2 119888)

119906(3 1198861) gt 119906(3 1198863) gt 119906(3 1198862) gt 119906(3 119888)

Figure 2 The normal form 1198921of the positional game structures G

1from Figure 1 Each player has only two strategies to terminate (119905) or

proceed (119901) Hence 1198921is represented by a 2 times 2 times 2 table each entry of which contains 3 terminals corresponding to the 3 potential initial

positions V1 V2 V3of G1 The rows and columns are the strategies of the players 1 and 2 while two strategies of the player 3 are the left and

right 2 times 2 subtables The corresponding game (1198921 119906) has no uniform NE whenever a utility function 119906 119868 times 119860 rarr R satisfies the constraints

1198801specified in the figure

is assigned to each strategy profile 119909 isin 119883 Thus a game form119892V0

119883 rarr 119860 is defined It is called the normal form of theinitialized positional game structureG

If the game structureG = (119866119863) is not initialized thenwerepeat the above construction for every initial position V

0isin

119881119881119879to obtain a play119901 = 119901(119909 V

0) outcome 119886 = 119892(119909 V

0) and

mapping 119892 119883 times (119881 119881119879) rarr 119860 which is the normal form

of G in this case In general we have 119892(119909 V0) = 119892V0

(119909) Forthe (noninitialized) game structures in Figure 1 their normalforms are given in Figures 2 and 3

Given also a payoff 119906 119868 times 119860 rarr R the pairs (119892V0 119906) and(119892 119906) define the games in the normal form for the above twocases

Of course these games can be also represented by thecorresponding real-valued mappings

119891V0 119868 times 119883 997888rarr R 119891 119868 times 119883 times (

119881

119881119879

) 997888rarr R (1)

where 119891V0(119894 119909) = 119891(119894 119909 V0) = 119906(119894 119892V0

(119909)) = 119906(119894 119892(119909 V0)) for

all 119894 isin 119868 119909 isin 119883 V0isin 119881 119881

119879

Remark 3 Yet it seems convenient to separate the game from119892 and utility function 119906

By this approach 119892 ldquotakes responsibility for structuralpropertiesrdquo of the game (119892 119906) that is the properties that holdfor any 119906

16 Nash Equilibria in Pure Strategies The concept of Nashequilibria is defined standardly [21 22] for the normal formgames

First let us consider the initialized case Given 119892V0

119883 rarr 119860 and 119906 119868 times 119860 rarr R a situation 119909 isin 119883 iscalled a Nash equilibrium (NE) in the normal form game(119892V0

119906) if 119891V0(119894 119909) ge 119891V0(119894 1199091015840) for each player 119894 isin 119868 and every

strategy profile 1199091015840 isin 119883 that can differ from 119909 only in the 119894thcomponent In other words no player 119894 isin 119868 can profit bychoosing a new strategy if all opponents keep their old strate-gies

In the noninitialized case the similar property is requiredfor each V

0isin 119881119881

119879 Given a payoff119891 119868times119883times(119881119881

119879) rarr R

a strategy profile 119909 isin 119883 is called a uniform NE if 119891(119894 119909 V0) ge

119891(119894 1199091015840 V0) for each 119894 isin 119868 every 1199091015840 defined as above and for

all V0isin 119881 119881

119879 too

Remark 4 In the literature the last concept is frequentlycalled a subgame perfect NE rather than a uniform NE Thisname is justified when the digraph 119866 = (119881 119864) is acyclic andeach vertex V isin 119881 can be reached from V

0 Indeed in this

case (119866119863 V 119906) is a subgame of (119866119863 V0 119906) for each V isin 119881

However if 119866 has a dicycle then any two its vertices V1015840 andV10158401015840 can be reached one from the other that is (119866119863 V1015840 119906)is a subgame of (119866119863 V10158401015840 119906) and vice versa Thus the nameuniform (or ergodic) NE seems more accurate

17 Uniformly Best Responses Again let us start with theinitialized case Given the normal form 119891V0

119868 times 119883 rarr R

of an initialized Chess-like game a player 119894 isin 119868 and a pair ofstrategy profiles 119909 1199091015840 such that 1199091015840 may differ from 119909 only inthe 119894th component we say that 1199091015840improves 119909 (for the player119894) if 119891V0(119894 119909) lt 119891V0

(119894 1199091015840) Let us underline that the inequality

is strict Furthermore by this definition a situation 119909 isin 119883

is a NE if and only if it can be improved by no player 119894 isin 119868in other words any sequence of improvements either can beextended or terminates in an NE

Given a player 119894 isin 119868 and situation 119909 = (119909119894| 119894 isin 119868) a

strategy 119909lowast119894isin 119883119894is called a best response (BR) of 119894 in 119909 if

119891V0(119894 119909lowast) ge 119891V0

(119894 1199091015840) for any 1199091015840 where 119909lowast and 1199091015840 are both

obtained from 119909 by replacement of its 119894th component 119909119894by

119909lowast

119894and 1199091015840

119894 respectively A BR 119909

lowast

119894is not necessarily unique

but the corresponding best achievable value 119891V0(119894 119909lowast) is of

course unique Moreover somewhat surprisingly such bestvalues can be achieved by aBR119909lowast

119894simultaneously for all initial

positions V0isin 119881 119881

119879 (See eg [1 4ndash6] of course this result

is well known inmuchmore general probabilistic setting seeeg textbooks [24ndash26])

4 GameTheory

119905119905119905 119905119905119901 119905119901119905 119905119901119901 119901119905119905 119901119905119901 119901119901119905 119901119901119901

119905119905119905

119905119905119901

119905119901119905

119905119901119901

119901119905119905

119901119905119901

119901119901119905

119901119901119901

1199001 1198866 gt 1198865 gt 1198862 gt 1198861 gt 1198863 gt 1198864 gt 119888 1199002 1198863 gt 1198862 gt 1198866 gt 1198864 gt 1198865 gt 119888 1198866 gt 1198861 gt 119888

119886111988621198863119886411988651198866 119886111988621198863119886411988651198861 119886111988621198863119886511988651198866 119886111988621198863119886511988651198861 119886111988631198863119886411988651198866 119886111988631198863119886411988651198861 119886111988631198863119886511988651198866 119886111988631198863119886511988651198861

119886111988621198863119886411988661198866 119886111988621198863119886411988611198861 119886111988621198863119886611988661198866 119886111988621198863119886111988611198861 119886111988631198863119886411988661198866 119886111988631198863119886411988611198861 119886111988631198863119886611988661198866 119886111988631198863119886111988611198861

119886111988621198864119886411988651198866 119886111988621198864119886411988651198861 119886111988621198865119886511988651198866 119886111988621198865119886511988651198861 119886111988641198864119886411988651198866 119886111988641198864119886411988651198861 119886111988651198865119886511988651198866 119886111988651198865119886511988651198861

119886111988621198864119886411988661198866 119886111988621198864119886411988611198861 119886111988621198866119886611988661198866 119886111988621198861119886111988611198861 119886111988641198864119886411988661198866 119886111988641198864119886411988611198861 119886111988661198866119886611988661198866 119886111988611198861119886111988611198861

119886211988621198863119886411988651198866 119886211988621198863119886411988651198862 119886211988621198863119886511988651198866 119886211988621198863119886511988651198862 119886311988631198863119886411988651198866 119886311988631198863119886411988651198863 119886311988631198863119886511988651198866 119886311988631198863119886511988651198863

119886211988621198863119886411988661198866 119886211988621198863119886411988621198862 119886211988621198863119886611988661198866 119886211988621198863119886211988621198862 119886311988631198863119886411988661198866 119886311988631198863119886411988631198863 119886311988631198863119886611988661198866 119886311988631198863119886311988631198863

119886211988621198864119886411988651198866 119886211988621198864119886411988651198862 119886211988621198865119886511988651198866 119886211988621198865119886511988651198862 119886411988641198864119886411988651198866 119886411988641198864119886411988651198864 119886511988651198865119886511988651198866 119886511988651198865119886511988651198865

119886211988621198864119886411988661198866 119886211988621198864119886411988621198862 119886211988621198866119886611988661198866 119886211988621198862119886211988621198862 119886411988641198864119886411988661198866 119886411988641198864119886411988641198864 119886611988661198866119886611988661198866 119888 119888 119888 119888 119888 119888

1198922

Figure 3 The normal form 1198922of the positional game structuresG

2from Figure 1 There are two players controlling 3 positions each Again

in every position there are only two options to terminate (119905) or proceed (119901) Hence in G2 each player has 8 strategies which are naturally

coded by the 3-letter words in the alphabet 119905 119901 Respectively 1198922is represented by the 8 times 8 table each entry of which contains 6 terminals

corresponding to the 6 (nonterminal) potential initial positions V1 V

6of G2 Again players 1 and 2 control the rows and columns

respectively The corresponding game (1198921 119906) has no uniform NE whenever a utility function 119906 119868 times 119860 rarr R satisfies the constraints 119880

2

specified under the table Indeed a (unique) uniformly best response of the player 1 (resp 2) to each strategy of 2 (resp 1) is shown by thewhite discs (resp black squares) Since the obtained two sets are disjoint no uniform NE exists in (119892

1 119906)

Theorem 5 Let 119891 119868 times 119883 times (119881 119881119879) rarr R be the normal

form of a (noninitialized) Chess-like game (119866119863 119906) Given aplayer 119894 isin 119868 and a situation 119909 isin 119883 there is a (pure positional)strategy 119909lowast

119894isin 119883119894which is a BR of 119894 in 119909 for all initial positions

V0isin 119881 119881

119879simultaneously

We will call such a strategy 119909lowast119894a uniformly BR of the

player 119894 in the situation 119909 Obviously the nonstrict inequality119891V(119894 119909) le 119891V(119894 119909

lowast) holds for each position V isin 119881 We will

say that 119909lowast119894improves 119909 if this inequality is strict 119891V0(119894 119909) lt

119891V0(119894 119909lowast) for at least one V

0isin 119881 This statement will

serve as the definition of a uniform improvement for thenoninitialized case Let us remark that by this definition asituation 119909 isin 119883 is a uniform NE if and only if 119909 can beuniformly improved by no player 119894 isin 119868 in other words anysequence of uniform improvements either can be extended orterminates in a uniform NE

For completeness let us repeat here the simple proof ofTheorem 5 suggested in [1]

Given a noninitialized Chess-like game G = (119866119863 119906) aplayer 119894 isin 119868 and a strategy profile 119909 isin 119883 in every positionV isin 119881 (119881

119894cup 119881119879) let us fix a move (V V1015840) in accordance with

119909 and delete all other moves Then let us order 119860 accordingto the preference 119906

119894= 119906(119894 lowast) Let 1198861 isin 119860 be a best outcome

(Note that theremight be several such outcomes and also that1198861= 119888 might hold) Let 1198811 denote the set of positions from

which player 119894 can reach 1198861 (in particular 1198861 isin 1198811) Let us

fix corresponding moves in 1198811 cap 119881119894 Obviously there is no

move to 1198861 from 119881 1198811 Moreover if 1198861 = 119888 then player 119894

cannot reach a dicycle beginning from 119881 1198811 in particular

the induced digraph 1198661= 119866[119881 119881

1] contains no dicycle

Then let us consider an outcome 1198862 that is the best for119894 in 119860 except maybe 1198861 and repeat the same arguments asabove for 119866

1and 1198862 and so forth This procedure will result

in a uniformly BR 119909lowast119894of 119894 in 119909 since the chosen moves of 119894 are

optimal independently of V0

18 Two Open Problems Related to Nash-Solvability of Initial-ized Chess-Like Game Structures Given an initialized gamestructureG = (119866119863 V

0) it is an open questionwhether anNE

(in pure positional strategies) exists for every utility function119906 In [4] the problemwas raised and solved in the affirmativefor two special cases |119868| le 2 or |119860| le 3 The last result wasstrengthened to |119860| le 4 in [7] More details can be found in[1] and in the last section of [6]

In general the above problem is still open even if weassume that 119888 is the worst outcome for all players

Yet if we additionally assume that G is play-once (ie|119881119894| = 1 for each 119894 isin 119868) then the answer is positive [4]

However in the next subsection we will show that it becomesnegative if we ask for the existence of a uniform NE ratherthan an initialized one

19 Chess-Like Games with a Unique Dicycle and withoutUniform Nash Equilibria in Pure Positional Strategies Letus consider two noninitialized Chess-like positional gamestructures G

1and G

2given in Figure 1 For 119895 = 1 2 the

corresponding digraph 119866119895= (119881119895 119864119895) consists of a unique

dicycle119862119895of length 3119895 and amatching connecting each vertex

V119895

ℓof 119862119895to a terminal 119886119895

ℓ where ℓ = 1 3119895 and 119895 =

1 2 The digraph 1198662is bipartite respectively G

2is a two-

person game structures in which two players take turns in

GameTheory 5

other words players 1 and 2 control positions V1 V3 V5and

V2 V4 V6 respectively In contrast G

1is a play-once three-

person game structure that is each player controls a uniqueposition In every nonterminal position V

119895

ℓthere are only two

moves one of them (119905) immediately terminates in 119886119895ℓ while

the other one (119901) proceeds to V119895ℓ+1

by convention we assume3119895 + 1 = 1

Remark 6 In Figure 1 the symbols 119886119895

ℓfor the terminal

positions are shown but V119895ℓfor the corresponding positions

of the dicycle are skipped moreover in Figures 1ndash3 we omitthe superscript 119895 in 119886119895

ℓ for simplicity and to save space

Thus in G1each player has two strategies coded by the

letters 119905 and 119901 while inG2each player has 8 strategies coded

by the 3-letter words in the alphabet 119905 119901 For example thestrategy (119905119901119905) of player 2 inG

2requires to proceed to V2

5from

V24and to terminate in 1198862

2from V2

2and in 1198862

6from V2

6

The corresponding normal game forms 1198921and 119892

2of size

2 times 2 times 2 and 8 times 8 are shown in Figures 2 and 3 respectivelySince both game structures are noninitialized each situationis a set of 2 and 6 terminals respectively These terminalscorrespond to the nonterminal positions of G

1and G

2 each

of which can serve as an initial positionA uniform NE free example for G

1was suggested in

[4] see also [1 8] Let us consider a family 1198801of the utility

functions defined by the following constraints

119906 (1 1198862) gt 119906 (1 119886

1) gt 119906 (1 119886

3) gt 119906 (1 119888)

119906 (2 1198863) gt 119906 (2 119886

2) gt 119906 (2 119886

1) gt 119906 (2 119888)

119906 (3 1198861) gt 119906 (3 119886

3) gt 119906 (3 119886

2) gt 119906 (3 119888)

(2)

In other words for each player 119894 isin 119868 = 1 2 3 toterminate is an average outcome it is better (worse) whenthe next (previous) player terminates finally if nobody doesthen the dicycle 119888 appears which is the worst outcome for allThe considered game has an improvement cycle of length 6which is shown in Figure 2 Indeed let player 1 terminatesat 1198861 while 2 and 3 proceed The corresponding situation

(1198861 1198861 1198861) can be improved by 2 to (119886

1 1198862 1198861) which in its

turn can be improved by 1 to (1198862 1198862 1198862) Repeating the similar

procedure two times more we obtain the improvement cycleshown in Figure 2

There are two more situations which result in (1198861 1198862 1198863)

and (119888 119888 119888) They appear when all three players terminate orproceed simultaneously Yet none of these two situations is anNE either Moreover each of them can be improved by everyplayer 119894 isin 119868 = 1 2 3

Thus the following negative result holds which we recallwithout proof from [4] see also [1]

Theorem 7 Game (G1 119906) has no uniform NE in pure strate-

gies whenever 119906 isin 1198801

We note that each player has positive payoffs This iswithout loss of generality as we can shift the payoffs by apositive constant without changing the game

A similar two-person uniform NE-free example wassuggested in [1] for G

2 Let us consider a family 119880

2of the

utility functions defined by the following constraints

119906 (1 1198866) gt 119906 (1 119886

5) gt 119906 (1 119886

2) gt 119906 (1 119886

1)

gt 119906 (1 1198863) gt 119906 (1 119886

4) gt 119906 (1 119888)

119906 (2 1198863) gt 119906 (2 119886

2) gt 119906 (2 119886

6)

gt 119906 (2 1198864) gt 119906 (2 119886

5) gt 119906 (2 119888)

119906 (2 1198866) gt 119906 (2 119886

1) gt 119906 (2 119888)

(3)

We claim that the Chess-like game (G2 119906) has no uniform

NE whenever 119906 isin 1198802

Let us remark that |1198802| = 3 and that 119888 is theworst outcome

for both players for all 119906 isin 1198802 To verify this let us consider

the normal form 1198922in Figure 3 By Theorem 5 there is a

uniformly BR of player 2 to each strategy of player 1 and viceversa It is not difficult to check that the obtained two setsof the BRs (which are denoted by the white discs and blacksquares in Figure 3) are disjoint Hence there is no uniformNE Furthermore it is not difficult to verify that the obtained16 situations induce an improvement cycle of length 10 andtwo improvement paths of lengths 2 and 4 that end in thiscycle

Theorem8 (see [1]) Game (G2 119906) has no uniformNE in pure

strategies whenever 119906 isin 1198802

The goal of the present paper is to demonstrate that theabove two game structures may have no uniform NE notonly in pure but also in mixed strategies Let us note that byNashrsquos theorem [21 22] NE in mixed strategies exist in anyinitialized game structure Yet this result cannot be extendedto the noninitialized game structure and uniform NE In thisresearch we are motivated by the results of [8 11]

110 Mixed and Independently Mixed Strategies Standardlya mixed strategy 119910

119894of a player 119894 isin 119868 is defined as a

probabilistic distribution over the set119883119894of his pure strategies

Furthermore 119910119894is called an independently mixed strategy if

119894 randomizes in his positions V isin 119881119894independently We

will denote by 119884119894and by 119885

119894sube 119884119894the sets of mixed and

independently mixed strategies of player 119894 isin 119868 respectively

Remark 9 Let us recall that the players are restricted totheir positional strategies and let us also note that the latterconcept is closely related to the so-called behavioral strategiesintroduced by Kuhn [19 20] Although Kuhn restrictedhimself to trees yet his construction can be extended todirected graphs too

Let us recall that a game structure is called play-once ifeach player is in control of a unique position For exampleG1is play-once Obviously the classes of mixed and inde-

pendently mixed strategies coincide for a play-once gamestructure However for G

2these two notion differ Each

player 119894 isin 119868 = 1 2 controls 3 positions and has 8

6 GameTheory

pure strategies Hence the set of mixed strategies 119884119894is of

dimension 7 while the set119885119894sube 119884119894of the independentlymixed

strategies is only 3-dimensional

2 Markovian and A Priori Realizations

For the independently mixed strategies we will consider twodifferent options

For every player 119894 isin 119868 let us consider a probabilitydistribution 119875

119894

V for all positions V isin 119881119894 which assigns

a probability 119901(V V1015840) to each move (V V1015840) from V isin 119881119894

standardly assuming

0 le 119901 (V V1015840) le 1 sum

V1015840isin119881

119901 (V V1015840) = 1

119901 (V V1015840) = 0 whenever (V V1015840) notin 119864

(4)

Now the limit distributions of the terminals 119860 =

1198861 119886

119898 119886infin can be defined in two ways which we will be

referred to as theMarkovian and a priori realizationsThe first approach is classical the limit distribution can

be found by solving a 119898 times 119898 system of linear equations seefor example [27] and also [26]

For example let us consider G1and let 119901

119895be the

probability to proceed in V119895for 119895 = 1 2 3 If 119901

1= 1199012=

1199013= 1 then obviously the play will cycle with probability 1

resulting in the limit distribution (0 0 0 1) for (1198861 1198862 1198863 119888)

Otherwise assuming that V1is the initial position we obtain

the limit distribution

(

1 minus 1199011

1 minus 119901111990121199013

1199011(1 minus 119901

2)

1 minus 119901111990121199013

11990111199012(1 minus 119901

3)

1 minus 119901111990121199013

0) (5)

Indeed positions V1 V2 V3are transient and the probabil-

ity of cycling forever is 0 whenever 119901111990121199013lt 1 Obviously

the sum of the above four probabilities is 1The Markovian approach assumes that for 119905 = 0 1

the move 119890(119905) = (V(119905) V(119905 + 1)) is chosen randomly inaccordance with the distribution 119875V(119905) and independently forall 119905 (furthermore V(0) = V

0is a fixed initial position) In

particular if the play comes to the same position again that isV = V(119905) = V(1199051015840) for some 119905 lt 1199051015840 then the moves 119890(119905) and 119890(1199051015840)may be distinct although they are chosen (independently)with the same distribution 119875V

The concept of a priori realization is based on the follow-ing alternative assumptions A move (V V1015840) is chosen accord-ing to 119875V independently for all V isin 119881 119881

119879 but only once

before the game starts Being chosen themove (V V1015840) is appliedwhenever the play comes at V By these assumptions eachinfinite play ℓ is a lasso that is it consists of an initial part(that might be empty) and an infinitely repeated dicycle 119888

ℓ

Alternatively ℓ may be finite that is it terminates in a 119881119879

In both cases ℓ begins in V0and the probability of ℓ is the

product of the probabilities of all its moves 119875ℓ= prod119890isinℓ119901(119890)

In this way we obtain a probability distribution on the setof lassos of the digraph In particular the effective payoff isdefined as the expected payoffs for the corresponding lassosLet us also note that (in contrast to the Markovian case)

the computation of limit distribution is not computationallyefficient since the set of plays may grow exponentially in sizeof the digraph No polynomial algorithm computing the limitdistribution is known for a priori realizations Returning toour exampleG

1 we obtain the following limit distribution

(1 minus 1199011 1199011(1 minus 119901

2) 11990111199012(1 minus 119901

3) 119901111990121199013)

for the outcomes (1198861 1198862 1198863 119888)

(6)

with initial position V1 The probability of outcome 119888 is

119901111990121199013 it is strictly positive whenever 119901

119894gt 0 for all 119894 isin 119868

Indeed in contrast to theMarkovian realization the cyclewillbe repeated infinitely whenever it appears once under a priorirealization

Remark 10 Thus solving the Chess-like games in the inde-pendently mixed strategies looks more natural under apriori (rather than Markovian) realizations Unfortunately itseems not that easy to suggest more applications of a priorirealizations and we have to acknowledge that the conceptof the Markovian realization is much more fruitful Let usalso note that playing in pure strategies can be viewed as aspecial case of both Markovian and a priori realizations withdegenerate probability distributions

As we already mentioned the mixed and independentlymixed strategies coincide for G

1since it is play-once Yet

these two classes of strategies differ inG2

3 Chess-Like Games with No Uniform NE

In the present paper we will strengthen Theorems 7 and 8showing that games (G

1 119906) and (G

2 119906) may fail to have an

NE (not only in pure but even) in mixed strategies as well asin the independentlymixed strategies under bothMarkovianand a priori realizations

For convenience let 119869 = 1 119898 denote the set ofindices of nonterminal positions We will refer to positionsgiving only these indices

Let us recall the definition of payoff function 119891V0(119894 119909) ofplayer 119894 for the initial position V

0and the strategy profile

119909 see Theorem 5 Let us extend this definition introducingthe payoff function for the mixed and independently mixedstrategies In both cases we define it as the expected payoffunder one of the above realizations and denote by 119865V0(119894 119901)where 119901 is an 119898-vector whose 119895th coordinate 119901

119895is the

probability of proceeding (not terminating) at position 119895 isin 119869

Remark 11 Let us observe that in both G1and G

2 the

payoff functions119865V0(119894 119901) 119894 isin 119868 are continuously differentiablefunctions of 119901

119895when 0 lt 119901

119895lt 1 for all 119895 isin 119869 for all players

119894 isin 119868 Hence if 119901 is a uniform NE such that 0 lt 119901119895lt 1 for all

119895 isin 119869 (under either a priori or Markovian realization) then

120597119865V0(119894 119901)

120597119901119895

= 0 forall119894 isin 119868 119895 isin 119869 V0isin 119881 (7)

In the next two sections we will construct games thathave no uniform NE under both a priori and Markovian

GameTheory 7

realizations Assuming that a uniform mixed NE exists wewill obtain a contradiction with (7) whenever 0 lt 119901

119895lt 1 for

all 119895 isin 119869

31 (G1119906) Examples The next lemma will be instrumental

in the proofs of the following two theorems

Lemma 12 The probabilities to proceed satisfy 0 lt 119901119895lt 1 for

all 119895 isin 119869 = 1 2 3 in any independently mixed uniform NEin game (G

1 119906) where 119906 isin 119880

1 and under both a priori and

Markovian realizations

Proof Let us assume indirectly that there is an (indepen-dently) mixed uniform NE under a priori realization with119901119895= 0 for some 119895 isin 119869 This would imply the existence

of an acyclic game with uniform NE in contradiction withTheorem 7 Now let us consider the case 119901

119895= 1 Due to the

circular symmetry of (G1 119906) we can choose any player say

119895 = 1 The preference list of player 3 is 119906(3 1198861) gt 119906(3 119886

3) gt

119906(3 1198862) gt 119906(3 119888) His most favorable outcome 119886

1 is not

achievable since 1199011= 1 Hence 119901

3= 0 because his second

best outcome is 1198863 Thus the game is reduced to an acyclic

one in contradiction withTheorem 7 again

Theorem 13 Game (G1 119906) has no uniform NE in indepen-

dently mixed strategies under a priori realization whenever119906 isin 119880

1

Proof To simplify our notation we denote by 119895+and 119895

minus

the following and preceding positions along the 3-cycle ofG1 respectively Assume indirectly that (119901

1 1199012 1199013) forms a

uniform NE and considers the effective payoff of player 1

119865119895(1 119901) = (1 minus 119901

119895) 119906 (1 119886

119895) + 119901119895(1 minus 119901

119895+) 119906 (1 119886

119895+)

+ 119901119895119901119895+(1 minus 119901

119895minus) 119906 (1 119886

119895minus) + 119901119895119901119895+119901119895minus119906 (1 119888)

(8)

where 119895 is the initial positionBy Lemma 12 we must have 0 lt 119901

119895lt 1 for 119895 isin 119869 =

1 2 3 Therefore (7) must hold Hence (120597119865119895(1 119901)120597119901

119895minus) =

119901119895119901119895+(119906(1 119888) minus 119906(1 119886

119895minus)) = 0 and 119901

119895119901119895+

= 0 follows since119906(1 119886119895minus) gt 119906(1 119888) Thus 119901

111990121199013= 0 in contradiction to our

assumption

Let us recall that for G1 independently mixed strategies

and mixed strategies are the sameNow let us consider the Markovian realization Game

(G1 119906)may have noNE inmixed strategies underMarkovian

realization either yet only for some special payoffs 119906 isin 1198801

Theorem 14 Game (G1 119906) with 119906 isin 119880

1 has no uniform NE

in independentlymixed strategies underMarkovian realizationif and only if 120583

112058321205833ge 1 where

1205831=

119906 (1 1198862) minus 119906 (1 119886

1)

119906 (1 1198861) minus 119906 (1 119886

3)

1205832=

119906 (2 1198863) minus 119906 (2 119886

2)

119906 (2 1198862) minus 119906 (2 119886

1)

1205833=

119906 (3 1198861) minus 119906 (3 119886

3)

119906 (3 1198863) minus 119906 (3 119886

2)

(9)

It is easy to verify that 120583119894gt 0 for 119894 = 1 2 3 whenever

119906 isin 1198801 Let us also note that in the symmetric case 120583

1= 1205832=

1205833= 120583 the above condition 120583

112058321205833ge 1 turns into 120583 ge 1

Proof Let 119901 = (1199011 1199012 1199013) be a uniform NE in the game

(G1 119906) underMarkovian realizationThen by Lemma 12 0 lt

119901119894lt 1 for 119894 isin 119868 = 1 2 3Thepayoff function of a player with

respect to the initial position that this player controls is givenby one of the next three formulas

1198651(1 119901)

=

(1 minus 1199011) 119906 (1 119886

1)+1199011(1 minus 119901

2) 119906 (1 119886

2)+11990111199012(1 minus 119901

3) 119906 (1 119886

3)

1 minus 119901111990121199013

1198652(2 119901)

=

(1 minus 1199012) 119906 (2 119886

2)+1199012(1 minus 119901

3) 119906 (2 119886

3)+11990121199013(1 minus 119901

1) 119906 (2 119886

1)

1 minus 119901111990121199013

1198653(3 119901)

=

(1 minus 1199013) 119906 (3 119886

3)+1199013(1 minus 119901

1) 119906 (3 119886

1)+11990131199011(1 minus 119901

2) 119906 (3 119886

2)

1 minus 119901111990121199013

(10)

By Lemma 12 (7) holds for any uniformNETherefore wehave

(1 minus 119901111990121199013)2 1205971198651

(1 119901)

1205971199011

= 1199012(1 minus 119901

3) 119906 (1 119886

3) + (119901

21199013minus 1) 119906 (1 119886

1)

+ (1 minus 1199012) 119906 (1 119886

2) = 0

(1 minus 119901111990121199013)2 1205971198652

(2 119901)

1205971199012

= 1199013(1 minus 119901

1) 119906 (2 119886

1) + (119901

11199013minus 1) 119906 (2 119886

2)

+ (1 minus 1199013) 119906 (2 119886

3) = 0

(1 minus 119901111990121199013)2 1205971198653

(3 119901)

1205971199013

= 1199011(1 minus 119901

2) 119906 (3 119886

2) + (119901

11199012minus 1) 119906 (3 119886

3)

+ (1 minus 1199011) 119906 (3 119886

1) = 0

(11)

Setting 120582119894= 120583119894+ 1 for 119894 = 1 2 3 we can transform the

above equations to the following form

1205821(1 minus 119901

2) = 1 minus 119901

21199013

1205822(1 minus 119901

3) = 1 minus 119901

11199013

1205823(1 minus 119901

1) = 1 minus 119901

11199012

(12)

8 GameTheory

Assuming 0 lt 119901119895lt 1 119895 isin 119869 and using successive elimination

we uniquely express 119901 via 120582 as follows

0 lt 1199011=

1205822+ 1205823minus 12058211205822minus 12058221205823+ 120582112058221205823minus 1

12058211205823minus 1205821+ 1

lt 1

0 lt 1199012=


12058211205822minus 1205822+ 1

lt 1

0 lt 1199013=


12058221205823minus 1205823+ 1

lt 1

(13)

Interestingly all three 119901119895lt 1 inequalities are equivalent

with the condition (1205821minus1)(120582

2minus1)(120582

3minus1) lt 1 that is120583

112058321205833lt

1 which completes the proof

32 (G2119906) Examples Here we will show that (G

2 119906) may

have no uniform NE for both Markovian and a priorirealizations in independently mixed strategies whenever119906 isin 119880

2 As for the mixed (unlike the independently mixed)

strategies we obtain NE-free examples only for some (not forall) 119906 isin 119880

2

We begin with extending Lemma 12 to game (G2 119906) and

119906 isin 1198802as follows


all 119895 isin 119869 = 1 2 3 4 5 6 in any independently mixed uniformNE in game (G

2 119906) where 119906 isin 119880

2 and under both a priori

and Markovian realizations

Proof To prove that 119901119895lt 1 for all 119895 isin 119869 let us consider the

following six cases

(i) If 1199011= 1 then player 2 will proceed at position 6 as

1198862gt 1198866in 1198802 implying 119901

6= 1

(ii) If 1199012= 1 then either 119901

1= 0 or 119901

3= 1 as player 1

prefers 1198861to 1198863

(iii) If 1199013= 1 then 119901

2= 0 as player 2 cannot achieve his

best outcome of 1198863 while 119886

2is his second best one

(iv) If 1199014= 1 then 119901

3= 1 as player 1rsquos worst outcome is

1198863in the current situation

(v) If 1199015= 1 then 119901

4= 1 as player 2 prefers 119886

6to 1198864

(vi) If 1199016= 1 then 119901

5= 0 as player 1rsquos best outcome is 119886

5

now

It is easy to verify that by the above implications in all sixcases at least one of the proceeding probabilities should be 0in contradiction toTheorem 8

Let us show that the game (G2 119906) might have no NE in

independently mixed strategies under both Markovian and apriori realizations Let us consider the Markovian one first

Theorem 16 Game (G2 119906) has no uniform NE in the inde-

pendently mixed strategies under Markovian realization for all119906 isin 119880

2

Proof Let us consider the uniform NE conditions for player2 Lemma 15 implies that (7) must be satisfied Applying it tothe partial derivatives with respect to 119901

4and 119901

6we obtain

(1 minus 119901111990121199013119901411990151199016)2

11990111199012119901311990141199015

1205971198651(2 119901)

1205971199016

= ((1 minus 1199011) 119906 (2 119886

1) + 1199011(1 minus 119901

2) 119906 (2 119886

2)

+ 11990111199012(1 minus 119901

3) 119906 (2 119886

3)

+ 119901111990121199013(1 minus 119901

4) 119906 (2 119886

4)

+ 1199011119901211990131199014(1 minus 119901

5) 119906 (2 119886

5)

minus (1 minus 11990111199012119901311990141199015) 119906 (2 119886

6)) = 0

(1 minus 119901111990121199013119901411990151199016)2

11990111199012119901311990151199016

1205971198655(2 119901)

1205971199014

= (1 minus 1199015) 119906 (2 119886

5) + 1199015(1 minus 119901

6) 119906 (2 119886

6)

+ 11990151199016(1 minus 119901

1) 119906 (2 119886

1) + 119901511990161199011(1 minus 119901

2) 119906 (2 119886

2)

+ 1199015119901611990111199012(1 minus 119901

3) 119906 (2 119886

3)

minus (1 minus 11990111199012119901311990151199016) 119906 (2 119886

4) = 0

(14)

Let us multiply the first equation by 11990151199016and subtract it

from the second one yielding

(1 minus 119901111990121199013119901411990151199016) [minus119906 (2 119886

4) + (1 minus 119901

5) 119906 (2 119886

5)

+1199015119906 (2 119886

6)] = 0

(15)

or equivalently 119906(2 1198864) minus (1 minus 119901

5)119906(2 119886

5) minus 1199015119906(2 1198866) = 0

From this equation we find

1199015=

119906 (2 1198864) minus 119906 (2 119886

5)

119906 (2 1198866) minus 119906 (2 119886

5)

(16)

Furthermore the condition 0 lt 1199015lt 1 implies that either

119906(2 1198865) lt 119906(2 119886

4) lt 119906(2 119886

6) or 119906(2 119886

5) gt 119906(2 119886

4) gt

119906(2 1198866) Both orders contradict the preference list 119880

2 thus

completing the proof

Now let us consider the case of a priori realization


dentlymixed strategies under a priori realization for all 119906 isin 1198802

Proof Let us assume indirectly that 119901 =

(1199011 1199012 1199013 1199014 1199015 1199016) form a uniform NE Let us consider

the effective payoff of the player 1 with respect to the initialposition 2

1198652(1 119901) = (1 minus 119901

2) 119906 (1 119886

2) + 1199012(1 minus 119901

3) 119906 (1 119886

3)

+ 11990121199013(1 minus 119901

4) 119906 (1 119886

4)

GameTheory 9

+ 119901211990131199014(1 minus 119901

5) 119906 (1 119886

5)

+ 1199012119901311990141199015(1 minus 119901

6) 119906 (1 119886

6)

+ 11990121199013119901411990151199016(1 minus 119901

1) 119906 (1 119886

1)

(17)By Lemma 15 we have 0 lt 119901

119895lt 1 for 119895 isin 119869 = 1 2 3 4 5 6

Hence (7) must hold in particular (1205971198652(1 119901)120597119901

1) = 0 and

since 119906 isin 1198802is positive we obtain 119901

21199013119901411990151199016= 0 that is a

contradiction

The last result can be extended from the independentlymixed to mixed strategies However the correspondingexample is constructed not for all but only for some 119906 isin 119880

2

Theorem 18 The game (G2 119906) has no uniform NE in mixed

strategies at least for some 119906 isin 1198802

Proof Let us recall that there are two players inG2controling

three positions each and there are two possible moves inevery position Thus each player has eight pure strategiesStandardly the mixed strategies are defined as probabilitydistributions on the set of the pure strategies that is 119909 119910 isin

S8 where 119911 = (119911

1 119911

8) isin S8if and only if sum8

119894=1119911119894= 1 and

119911 ge 0Furthermore let us denote by 119886

119896119897(V0) the outcome of the

game beginning in the initial position V0isin 119881 in case when

player 1 chooses his pure strategy 119896 and player 2 chooses herpure strategy 119897 where 119896 119897 isin 1 8

Given a utility function 119906 119868 times119860 rarr R if a pair of mixedstrategies 119909 119910 isin S

8form a uniform NE then

8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0))

= 119911V0 if 119910

119897gt 0

le 119911V0 otherwise

(18)

must hold for some 119911V0 value for all initial positions V0isin

119881 Indeed otherwise player 2 would change the probabilitydistribution 119910 to get a better value Let 119878 = 119894 | 119910

119894gt 0 denote

the set of indices of all positive components of119910 isin S8 By (19)

there exists a subset 119878 sube 1 119899 such that the next systemis feasible

8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0)) = 119911V0

forall119897 isin 119878

8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0)) le 119911V0

forall119897 notin 119878

8

sum

119896=1

119909119896= 1

119909119896ge 0 forall119896 = 1 8

119911V0unrestricted forallV

0isin 119881

(19)

Then let us consider for example a utility function 119906 isin

1198802with the following payoffs of player 2

119906 (2 1198861) = 43 119906 (2 119886

2) = 81 119906 (2 119886

3) = 93

119906 (2 1198864) = 50 119906 (2 119886

5) = 15 119906 (2 119886

1) = 80

119906 (2 119888) = 0

(20)

We verified that (19) is infeasible for all subsets 119878 sube 1 8such that |119878| ge 2 Since for any119906 isin 119880

2there is no pure strategy

NE either we obtain a contradiction

33 Concluding Remarks

Remark 19 In the last two theorems in contrast withTheorem 14 uniform NE exist for no 119906 isin 119880

2

Remark 20 Let us note that Nashrsquos results [21 22] guarantee-ing the existence of an NE in mixed strategies for any normalform games are applicable in case of a fixed initial positionYet our results show that Nashrsquos theorem in general does notextend to the case of uniform NE except for the 119899-personacyclic case [12 19 20] and the two-person zero sum cases

Remark 21 It seems that the same holds for all 119906 isin 1198802

We tested (19) for many randomly chosen 119906 isin 1198802and

encountered infeasibility for all 119878 sube 1 8 such that |119878| ge2 Yet we have no proof and it still remains open whether forany 119906 isin 119880

2there is no NE in mixed strategies

Remark 22 Finally let us note that for an arbitrary Chess-likegame structure (not only for G

1and G

2) in independently

mixed strategies under both the Markovian and a priorirealizations for any 119894 isin 119868 and 119896 119897 isin 119869 the ratio (120597119865

119897(119894 119901)

120597119901119894)(120597119865119896(119894 119901)120597119901

119894) = 119875(119894 119896 119897) is a positive constant

Acknowledgments

The first and third authors acknowledge the partial supportby the NSF Grants IIS-1161476 and also CMMI-0856663The second author is thankful to Janos Flesch for helpfuldiscussions All author are also thankful to an anonymousreviewer for many helpful remarks and suggestions

References

[1] E Boros K Elbassioni V Gurvich and K Makino ldquoOn Nashequilibria and improvement cycles in pure positional strategiesfor Chess-like and Backgammon-like n-person gamesrdquoDiscreteMathematics vol 312 no 4 pp 772ndash788 2012

[2] D Andersson V Gurvich and T D Hansen ldquoOn acyclicity ofgames with cyclesrdquo Discrete Applied Mathematics vol 158 no10 pp 1049ndash1063 2010

[3] D Andersson V Gurvich and T D Hansen ldquoOn acyclicity ofgames with cyclesrdquo in Algorithmic Aspects in Information andManagement vol 5564 pp 15ndash28 2009

[4] E Boros and V Gurvich ldquoOn Nash-solvability in pure station-ary strategies of finite games with perfect information whichmay have cyclesrdquo Mathematical Social Sciences vol 46 no 2pp 207ndash241 2003

[5] E Boros and V Gurvich ldquoWhy chess and backgammon can besolved in pure positional uniformly optimal strategiesrdquo RUT-COR Research Report 21-2009 Rutgers University

[6] E Boros V Gurvich K Makino and W Shao ldquoNash-solvabletwo-person symmetric cycle game formsrdquo Discrete AppliedMathematics vol 159 no 15 pp 1461ndash1487 2011

10 GameTheory

[7] E Boros and R Rand ldquoTerminal games with three terminalshave proper Nash equilibriardquo RUTCOR Research Report RRR-22-2009 Rutgers University

[8] J Flesch J Kuipers G Shoenmakers and O J Vrieze ldquoSub-game-perfect equilibria in free transition gamesrdquo ResearchMemorandum RM08027 University of Maastricht Maas-tricht The Netherlands 2008

[9] J Flesch J Kuipers G Shoenmakers and O J Vrieze ldquoSub-game-perfection equilibria in stochastic games with perfectinformation and recursive payosrdquo Research MemorandumRM08041 University of Maastricht Maastricht The Nether-lands 2008

[10] J Kuipers J Flesch G Schoenmakers and K Vrieze ldquoPuresubgame-perfect equilibria in free transition gamesrdquo EuropeanJournal of Operational Research vol 199 no 2 pp 442ndash4472009

[11] J Flesch J Kuipers G Schoenmakers and K Vrieze ldquoSubgameperfection in positive recursive games with perfect informa-tionrdquoMathematics of Operations Research vol 35 no 1 pp 193ndash207 2010

[12] DGale ldquoA theory ofN-person gameswith perfect informationrdquoProceedings of the National Academy of Sciences vol 39 no 6pp 496ndash501 1953

[13] V A Gurvich ldquoOn theory of multistep gamesrdquoUSSR Computa-tional Mathematics and Mathematical Physics vol 13 no 6 pp143ndash161 1973

[14] V A Gurvich ldquoThe solvability of positional games in purestrategiesrdquo USSR Computational Mathematics and Mathemat-ical Physics vol 15 no 2 pp 74ndash87 1975

[15] V Gurvich ldquoEquilibrium in pure strategiesrdquo Soviet Mathemat-ics vol 38 no 3 pp 597ndash602 1989

[16] V Gurvich ldquoA stochastic game with complete information andwithout equilibrium situations in pure stationary strategiesrdquoRussian Mathematical Surveys vol 43 no 2 pp 171ndash172 1988

[17] V Gurvich ldquoA theorem on the existence of equilibrium situ-ations in pure stationary strategies for ergodic extensions of(2 times 119896) bimatrix gamesrdquo Russian Mathematical Surveys vol 45no 4 pp 170ndash172 1990

[18] V Gurvich ldquoSaddle point in pure strategiesrdquo Russian Academyof ScienceDokladyMathematics vol 42 no 2 pp 497ndash501 1990

[19] H Kuhn ldquoExtensive gamesrdquo Proceedings of the NationalAcademy of Sciences vol 36 pp 286ndash295 1950

[20] H Kuhn ldquoExtensive games and the problems of informationrdquoAnnals of Mathematics Studies vol 28 pp 193ndash216 1953

[21] J Nash ldquoEquilibrium points in n-person gamesrdquo Proceedings ofthe National Academy of Sciences vol 36 no 1 pp 48ndash49 1950

[22] J Nash ldquoNon-cooperative gamesrdquo Annals of Mathematics vol54 no 2 pp 286ndash295 1951

[23] A Condon ldquoAn algorithm for simple stochastic gamesrdquo inAdvances in Computational Complexity Theory vol 13 ofDIMACS series in discrete mathematics and theoretical computerscience 1993

[24] I V Romanovsky ldquoOn the solvability of Bellmanrsquos functionalequation for a Markovian decision processrdquo Journal of Mathe-matical Analysis and Applications vol 42 no 2 pp 485ndash4981973

[25] R A Howard Dynamic Programming and Markov ProcessesThe MIT Press 1960

[26] H Mine and S Osaki Markovian Decision Process AmericanElsevier New York NY USA 1970

[27] J G Kemeny and J L Snell Finite Markov Chains Springer1960

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


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Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 GameTheory

1

3

2

1

2

1

2

1

21198861

1198862

1198863

1199701

1199702

1198861

1198862

1198863

1198864

1198865

1198866

Figure 1 Two Chess-like game structuresG1andG

2 InG

1 there are 3 players controlling one position each (ieG

1is play-once) while in

G2there are only two players who alternate turns hence each of them controls 3 positions V

1 V3 V5are controlled by player 1 and V

2 V4 V6by

player 2 In each position Vℓ the corresponding player has only two options (119901) to proceed to V

ℓ+1and (119905) to terminate at 119886

ℓ where ℓ isin 1 2 3

(V4= V1) inG

1and ℓ isin 1 6 (V

7= V1) inG

2 To save space we show only symbols 119886

ℓ while the corresponding vertex names are omitted

An initial position V0isin 119881 may be fixed The triplet

(119866119863 V0) or pair G = (119866119863) is called a Chess-like positional

game structure (or just a game structure for short) initializedor noninitialized respectively By default we assume that it isnot initialized

Two examples of (noninitialized) game structuresG1and

G2are given in Figure 1

13 Plays Outcomes Preferences and Payoffs Given an ini-tialized positional game structure (119866119863 V

0) a play is defined

as a directed path that begins in V0and either ends in a

terminal position 119886 isin 119881119879or is infinite In this paper we

assume that all infinite plays form one outcome 119886infin

(or 119888)in addition to the standard Terminal outcomes of 119881

119879 (In [5]

this condition was referred to as AIPFOOT)A utility (or payoff) function is amapping 119906 119868times119860 rarr R

whose value 119906(119894 119886) is interpreted as a profit of the player 119894 isin119868 = 1 119899 in case of the outcome 119886 isin 119860 = 119886

1 119886

119898 119886infin

A payoff is called zero-sum ifsum119894isin119868119906(119894 119886) = 0 for every 119886 isin

119860 Two-person zero sum games are important For examplethe standard Chess and Backgammon are two-person zero-sum games in which every infinite play is a draw 119906(1 119886

infin) =

119906(2 119886infin) = 0 It is easy to realize that 119906(119894 119886

infin) = 0 can be

assumed for all players 119894 isin 119868 without any loss of generalityAnother important class of payoffs is defined by the

condition 119906(119894 119886infin) lt 119906(119894 119886) for all 119894 isin 119868 and 119886 isin 119881

119879 in other

words the infinite outcome 119886infin

is ranked as the worst one byall players Several possible motivations for this assumptionare discussed in [4 5]

A quadruple (119866119863 V0 119906) and triplet (119866119863 119906) will be

called a Chess-like game initialized and noninitializedrespectively

Remark 2 From the other side the Chess-like games can beviewed as the transition-free deterministic stochastic gameswith perfect information see for example [8ndash11]

In these games every nonterminal position is controlledby a player 119894 isin 119868 and the local reward 119903(119894 119890) is 0 for each player119894 isin 119868 and move 119890 unless 119890 = (V1015840 V) is a terminal move that isV isin 119881119879 Obviously in the considered case all infinite plays are

equivalent since the effective payoff is 0 for every such playFurthermore obviously 119886

infinis the worst outcome for a player

119894 isin 119868 if and only if 119903(119894 119890) gt 0 for every terminal move 119890If |119868| = 119899 = 119898 = |119860| = 2 then the zero-sum Chess-like

games turn into a subclass of the so-called simple stochasticgames which were introduced by Condon in [23]

14 Pure Positional Strategies Given game structure G =

(119866119863) a (pure positional) strategy 119909119894of a player 119894 isin 119868 is a

mapping 119909119894 119881119894rarr 119864119894that assigns to each position V isin 119881

119894a

move (V V1015840) from this positionThe concept of mixed strategies will be considered in

Section 110 till then only pure strategies are consideredMoreover in this paper we restrict the players to theirpositional (pure) strategies In other words the move (V V1015840)of a player 119894 isin 119868 in a position V isin 119881

119894depends only on the

position V itself not on the preceding positions or movesLet 119883

119894be the set of all strategies of a player 119894 isin 119868 and

119883 = prod119894isin119868119883119894be the direct product of these sets An element

119909 = 1199091 119909

119899 isin 119883 is called a strategy profile or situation

15 Normal Forms A positional game structure can berepresented in the normal (or strategic) form

Let us begin with the initialized case Given a gamestructure G = (119866119863 V

0) and a strategy profile 119909 isin 119883 a play

119901(119909) is uniquely defined by the following rules it begins inV0and in each position V isin 119881

119894proceeds with the arc (V V1015840)

determined by the strategy 119909119894 Obviously 119901(119909) either ends

in a terminal position 119886 isin 119881119879or 119901(119909) is infinite In the

latter case 119901(119909) is a lasso that is it consists of an initial partand a directed cycle (dicycle) repeated infinitely This holdsbecause all players are restricted to their positional strategiesIn either case an outcome 119886 = 119892(119909) isin 119860 = 119886

1 119886

119898 119886infin

GameTheory 3

1

2

3

1198921

119905

119901

119901119905

119905

119901

119901

119901

119905

1199051198861 1198862 1198863 1198861 1198863 1198863

1198862 1198862 1198863 1198863 1198863 1198863

1198861 1198862 1198861 1198861 1198861 1198861

1198862 1198862 1198862 119888 119888 119888

119906(1 1198862) gt 119906(1 1198861) gt 119906(1 1198863) gt 119906(1 119888)

119906(2 1198863) gt 119906(2 1198862) gt 119906(2 1198861) gt 119906(2 119888)

119906(3 1198861) gt 119906(3 1198863) gt 119906(3 1198862) gt 119906(3 119888)










0isin

119881119881119879to obtain a play119901 = 119901(119909 V

0) outcome 119886 = 119892(119909 V

0) and







119881

119881119879

) 997888rarr R (1)

where 119891V0(119894 119909) = 119891(119894 119909 V0) = 119906(119894 119892V0

(119909)) = 119906(119894 119892(119909 V0)) for


119879









0isin 119881119881


119879) rarr R



all V0isin 119881 119881

119879 too


0 Indeed in this




119868 times 119883 rarr R







119891V0(119894 119909lowast) ge 119891V0



119909lowast

119894and 1199091015840


lowast








4 GameTheory

119905119905119905 119905119905119901 119905119901119905 119905119901119901 119901119905119905 119901119905119901 119901119901119905 119901119901119901

119905119905119905

119905119905119901

119905119901119905

119905119901119901

119901119905119905

119901119905119901

119901119901119905

119901119901119901


119886111988621198863119886411988651198866 119886111988621198863119886411988651198861 119886111988621198863119886511988651198866 119886111988621198863119886511988651198861 119886111988631198863119886411988651198866 119886111988631198863119886411988651198861 119886111988631198863119886511988651198866 119886111988631198863119886511988651198861

119886111988621198863119886411988661198866 119886111988621198863119886411988611198861 119886111988621198863119886611988661198866 119886111988621198863119886111988611198861 119886111988631198863119886411988661198866 119886111988631198863119886411988611198861 119886111988631198863119886611988661198866 119886111988631198863119886111988611198861

119886111988621198864119886411988651198866 119886111988621198864119886411988651198861 119886111988621198865119886511988651198866 119886111988621198865119886511988651198861 119886111988641198864119886411988651198866 119886111988641198864119886411988651198861 119886111988651198865119886511988651198866 119886111988651198865119886511988651198861

119886111988621198864119886411988661198866 119886111988621198864119886411988611198861 119886111988621198866119886611988661198866 119886111988621198861119886111988611198861 119886111988641198864119886411988661198866 119886111988641198864119886411988611198861 119886111988661198866119886611988661198866 119886111988611198861119886111988611198861

119886211988621198863119886411988651198866 119886211988621198863119886411988651198862 119886211988621198863119886511988651198866 119886211988621198863119886511988651198862 119886311988631198863119886411988651198866 119886311988631198863119886411988651198863 119886311988631198863119886511988651198866 119886311988631198863119886511988651198863

119886211988621198863119886411988661198866 119886211988621198863119886411988621198862 119886211988621198863119886611988661198866 119886211988621198863119886211988621198862 119886311988631198863119886411988661198866 119886311988631198863119886411988631198863 119886311988631198863119886611988661198866 119886311988631198863119886311988631198863

119886211988621198864119886411988651198866 119886211988621198864119886411988651198862 119886211988621198865119886511988651198866 119886211988621198865119886511988651198862 119886411988641198864119886411988651198866 119886411988641198864119886411988651198864 119886511988651198865119886511988651198866 119886511988651198865119886511988651198865

119886211988621198864119886411988661198866 119886211988621198864119886411988621198862 119886211988621198866119886611988661198866 119886211988621198862119886211988621198862 119886411988641198864119886411988661198866 119886411988641198864119886411988641198864 119886611988661198866119886611988661198866 119888 119888 119888 119888 119888 119888

1198922








2


1 119906)




V0isin 119881 119881
































1and G




V119895

ℓof 119862119895to a terminal 119886119895

ℓ where ℓ = 1 3119895 and 119895 =


2is a two-


GameTheory 5





119895






ℓfor the terminal








5from


2from V2

2and in 1198862

6from V2

6


2of size


1and G

2 each


1was suggested in



119906 (1 1198862) gt 119906 (1 119886

1) gt 119906 (1 119886

3) gt 119906 (1 119888)

119906 (2 1198863) gt 119906 (2 119886

2) gt 119906 (2 119886

1) gt 119906 (2 119888)

119906 (3 1198861) gt 119906 (3 119886

3) gt 119906 (3 119886

2) gt 119906 (3 119888)

(2)



1 1198862 1198861) which in its











2of the


119906 (1 1198866) gt 119906 (1 119886

5) gt 119906 (1 119886

2) gt 119906 (1 119886

1)

gt 119906 (1 1198863) gt 119906 (1 119886

4) gt 119906 (1 119888)

119906 (2 1198863) gt 119906 (2 119886

2) gt 119906 (2 119886

6)

gt 119906 (2 1198864) gt 119906 (2 119886

5) gt 119906 (2 119888)

119906 (2 1198866) gt 119906 (2 119886

1) gt 119906 (2 119888)

(3)























6 GameTheory







119894



standardly assuming

0 le 119901 (V V1015840) le 1 sum

V1015840isin119881

119901 (V V1015840) = 1


(4)


1198861 119886





119895be the


1= 1199012=





(

1 minus 1199011

1 minus 119901111990121199013

1199011(1 minus 119901

2)

1 minus 119901111990121199013

11990111199012(1 minus 119901

3)

1 minus 119901111990121199013

0) (5)










ℓ







(1 minus 1199011 1199011(1 minus 119901

2) 11990111199012(1 minus 119901

3) 119901111990121199013)

for the outcomes (1198861 1198862 1198863 119888)

(6)



119894gt 0 for all 119894 isin 119868








1 119906) and (G







119895is the



2 the


119895when 0 lt 119901




120597119865V0(119894 119901)

120597119901119895



GameTheory 7


119895lt 1 for

all 119895 isin 119869





1 119906) where 119906 isin 119880








3) gt


1 is not







1


minus


1 1199012 1199013) forms a


119865119895(1 119901) = (1 minus 119901

119895) 119906 (1 119886

119895) + 119901119895(1 minus 119901

119895+) 119906 (1 119886

119895+)

+ 119901119895119901119895+(1 minus 119901

119895minus) 119906 (1 119886

119895minus) + 119901119895119901119895+119901119895minus119906 (1 119888)

(8)


119895lt 1 for 119895 isin 119869 =


119895minus) =

119901119895119901119895+(119906(1 119888) minus 119906(1 119886

119895minus)) = 0 and 119901

119895119901119895+



assumption






1 has no uniform NE


112058321205833ge 1 where

1205831=

119906 (1 1198862) minus 119906 (1 119886

1)

119906 (1 1198861) minus 119906 (1 119886

3)

1205832=

119906 (2 1198863) minus 119906 (2 119886

2)

119906 (2 1198862) minus 119906 (2 119886

1)

1205833=

119906 (3 1198861) minus 119906 (3 119886

3)

119906 (3 1198863) minus 119906 (3 119886

2)

(9)



1= 1205832=


112058321205833ge 1 turns into 120583 ge 1





1198651(1 119901)

=

(1 minus 1199011) 119906 (1 119886

1)+1199011(1 minus 119901

2) 119906 (1 119886

2)+11990111199012(1 minus 119901

3) 119906 (1 119886

3)

1 minus 119901111990121199013

1198652(2 119901)

=

(1 minus 1199012) 119906 (2 119886

2)+1199012(1 minus 119901

3) 119906 (2 119886

3)+11990121199013(1 minus 119901

1) 119906 (2 119886

1)

1 minus 119901111990121199013

1198653(3 119901)

=

(1 minus 1199013) 119906 (3 119886

3)+1199013(1 minus 119901

1) 119906 (3 119886

1)+11990131199011(1 minus 119901

2) 119906 (3 119886

2)

1 minus 119901111990121199013

(10)


(1 minus 119901111990121199013)2 1205971198651

(1 119901)

1205971199011

= 1199012(1 minus 119901

3) 119906 (1 119886

3) + (119901

21199013minus 1) 119906 (1 119886

1)

+ (1 minus 1199012) 119906 (1 119886

2) = 0

(1 minus 119901111990121199013)2 1205971198652

(2 119901)

1205971199012

= 1199013(1 minus 119901

1) 119906 (2 119886

1) + (119901

11199013minus 1) 119906 (2 119886

2)

+ (1 minus 1199013) 119906 (2 119886

3) = 0

(1 minus 119901111990121199013)2 1205971198653

(3 119901)

1205971199013

= 1199011(1 minus 119901

2) 119906 (3 119886

2) + (119901

11199012minus 1) 119906 (3 119886

3)

+ (1 minus 1199011) 119906 (3 119886

1) = 0

(11)



1205821(1 minus 119901

2) = 1 minus 119901

21199013

1205822(1 minus 119901

3) = 1 minus 119901

11199013

1205823(1 minus 119901

1) = 1 minus 119901

11199012

(12)

8 GameTheory



0 lt 1199011=


12058211205823minus 1205821+ 1

lt 1

0 lt 1199012=


12058211205822minus 1205822+ 1

lt 1

0 lt 1199013=


12058221205823minus 1205823+ 1

lt 1

(13)



2minus1)(120582


112058321205833lt



2 119906) may




2





2 119906) where 119906 isin 119880




following six cases


1198862gt 1198866in 1198802 implying 119901

6= 1

(ii) If 1199012= 1 then either 119901

1= 0 or 119901

3= 1 as player 1

prefers 1198861to 1198863

(iii) If 1199013= 1 then 119901




(iv) If 1199014= 1 then 119901



(v) If 1199015= 1 then 119901


6to 1198864

(vi) If 1199016= 1 then 119901


5

now






2


4and 119901

6we obtain

(1 minus 119901111990121199013119901411990151199016)2

11990111199012119901311990141199015

1205971198651(2 119901)

1205971199016

= ((1 minus 1199011) 119906 (2 119886

1) + 1199011(1 minus 119901

2) 119906 (2 119886

2)

+ 11990111199012(1 minus 119901

3) 119906 (2 119886

3)

+ 119901111990121199013(1 minus 119901

4) 119906 (2 119886

4)

+ 1199011119901211990131199014(1 minus 119901

5) 119906 (2 119886

5)

minus (1 minus 11990111199012119901311990141199015) 119906 (2 119886

6)) = 0

(1 minus 119901111990121199013119901411990151199016)2

11990111199012119901311990151199016

1205971198655(2 119901)

1205971199014

= (1 minus 1199015) 119906 (2 119886

5) + 1199015(1 minus 119901

6) 119906 (2 119886

6)

+ 11990151199016(1 minus 119901

1) 119906 (2 119886

1) + 119901511990161199011(1 minus 119901

2) 119906 (2 119886

2)

+ 1199015119901611990111199012(1 minus 119901

3) 119906 (2 119886

3)

minus (1 minus 11990111199012119901311990151199016) 119906 (2 119886

4) = 0

(14)



(1 minus 119901111990121199013119901411990151199016) [minus119906 (2 119886

4) + (1 minus 119901

5) 119906 (2 119886

5)

+1199015119906 (2 119886

6)] = 0

(15)


5)119906(2 119886

5) minus 1199015119906(2 1198866) = 0


1199015=

119906 (2 1198864) minus 119906 (2 119886

5)

119906 (2 1198866) minus 119906 (2 119886

5)

(16)


119906(2 1198865) lt 119906(2 119886

4) lt 119906(2 119886

6) or 119906(2 119886

5) gt 119906(2 119886

4) gt


2 thus








1198652(1 119901) = (1 minus 119901

2) 119906 (1 119886

2) + 1199012(1 minus 119901

3) 119906 (1 119886

3)

+ 11990121199013(1 minus 119901

4) 119906 (1 119886

4)

GameTheory 9

+ 119901211990131199014(1 minus 119901

5) 119906 (1 119886

5)

+ 1199012119901311990141199015(1 minus 119901

6) 119906 (1 119886

6)

+ 11990121199013119901411990151199016(1 minus 119901

1) 119906 (1 119886

1)


119895lt 1 for 119895 isin 119869 = 1 2 3 4 5 6


1) = 0 and


21199013119901411990151199016= 0 that is a

contradiction


2





S8 where 119911 = (119911

1 119911


119894=1119911119894= 1 and







8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0))

= 119911V0 if 119910

119897gt 0


(18)



119894gt 0 denote



8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0)) = 119911V0


8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0)) le 119911V0


8

sum

119896=1

119909119896= 1

119909119896ge 0 forall119896 = 1 8


0isin 119881

(19)



119906 (2 1198861) = 43 119906 (2 119886

2) = 81 119906 (2 119886

3) = 93

119906 (2 1198864) = 50 119906 (2 119886

5) = 15 119906 (2 119886

1) = 80

119906 (2 119888) = 0

(20)






2







1and G

2) in independently


119897(119894 119901)

120597119901119894)(120597119865119896(119894 119901)120597119901


Acknowledgments


References







10 GameTheory





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









GameTheory 3

1

2

3

1198921

119905

119901

119901119905

119905

119901

119901

119901

119905

1199051198861 1198862 1198863 1198861 1198863 1198863

1198862 1198862 1198863 1198863 1198863 1198863

1198861 1198862 1198861 1198861 1198861 1198861

1198862 1198862 1198862 119888 119888 119888

119906(1 1198862) gt 119906(1 1198861) gt 119906(1 1198863) gt 119906(1 119888)

119906(2 1198863) gt 119906(2 1198862) gt 119906(2 1198861) gt 119906(2 119888)

119906(3 1198861) gt 119906(3 1198863) gt 119906(3 1198862) gt 119906(3 119888)










0isin

119881119881119879to obtain a play119901 = 119901(119909 V

0) outcome 119886 = 119892(119909 V

0) and







119881

119881119879

) 997888rarr R (1)

where 119891V0(119894 119909) = 119891(119894 119909 V0) = 119906(119894 119892V0

(119909)) = 119906(119894 119892(119909 V0)) for


119879









0isin 119881119881


119879) rarr R



all V0isin 119881 119881

119879 too


0 Indeed in this




119868 times 119883 rarr R







119891V0(119894 119909lowast) ge 119891V0



119909lowast

119894and 1199091015840


lowast








4 GameTheory

119905119905119905 119905119905119901 119905119901119905 119905119901119901 119901119905119905 119901119905119901 119901119901119905 119901119901119901

119905119905119905

119905119905119901

119905119901119905

119905119901119901

119901119905119905

119901119905119901

119901119901119905

119901119901119901


119886111988621198863119886411988651198866 119886111988621198863119886411988651198861 119886111988621198863119886511988651198866 119886111988621198863119886511988651198861 119886111988631198863119886411988651198866 119886111988631198863119886411988651198861 119886111988631198863119886511988651198866 119886111988631198863119886511988651198861

119886111988621198863119886411988661198866 119886111988621198863119886411988611198861 119886111988621198863119886611988661198866 119886111988621198863119886111988611198861 119886111988631198863119886411988661198866 119886111988631198863119886411988611198861 119886111988631198863119886611988661198866 119886111988631198863119886111988611198861

119886111988621198864119886411988651198866 119886111988621198864119886411988651198861 119886111988621198865119886511988651198866 119886111988621198865119886511988651198861 119886111988641198864119886411988651198866 119886111988641198864119886411988651198861 119886111988651198865119886511988651198866 119886111988651198865119886511988651198861

119886111988621198864119886411988661198866 119886111988621198864119886411988611198861 119886111988621198866119886611988661198866 119886111988621198861119886111988611198861 119886111988641198864119886411988661198866 119886111988641198864119886411988611198861 119886111988661198866119886611988661198866 119886111988611198861119886111988611198861

119886211988621198863119886411988651198866 119886211988621198863119886411988651198862 119886211988621198863119886511988651198866 119886211988621198863119886511988651198862 119886311988631198863119886411988651198866 119886311988631198863119886411988651198863 119886311988631198863119886511988651198866 119886311988631198863119886511988651198863

119886211988621198863119886411988661198866 119886211988621198863119886411988621198862 119886211988621198863119886611988661198866 119886211988621198863119886211988621198862 119886311988631198863119886411988661198866 119886311988631198863119886411988631198863 119886311988631198863119886611988661198866 119886311988631198863119886311988631198863

119886211988621198864119886411988651198866 119886211988621198864119886411988651198862 119886211988621198865119886511988651198866 119886211988621198865119886511988651198862 119886411988641198864119886411988651198866 119886411988641198864119886411988651198864 119886511988651198865119886511988651198866 119886511988651198865119886511988651198865

119886211988621198864119886411988661198866 119886211988621198864119886411988621198862 119886211988621198866119886611988661198866 119886211988621198862119886211988621198862 119886411988641198864119886411988661198866 119886411988641198864119886411988641198864 119886611988661198866119886611988661198866 119888 119888 119888 119888 119888 119888

1198922








2


1 119906)




V0isin 119881 119881
































1and G




V119895

ℓof 119862119895to a terminal 119886119895

ℓ where ℓ = 1 3119895 and 119895 =


2is a two-


GameTheory 5





119895






ℓfor the terminal








5from


2from V2

2and in 1198862

6from V2

6


2of size


1and G

2 each


1was suggested in



119906 (1 1198862) gt 119906 (1 119886

1) gt 119906 (1 119886

3) gt 119906 (1 119888)

119906 (2 1198863) gt 119906 (2 119886

2) gt 119906 (2 119886

1) gt 119906 (2 119888)

119906 (3 1198861) gt 119906 (3 119886

3) gt 119906 (3 119886

2) gt 119906 (3 119888)

(2)



1 1198862 1198861) which in its











2of the


119906 (1 1198866) gt 119906 (1 119886

5) gt 119906 (1 119886

2) gt 119906 (1 119886

1)

gt 119906 (1 1198863) gt 119906 (1 119886

4) gt 119906 (1 119888)

119906 (2 1198863) gt 119906 (2 119886

2) gt 119906 (2 119886

6)

gt 119906 (2 1198864) gt 119906 (2 119886

5) gt 119906 (2 119888)

119906 (2 1198866) gt 119906 (2 119886

1) gt 119906 (2 119888)

(3)























6 GameTheory







119894



standardly assuming

0 le 119901 (V V1015840) le 1 sum

V1015840isin119881

119901 (V V1015840) = 1


(4)


1198861 119886





119895be the


1= 1199012=





(

1 minus 1199011

1 minus 119901111990121199013

1199011(1 minus 119901

2)

1 minus 119901111990121199013

11990111199012(1 minus 119901

3)

1 minus 119901111990121199013

0) (5)










ℓ







(1 minus 1199011 1199011(1 minus 119901

2) 11990111199012(1 minus 119901

3) 119901111990121199013)

for the outcomes (1198861 1198862 1198863 119888)

(6)



119894gt 0 for all 119894 isin 119868








1 119906) and (G







119895is the



2 the


119895when 0 lt 119901




120597119865V0(119894 119901)

120597119901119895



GameTheory 7


119895lt 1 for

all 119895 isin 119869





1 119906) where 119906 isin 119880








3) gt


1 is not







1


minus


1 1199012 1199013) forms a


119865119895(1 119901) = (1 minus 119901

119895) 119906 (1 119886

119895) + 119901119895(1 minus 119901

119895+) 119906 (1 119886

119895+)

+ 119901119895119901119895+(1 minus 119901

119895minus) 119906 (1 119886

119895minus) + 119901119895119901119895+119901119895minus119906 (1 119888)

(8)


119895lt 1 for 119895 isin 119869 =


119895minus) =

119901119895119901119895+(119906(1 119888) minus 119906(1 119886

119895minus)) = 0 and 119901

119895119901119895+



assumption






1 has no uniform NE


112058321205833ge 1 where

1205831=

119906 (1 1198862) minus 119906 (1 119886

1)

119906 (1 1198861) minus 119906 (1 119886

3)

1205832=

119906 (2 1198863) minus 119906 (2 119886

2)

119906 (2 1198862) minus 119906 (2 119886

1)

1205833=

119906 (3 1198861) minus 119906 (3 119886

3)

119906 (3 1198863) minus 119906 (3 119886

2)

(9)



1= 1205832=


112058321205833ge 1 turns into 120583 ge 1





1198651(1 119901)

=

(1 minus 1199011) 119906 (1 119886

1)+1199011(1 minus 119901

2) 119906 (1 119886

2)+11990111199012(1 minus 119901

3) 119906 (1 119886

3)

1 minus 119901111990121199013

1198652(2 119901)

=

(1 minus 1199012) 119906 (2 119886

2)+1199012(1 minus 119901

3) 119906 (2 119886

3)+11990121199013(1 minus 119901

1) 119906 (2 119886

1)

1 minus 119901111990121199013

1198653(3 119901)

=

(1 minus 1199013) 119906 (3 119886

3)+1199013(1 minus 119901

1) 119906 (3 119886

1)+11990131199011(1 minus 119901

2) 119906 (3 119886

2)

1 minus 119901111990121199013

(10)


(1 minus 119901111990121199013)2 1205971198651

(1 119901)

1205971199011

= 1199012(1 minus 119901

3) 119906 (1 119886

3) + (119901

21199013minus 1) 119906 (1 119886

1)

+ (1 minus 1199012) 119906 (1 119886

2) = 0

(1 minus 119901111990121199013)2 1205971198652

(2 119901)

1205971199012

= 1199013(1 minus 119901

1) 119906 (2 119886

1) + (119901

11199013minus 1) 119906 (2 119886

2)

+ (1 minus 1199013) 119906 (2 119886

3) = 0

(1 minus 119901111990121199013)2 1205971198653

(3 119901)

1205971199013

= 1199011(1 minus 119901

2) 119906 (3 119886

2) + (119901

11199012minus 1) 119906 (3 119886

3)

+ (1 minus 1199011) 119906 (3 119886

1) = 0

(11)



1205821(1 minus 119901

2) = 1 minus 119901

21199013

1205822(1 minus 119901

3) = 1 minus 119901

11199013

1205823(1 minus 119901

1) = 1 minus 119901

11199012

(12)

8 GameTheory



0 lt 1199011=


12058211205823minus 1205821+ 1

lt 1

0 lt 1199012=


12058211205822minus 1205822+ 1

lt 1

0 lt 1199013=


12058221205823minus 1205823+ 1

lt 1

(13)



2minus1)(120582


112058321205833lt



2 119906) may




2





2 119906) where 119906 isin 119880




following six cases


1198862gt 1198866in 1198802 implying 119901

6= 1

(ii) If 1199012= 1 then either 119901

1= 0 or 119901

3= 1 as player 1

prefers 1198861to 1198863

(iii) If 1199013= 1 then 119901




(iv) If 1199014= 1 then 119901



(v) If 1199015= 1 then 119901


6to 1198864

(vi) If 1199016= 1 then 119901


5

now






2


4and 119901

6we obtain

(1 minus 119901111990121199013119901411990151199016)2

11990111199012119901311990141199015

1205971198651(2 119901)

1205971199016

= ((1 minus 1199011) 119906 (2 119886

1) + 1199011(1 minus 119901

2) 119906 (2 119886

2)

+ 11990111199012(1 minus 119901

3) 119906 (2 119886

3)

+ 119901111990121199013(1 minus 119901

4) 119906 (2 119886

4)

+ 1199011119901211990131199014(1 minus 119901

5) 119906 (2 119886

5)

minus (1 minus 11990111199012119901311990141199015) 119906 (2 119886

6)) = 0

(1 minus 119901111990121199013119901411990151199016)2

11990111199012119901311990151199016

1205971198655(2 119901)

1205971199014

= (1 minus 1199015) 119906 (2 119886

5) + 1199015(1 minus 119901

6) 119906 (2 119886

6)

+ 11990151199016(1 minus 119901

1) 119906 (2 119886

1) + 119901511990161199011(1 minus 119901

2) 119906 (2 119886

2)

+ 1199015119901611990111199012(1 minus 119901

3) 119906 (2 119886

3)

minus (1 minus 11990111199012119901311990151199016) 119906 (2 119886

4) = 0

(14)



(1 minus 119901111990121199013119901411990151199016) [minus119906 (2 119886

4) + (1 minus 119901

5) 119906 (2 119886

5)

+1199015119906 (2 119886

6)] = 0

(15)


5)119906(2 119886

5) minus 1199015119906(2 1198866) = 0


1199015=

119906 (2 1198864) minus 119906 (2 119886

5)

119906 (2 1198866) minus 119906 (2 119886

5)

(16)


119906(2 1198865) lt 119906(2 119886

4) lt 119906(2 119886

6) or 119906(2 119886

5) gt 119906(2 119886

4) gt


2 thus








1198652(1 119901) = (1 minus 119901

2) 119906 (1 119886

2) + 1199012(1 minus 119901

3) 119906 (1 119886

3)

+ 11990121199013(1 minus 119901

4) 119906 (1 119886

4)

GameTheory 9

+ 119901211990131199014(1 minus 119901

5) 119906 (1 119886

5)

+ 1199012119901311990141199015(1 minus 119901

6) 119906 (1 119886

6)

+ 11990121199013119901411990151199016(1 minus 119901

1) 119906 (1 119886

1)


119895lt 1 for 119895 isin 119869 = 1 2 3 4 5 6


1) = 0 and


21199013119901411990151199016= 0 that is a

contradiction


2





S8 where 119911 = (119911

1 119911


119894=1119911119894= 1 and







8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0))

= 119911V0 if 119910

119897gt 0


(18)



119894gt 0 denote



8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0)) = 119911V0


8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0)) le 119911V0


8

sum

119896=1

119909119896= 1

119909119896ge 0 forall119896 = 1 8


0isin 119881

(19)



119906 (2 1198861) = 43 119906 (2 119886

2) = 81 119906 (2 119886

3) = 93

119906 (2 1198864) = 50 119906 (2 119886

5) = 15 119906 (2 119886

1) = 80

119906 (2 119888) = 0

(20)






2







1and G

2) in independently


119897(119894 119901)

120597119901119894)(120597119865119896(119894 119901)120597119901


Acknowledgments


References







10 GameTheory





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









4 GameTheory

119905119905119905 119905119905119901 119905119901119905 119905119901119901 119901119905119905 119901119905119901 119901119901119905 119901119901119901

119905119905119905

119905119905119901

119905119901119905

119905119901119901

119901119905119905

119901119905119901

119901119901119905

119901119901119901


119886111988621198863119886411988651198866 119886111988621198863119886411988651198861 119886111988621198863119886511988651198866 119886111988621198863119886511988651198861 119886111988631198863119886411988651198866 119886111988631198863119886411988651198861 119886111988631198863119886511988651198866 119886111988631198863119886511988651198861

119886111988621198863119886411988661198866 119886111988621198863119886411988611198861 119886111988621198863119886611988661198866 119886111988621198863119886111988611198861 119886111988631198863119886411988661198866 119886111988631198863119886411988611198861 119886111988631198863119886611988661198866 119886111988631198863119886111988611198861

119886111988621198864119886411988651198866 119886111988621198864119886411988651198861 119886111988621198865119886511988651198866 119886111988621198865119886511988651198861 119886111988641198864119886411988651198866 119886111988641198864119886411988651198861 119886111988651198865119886511988651198866 119886111988651198865119886511988651198861

119886111988621198864119886411988661198866 119886111988621198864119886411988611198861 119886111988621198866119886611988661198866 119886111988621198861119886111988611198861 119886111988641198864119886411988661198866 119886111988641198864119886411988611198861 119886111988661198866119886611988661198866 119886111988611198861119886111988611198861

119886211988621198863119886411988651198866 119886211988621198863119886411988651198862 119886211988621198863119886511988651198866 119886211988621198863119886511988651198862 119886311988631198863119886411988651198866 119886311988631198863119886411988651198863 119886311988631198863119886511988651198866 119886311988631198863119886511988651198863

119886211988621198863119886411988661198866 119886211988621198863119886411988621198862 119886211988621198863119886611988661198866 119886211988621198863119886211988621198862 119886311988631198863119886411988661198866 119886311988631198863119886411988631198863 119886311988631198863119886611988661198866 119886311988631198863119886311988631198863

119886211988621198864119886411988651198866 119886211988621198864119886411988651198862 119886211988621198865119886511988651198866 119886211988621198865119886511988651198862 119886411988641198864119886411988651198866 119886411988641198864119886411988651198864 119886511988651198865119886511988651198866 119886511988651198865119886511988651198865

119886211988621198864119886411988661198866 119886211988621198864119886411988621198862 119886211988621198866119886611988661198866 119886211988621198862119886211988621198862 119886411988641198864119886411988661198866 119886411988641198864119886411988641198864 119886611988661198866119886611988661198866 119888 119888 119888 119888 119888 119888

1198922








2


1 119906)




V0isin 119881 119881
































1and G




V119895

ℓof 119862119895to a terminal 119886119895

ℓ where ℓ = 1 3119895 and 119895 =


2is a two-


GameTheory 5





119895






ℓfor the terminal








5from


2from V2

2and in 1198862

6from V2

6


2of size


1and G

2 each


1was suggested in



119906 (1 1198862) gt 119906 (1 119886

1) gt 119906 (1 119886

3) gt 119906 (1 119888)

119906 (2 1198863) gt 119906 (2 119886

2) gt 119906 (2 119886

1) gt 119906 (2 119888)

119906 (3 1198861) gt 119906 (3 119886

3) gt 119906 (3 119886

2) gt 119906 (3 119888)

(2)



1 1198862 1198861) which in its











2of the


119906 (1 1198866) gt 119906 (1 119886

5) gt 119906 (1 119886

2) gt 119906 (1 119886

1)

gt 119906 (1 1198863) gt 119906 (1 119886

4) gt 119906 (1 119888)

119906 (2 1198863) gt 119906 (2 119886

2) gt 119906 (2 119886

6)

gt 119906 (2 1198864) gt 119906 (2 119886

5) gt 119906 (2 119888)

119906 (2 1198866) gt 119906 (2 119886

1) gt 119906 (2 119888)

(3)























6 GameTheory







119894



standardly assuming

0 le 119901 (V V1015840) le 1 sum

V1015840isin119881

119901 (V V1015840) = 1


(4)


1198861 119886





119895be the


1= 1199012=





(

1 minus 1199011

1 minus 119901111990121199013

1199011(1 minus 119901

2)

1 minus 119901111990121199013

11990111199012(1 minus 119901

3)

1 minus 119901111990121199013

0) (5)










ℓ







(1 minus 1199011 1199011(1 minus 119901

2) 11990111199012(1 minus 119901

3) 119901111990121199013)

for the outcomes (1198861 1198862 1198863 119888)

(6)



119894gt 0 for all 119894 isin 119868








1 119906) and (G







119895is the



2 the


119895when 0 lt 119901




120597119865V0(119894 119901)

120597119901119895



GameTheory 7


119895lt 1 for

all 119895 isin 119869





1 119906) where 119906 isin 119880








3) gt


1 is not







1


minus


1 1199012 1199013) forms a


119865119895(1 119901) = (1 minus 119901

119895) 119906 (1 119886

119895) + 119901119895(1 minus 119901

119895+) 119906 (1 119886

119895+)

+ 119901119895119901119895+(1 minus 119901

119895minus) 119906 (1 119886

119895minus) + 119901119895119901119895+119901119895minus119906 (1 119888)

(8)


119895lt 1 for 119895 isin 119869 =


119895minus) =

119901119895119901119895+(119906(1 119888) minus 119906(1 119886

119895minus)) = 0 and 119901

119895119901119895+



assumption






1 has no uniform NE


112058321205833ge 1 where

1205831=

119906 (1 1198862) minus 119906 (1 119886

1)

119906 (1 1198861) minus 119906 (1 119886

3)

1205832=

119906 (2 1198863) minus 119906 (2 119886

2)

119906 (2 1198862) minus 119906 (2 119886

1)

1205833=

119906 (3 1198861) minus 119906 (3 119886

3)

119906 (3 1198863) minus 119906 (3 119886

2)

(9)



1= 1205832=


112058321205833ge 1 turns into 120583 ge 1





1198651(1 119901)

=

(1 minus 1199011) 119906 (1 119886

1)+1199011(1 minus 119901

2) 119906 (1 119886

2)+11990111199012(1 minus 119901

3) 119906 (1 119886

3)

1 minus 119901111990121199013

1198652(2 119901)

=

(1 minus 1199012) 119906 (2 119886

2)+1199012(1 minus 119901

3) 119906 (2 119886

3)+11990121199013(1 minus 119901

1) 119906 (2 119886

1)

1 minus 119901111990121199013

1198653(3 119901)

=

(1 minus 1199013) 119906 (3 119886

3)+1199013(1 minus 119901

1) 119906 (3 119886

1)+11990131199011(1 minus 119901

2) 119906 (3 119886

2)

1 minus 119901111990121199013

(10)


(1 minus 119901111990121199013)2 1205971198651

(1 119901)

1205971199011

= 1199012(1 minus 119901

3) 119906 (1 119886

3) + (119901

21199013minus 1) 119906 (1 119886

1)

+ (1 minus 1199012) 119906 (1 119886

2) = 0

(1 minus 119901111990121199013)2 1205971198652

(2 119901)

1205971199012

= 1199013(1 minus 119901

1) 119906 (2 119886

1) + (119901

11199013minus 1) 119906 (2 119886

2)

+ (1 minus 1199013) 119906 (2 119886

3) = 0

(1 minus 119901111990121199013)2 1205971198653

(3 119901)

1205971199013

= 1199011(1 minus 119901

2) 119906 (3 119886

2) + (119901

11199012minus 1) 119906 (3 119886

3)

+ (1 minus 1199011) 119906 (3 119886

1) = 0

(11)



1205821(1 minus 119901

2) = 1 minus 119901

21199013

1205822(1 minus 119901

3) = 1 minus 119901

11199013

1205823(1 minus 119901

1) = 1 minus 119901

11199012

(12)

8 GameTheory



0 lt 1199011=


12058211205823minus 1205821+ 1

lt 1

0 lt 1199012=


12058211205822minus 1205822+ 1

lt 1

0 lt 1199013=


12058221205823minus 1205823+ 1

lt 1

(13)



2minus1)(120582


112058321205833lt



2 119906) may




2





2 119906) where 119906 isin 119880




following six cases


1198862gt 1198866in 1198802 implying 119901

6= 1

(ii) If 1199012= 1 then either 119901

1= 0 or 119901

3= 1 as player 1

prefers 1198861to 1198863

(iii) If 1199013= 1 then 119901




(iv) If 1199014= 1 then 119901



(v) If 1199015= 1 then 119901


6to 1198864

(vi) If 1199016= 1 then 119901


5

now






2


4and 119901

6we obtain

(1 minus 119901111990121199013119901411990151199016)2

11990111199012119901311990141199015

1205971198651(2 119901)

1205971199016

= ((1 minus 1199011) 119906 (2 119886

1) + 1199011(1 minus 119901

2) 119906 (2 119886

2)

+ 11990111199012(1 minus 119901

3) 119906 (2 119886

3)

+ 119901111990121199013(1 minus 119901

4) 119906 (2 119886

4)

+ 1199011119901211990131199014(1 minus 119901

5) 119906 (2 119886

5)

minus (1 minus 11990111199012119901311990141199015) 119906 (2 119886

6)) = 0

(1 minus 119901111990121199013119901411990151199016)2

11990111199012119901311990151199016

1205971198655(2 119901)

1205971199014

= (1 minus 1199015) 119906 (2 119886

5) + 1199015(1 minus 119901

6) 119906 (2 119886

6)

+ 11990151199016(1 minus 119901

1) 119906 (2 119886

1) + 119901511990161199011(1 minus 119901

2) 119906 (2 119886

2)

+ 1199015119901611990111199012(1 minus 119901

3) 119906 (2 119886

3)

minus (1 minus 11990111199012119901311990151199016) 119906 (2 119886

4) = 0

(14)



(1 minus 119901111990121199013119901411990151199016) [minus119906 (2 119886

4) + (1 minus 119901

5) 119906 (2 119886

5)

+1199015119906 (2 119886

6)] = 0

(15)


5)119906(2 119886

5) minus 1199015119906(2 1198866) = 0


1199015=

119906 (2 1198864) minus 119906 (2 119886

5)

119906 (2 1198866) minus 119906 (2 119886

5)

(16)


119906(2 1198865) lt 119906(2 119886

4) lt 119906(2 119886

6) or 119906(2 119886

5) gt 119906(2 119886

4) gt


2 thus








1198652(1 119901) = (1 minus 119901

2) 119906 (1 119886

2) + 1199012(1 minus 119901

3) 119906 (1 119886

3)

+ 11990121199013(1 minus 119901

4) 119906 (1 119886

4)

GameTheory 9

+ 119901211990131199014(1 minus 119901

5) 119906 (1 119886

5)

+ 1199012119901311990141199015(1 minus 119901

6) 119906 (1 119886

6)

+ 11990121199013119901411990151199016(1 minus 119901

1) 119906 (1 119886

1)


119895lt 1 for 119895 isin 119869 = 1 2 3 4 5 6


1) = 0 and


21199013119901411990151199016= 0 that is a

contradiction


2





S8 where 119911 = (119911

1 119911


119894=1119911119894= 1 and







8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0))

= 119911V0 if 119910

119897gt 0


(18)



119894gt 0 denote



8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0)) = 119911V0


8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0)) le 119911V0


8

sum

119896=1

119909119896= 1

119909119896ge 0 forall119896 = 1 8


0isin 119881

(19)



119906 (2 1198861) = 43 119906 (2 119886

2) = 81 119906 (2 119886

3) = 93

119906 (2 1198864) = 50 119906 (2 119886

5) = 15 119906 (2 119886

1) = 80

119906 (2 119888) = 0

(20)






2







1and G

2) in independently


119897(119894 119901)

120597119901119894)(120597119865119896(119894 119901)120597119901


Acknowledgments


References







10 GameTheory





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









GameTheory 5





119895






ℓfor the terminal








5from


2from V2

2and in 1198862

6from V2

6


2of size


1and G

2 each


1was suggested in



119906 (1 1198862) gt 119906 (1 119886

1) gt 119906 (1 119886

3) gt 119906 (1 119888)

119906 (2 1198863) gt 119906 (2 119886

2) gt 119906 (2 119886

1) gt 119906 (2 119888)

119906 (3 1198861) gt 119906 (3 119886

3) gt 119906 (3 119886

2) gt 119906 (3 119888)

(2)



1 1198862 1198861) which in its











2of the


119906 (1 1198866) gt 119906 (1 119886

5) gt 119906 (1 119886

2) gt 119906 (1 119886

1)

gt 119906 (1 1198863) gt 119906 (1 119886

4) gt 119906 (1 119888)

119906 (2 1198863) gt 119906 (2 119886

2) gt 119906 (2 119886

6)

gt 119906 (2 1198864) gt 119906 (2 119886

5) gt 119906 (2 119888)

119906 (2 1198866) gt 119906 (2 119886

1) gt 119906 (2 119888)

(3)























6 GameTheory







119894



standardly assuming

0 le 119901 (V V1015840) le 1 sum

V1015840isin119881

119901 (V V1015840) = 1


(4)


1198861 119886





119895be the


1= 1199012=





(

1 minus 1199011

1 minus 119901111990121199013

1199011(1 minus 119901

2)

1 minus 119901111990121199013

11990111199012(1 minus 119901

3)

1 minus 119901111990121199013

0) (5)










ℓ







(1 minus 1199011 1199011(1 minus 119901

2) 11990111199012(1 minus 119901

3) 119901111990121199013)

for the outcomes (1198861 1198862 1198863 119888)

(6)



119894gt 0 for all 119894 isin 119868








1 119906) and (G







119895is the



2 the


119895when 0 lt 119901




120597119865V0(119894 119901)

120597119901119895



GameTheory 7


119895lt 1 for

all 119895 isin 119869





1 119906) where 119906 isin 119880








3) gt


1 is not







1


minus


1 1199012 1199013) forms a


119865119895(1 119901) = (1 minus 119901

119895) 119906 (1 119886

119895) + 119901119895(1 minus 119901

119895+) 119906 (1 119886

119895+)

+ 119901119895119901119895+(1 minus 119901

119895minus) 119906 (1 119886

119895minus) + 119901119895119901119895+119901119895minus119906 (1 119888)

(8)


119895lt 1 for 119895 isin 119869 =


119895minus) =

119901119895119901119895+(119906(1 119888) minus 119906(1 119886

119895minus)) = 0 and 119901

119895119901119895+



assumption






1 has no uniform NE


112058321205833ge 1 where

1205831=

119906 (1 1198862) minus 119906 (1 119886

1)

119906 (1 1198861) minus 119906 (1 119886

3)

1205832=

119906 (2 1198863) minus 119906 (2 119886

2)

119906 (2 1198862) minus 119906 (2 119886

1)

1205833=

119906 (3 1198861) minus 119906 (3 119886

3)

119906 (3 1198863) minus 119906 (3 119886

2)

(9)



1= 1205832=


112058321205833ge 1 turns into 120583 ge 1





1198651(1 119901)

=

(1 minus 1199011) 119906 (1 119886

1)+1199011(1 minus 119901

2) 119906 (1 119886

2)+11990111199012(1 minus 119901

3) 119906 (1 119886

3)

1 minus 119901111990121199013

1198652(2 119901)

=

(1 minus 1199012) 119906 (2 119886

2)+1199012(1 minus 119901

3) 119906 (2 119886

3)+11990121199013(1 minus 119901

1) 119906 (2 119886

1)

1 minus 119901111990121199013

1198653(3 119901)

=

(1 minus 1199013) 119906 (3 119886

3)+1199013(1 minus 119901

1) 119906 (3 119886

1)+11990131199011(1 minus 119901

2) 119906 (3 119886

2)

1 minus 119901111990121199013

(10)


(1 minus 119901111990121199013)2 1205971198651

(1 119901)

1205971199011

= 1199012(1 minus 119901

3) 119906 (1 119886

3) + (119901

21199013minus 1) 119906 (1 119886

1)

+ (1 minus 1199012) 119906 (1 119886

2) = 0

(1 minus 119901111990121199013)2 1205971198652

(2 119901)

1205971199012

= 1199013(1 minus 119901

1) 119906 (2 119886

1) + (119901

11199013minus 1) 119906 (2 119886

2)

+ (1 minus 1199013) 119906 (2 119886

3) = 0

(1 minus 119901111990121199013)2 1205971198653

(3 119901)

1205971199013

= 1199011(1 minus 119901

2) 119906 (3 119886

2) + (119901

11199012minus 1) 119906 (3 119886

3)

+ (1 minus 1199011) 119906 (3 119886

1) = 0

(11)



1205821(1 minus 119901

2) = 1 minus 119901

21199013

1205822(1 minus 119901

3) = 1 minus 119901

11199013

1205823(1 minus 119901

1) = 1 minus 119901

11199012

(12)

8 GameTheory



0 lt 1199011=


12058211205823minus 1205821+ 1

lt 1

0 lt 1199012=


12058211205822minus 1205822+ 1

lt 1

0 lt 1199013=


12058221205823minus 1205823+ 1

lt 1

(13)



2minus1)(120582


112058321205833lt



2 119906) may




2





2 119906) where 119906 isin 119880




following six cases


1198862gt 1198866in 1198802 implying 119901

6= 1

(ii) If 1199012= 1 then either 119901

1= 0 or 119901

3= 1 as player 1

prefers 1198861to 1198863

(iii) If 1199013= 1 then 119901




(iv) If 1199014= 1 then 119901



(v) If 1199015= 1 then 119901


6to 1198864

(vi) If 1199016= 1 then 119901


5

now






2


4and 119901

6we obtain

(1 minus 119901111990121199013119901411990151199016)2

11990111199012119901311990141199015

1205971198651(2 119901)

1205971199016

= ((1 minus 1199011) 119906 (2 119886

1) + 1199011(1 minus 119901

2) 119906 (2 119886

2)

+ 11990111199012(1 minus 119901

3) 119906 (2 119886

3)

+ 119901111990121199013(1 minus 119901

4) 119906 (2 119886

4)

+ 1199011119901211990131199014(1 minus 119901

5) 119906 (2 119886

5)

minus (1 minus 11990111199012119901311990141199015) 119906 (2 119886

6)) = 0

(1 minus 119901111990121199013119901411990151199016)2

11990111199012119901311990151199016

1205971198655(2 119901)

1205971199014

= (1 minus 1199015) 119906 (2 119886

5) + 1199015(1 minus 119901

6) 119906 (2 119886

6)

+ 11990151199016(1 minus 119901

1) 119906 (2 119886

1) + 119901511990161199011(1 minus 119901

2) 119906 (2 119886

2)

+ 1199015119901611990111199012(1 minus 119901

3) 119906 (2 119886

3)

minus (1 minus 11990111199012119901311990151199016) 119906 (2 119886

4) = 0

(14)



(1 minus 119901111990121199013119901411990151199016) [minus119906 (2 119886

4) + (1 minus 119901

5) 119906 (2 119886

5)

+1199015119906 (2 119886

6)] = 0

(15)


5)119906(2 119886

5) minus 1199015119906(2 1198866) = 0


1199015=

119906 (2 1198864) minus 119906 (2 119886

5)

119906 (2 1198866) minus 119906 (2 119886

5)

(16)


119906(2 1198865) lt 119906(2 119886

4) lt 119906(2 119886

6) or 119906(2 119886

5) gt 119906(2 119886

4) gt


2 thus








1198652(1 119901) = (1 minus 119901

2) 119906 (1 119886

2) + 1199012(1 minus 119901

3) 119906 (1 119886

3)

+ 11990121199013(1 minus 119901

4) 119906 (1 119886

4)

GameTheory 9

+ 119901211990131199014(1 minus 119901

5) 119906 (1 119886

5)

+ 1199012119901311990141199015(1 minus 119901

6) 119906 (1 119886

6)

+ 11990121199013119901411990151199016(1 minus 119901

1) 119906 (1 119886

1)


119895lt 1 for 119895 isin 119869 = 1 2 3 4 5 6


1) = 0 and


21199013119901411990151199016= 0 that is a

contradiction


2





S8 where 119911 = (119911

1 119911


119894=1119911119894= 1 and







8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0))

= 119911V0 if 119910

119897gt 0


(18)



119894gt 0 denote



8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0)) = 119911V0


8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0)) le 119911V0


8

sum

119896=1

119909119896= 1

119909119896ge 0 forall119896 = 1 8


0isin 119881

(19)



119906 (2 1198861) = 43 119906 (2 119886

2) = 81 119906 (2 119886

3) = 93

119906 (2 1198864) = 50 119906 (2 119886

5) = 15 119906 (2 119886

1) = 80

119906 (2 119888) = 0

(20)






2







1and G

2) in independently


119897(119894 119901)

120597119901119894)(120597119865119896(119894 119901)120597119901


Acknowledgments


References







10 GameTheory





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









6 GameTheory







119894



standardly assuming

0 le 119901 (V V1015840) le 1 sum

V1015840isin119881

119901 (V V1015840) = 1


(4)


1198861 119886





119895be the


1= 1199012=





(

1 minus 1199011

1 minus 119901111990121199013

1199011(1 minus 119901

2)

1 minus 119901111990121199013

11990111199012(1 minus 119901

3)

1 minus 119901111990121199013

0) (5)










ℓ







(1 minus 1199011 1199011(1 minus 119901

2) 11990111199012(1 minus 119901

3) 119901111990121199013)

for the outcomes (1198861 1198862 1198863 119888)

(6)



119894gt 0 for all 119894 isin 119868








1 119906) and (G







119895is the



2 the


119895when 0 lt 119901




120597119865V0(119894 119901)

120597119901119895



GameTheory 7


119895lt 1 for

all 119895 isin 119869





1 119906) where 119906 isin 119880








3) gt


1 is not







1


minus


1 1199012 1199013) forms a


119865119895(1 119901) = (1 minus 119901

119895) 119906 (1 119886

119895) + 119901119895(1 minus 119901

119895+) 119906 (1 119886

119895+)

+ 119901119895119901119895+(1 minus 119901

119895minus) 119906 (1 119886

119895minus) + 119901119895119901119895+119901119895minus119906 (1 119888)

(8)


119895lt 1 for 119895 isin 119869 =


119895minus) =

119901119895119901119895+(119906(1 119888) minus 119906(1 119886

119895minus)) = 0 and 119901

119895119901119895+



assumption






1 has no uniform NE


112058321205833ge 1 where

1205831=

119906 (1 1198862) minus 119906 (1 119886

1)

119906 (1 1198861) minus 119906 (1 119886

3)

1205832=

119906 (2 1198863) minus 119906 (2 119886

2)

119906 (2 1198862) minus 119906 (2 119886

1)

1205833=

119906 (3 1198861) minus 119906 (3 119886

3)

119906 (3 1198863) minus 119906 (3 119886

2)

(9)



1= 1205832=


112058321205833ge 1 turns into 120583 ge 1





1198651(1 119901)

=

(1 minus 1199011) 119906 (1 119886

1)+1199011(1 minus 119901

2) 119906 (1 119886

2)+11990111199012(1 minus 119901

3) 119906 (1 119886

3)

1 minus 119901111990121199013

1198652(2 119901)

=

(1 minus 1199012) 119906 (2 119886

2)+1199012(1 minus 119901

3) 119906 (2 119886

3)+11990121199013(1 minus 119901

1) 119906 (2 119886

1)

1 minus 119901111990121199013

1198653(3 119901)

=

(1 minus 1199013) 119906 (3 119886

3)+1199013(1 minus 119901

1) 119906 (3 119886

1)+11990131199011(1 minus 119901

2) 119906 (3 119886

2)

1 minus 119901111990121199013

(10)


(1 minus 119901111990121199013)2 1205971198651

(1 119901)

1205971199011

= 1199012(1 minus 119901

3) 119906 (1 119886

3) + (119901

21199013minus 1) 119906 (1 119886

1)

+ (1 minus 1199012) 119906 (1 119886

2) = 0

(1 minus 119901111990121199013)2 1205971198652

(2 119901)

1205971199012

= 1199013(1 minus 119901

1) 119906 (2 119886

1) + (119901

11199013minus 1) 119906 (2 119886

2)

+ (1 minus 1199013) 119906 (2 119886

3) = 0

(1 minus 119901111990121199013)2 1205971198653

(3 119901)

1205971199013

= 1199011(1 minus 119901

2) 119906 (3 119886

2) + (119901

11199012minus 1) 119906 (3 119886

3)

+ (1 minus 1199011) 119906 (3 119886

1) = 0

(11)



1205821(1 minus 119901

2) = 1 minus 119901

21199013

1205822(1 minus 119901

3) = 1 minus 119901

11199013

1205823(1 minus 119901

1) = 1 minus 119901

11199012

(12)

8 GameTheory



0 lt 1199011=


12058211205823minus 1205821+ 1

lt 1

0 lt 1199012=


12058211205822minus 1205822+ 1

lt 1

0 lt 1199013=


12058221205823minus 1205823+ 1

lt 1

(13)



2minus1)(120582


112058321205833lt



2 119906) may




2





2 119906) where 119906 isin 119880




following six cases


1198862gt 1198866in 1198802 implying 119901

6= 1

(ii) If 1199012= 1 then either 119901

1= 0 or 119901

3= 1 as player 1

prefers 1198861to 1198863

(iii) If 1199013= 1 then 119901




(iv) If 1199014= 1 then 119901



(v) If 1199015= 1 then 119901


6to 1198864

(vi) If 1199016= 1 then 119901


5

now






2


4and 119901

6we obtain

(1 minus 119901111990121199013119901411990151199016)2

11990111199012119901311990141199015

1205971198651(2 119901)

1205971199016

= ((1 minus 1199011) 119906 (2 119886

1) + 1199011(1 minus 119901

2) 119906 (2 119886

2)

+ 11990111199012(1 minus 119901

3) 119906 (2 119886

3)

+ 119901111990121199013(1 minus 119901

4) 119906 (2 119886

4)

+ 1199011119901211990131199014(1 minus 119901

5) 119906 (2 119886

5)

minus (1 minus 11990111199012119901311990141199015) 119906 (2 119886

6)) = 0

(1 minus 119901111990121199013119901411990151199016)2

11990111199012119901311990151199016

1205971198655(2 119901)

1205971199014

= (1 minus 1199015) 119906 (2 119886

5) + 1199015(1 minus 119901

6) 119906 (2 119886

6)

+ 11990151199016(1 minus 119901

1) 119906 (2 119886

1) + 119901511990161199011(1 minus 119901

2) 119906 (2 119886

2)

+ 1199015119901611990111199012(1 minus 119901

3) 119906 (2 119886

3)

minus (1 minus 11990111199012119901311990151199016) 119906 (2 119886

4) = 0

(14)



(1 minus 119901111990121199013119901411990151199016) [minus119906 (2 119886

4) + (1 minus 119901

5) 119906 (2 119886

5)

+1199015119906 (2 119886

6)] = 0

(15)


5)119906(2 119886

5) minus 1199015119906(2 1198866) = 0


1199015=

119906 (2 1198864) minus 119906 (2 119886

5)

119906 (2 1198866) minus 119906 (2 119886

5)

(16)


119906(2 1198865) lt 119906(2 119886

4) lt 119906(2 119886

6) or 119906(2 119886

5) gt 119906(2 119886

4) gt


2 thus








1198652(1 119901) = (1 minus 119901

2) 119906 (1 119886

2) + 1199012(1 minus 119901

3) 119906 (1 119886

3)

+ 11990121199013(1 minus 119901

4) 119906 (1 119886

4)

GameTheory 9

+ 119901211990131199014(1 minus 119901

5) 119906 (1 119886

5)

+ 1199012119901311990141199015(1 minus 119901

6) 119906 (1 119886

6)

+ 11990121199013119901411990151199016(1 minus 119901

1) 119906 (1 119886

1)


119895lt 1 for 119895 isin 119869 = 1 2 3 4 5 6


1) = 0 and


21199013119901411990151199016= 0 that is a

contradiction


2





S8 where 119911 = (119911

1 119911


119894=1119911119894= 1 and







8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0))

= 119911V0 if 119910

119897gt 0


(18)



119894gt 0 denote



8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0)) = 119911V0


8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0)) le 119911V0


8

sum

119896=1

119909119896= 1

119909119896ge 0 forall119896 = 1 8


0isin 119881

(19)



119906 (2 1198861) = 43 119906 (2 119886

2) = 81 119906 (2 119886

3) = 93

119906 (2 1198864) = 50 119906 (2 119886

5) = 15 119906 (2 119886

1) = 80

119906 (2 119888) = 0

(20)






2







1and G

2) in independently


119897(119894 119901)

120597119901119894)(120597119865119896(119894 119901)120597119901


Acknowledgments


References







10 GameTheory





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









GameTheory 7


119895lt 1 for

all 119895 isin 119869





1 119906) where 119906 isin 119880








3) gt


1 is not







1


minus


1 1199012 1199013) forms a


119865119895(1 119901) = (1 minus 119901

119895) 119906 (1 119886

119895) + 119901119895(1 minus 119901

119895+) 119906 (1 119886

119895+)

+ 119901119895119901119895+(1 minus 119901

119895minus) 119906 (1 119886

119895minus) + 119901119895119901119895+119901119895minus119906 (1 119888)

(8)


119895lt 1 for 119895 isin 119869 =


119895minus) =

119901119895119901119895+(119906(1 119888) minus 119906(1 119886

119895minus)) = 0 and 119901

119895119901119895+



assumption






1 has no uniform NE


112058321205833ge 1 where

1205831=

119906 (1 1198862) minus 119906 (1 119886

1)

119906 (1 1198861) minus 119906 (1 119886

3)

1205832=

119906 (2 1198863) minus 119906 (2 119886

2)

119906 (2 1198862) minus 119906 (2 119886

1)

1205833=

119906 (3 1198861) minus 119906 (3 119886

3)

119906 (3 1198863) minus 119906 (3 119886

2)

(9)



1= 1205832=


112058321205833ge 1 turns into 120583 ge 1





1198651(1 119901)

=

(1 minus 1199011) 119906 (1 119886

1)+1199011(1 minus 119901

2) 119906 (1 119886

2)+11990111199012(1 minus 119901

3) 119906 (1 119886

3)

1 minus 119901111990121199013

1198652(2 119901)

=

(1 minus 1199012) 119906 (2 119886

2)+1199012(1 minus 119901

3) 119906 (2 119886

3)+11990121199013(1 minus 119901

1) 119906 (2 119886

1)

1 minus 119901111990121199013

1198653(3 119901)

=

(1 minus 1199013) 119906 (3 119886

3)+1199013(1 minus 119901

1) 119906 (3 119886

1)+11990131199011(1 minus 119901

2) 119906 (3 119886

2)

1 minus 119901111990121199013

(10)


(1 minus 119901111990121199013)2 1205971198651

(1 119901)

1205971199011

= 1199012(1 minus 119901

3) 119906 (1 119886

3) + (119901

21199013minus 1) 119906 (1 119886

1)

+ (1 minus 1199012) 119906 (1 119886

2) = 0

(1 minus 119901111990121199013)2 1205971198652

(2 119901)

1205971199012

= 1199013(1 minus 119901

1) 119906 (2 119886

1) + (119901

11199013minus 1) 119906 (2 119886

2)

+ (1 minus 1199013) 119906 (2 119886

3) = 0

(1 minus 119901111990121199013)2 1205971198653

(3 119901)

1205971199013

= 1199011(1 minus 119901

2) 119906 (3 119886

2) + (119901

11199012minus 1) 119906 (3 119886

3)

+ (1 minus 1199011) 119906 (3 119886

1) = 0

(11)



1205821(1 minus 119901

2) = 1 minus 119901

21199013

1205822(1 minus 119901

3) = 1 minus 119901

11199013

1205823(1 minus 119901

1) = 1 minus 119901

11199012

(12)

8 GameTheory



0 lt 1199011=


12058211205823minus 1205821+ 1

lt 1

0 lt 1199012=


12058211205822minus 1205822+ 1

lt 1

0 lt 1199013=


12058221205823minus 1205823+ 1

lt 1

(13)



2minus1)(120582


112058321205833lt



2 119906) may




2





2 119906) where 119906 isin 119880




following six cases


1198862gt 1198866in 1198802 implying 119901

6= 1

(ii) If 1199012= 1 then either 119901

1= 0 or 119901

3= 1 as player 1

prefers 1198861to 1198863

(iii) If 1199013= 1 then 119901




(iv) If 1199014= 1 then 119901



(v) If 1199015= 1 then 119901


6to 1198864

(vi) If 1199016= 1 then 119901


5

now






2


4and 119901

6we obtain

(1 minus 119901111990121199013119901411990151199016)2

11990111199012119901311990141199015

1205971198651(2 119901)

1205971199016

= ((1 minus 1199011) 119906 (2 119886

1) + 1199011(1 minus 119901

2) 119906 (2 119886

2)

+ 11990111199012(1 minus 119901

3) 119906 (2 119886

3)

+ 119901111990121199013(1 minus 119901

4) 119906 (2 119886

4)

+ 1199011119901211990131199014(1 minus 119901

5) 119906 (2 119886

5)

minus (1 minus 11990111199012119901311990141199015) 119906 (2 119886

6)) = 0

(1 minus 119901111990121199013119901411990151199016)2

11990111199012119901311990151199016

1205971198655(2 119901)

1205971199014

= (1 minus 1199015) 119906 (2 119886

5) + 1199015(1 minus 119901

6) 119906 (2 119886

6)

+ 11990151199016(1 minus 119901

1) 119906 (2 119886

1) + 119901511990161199011(1 minus 119901

2) 119906 (2 119886

2)

+ 1199015119901611990111199012(1 minus 119901

3) 119906 (2 119886

3)

minus (1 minus 11990111199012119901311990151199016) 119906 (2 119886

4) = 0

(14)



(1 minus 119901111990121199013119901411990151199016) [minus119906 (2 119886

4) + (1 minus 119901

5) 119906 (2 119886

5)

+1199015119906 (2 119886

6)] = 0

(15)


5)119906(2 119886

5) minus 1199015119906(2 1198866) = 0


1199015=

119906 (2 1198864) minus 119906 (2 119886

5)

119906 (2 1198866) minus 119906 (2 119886

5)

(16)


119906(2 1198865) lt 119906(2 119886

4) lt 119906(2 119886

6) or 119906(2 119886

5) gt 119906(2 119886

4) gt


2 thus








1198652(1 119901) = (1 minus 119901

2) 119906 (1 119886

2) + 1199012(1 minus 119901

3) 119906 (1 119886

3)

+ 11990121199013(1 minus 119901

4) 119906 (1 119886

4)

GameTheory 9

+ 119901211990131199014(1 minus 119901

5) 119906 (1 119886

5)

+ 1199012119901311990141199015(1 minus 119901

6) 119906 (1 119886

6)

+ 11990121199013119901411990151199016(1 minus 119901

1) 119906 (1 119886

1)


119895lt 1 for 119895 isin 119869 = 1 2 3 4 5 6


1) = 0 and


21199013119901411990151199016= 0 that is a

contradiction


2





S8 where 119911 = (119911

1 119911


119894=1119911119894= 1 and







8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0))

= 119911V0 if 119910

119897gt 0


(18)



119894gt 0 denote



8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0)) = 119911V0


8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0)) le 119911V0


8

sum

119896=1

119909119896= 1

119909119896ge 0 forall119896 = 1 8


0isin 119881

(19)



119906 (2 1198861) = 43 119906 (2 119886

2) = 81 119906 (2 119886

3) = 93

119906 (2 1198864) = 50 119906 (2 119886

5) = 15 119906 (2 119886

1) = 80

119906 (2 119888) = 0

(20)






2







1and G

2) in independently


119897(119894 119901)

120597119901119894)(120597119865119896(119894 119901)120597119901


Acknowledgments


References







10 GameTheory





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









8 GameTheory



0 lt 1199011=


12058211205823minus 1205821+ 1

lt 1

0 lt 1199012=


12058211205822minus 1205822+ 1

lt 1

0 lt 1199013=


12058221205823minus 1205823+ 1

lt 1

(13)



2minus1)(120582


112058321205833lt



2 119906) may




2





2 119906) where 119906 isin 119880




following six cases


1198862gt 1198866in 1198802 implying 119901

6= 1

(ii) If 1199012= 1 then either 119901

1= 0 or 119901

3= 1 as player 1

prefers 1198861to 1198863

(iii) If 1199013= 1 then 119901




(iv) If 1199014= 1 then 119901



(v) If 1199015= 1 then 119901


6to 1198864

(vi) If 1199016= 1 then 119901


5

now






2


4and 119901

6we obtain

(1 minus 119901111990121199013119901411990151199016)2

11990111199012119901311990141199015

1205971198651(2 119901)

1205971199016

= ((1 minus 1199011) 119906 (2 119886

1) + 1199011(1 minus 119901

2) 119906 (2 119886

2)

+ 11990111199012(1 minus 119901

3) 119906 (2 119886

3)

+ 119901111990121199013(1 minus 119901

4) 119906 (2 119886

4)

+ 1199011119901211990131199014(1 minus 119901

5) 119906 (2 119886

5)

minus (1 minus 11990111199012119901311990141199015) 119906 (2 119886

6)) = 0

(1 minus 119901111990121199013119901411990151199016)2

11990111199012119901311990151199016

1205971198655(2 119901)

1205971199014

= (1 minus 1199015) 119906 (2 119886

5) + 1199015(1 minus 119901

6) 119906 (2 119886

6)

+ 11990151199016(1 minus 119901

1) 119906 (2 119886

1) + 119901511990161199011(1 minus 119901

2) 119906 (2 119886

2)

+ 1199015119901611990111199012(1 minus 119901

3) 119906 (2 119886

3)

minus (1 minus 11990111199012119901311990151199016) 119906 (2 119886

4) = 0

(14)



(1 minus 119901111990121199013119901411990151199016) [minus119906 (2 119886

4) + (1 minus 119901

5) 119906 (2 119886

5)

+1199015119906 (2 119886

6)] = 0

(15)


5)119906(2 119886

5) minus 1199015119906(2 1198866) = 0


1199015=

119906 (2 1198864) minus 119906 (2 119886

5)

119906 (2 1198866) minus 119906 (2 119886

5)

(16)


119906(2 1198865) lt 119906(2 119886

4) lt 119906(2 119886

6) or 119906(2 119886

5) gt 119906(2 119886

4) gt


2 thus








1198652(1 119901) = (1 minus 119901

2) 119906 (1 119886

2) + 1199012(1 minus 119901

3) 119906 (1 119886

3)

+ 11990121199013(1 minus 119901

4) 119906 (1 119886

4)

GameTheory 9

+ 119901211990131199014(1 minus 119901

5) 119906 (1 119886

5)

+ 1199012119901311990141199015(1 minus 119901

6) 119906 (1 119886

6)

+ 11990121199013119901411990151199016(1 minus 119901

1) 119906 (1 119886

1)


119895lt 1 for 119895 isin 119869 = 1 2 3 4 5 6


1) = 0 and


21199013119901411990151199016= 0 that is a

contradiction


2





S8 where 119911 = (119911

1 119911


119894=1119911119894= 1 and







8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0))

= 119911V0 if 119910

119897gt 0


(18)



119894gt 0 denote



8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0)) = 119911V0


8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0)) le 119911V0


8

sum

119896=1

119909119896= 1

119909119896ge 0 forall119896 = 1 8


0isin 119881

(19)



119906 (2 1198861) = 43 119906 (2 119886

2) = 81 119906 (2 119886

3) = 93

119906 (2 1198864) = 50 119906 (2 119886

5) = 15 119906 (2 119886

1) = 80

119906 (2 119888) = 0

(20)






2







1and G

2) in independently


119897(119894 119901)

120597119901119894)(120597119865119896(119894 119901)120597119901


Acknowledgments


References







10 GameTheory





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









GameTheory 9

+ 119901211990131199014(1 minus 119901

5) 119906 (1 119886

5)

+ 1199012119901311990141199015(1 minus 119901

6) 119906 (1 119886

6)

+ 11990121199013119901411990151199016(1 minus 119901

1) 119906 (1 119886

1)


119895lt 1 for 119895 isin 119869 = 1 2 3 4 5 6


1) = 0 and


21199013119901411990151199016= 0 that is a

contradiction


2





S8 where 119911 = (119911

1 119911


119894=1119911119894= 1 and







8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0))

= 119911V0 if 119910

119897gt 0


(18)



119894gt 0 denote



8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0)) = 119911V0


8

sum

119896=1

119909119896119906 (2 119886

119896119897(V0)) le 119911V0


8

sum

119896=1

119909119896= 1

119909119896ge 0 forall119896 = 1 8


0isin 119881

(19)



119906 (2 1198861) = 43 119906 (2 119886

2) = 81 119906 (2 119886

3) = 93

119906 (2 1198864) = 50 119906 (2 119886

5) = 15 119906 (2 119886

1) = 80

119906 (2 119888) = 0

(20)






2







1and G

2) in independently


119897(119894 119901)

120597119901119894)(120597119865119896(119894 119901)120597119901


Acknowledgments


References







10 GameTheory





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









10 GameTheory





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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