topic 12 game theory

8/19/2019 Topic 12 Game Theory

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Topic 12

Game Theory


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Topics to be Discussed

• Game Theory

• Dominant Strategy, Nash Equilibrium, and MaximinStrategy

• The Prisoner’s Dilemma

• Cartel Cheating


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Game Theory• Game is any situation in whih players !the "artii"ants# ma$e

strategi deisions%▫ Ex& 'irms om"eting with eah other by setting "ries, grou" o'

onsumers bidding against eah other in an aution• Game theory tries to determine o"timal strategy 'or eah

"layer%• Game theory is the study o' how "eo"le ma$e deisions in

situations in whih attaining their goals de"ends on their

interations with others%• (t is a sub'iled o' eonomis that analy)es the hoies made by

ri*al 'irms, "eo"le, and e*en go*ernment when they are tryingto maximi)e their own well+being while antii"ating and

reating to the ations o' others in their en*ironment%


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Some important Game Theory TermsPlayers& deision ma$ers, or more oligo"olists%

-tions& all the "ossible mo*es a "layer an ma$e%

(n'ormation& how muh eah "layers $nows at eah "oint in the game%

strategies& rules telling eah "layer whih ation to

hoose at eah "oint in the game%


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Some Important Game Theory Terms

Payo''& "ro'it, ex"eted "ro'it, or utility the "layerreei*e a'ter all the "layers ha*e "i$ed strategies and

the games has been "layed out%

.utomes& all "ossible results o' all the "ossibleresults o' the strategies that the "layers an selet/ thatis, the set o' all "ossible game "ayo''%

Equilibrium& strategi ombination onsist o' the best

strategy 'or eah "layer in the game%


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Gaming and Strategic Decisions

• 0(' ( belie*e that my om"etitors are rational and atto maximi)e their own "ro'its, how should ( ta$e their

beha*ior into aount when ma$ing my own "ro'it+

maximi)ing deisions12


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Noncooperative v. Cooperative Games• Coo"erati*e Game

▫ Players negotiate binding ontrats that allow them to "lan 3oint strategies

Exam"le& 4uyer and seller negotiating the "rie o' a goodor ser*ie or a 3oint *enture by two 'irms !i%e% Miroso'tand -""le#

4inding ontrats are "ossible

• Coo"erati*e equilibrium is one in whih "layers in agame oo"erate to inrease their mutual "ayo''%

• Collusion is an agreement among 'irms to ollude,eg% harging the same "rie or other wise not to

om"ete%


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Noncooperative v. Cooperative Games

• Nonoo"erati*e Game▫ Negotiation and en'orement o' binding ontrats between

"layers is not "ossible Exam"le& Two om"eting 'irms assuming the others beha*ior

determine, inde"endently, "riing and ad*ertising strategy to gain

mar$et share

4inding ontrats are not "ossible

• Non+oo"erati*e equilibrium is one in whih "layersdon’t oo"erate but "ursue their own sel'+interest%


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Noncooperative v. Cooperative Games

• 0The strategy design is based on understanding youro""onent’s "oint o' *iew, and !assuming you

o""onent is rational# deduing how he or she is li$ely

to res"ond to your ations2

It is essential to understand your opponent’s point of view and to deduce his or her

likely responses to your actions.

Note that the 'undamental di''erene between oo"erati*e and nonoo"erati*e

games lies in the contracting possibilities. (n oo"erati*e games, binding ontrats are

"ossible/ in nonoo"erati*e games, they are not%


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Dominant Strategies

• Dominant Strategy is one that is o"timalno matter what an o""onent does%

• - dominant strategy is one that is best 'or a "layer, no

matter what strategies other "layers use%▫ -n Exam"le - 5 4 sell om"eting "roduts They are deiding whether to underta$e ad*ertising

am"aigns Their deision is interde"endene on other 'irm

deision%


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Payoff atri! for "dvertising Game# $"%&'

Firm

A

"dvertise

Don(t

"dvertise

"dvertise

Don(t"dvertise

Firm B

1)% * 1*% )

1)% 2+% ,

Dominant Strategies

- 4

-

-

-4

4

4

-hat strategy shoud each firm choose/


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Dominant Strategies6irm -&• (' 'irm 4 does ad*ertise, 6irm - will earn a "ro'it o'

78 i' it also ad*ertise and 9 i' it doesn’t%• Thus, 'irm - should ad*ertise i' 'irm 4 ad*ertise%• (' 'irm 4 doesn’t ad*ertise, 'irm - would earn "ro'it

o' 7: i' it ad*ertise and 78 i' it doesn’t%

• Thus, 'irm - should ad*ertise whether 'irm 4ad*ertise or not

"dvertising is the dominant strategy for 0irm "


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Dominant Strategies6irm 4&

• (' 'irm - does ad*ertise, 6irm 4 will earn a "ro'it o' : i'

it also ad*ertise and 8 i' it doesn’t%• Thus, 'irm 4 should ad*ertise i' 'irm - ad*ertise%

• (' 'irm - doesn’t ad*ertise, 'irm 4 would earn "ro'it o'; i' it ad*ertise and i' it doesn’t%

• Thus, 'irm 4 should ad*ertise whether 'irm - ad*ertiseor not

"dvertising is the dominant strategy for 0irm &


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Payoff atri! for "dvertising Game

• .bser*ations

▫ -& regardless o' 4,

ad*ertising is the best▫ 4& regardless o' -,

ad*ertising is best

Firm A

"dvertise

Don(t

"dvertise

"dvertise

Don(t

"dvertise

Firm B

1)% * 1*% )

1)% 2+% ,&oth firms ith advertise.


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Payoff atri! for "dvertising Game• .bser*ations

▫ Dominant strategy

'or - 5 4 is toad*ertise

▫ Do not worry aboutthe other "layer

▫ Equilibrium indominant strategy

Firm A

"dvertiseDon(t

"dvertise

"dvertise

Don(t

"dvertise

Firm B

1)% * 1*% )

1)% 2+% ,


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Dominant Strategies

• Equilibrium in dominant strategies▫ .utome o' a game in whih eah 'irm is doing the best it an regardless o' what its om"etitors are doing

▫ ."timal strategy is determined without worrying aboutations o' other "layers

•


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Dominant Strategies

• Game =ithout Dominant Strategy▫ The o"timal deision o' a "layer without a dominant

strategy will de"end on what the other "layer does%

▫ >e*ising the "ayo'' matrix we an see a situationwhere no dominant strategy exists


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1)% * 1*% )

2)% 2+% ,

F i r m

A

"dvertise

Don(t

"dvertise

"dvertise

Don(t"dvertise

Firm B

odified "dvertising Game

-

-

-

-

4

4

4

4



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1)% * 1*% )

2)% 2+% ,

Firm A

"dvertise

Don(t

"dvertise

"dvertise

Don(t

"dvertise

Firm Bodified "dvertising Game• .bser*ations

▫ -& No dominantstrategy/ de"ends on

4’s ations

▫ 4& Dominant strategyis to -d*ertise

▫ 6irm - determines 4’s

dominant strategy andma$es its deision

aordingly


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Nash 3uiibrium 4evisited

1)% * 1*% )

2)% 2+% ,

Firm A

"dvertise

Don(t

"dvertise

"dvertise

Don(t

"dvertise

Firm BIn order for firm " todetermine hether to advertise%firm " must first try todetermine hat firm & i do.

If firm & advertises% firm "earns a profit of 1) if itadvertises and + if it does not.

If firm & does not advertise%

firm " earns a profit of 1* if itadvertises and 2) if it does not.

Thus% firm " shoud advertise iffirm & advertise% and it shoudnot advertise if firm & doesn(t.


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1)% * 1*% )

2)% 2+% ,

Firm A

"dvertise

Don(t

"dvertise

"dvertise

Don(t

"dvertise

Firm B0irm " has to determines &(sdominant strategy and ma5es

its decision accordingy.

0irm &(s dominant strategy is

to advertise% therefore% theoptima strategy for firm " is

aso to advertise. This is Nash

e3uiibrium.

6ny hen each payer has

chose its optima strategy

given the strategy of the

other payer do e have

Nash e3uiibrium%


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• - Nash equilibrium is a situation in whih eah "layer hooses the best strategy, in light o' thestrategies hosen by the other "layer or "layers

• - dominant strategy is stable, but in many games one or more "arty does not ha*e a dominant strategy%

• - more general equilibrium one"t is the Nash Equilibrium introdued in ha"ter 78▫ - set o' strategies !or ations# suh that eah "layer is doing the best

it an gi*en the ations o' its o""onents• None o' the "layers ha*e inenti*e to de*iate 'rom its Nashstrategy, there'ore it is stable▫ (n the Cournot model, eah 'irm sets its own "rie assuming the

other 'irms out"uts are 'ixed% Cournot equilibrium is a Nash

Equilibrium


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• Dominant Strategy▫ 0(’m doing the best ( an no matter what you do%

?ou’re doing the best you an no matter what ( do%2• Nash Equilibrium

▫ 0(’m doing the best ( an gi*en what you are doing%?ou’re doing the best you an gi*en what ( am doing%2

• Dominant strategy is s"eial ase o' Nashequilibrium


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• Two ereal om"anies 'ae a mar$et in whih twonew ty"es o' ereal an be suess'ully introdued

"ro*ided eah ty"e is introdued by only one 'irm

• Produt Choie Problem▫ Mar$et 'or one "roduer o' ris"y ereal

▫ Mar$et 'or one "roduer o' sweet ereal

▫ Eah 'irm only has the resoures to introdue one ereal

▫ Nonoo"erati*e


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Produt Choie Problem

F i r m

1

Crispy Seet

Crispy

Seet

Firm 2

7*% 7* 1)% 1)

7*% 7*1)% 1)



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Product Choice Probem

Firm 1

Crispy Seet

Crispy

Seet

Firm 2

7*% 7* 1)% 1)

7*% 7*1)% 1)

If firm 1 hears firm 2 isintroducing a ne seetcerea% its best action is toma5e crispy

&ottom eft corner is Nashe3uiibrium

-hat is other Nash3uiibrium/8pper right7hand corner.


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&"C9 :6C"TI6N G"

0IG84 1;.1

T9 &"C9 :6C"TI6N G"

?ou !Y # and a om"etitor !C # "lan to sell so't drin$s on a beah%

(' sunbathers are s"read e*enly aross the beah and will wal$ to the losest *endor,the two o' you will loate next to eah other at the enter o' the beah% This is the only

Nash equilibrium%

(' your om"etitor loated at "oint A, you would want to mo*e until you were 3ust to

the le't, where you ould a"ture three+'ourths o' all sales%

4ut your om"etitor would then want to mo*e ba$ to the enter, and you would do thesame%


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Nash 3uiibrium 4evisited• Maximin Strategies @ maximi)ing minimum gain• Senario

▫ Two 'irms om"ete selling 'ile+enry"tion so'tware▫ They both use the same enry"tion standard !'ilesenry"ted by one so'tware an be read by the other +ad*antage to onsumers#

▫ 6irm 7 has a muh larger mar$et share than 6irm ▫ 4oth are onsidering in*esting in a new enry"tion

standard


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a!imin Strategy

F i r m 1

Don(t invest Invest

Firm 2

)% ) 71)% 1)

2)% 1)71))% )

Don(tinvest

Invest



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Maximin Strategy

• .bser*ations▫ (' Firm does not

in*est, Firm ! inurs

signi'iant losses !+788#

▫ Firm ! might "laydon’t in*est

Minimi)e losses to 78 @ ma!imin strategy

Firm 1

Don(t invest Invest

Firm 2

)% ) 71)% 1)

2)% 1)71))% )

Don(t invest

Invest


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a!imin Strategy

• (' both are rational and in'ormed▫ 4oth 'irms in*est

▫ Nash equilibrium

• (' Player is not rational or om"letely in'ormed▫ 6irm 7’s maximin strategy is to not in*est

▫ 6irm ’s maximin strategy is to in*est%▫ (' 6irm 7 $nows 6irm is using a maximin strategy,6irm 7 would in*est


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a!imin Strategy

• (' 'irm 7 is unsure about what 'irm will do, it anassign "robabilities to eah "ossible ation

▫ Could use a strategy that maximi)es its ex"eted "ayo''

▫ 6irm 7’s strategy de"ends ritially on its assessment o' "robabilities 'or 'irm


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Prisoners( Diemma

P r i s o

n e

r A

Confess Don(t Confess

Confess

Don(t

Confess

Prisoner B

*% * 1% 1)

2% 21)% 1

-oud you choose to confess/


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Prisoners( Diemma• =hat is the dominant strategy 'or eah "risoner1

• =hat is the Nash Equilibrium1

• =hat is maximin strategy1

The ideal outcome is one in which neither prisoner confesses, so that both get

two years in prison.Confessing, however, is a dominant strategy for each prisoner—it yields ahigher payoff regardless of the strategy of the other prisoner.

Dominant strategies are also maximin strategies. The outcome in which bothprisoners confess is both a ash e!uilibrium and a maximin solution. Thus, in a

very strong sense, it is rational for each prisoner to confess.


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Prisoners( Diemma

• - situation in whih "ursuing dominant strategiesresults in non+oo"eration that lea*es e*eryone worse

o''%• The one"t o' the "risoner’s dilemma an be used to

analy)e "rie and non+"rie om"etition inoligo"olisti mar$ets, as well as the inenti*e to heat

in a artel !i%e%, the tendeny to seretly ut "ries orto sell more than the alloated quota%


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Prisoners( Diemma

F i r m

A

:o Price 9igh Price

:o Price

9igh Price

Firm B

2% 2 *% 1

;% ;1% *



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Prisoners( Diemma6irm -&• (' 'irm 4 harged a low "rie !say A9#, 'irm - would

earn a "ro'it o' i' it also harged the low "rie !A9# and7 i' it harge a high "rie !say, A;#%

• (' 'irm 4 harged the high "rie !A;#, 'irm - would earna "ro'it o' : i' it harged the low "rie and B i' it hargedthe high "rie%

• Thus, 'irm - should ado"t its dominant strategy o'harging the low "rie%


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Prisoners( Diemma• 6irm 4&• (' 'irm - harged a low "rie !say A9#, 'irm 4 would

earn a "ro'it o' i' it also harged the low "rie !A9# and

7 i' it harge a high "rie !say, A;#%• (' 'irm - harged the high "rie !A;#, 'irm 4 would earn

a "ro'it o' : i' it harged the low "rie and B i' it hargedthe high "rie%

• Thus, 'irm 4 should ado"t its dominant strategy o'harging the low "rie%


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Prisoners( Diemma

F i r m

A

:o Price 9igh Price

:o Price

9igh Price

Firm B

2% 2 *% 1

;% ;1% *


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Prisoners( Diemma

•


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Prisoners( Diemma

F i r m

A

:o Price 9igh Price

:o Price

9igh Price

Firm B

2% 2 *% 1

;% ;1% *


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Prisoners( Diemma• Su""ose that 'irm - harged the high "rie with the

ex"etation that 'irm 4 would also harge the high "iee

!so that eah 'irm would earn a "ro'it o' B#%

• Gi*en that 'irm - has harged the higher, howe*er, 'irm4 now has an inenti*e to harge the low "rie, beause

by doing so it an inrease its "ro'its to :%


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Prisoners( Diemma

F i r m

A

:o Price 9igh Price

:o Price

9igh Price

Firm B

2% 2 *% 1

;% ;1% *


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Prisoners( Diemma

• Su""ose that 'irm 4 harged the high "rie with theex"etation that 'irm - would also harge the high "iee

!so that eah 'irm would earn a "ro'it o' B#%

• Gi*en that 'irm 4 has harged the higher, howe*er, 'irm- now has an inenti*e to harge the low "rie, beause

by doing so it an inrease its "ro'its to :%


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Prisoners( Diemma

• The net result is that eah 'irm harges the low "rie andearns a "ro'it o' only %

• .nly i' the two 'irms oo"erate and both harge the high

"rie will they earn the highest "ro'it o' B !ando*erome their dilemma#

• The one"t o' the "risoner’s dilemma an be used toanaly)e "rie and non+"rie om"etition in oligo"olisti

mar$ets, as well as the inenti*e to heat in a artel !i%e%,the tendeny to seretly ut "ries or to sell more thanthe alloated quota%


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Can 0irms scape the Prisoner(s Diemma/

Price eadership - 'orm o' im"liit ollusion in whih one

'irm in an oligo"oly announes a "rie hange and the other

'irms in the industry math the hange%

Eg% through 78s, General Motors would announe a "rie

hange at the beginning i' a model year, and 6ord and Chrysler

would math GM’s "rie hange%


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Carte Cheating

F i

r m

A

Cheat Don(t cheat

Cheat

Don(t cheat

Firm B

2% 2 *% 1

;% ;1% *


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Carte Cheating• Eah 'irm ado"ts its dominant strategy o' heating earns

a "ro'it o' %• 4uy not heating, eah member o' the Cartel would earn

the highest "ro'it o' B%• The Cartel members then 'ae the "risoners’ dilemma%• - Cartel an "re*ent or redue the "robability o'

heating by monitoring the sales o' eah member and "unishing heaters%

topic 12 game theory

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