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EC487 Advanced Microeconomics, Part I:Lecture 10
Leonardo Felli
32L.LG.04
1 December 2017
Repeated Games
I This is the class of dynamic games which is best understoodin game theory.
I Players face in each period the same normal form stage game.
I Players’ payoffs are a weighted discounted average of thepayoffs players receive in every stage game.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 2 / 66
Repeated Games (cont’d)
Main point of the analysis:
I players’ overall payoffs depend on the present and the futurestage game payoffs,
I it is possible that the threat of a lower future payoff mayinduce a player at present to choose a strategy different fromthe stage game best reply.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 3 / 66
Example: the repeated prisoner dilemma
I Stage game:1\2 C D
C 1, 1 −1, 2
D 2,−1 0, 0
I Per period payoff depends on current action: gi (at) .
I Players’ common discount factor δ.
I It is convenient to label the first period t = 0.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 4 / 66
Repeated Prisoner Dilemma (cont’d)
I Since we are going to compare the equilibrium payoffs fordifferent time horizons we need to re-normalize the payoffs sothat they are comparable.
I The average discounted payoff for a T -periods game is:
Π =1− δ
1− δTT−1∑t=0
δtgi (at)
I Clearly if gi (at) = 1
Π =1− δ
1− δTT−1∑t=0
δt =
(1− δ
1− δT
)(1− δT
1− δ
)= 1
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 5 / 66
Finitely Repeated Prisoner Dilemma
I Assume first that the prisoners’ dilemma game is repeated afinite number of times.
I Nash equilibrium payoffs of the stage game: (0, 0).
I Subgame Perfect equilibrium strategies: each player choosesaction D independently of the period and the action the otherplayer chose in the past.
1\2 C D
C 1, 1 −1, 2
D 2,−1 0, 0
Proof: backward induction.
I Subgame Perfection seems to prevent any gain from repeated,but finite interaction, but...
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 6 / 66
Finitely Repeated Game
I Consider a different finitely repeated game.
I Stage game:L C R
T 1, 1 5, 0 0, 0
M 0, 5 4, 4 0, 0
B 0, 0 0, 0 3, 3
I Nash equilibria of the stage game: (T , L) and (B,R).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 7 / 66
Finitely Repeated Game (cont’d)
Assume the game is played twice and consider the followingstrategies:
Player 1:
I play M in the first period;
I in the second period play B if the observed outcome is (M,C );
I in the second period play T if the observed outcome is not(M,C );
Player 2:
I play C in the first period;
I in the second period play R if the observed outcome is (M,C );
I in the second period play L if the observed outcome is not(M,C );
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 8 / 66
Finitely Repeated Game (cont’d)
Proposition
If δ ≥ 12 then these strategies are a subgame perfect equilibrium of
the game.
L C R
T 1, 1 5, 0 0, 0
M 0, 5 4, 4 0, 0
B 0, 0 0, 0 3, 3
Proof: Backward induction: in the last period the strategiesprescribe a Nash equilibrium. In the first period both player 1 andplayer 2 conform to the strategies if and only if:(
1− δ1− δ2
)[4 + δ 3] =
4 + δ 3
1 + δ≥(
1− δ1− δ2
)[5 + δ] =
5 + δ
1 + δ
The inequality is satisfied for δ ≥ 12 .
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 9 / 66
Infinitely Repeated Prisoner Dilemma
I Consider now the the infinitely repeated prisoner dilemma:T = +∞.
I Stage game:1\2 C D
C 1, 1 −1, 2
D 2,−1 0, 0
Proposition
Both player choosing strategy D in every period is an SPE of therepeated game.
I Proof: by one deviation principle. Notice that an infinitelyrepeated game is continuous at infinity.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 10 / 66
Infinitely Repeated Prisoner Dilemma (cont’d)
Proposition
The (D,D) equilibrium is the only equilibrium if we restrictplayers’ strategies to be history independent.
Proposition
If δ ≥ 12 then the following strategy profile (σA, σB) is a SPE of
the repeated game:
I Player i chooses C in the first period.
I Player i continues to choose C as long as no player haschosen D in any previous period.
I Player i will choose D if a player has chosen D in the past(for the rest of the game).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 11 / 66
Infinitely Repeated Prisoner Dilemma (cont’d)
Proof: If a player i conforms to the prescribed strategies the payoffis 1.
If a player deviates in one period and conforms to the prescribedstrategy from there on (one deviation principle) the continuationpayoff is:
(1− δ)(2 + 0 + . . .) = (1− δ) 2
If δ ≥ 12 then
1 ≥ (1− δ) 2.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 12 / 66
Infinitely Repeated Prisoner Dilemma (cont’d)
We still need to check that in the subgame in which both playersare choosing D neither player wants to deviate.
However, choosing D in every period is a SPE of the entire gamehence it is a SPE of the (punishment) subgames.
Notice that using this type of strategies not only choosing (C ,C )in every period is a SPE outcome, a big number of other SPEoutcomes are also achievable.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 13 / 66
Infinitely Repeated Prisoner Dilemma (cont’d)
Indeed there exists a Folk Theorem.
6
-
q
q
HHHHHHHHAAAAAAAAAHH
HHHH
HHAAAAAAAAA
(1, 1)
(0, 0)
(−1, 2)
(2,−1)
Π2
Π1
AAAAAHH
HH qq
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 14 / 66
General repeated normal form game
Definition
Let G be a given stage game: a normal form game
G ={N,Ai , gi (a
t)}
Definition
Let G∞ be the infinitely repeated game associated with the stagegame above:
G∞ = {N,H,P,Ui (σ)}
such that:
I H =⋃∞
t=0 At where A0 = ∅;
I P(h) = N for every h ∈ H −Z;
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 15 / 66
Repeated Normal form Game (cont’d)
I The payoffs for the game G∞ in the case δ < 1 are:
Ui (σ) = (1− δ)∞∑t=0
δtgi (σt(ht))
I Denote ht the history known to the players at the beginningof period t: ht = {a0, a1, . . . , at−1}.
I Let Ht = At−1 to be the space of all possible period thistories.
I A pure strategy for player i ∈ {1, 2} in the game G∞ is thenthe infinite sequence of mappings: {sti }∞t=0 such thatsti : Ht → Ai .
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 16 / 66
Repeated Normal form Game (cont’d)
In general we will allow players to mix in every possible stagegame: ∆i (Ai ) set of probability distributions on Ai .
A behavioral mixed strategy in this environment is instead aninfinite sequence of mappings: {σti }∞t=0 such thatσti : Ht → ∆(Ai ).
Notice that mixed strategies cannot depend on past mixedstrategies by the opponents but only on their realizations.
The payoffs for the game G∞ in the case δ < 1 are:
Ui = Eσ(1− δ)∞∑t=0
δtgi (σt(ht))
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 17 / 66
Repeated Normal form Game (cont’d)
I Notice that Eσ(·) is the expectation with respect to thedistribution over the infinite histories generated by the profileof mixed behavioral strategies {σti }∞t=0.
I Notice that this specification of payoffs allows us toreinterpret the discount factor δ as:
I the probability that the game will be played in the followingperiod, where these probabilities are assumed to beindependent across periods.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 18 / 66
Repeated Normal form Game (cont’d)
We allow the players to coordinate their strategies through the useof a public randomizing device whose realization in period t is ωt .
Therefore a period t history for player i is:
ht = {a0, . . . , at−1;ω0, . . . , ωt}.
Proposition
If α∗ is a NE strategy profile for the stage game G , then thestrategy:
“each player i plays α∗i independently of the history ofplay”
are a NE and a SPE of the infinitely repeated game G∞(δ).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 19 / 66
Repeated Normal form Game (cont’d)
I The proof that the strategies above are a Subgame Perfectequilibrium of the game G∞(δ) is easily obtained by usingone-deviation-only principle.
I In any given period consider the deviation in the immediateperiod and then let the players continue playing theequilibrium strategies.
I Then any deviation cannot be profitable: it is a deviation fromthe Nash equilibrium action of stage game G .
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 20 / 66
Repeated Normal form Game (cont’d)
Assume now that the stage game G has n NE{αj ,∗}n
j=1.
Proposition
Then, for any map j(t) from time periods into an index of the NE{αj ,∗}n
j=1, the strategies:
“each player i plays αj(t),∗i in period t”
are a SPE of the game G∞(δ).
These SPE strategies are history independent. Therefore eachplayer’s best response in every period t is to play the stage gamebest response in t: today’s decision does not affect the future play.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 21 / 66
Repeated Normal form Game (cont’d)
I In other words, playing repeatedly the stage game G does notreduce the set of equilibrium payoffs.
I To be able to move to the Folk Theorem we first need todefine an area known as the set of feasible and individuallyrational payoffs.
I Consider as an example the following battle of sexes game G :
1\2 B F
B 1, 2 0, 0
F 0, 0 2, 1
I Assume this game is repeated an infinite number of timesG∞(δ).
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 22 / 66
Repeated Normal form Game (cont’d)
I We first need to define the set of feasible payoffs of G∞.
I Recall that when choosing actions in every period players cancoordinate using a public randomizing device.
I This implies that: every payoff associated with a pure strategyprofile a can be achieved: (1, 2), (0, 0), (2, 1).
I It also implies that every payoff generated by any linear andconvex combination of the pure strategy profile can beachieved.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 23 / 66
Repeated Normal form Game (cont’d)
I In general these payoffs are all in the convex hull of thepayoffs associated with the pure strategy profiles.
I This is the smallest convex set that includes the payoffsassociated with the pure strategy profiles.
I Formally:
V = convex hull {π | πi = gi (a) ∀a ∈ A}
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 24 / 66
Repeated Normal form Game (cont’d)
Graphically:
6
-��������@@@@��������q
(0, 0)
(1, 2)
(2, 1)
π2
π1
V
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 25 / 66
Repeated Normal form Game (cont’d)
I We now define the set of individually rational payoffs of G∞.
I We first need to define the minmaxing payoff of each player.
I The minmaxing payoff to player 1 is the lowest payoff thatplayer 2 can impose on player 1.
I Given that player 1 is rational this is the lowest payoff amongthe ones that are player 1’s best reply to player 2 strategies.
I This payoff is a best reply for player 1 since he is trying toachieve the best for himself, given his rationality.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 26 / 66
Repeated Normal form Game (cont’d)
I In other words, among the best reply payoffs for player 1,player 2 chooses her strategy that minimizes these payoffs.
I In the battle of sexes game:
1\2 B F
B 1, 2 0, 0
F 0, 0 2, 1
I the minmax payoff for player 1 is π1 = 1.
I the minmax payoff for player 2 is π2 = 1.
I In general:
πi = minα−i
[maxαi
gi (αi , α−i )
]
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 27 / 66
Repeated Normal form Game (cont’d)
I Denote mi−i the profile of minimax strategies for players −i if
they minmax player i .
I This is the lowest payoff player i ’s opponents can hold player ito by choice of α−i .
Definition
A payoff πi for player i is individually rational if and only if:
πi ≥ πi = minα−i
[maxαi
gi (αi , α−i )
]
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 28 / 66
Repeated Normal form Game (cont’d)
Definition
We define the set of individual rational payoffs to be the set ofpayoffs that give to each player a payoff
I = {(πi , π−i ) | πi ≥ πi}
The relevant set for us is the set of feasible and individuallyrational payoffs:
V = I ∩ V
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 29 / 66
Repeated Normal form Game (16)
The region of feasible and individually rational payoffs V:
6
-����������@@@@@����������q
(0, 0)
(1, 2)
(2, 1)
π2
π1
q @@@@@q
qqq
qπ2
π1
6
(1, 1)
V
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 30 / 66
Repeated Normal form Game (cont’d)
I Consider the following game:
1\2 L R
U −2, 2 1,−2
M 1,−2 −1, 2
D 0, 1 0, 1
I Restricting attention to pure strategies then we obtain:
I m21 = D and π2 = 1;
I m12 ∈ {L,R} and π1 = 1 .
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 31 / 66
Repeated Normal form Game (cont’d)
I Consider mixed strategies: assume that player 2 randomizeswith probability q on L.
I Then player 1’s expected payoffs for every possible strategychoice are:
Π1(U, q) = 1− 3q
Π1(M, q) = 2q − 1
Π1(D, q) = 0
I This implies that m12 = q ∈
[13 ,
12
]and Π1 = 0.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 32 / 66
Repeated Normal form Game (cont’d)
I Assume that player 1 randomizes with probability pU on Uand probability pM on M.
I Then player 2’s expected payoffs for every possible strategychoice are:
Π2(pU , pM , L) = 2 (pU − pM) + 1− pU − pM
Π2(pU , pM ,R) = 2 (pM − pU) + 1− pU − pM
I This implies that m21 = (pU , pM) =
(12 ,
12
)and that Π2 = 0.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 33 / 66
Folk Theorem
I Consider a general finite normal form stage game:
G = {N;Ai , gi (a),∀i ∈ N}
I and the dynamic game that consist of the infinitely repeatedplay of the game G when players’ discount factor is δ:
G∞(δ).
I The payoffs of the infinitely repeated game are:
πi = (1− δ)∞∑t=0
δt gi (ai , a−i )
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 34 / 66
Subgame Perfect Folk Theorem
Theorem (Subgame Perfect Folk Theorem – Fudenberg andMaskin (1986))
Consider a stage game such that dim(V) = #N, where #Ndenotes the number of players and V denotes the set of feasibleand individually rational payoffs.
Then for any v ∈ V such that (vi > πi ), there exists a δv such thatfor every δ ≥ δv there exists a SPE of G∞(δ) with payoff vector v .
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 35 / 66
Subgame Perfect Folk Theorem (cont’d)
I Notice that the extra condition dim(V) = #N is not tight, inparticular the theorem can be proved when
dim(V) = #N − 1.
I The dimensionality assumption is for example satisfied in thefollowing example.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 36 / 66
Subgame Perfect Folk Theorem: Example
I Let G be the following finite normal form game:
L R
U 2, 1 0, 2
D 0, 0 −1,−1
I Consider the dynamic game G∞(δ).
I The minmax payoff for both players in pure and mixedstrategies are:
π1 = 0 π2 = 0.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 37 / 66
Subgame Perfect Folk Theorem: Example (cont’d)
I The set V satisfies the dimensionality assumption dim(V) = 2:
6
-
(0, 2)
(0, 0)
(−1,−1)
(2, 1)
π2
π1
Π2
Π1
V
HHHHHHHH
q
������������������������
������HH
HHHH
HH
qHH
HHHH
HHq
q
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 38 / 66
Subgame Perfect Folk Theorem: Proof
Proof:
I For simplicity we focus on the case in which there exists apure action profile a such that
g(a) = v .
I Assume first that the minmax action profile mi−i for every
i ∈ N is also a pure strategy so that any deviation fromminmax behavior is easy to detect.
I Choose v ′ ∈ int(V) — recall that (vi > πi ) — such that:
πi < v ′i < vi ∀i ∈ N
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 39 / 66
Subgame Perfect Folk Theorem: Proof (cont’d)
I Choose also an ε > 0 and a
v ′(i) = (v ′1 + ε, . . . , v ′i−1 + ε, v ′i , v′i+1 + ε, . . . , v ′I + ε)
such that:v ′(i) ∈ V ∀i ∈ N.
I Notice that the role of the full-dimensionality assumption is toassure that there exists a v ′(i) for all i and for some ε and v ′.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 40 / 66
Subgame Perfect Folk Theorem: Proof (cont’d)
I Once again for simplicity assume that for every i ∈ N thereexists an action profile a(i) such that
g(a(i)) = v ′(i).
I Further denote w ji = gi (m
j) player i ’s payoff when minmaxingplayer j .
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 41 / 66
Subgame Perfect Folk Theorem: Proof (cont’d)
I Finally, choose n such that
maxa
gi (a) + nπi < mina
gi (a) + nv ′i
or
n >maxa
gi (a)−mina
gi (a)
v ′i − πi.
I Clearly there exists an n satisfying the condition above beingthe numerator bounded above and the denominator boundedbelow.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 42 / 66
Subgame Perfect Folk Theorem: Proof (cont’d)
I We label n the length of a punishment.
I To understand the condition above notice that for δ close to 1:
(1− δn) ' (1− δ)n.
I Consider now the following strategy profile.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 43 / 66
Subgame Perfect Folk Theorem: Proof (cont’d)
1. The play starts in Phase I.
Phase I: play the action profile a, (g(a) = v).
2. The play remains in Phase I so long as in each period:
I either the realized action is a
I or the realized action differs from a in two or morecomponents.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 44 / 66
Subgame Perfect Folk Theorem: Proof (cont’d)
3. If a single player j deviates from a then the play moves toPhase IIj .
Phase IIj : play mj each period.
4. The play stays in Phase IIj for n periods so long as in eachperiod:
I either the realized action is mj
I or the realized action differs from mj in two or morecomponents.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 45 / 66
Subgame Perfect Folk Theorem: Proof (cont’d)
5. After n subsequent periods in Phase IIj the play switches toPhase IIIj .
Phase IIIj : play a(j).
6. If during Phase IIj a single player i ’s action differs from mji
begin Phase IIi .
7. The play stays in Phase IIIj so long as in each period:
I either the realized action is a(j)
I or the realized action differs from a(j) in two or morecomponents.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 46 / 66
Subgame Perfect Folk Theorem: Proof (cont’d)
8. If during Phase IIIj a single player i ’s action differs from ai (j)then begin Phase IIi .
I Using one-deviation-principle we check now that no player hasan incentive to deviate from the prescribed action in anysubgame.
I Clearly each phase corresponds to a different type of propersubgame.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 47 / 66
Subgame Perfect Folk Theorem: Proof (cont’d)
Consider Phase I.
I By conforming player i receives payoff vi while by deviating hecannot receive a payoff higher than:
πi = (1− δ) maxa
gi (a) + δ[(1− δn)πi + δnv ′i
]I Since by construction vi > v ′i for δ sufficiently close to 1:πi < vi .
I Notice indeed that if δ = 0 then πi = maxa
gi (a) and if δ = 1
then πi = v ′i .
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 48 / 66
Subgame Perfect Folk Theorem: Proof (cont’d)
Consider Phase IIIj , j 6= i .
I By conforming player i receives payoff v ′i + ε while bydeviating he cannot receive more than:
πi = (1− δ) maxa
gi (a) + δ[(1− δn)πi + δnv ′i
]
I Payoff πi < v ′i + ε for δ sufficiently close to 1.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 49 / 66
Subgame Perfect Folk Theorem: Proof (cont’d)
Consider Phase IIIi .
I By conforming player i receives payoff v ′i while by deviating hecannot receive more than:
πi = (1− δ) maxa
gi (a) + δ[(1− δn)πi + δnv ′i
]I Indeed we need:
v ′i > (1− δ) maxa
gi (a) + δ[(1− δn)πi + δnv ′i
]I That can be re-written as:
(1− δn+1)v ′i > (1− δ) maxa
gi (a) + δ (1− δn)πi
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 50 / 66
Subgame Perfect Folk Theorem: Proof (cont’d)
I Using the approximation (1− δn) ' (1− δ)n we get:
(n + 1)v ′i > maxa
gi (a) + δnπi
I Since v ′i > mina gi (a) and δ < 1 the following is a sufficientcondition for the inequality above:
mina
gi (a) + nv ′i > maxa
gi (a) + nπi
I Clearly from the definition of n for δ sufficiently close to 1
v ′i > πi
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 51 / 66
Subgame Perfect Folk Theorem: Proof (cont’d)
Consider Phase IIj , j 6= i .
I If n′ periods remaining in Phase IIj player i ’s payoff byconforming is
ui =(
1− δn′)w ji + δn
′(v ′i + ε)
I while by deviating he cannot obtain more than:
πi = (1− δ) maxa
gi (a) + δ[(1− δn) v i + δnv ′i
]I Notice that for δ sufficiently close to 1
ui > πi
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 52 / 66
Subgame Perfect Folk Theorem: Proof (cont’d)
Finally consider Phase IIi .
I If n′ < n periods remain in Phase IIi player i ’s payoff byconforming is
u′i =(
1− δn′)πi + δn
′v ′i
I while by deviating:
π′i = (1− δn)πi + δnv ′i
I Clearly u′i > π′i .
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 53 / 66
Application of Repeated Games: Cartels
I Consider two firms repeatedly involved in a Cournot Duopolyfor an infinite number of periods.
I Both firms produce a perfectly homogeneous good with costfunctions:
c(qi ) = c qi ∀i ∈ {1, 2}.
I and inverse demand function:
P(q1 + q2) = a− (q1 + q2)
where c < a.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 54 / 66
Application of Repeated Games: Cartels (cont’d)
I The two firms’ profit functions are:
Π1(q1, q2) = q1 [a− (q1 + q2)− c]
Π2(q1, q2) = q2 [a− (q1 + q2)− c]
I The stage game equilibrium choices (q1, q2) are:
maxq1∈R+
q1 [a− (q1 + q2)− c]
maxq2∈R+
q2 [a− (q1 + q2)− c]
I which is the solution to the following problem:
qc1 =1
2(a− qc2 − c) qc2 =
1
2(a− qc1 − c) .
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 55 / 66
Application of Repeated Games: Cartels (cont’d)
I This solution is:
qc1 = qc2 =(a− c)
3
I with profits:
πc1 = πc2 =(a− c)2
9
I Consider now a single firm that is a monopolist in this marketand produces a quantity Q.
I This firm profit maximization problem is:
maxQ∈R+
Q [a− Q − c]
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 56 / 66
Application of Repeated Games: Cartels (cont’d)
I The first order conditions are then:
a− 2Q − c = 0
I or the monopolist quantity:
Qm =(a− c)
2, Πm =
(a− c)2
4
I Assume now that the two firms, without any explicit deal,decide each to produce half of the monopolist quantity:
qm =1
2Qm =
(a− c)
4
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 57 / 66
Application of Repeated Games: Cartels (cont’d)
I Each firm’s profit in this case is:
πm1 = πm2 =(a− c)2
8
I Notice that clearly:
πci =(a− c)2
9< πmi =
(a− c)2
8
I The quantity qm does dominate qci for both firms.
I However, if one of the firm, say firm 1, produces quantity
qm1 =(a− c)
4
then firm 2 can gain by choosing a different quantity.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 58 / 66
Application of Repeated Games: Cartels (cont’d)
I In particular, if firm 2 chooses the quantity:
q̄2 =(a− qm − c)
2=
3 (a− c)
8
I Then firm 2’s profit is:
π̄2 =9 (a− c)2
64
I which clearly is:
π̄2 =9 (a− c)2
64> πm =
(a− c)2
8
I This is the reason why for both firms to choose (qm1 , qm2 ) is
not a Nash equilibrium of the Cournot model.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 59 / 66
Application of Repeated Games: Cartels (cont’d)
I Assume however that the two firms compete for an infinitenumber of periods.
I Consider the following strategies:
I Firm 1:
I choose quantity qm1 in the first period;
I in every subsequent period choose quantity qm1 if the observedoutcome in the previous period is (qm1 , q
m2 );
I in every subsequent period choose quantity qc1 if in theprevious period you observe that firm 2 chose quantity q̄2;
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 60 / 66
Application of Repeated Games: Cartels (cont’d)
I Firm 2:
I choose quantity qm2 in the first period;
I in every subsequent period choose quantity qm2 if the observedoutcome in the previous period is (qm1 , q
m2 );
I in every subsequent period choose quantity qc2 if in theprevious period you observe that firm 2 chose quantity q̄1;
I Recall that the average discounted payoff of each firm is:
(1− δ)∞∑t=0
δtπi (t)
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 61 / 66
Application of Repeated Games: Cartels (cont’d)
I These strategies do not require an explicit agreement betweenthe two firms provided each firm believes the other firmbehaves this way.
I Question: for which δ neither firm wants to deviate fromthese strategies?
I Consider firm i :
πmi ≥ (1− δ)π̄i + δπci
or(a− c)2
8≥ (1− δ)
9 (a− c)2
64+ δ
(a− c)2
9
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 62 / 66
Application of Repeated Games: Cartels (cont’d)
I which is satisfied if and only if:
δ ≥ 9
17
I Moreover no firm has an incentive to deviate from punishmentstrategies since (qc1 , q
c2) is a Nash equilibrium of the Cournot
stage game.
I Therefore the cartel behaviour described by the strategiesabove is a Subgame Perfect equilibrium of the infinitelyrepeated game if and only if δ ≥ 9/17.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 63 / 66
Subgame Perfect Folk Theorem: Comment
I Notice that in the theory of repeated games there does notexists a commonly accepted theory predicting that the playerwill play an equilibrium whose payoff is on the Pareto-frontierof V.
I In other words nothing guarantees that the outcome will beon the Pareto frontier.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 64 / 66
Subgame Perfect Folk Theorem: Comment (cont’d)
Indeed:
6
-
(0, 2)
(0, 0)
(−1,−1)
(2, 1)
π2
π1
Π2
Π1
V
HHHHHHHH
q
������������������������
������HH
HHHH
HH
qHH
HHHH
HHq
q
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 65 / 66
Subgame Perfect Folk Theorem: Comment (cont’d)
Overall the Folk Theorem warns us to use caution when arguingthat the best way of making predictions in a strategic setting is byusing Nash and even Subgame Perfect equilibria.
I Office hours in the next two weeks:
Tuesday, Dec. 5 and 12: 11:00-13:00 am
I Exam scheduled for:
Thursday, January 4, 2018 at 14:30p.m.
Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part II 1 December 2017 66 / 66