M9302 Mathematical Models in Economics
Instructor: Georgi Burlakov
2.5.Repeated Games
Lecture 4
07.04.2011
OPVK_MU.tif

o
oThe aim of the forth lecture is to describe a special subclass of dynamic games of complete and
perfect information called repeated games
o
oKey question: Can threats and promises about future behavior influence current behavior in
repeated relationships?
Repeated Games

o
oLet G = {A1,…,An; u1,…,un} denote a static game of complete information in which player 1 through
player n simultaneously choose actions a1 through an
o from the action spaces A1 through An.
oRespectively, the payoffs are u(a,…,a) through u(a,…,a)
oAllow for any finite number of repetitions.
oThen, G is called the stage game of the repeated game
Repeated Games

o
oFinitely repeated game: Given a stage game G, let G(T) denote the finitely repeated game in which
G is played T times, with the outcomes of all preceding plays observed before the next play begins.
oThe payoffs for G(T) are simply the sum of the payoffs from the T stage games.
Finitely Repeated Game

o
oIn the finitely repeated game G(T), a subgame beginning at stage t+1 is the repeated game in which
G is played T-t times, denoted G(T-t).
oThere are many subgames that begin in stage t+1, one for each of the possible histories of play
through stage t.
oThe tth stage of a repeated game (t<T) is not a subgame of the repeated game.
o
Finitely Repeated Game

Example: 2-stage Students’ Dilemma
o
oGrading Policy:
o
othe students over the average have a STRONG PASS (Grade A, or 1),
othe ones with average performance get a WEAK PASS (Grade C, or 3) and
owho is under the average
o FAIL (Grade F, or 5).

Example: 2-stage Students’ Dilemma
o
oLeisure Rule: HARD study schedule devotes twice more time (leisure = 1) to studying than the EASY
one (leisure = 2).
Player i’s
 choice
Others’ choice
LEISURE
GRADE
Player i’ payoff
Easy
All Easy
2
3
-1
At least one Hard
Hard
At least one Easy
All Hard
Player i’s
 choice
Others’ choice
LEISURE
GRADE
Player i’ payoff
Easy
All Easy
At least one Hard
Hard
At least one Easy
1
1
0
All Hard
Player i’s
 choice
Others’ choice
LEISURE
GRADE
Player i’ payoff
Easy
All Easy
At least one Hard
2
5
-3
Hard
At least one Easy
All Hard
Player i’s
 choice
Others’ choice
LEISURE
GRADE
Player i’ payoff
Easy
All Easy
At least one Hard
Hard
At least one Easy
All Hard
1
3
-2
Player i’s
 choice
Others’ choice
LEISURE
GRADE
Player i’ payoff
Easy
All Easy
At least one Hard
Hard
At least one Easy
All Hard

Example: 2-stage Students’ Dilemma
o
oBi-matrix of payoffs:
o
o
o
o
o
oRepeat the stage game twice!

Easy
Hard
Easy
-1,-1
-3,0
Hard
0,-3
-2,-2
YOU
OTHERS

Example: 2-stage Students’ Dilemma
o
o
o
o
o
o

Easy
Hard
Easy
-1,-1
-3,0
Hard
0,-3
-2,-2
YOU
OTHERS

Easy
Hard
Easy
-3,-3
-5,-2
Hard
-2,-5
-4,-4
YOU
OTHERS
Stage 2:
Stage 1:

Finitely Repeated Game
o
oProposition: If the stage game G has a unique Nash equilibrium then, for any finite T, the
repeated game G(T) has a unique subgame-perfect outcome:
o
nThe Nash equilibrium of G is played in every stage.

Finitely Repeated Game
o
oWhat if the stage game has no unique solution?
nIf G = {A1,…,An; u1,…,un} is a static game of complete information with multiple Nash equilibria,
there may be subgame-perfect outcomes of the repeated game G(T) in which the outcome in stage t<T
is not a Nash equilibrium in G.
o

o
oGrading Policy:
o
othe students over the average have a STRONG PASS (Grade A, or 1),
othe ones with average performance get a PASS (Grade B, or 2) and
owho is under the average
o FAIL (Grade F, or 5).
Example: 2-stage Students’ Dilemma - 2

Example: 2-stage Students’ Dilemma - 2
o
oLeisure Rule: HARD study schedule devotes all their time (leisure = 0) to studying than the EASY
one (leisure = 2).
Player i’s
 choice
Others’ choice
LEISURE
GRADE
Player i’ payoff
Easy
All Easy
2
At least one Hard
2
5
-3
Hard
At least one Easy
0
1
-1
All Hard
0
2
0
2
-2

Example: 2-stage Students’ Dilemma - 2
o
o
o
o
o
o

Easy
Hard
Easy
0,0
-3,-1
Hard
-1,-3
-2,-2
YOU
OTHERS
Suppose each player’s strategy is:
•Play Easy in the 2nd stage if the 1st stage outcome is
(Easy, Easy)
•Play Hard in the 2nd stage for any other 1st stage outcome

Example: 2-stage Students’ Dilemma - 2
o
o
o
o
o
o

Easy
Hard
Easy
0,0
-3,-1
Hard
-1,-3
-2,-2
YOU
OTHERS

Easy
Hard
Easy
0,0
-5,-3
Hard
-3,-5
-4,-4
YOU
OTHERS
Stage 2:
Stage 1:

Example: 2-stage Students’ Dilemma - 2
o
o
o
o
o
o

Easy
Hard
Easy
0,0
-5,-3
Hard
-3,-5
-4,-4
YOU
OTHERS
Stage 1:
The threat of player i to punish in the 2nd stage player j’s cheating in the 1st stage is not
credible.

Example: 2-stage Students’ Dilemma - 2
o
o
o
o
o
o

Easy
Hard
Easy
0,0
-3,-1
Hard
-1,-3
-2,-2
YOU
OTHERS
Stage 1:
{Easy, Easy} Pareto-dominates {Hard, Hard}
in the second stage. There is space for re-negotiation because punishment hurts punisher as well.

Example: 2-stage Students’ Dilemma - 2
o
o
o
o
o
o

Easy
Hard
C
O
Easy
0,0
-3,-1
-3,-3
-3,-3
Hard
-1,-3
-2,-2
-3,-3
-3,3
C
-3,-3
-3,-3
0,-2.5
-3,-3
O
-3,-3
-3,-3
-3,-3
-2.5,0
YOU
OTHERS
Add 2 more actions and suppose each player’s strategy is:
•Play Easy in the 2nd stage if the 1st stage outcome is (E, E)
•Play C in the 2nd stage if the 1st stage outcome is (E, w≠E)
•Play O in the 2nd stage if the 1st stage is (y≠E,z=E/H/C/O)
•Outcomes{C,C} and {O,O} are on the Pareto frontier.

o
oConclusion: Credible threats or promises about future behavior which leave no space for
negotiation (Pareto improvement) in the final stage can influence current behavior in a finite
repeated game.
o
Finitely Repeated Game

Infinitely Repeated Game
oGiven a stage-game G, let G(∞,δ) denote the infinitely repeated game in which G is repeated
forever and the players share the discount factor δ.
oFor each t, the outcomes of the t-1 preceding plays of G are observed.
oEach player’s payoff in G(∞,δ) is the present value of the player’s payoffs from the infinite
sequence of stage games

Infinitely Repeated Game
o
o
oThe history of play through stage t – in the finitely repeated game G(T) or the infinitely
repeated game G(∞,δ) – is the record of the player’s choices in stages 1 through t.

oStrategy /in a repeated game/ - the sequence of actions the player will take in each stage, for
each possible history of play through the previous stage.
oSubgame /in a repeated game/ - the piece of the game that remains to be played beginning at any
point at which the complete history of the game thus far is common knowledge among the players.
o
Infinitely Repeated Game

o
oAs in the finite-horizon case, there are as many subgames beginning at stage t+1 of G(∞,δ) as
there are possible histories through stage t.
oIn the infinitely repeated game G(∞,δ), each subgame beginning at stage t+1 is identical to the
original game.
o
Infinitely Repeated Game

oHow to compute the player’s payoff of an infinitely repeated game?
nSimply summing the payoffs of all stage-games does not provide a useful measure
nPresent value of the infinite sequence of payoffs:
o
o
Infinitely Repeated Game

o
oKey result: Even when the stage game has a unique Nash equilibrium it does not need to be present
in every stage of a SGP outcome of the infinitely repeated game.
oThe result follows the argument of the analysis of the 2-stage repeated game with credible
punishment.
Infinitely Repeated Game

Infinitely Repeated Game

Easy
Hard
C
O
Easy
0,0
-3,-1
-3,-3
-3,-3
Hard
-1,-3
-2,-2
-3,-3
-3,3
C
-3,-3
-3,-3
0,-2.5
-3,-3
O
-3,-3
-3,-3
-3,-3
-2.5,0
YOU
OTHERS
Instead of adding artificial equilibria that brings higher payoff tomorrow, the Pareto dominant
action is played.

Example: Infinite Students’ Dilemma
o
oGrading Policy:
o
othe students over the average have a STRONG PASS (Grade A, or 1),
othe ones with average performance get a WEAK PASS (Grade C, or 3) and
owho is under the average
o FAIL (Grade F, or 5).

Example: Infinite Students’ Dilemma
o
oLeisure Rule: HARD study schedule devotes twice more time (leisure = 1) to studying than the EASY
one (leisure = 2).
Player i’s
 choice
Others’ choice
LEISURE
GRADE
Player i’ payoff
Easy
All Easy
2
3
-1
At least one Hard
Hard
At least one Easy
All Hard
Player i’s
 choice
Others’ choice
LEISURE
GRADE
Player i’ payoff
Easy
All Easy
At least one Hard
Hard
At least one Easy
1
1
0
All Hard
Player i’s
 choice
Others’ choice
LEISURE
GRADE
Player i’ payoff
Easy
All Easy
At least one Hard
2
5
-3
Hard
At least one Easy
All Hard
Player i’s
 choice
Others’ choice
LEISURE
GRADE
Player i’ payoff
Easy
All Easy
At least one Hard
Hard
At least one Easy
All Hard
1
3
-2
Player i’s
 choice
Others’ choice
LEISURE
GRADE
Player i’ payoff
Easy
All Easy
At least one Hard
Hard
At least one Easy
All Hard

Example: Infinite Students’ Dilemma
o
oBi-matrix of payoffs:
o
o
o
o
o
oRepeat the stage game infinitely!

Easy
Hard
Easy
-1,-1
-3,0
Hard
0,-3
-2,-2
YOU
OTHERS

Example: Infinite Students’ Dilemma
oConsider the following trigger strategy:
nPlay Easy in the 1st stage.
nIn the tth stage if the outcome of all t-1 preceding stages has been (E, E) then play Easy,
nOtherwise, play Hard in the tth stage.
oNeed to define δ for which the trigger strategy is SGPNE.

Example: Infinite Students’ Dilemma
o
oSubgames could be grouped into 2 classes:
nSubgames in which the outcome of at least one earlier stage differs from (E,E) – trigger strategy
fails to induce cooperation
nSubgames in which all the outcomes of the earlier stages have been (E,E) – trigger strategy
induces cooperation

Example: Infinite Students’ Dilemma
oIf HARD is played in the 1st stage total payoff is:
o
o
oIf EASY is played in the 1st stage, let the present discounted value be V:
o

Example: Infinite Students’ Dilemma
oIn order to have a SGPE where (E, E) is played in all the stages till infinity the following
inequality must hold:
o
o
oAfter substituting for V we get the following condition on δ:
o

Folk’s Theorem
oIn order to generalize the result of the SD to hold for all infinitely repeated games, several key
terms need to be introduced:
oThe payoffs (x1,…,xn) are called feasible in the stage game G if they are a convex (i.e. weighted
average, with weights from 0 to1) combination of the pure-strategy payoffs of G.

Folk’s Theorem
o
oThe average payoff from an infinite sequence of stage-game payoffs is the payoff that would have
to be received in every stage so as to yield the same present value as the player’s infinite
sequence of stage-game payoffs.
oGiven the discount factor δ, the average payoff of the infinite sequence of payoffs
o     is:

Folk’s Theorem
oFolk’s Theorem (Friedman 1971): Let G be a finite, static game of complete information. Let
(e1,…,en) denote the payoffs from a Nash Equilibrium of G, and let (x1,…,xn) denote any other
feasible payoffs from G.
oIf xi > ei for every player i and if δ is sufficiently close to 1, then there exists a
subgame-perfect Nash equilibrium of the infinitely repeated game G(∞,δ) that achieves (x1,…,xn) as
the average payoff.
o

Folk’s Theorem
o
oReservation payoff ri – the largest payoff player i can guarantee receiving, no matter what the
other players do.
oIt must be that           , since if ri were greater than ei, it would not be a best response for
player i to play her Nash equilibrium strategy.
oIn SD, ri = ei but in the Cournot Duopoly Game (and typically) ri < ei

Folk’s Theorem
o
oFolk’s Theorem (Fudenber & Maskin 1986): If (x1, x2) is a feasible payoff from G, with xi>ri for
each i, then for δ sufficiently close to 1, there exists a SGPNE of G(∞,δ) that achieves (x1, x2)
as the average payoff even if xi<ei for one or both of the players.

Folk’s Theorem
o
oWhat if δ is close to 0?
n1st Approach: After deviation follow the trigger strategy and play the stage-game equilibrium.
n2nd Approach (Abreu 1988): After deviation play the N.E. that yields the lowest payoff of all N.E.
Average strategy can be lower than the one of the 1st approach if switching to stage game is not
the strongest credible punishment.

Summary
oKey question that stays behind repeated games is whether threats or promises about future behavior
can affect current behavior in repeated relationships.
oIn finite games, if the stage game has a unique Nash Equilibrium, repetition makes the threat of
deviation credible.
oIf stage game has multiple equilibria however there could be a space for negotiating the
punishment in the next stage after deviation.

Summary
oIn infinitely repeated games, even when the stage game has a unique Nash equilibrium it does not
need to be present in every stage of a SGP outcome of the infinitely repeated game.
oFolk’s theorem implies that if there is a set of feasible payoffs that are larger than the payoffs
from the stage game Nash equilibrium, and the discount factor is close to one, there is a SGPNE at
which the set of higher feasible payoffs is achieved as an average payoff.

Summary
oExtension of the Folk’s theorem implies that for 2-player infinitely repeated game if there is a
set of feasible payoffs that exceed the reservation ones, the outcome that yields these feasible
payoffs as an average payoff could constitute a  SGPNE even if they are smaller than the payoffs
from the stage game N.E., provided that the discount factor is close to 1.
oIf the discount factor is close to 0, an alternative strategy to the trigger one (where the stage
game equilibrium is played after deviation) is to play instead the N.E. that yields the lowest
payoff of all N.E. This might be stronger credible punishment.