[PPT] - Uniform Folk Theorems in Repeated Anonymous Random Matching Games PowerPoint Presentation

SLIDE 1

Uniform Folk Theorems in Repeated Anonymous Random Matching Games

Joyee Deb1 Julio Gonz´ alez-D´ ıaz2 J´ erˆ

me Renault3

1Stern School of Business, New York University

2Dept. of Statistics & Operations Research, University of Santiago de Compostela

3Toulouse School of Economics, Universit´

e Toulouse I

November 2013

SLIDE 2

Introduction Model and Preliminaries Results Conclusion

Cooperation among Strangers?

Think of two communities interacting repeatedly with one another.
A community of buyers and a community of sellers.
Every period, buyers meet sellers to conduct pair-wise trades.
Partners change randomly over time.
If the communities are large, it is possible that players...
Are unable to observe everyone else’s actions,
Do not observe each other’s true identities or recognize each other

(‘anonymous’ players).

Examples: Internet, Trading via third-parties, Inter-regional trade fairs of

medieval Europe

With limited information available about identities and past play, will

players “act in good faith” or “cheat”?

SLIDE 3

Introduction Model and Preliminaries Results Conclusion

Cooperation in an anonymous random matching game?

Formally, we study infinitely Repeated Anonymous Random Matching

Games (RARMG).

Two communities of players.
At each period, each player is randomly matched to an opponent from the

rival community to play a stage-game.

Matching is anonymous.
Players observe only the actions played in their own matches.

SLIDE 4

Introduction Model and Preliminaries Results Conclusion

Cooperation in an anonymous random matching game?

Formally, we study infinitely Repeated Anonymous Random Matching

Games (RARMG).

Two communities of players.
At each period, each player is randomly matched to an opponent from the

rival community to play a stage-game.

Matching is anonymous.
Players observe only the actions played in their own matches.
Can cooperation be achieved in this worst-case scenario?
What is the set of payoffs that can be sustained in equilibrium?
Can we sustain payoffs beyond the static Nash equilibrium of the

stage-game?

SLIDE 5

Introduction Model and Preliminaries Results Conclusion

The classic folk theorems do not apply

In this setting the standard folk theorems do not apply:

SLIDE 6

Introduction Model and Preliminaries Results Conclusion

The classic folk theorems do not apply

In this setting the standard folk theorems do not apply:

Deviations not publicly observed. So, hard to coordinate punishments.
If I face a deviation, I don’t know whom to punish.
Difficult to provide incentives to punish since, players may not be able to

differentiate a punishment from a deviation.

SLIDE 7

Introduction Model and Preliminaries Results Conclusion

What do we already know?

❄ ❄

more information

SLIDE 8

Introduction Model and Preliminaries Results Conclusion

What do we already know?

No information processing ❄ ❄

more information

SLIDE 9

Introduction Model and Preliminaries Results Conclusion

What do we already know?

No information processing

Kandori 1992 (Efficiency in some PD games)
Ellison 1994 (Efficiency in any PD)
Deb and Gonz´

alez-D´ ıaz (2011) (Class of games beyond PD)

❄ ❄

more information

SLIDE 10

Introduction Model and Preliminaries Results Conclusion

What do we already know?

No information processing

Kandori 1992 (Efficiency in some PD games)
Ellison 1994 (Efficiency in any PD)
Deb and Gonz´

alez-D´ ıaz (2011) (Class of games beyond PD)

Local information processing

Kandori 1992 (Folk Theorem in restricted class of games)
Okuno-Fujiwara and Postlewaite 1995 (Folk Theorem)
Dal B´
2007 (Folk Theorem)

❄ ❄

more information

SLIDE 11

Introduction Model and Preliminaries Results Conclusion

What do we already know?

No information processing

Kandori 1992 (Efficiency in some PD games)
Ellison 1994 (Efficiency in any PD)
Deb and Gonz´

alez-D´ ıaz (2011) (Class of games beyond PD)

Observing partner’s past play

Takahashi 2007 (Cooperation beyond PD)

Local information processing

Kandori 1992 (Folk Theorem in restricted class of games)
Okuno-Fujiwara and Postlewaite 1995 (Folk Theorem)
Dal B´
2007 (Folk Theorem)

❄ ❄

more information

SLIDE 12

Introduction Model and Preliminaries Results Conclusion

What do we already know?

No information processing

Kandori 1992 (Efficiency in some PD games)
Ellison 1994 (Efficiency in any PD)
Deb and Gonz´

alez-D´ ıaz (2011) (Class of games beyond PD)

Unverifiable information (cheap talk)

Deb 2008 (General Folk Theorem for N-player games)

Observing partner’s past play

Takahashi 2007 (Cooperation beyond PD)

Local information processing

Kandori 1992 (Folk Theorem in restricted class of games)
Okuno-Fujiwara and Postlewaite 1995 (Folk Theorem)
Dal B´
2007 (Folk Theorem)

❄ ❄

more information

SLIDE 13

Introduction Model and Preliminaries Results Conclusion

What do we already know?

No information processing

Kandori 1992 (Efficiency in some PD games)
Ellison 1994 (Efficiency in any PD)
Deb and Gonz´

alez-D´ ıaz (2011) (Class of games beyond PD)

This paper

Unverifiable information (cheap talk)

Deb 2008 (General Folk Theorem for N-player games)

Observing partner’s past play

Takahashi 2007 (Cooperation beyond PD)

Local information processing

Kandori 1992 (Folk Theorem in restricted class of games)
Okuno-Fujiwara and Postlewaite 1995 (Folk Theorem)
Dal B´
2007 (Folk Theorem)

❄ ❄

more information

SLIDE 14

Introduction Model and Preliminaries Results Conclusion

What do we do in this paper?

Consider a general class of games in the RARMG setting, and establish

possibility results.

SLIDE 15

Introduction Model and Preliminaries Results Conclusion

What do we do in this paper?

Consider a general class of games in the RARMG setting, and establish

possibility results.

How?

SLIDE 16

Introduction Model and Preliminaries Results Conclusion

What do we do in this paper?

Consider a general class of games in the RARMG setting, and establish

possibility results.

How?
Different solution concept, Strongly Uniform Equilibrium.
Players’ strategies are uniformly approximate best responses to their rivals’

strategies

Moreover, satisfy sequential rationality, global stability.

SLIDE 17

Introduction Model and Preliminaries Results Conclusion

What do we do in this paper?

Consider a general class of games in the RARMG setting, and establish

possibility results.

How?
Different solution concept, Strongly Uniform Equilibrium.
Players’ strategies are uniformly approximate best responses to their rivals’

strategies

Moreover, satisfy sequential rationality, global stability.

Use of approximate equilibria particularly well-suited in this setting

Skeptical view: Not new. Known to make things “easier.”

SLIDE 18

Introduction Model and Preliminaries Results Conclusion

What do we do in this paper?

Consider a general class of games in the RARMG setting, and establish

possibility results.

How?
Different solution concept, Strongly Uniform Equilibrium.
Players’ strategies are uniformly approximate best responses to their rivals’

strategies

Moreover, satisfy sequential rationality, global stability.

Use of approximate equilibria particularly well-suited in this setting

Skeptical view: Not new. Known to make things “easier.”
But approximate equilibria sensible in a community setting.
Equilibria typically interpreted as social norm.
Large literature in anthropology and theoretical biology argues that

evolution of cooperation and punishment as a social norm are a side-effect

f people’s tendency to conform.
Alternate model with “moral costs”?

SLIDE 19

Introduction Model and Preliminaries Results Conclusion

What do we do in this paper?

Preview of Results:

Three key results about cooperation in this setting:

SLIDE 20

Introduction Model and Preliminaries Results Conclusion

What do we do in this paper?

Preview of Results:

Three key results about cooperation in this setting:
A folk theorem for two communities.
A folk theorem for more than two communities.
A folk theorem for two communities with imperfect monitoring.

SLIDE 21

Introduction Model and Preliminaries Results Conclusion

What do we do in this paper?

Preview of Results:

Three key results about cooperation in this setting:
A folk theorem for two communities.
A folk theorem for more than two communities.
A folk theorem for two communities with imperfect monitoring.
Show that it is possible to some players to get payoffs outside the feasible,

individually rational payoff set.

SLIDE 22

Introduction Model and Preliminaries Results Conclusion

What do we do in this paper?

Preview of Results:

Three key results about cooperation in this setting:
A folk theorem for two communities.
A folk theorem for more than two communities.
A folk theorem for two communities with imperfect monitoring.
Show that it is possible to some players to get payoffs outside the feasible,

individually rational payoff set.

Highlight opportunity for correlated punishments.

SLIDE 23

Introduction Model and Preliminaries Results Conclusion

What do we do in this paper?

Preview of Results:

Three key results about cooperation in this setting:
A folk theorem for two communities.
A folk theorem for more than two communities.
A folk theorem for two communities with imperfect monitoring.
Show that it is possible to some players to get payoffs outside the feasible,

individually rational payoff set.

Highlight opportunity for correlated punishments.
Byproduct of analysis is a result about uniform equilibrium and strongly

uniform equilibrium in general games.

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Introduction Model and Preliminaries Results Conclusion

Outline

1

Introduction

2

Model and Preliminaries

3

Results

4

Conclusion

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Introduction Model and Preliminaries Results Conclusion

Model: Basics

Finite set of communities C = {1, . . . , |C|} with finite number of players

in each community M = {1, . . . , |M|}.

Finite action sets: A1, . . . , A|C|, A=

c∈C Ac.

Stage-game utility: u1, . . . , u|I|
Per-period utility: gi : AM → R denotes the expected (across all possible

matchings) payoff of i, given profile a ∈ AM. R is an upper bound on gi.

Repeated game utility: γT

1 , . . . , γT |C| :=(expected) undiscounted average

utilities up to period T. γδ

1 , . . . , γδ |I| the (expected) discounted average utilities.

We consider symmetric strategies
Feasible set: F = conv(u(A)).

SLIDE 26

Introduction Model and Preliminaries Results Conclusion

Model: Basics

In each period, players randomly matched (one from each community) to

play the stage-game

Matching is independent and uniform over time.
A player can observe only the action profiles in the games she is personally

engaged in. In particular,

Does not observe identities of her opponents.
Does not observe action profiles of any other pair of players.

SLIDE 27

Introduction Model and Preliminaries Results Conclusion

Model: Minmax Payoffs

SLIDE 28

Introduction Model and Preliminaries Results Conclusion

Model: Minmax Payoffs

Independent minmax:

For each i ∈ N, vi = mins−i ∈

j∈I\{ic} ∆(Aj ) maxai ∈Ai ui(ai, s−i).

IR = {x ∈ R|C| : x ≥ v} and FIR = F ∩ IR.

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Introduction Model and Preliminaries Results Conclusion

Model: Minmax Payoffs

Independent minmax:

For each i ∈ N, vi = mins−i ∈

j∈I\{ic} ∆(Aj ) maxai ∈Ai ui(ai, s−i).

IR = {x ∈ R|C| : x ≥ v} and FIR = F ∩ IR.

Correlated minmax:

wi = mins−i ∈∆(A−i ) maxai ∈Ai

a−i ∈A−i s−i(a−i)ui(ai, a−i).

IRC = {x ∈ R|C| : x ≥ w} and FIRC = F ∩ IRC.

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Introduction Model and Preliminaries Results Conclusion

Model: Minmax Payoffs

Independent minmax:

For each i ∈ N, vi = mins−i ∈

j∈I\{ic} ∆(Aj ) maxai ∈Ai ui(ai, s−i).

IR = {x ∈ R|C| : x ≥ v} and FIR = F ∩ IR.

Correlated minmax:

wi = mins−i ∈∆(A−i ) maxai ∈Ai

a−i ∈A−i s−i(a−i)ui(ai, a−i).

IRC = {x ∈ R|C| : x ≥ w} and FIRC = F ∩ IRC.

Repeated game minmax:
Player i ∈ N can be forced to xi ∈ R if, for each ε > 0, ∃ σ−i ∈ Σ−i and

T0 ∈ N such that, for each τi ∈ Σi and T ≥ T0, γT

i (τi , σ−i) ≤ xi + ε.

For each i ∈ N, v ∞

i

= inf{xi : i can be forced to xi}. IR∞ = {x ∈ R|C| : x ≥ v ∞} and FIR∞ = F ∩ IR∞.

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Introduction Model and Preliminaries Results Conclusion

Model: Minmax Payoffs

Independent minmax:

For each i ∈ N, vi = mins−i ∈

j∈I\{ic} ∆(Aj ) maxai ∈Ai ui(ai, s−i).

IR = {x ∈ R|C| : x ≥ v} and FIR = F ∩ IR.

Correlated minmax:

wi = mins−i ∈∆(A−i ) maxai ∈Ai

a−i ∈A−i s−i(a−i)ui(ai, a−i).

IRC = {x ∈ R|C| : x ≥ w} and FIRC = F ∩ IRC.

Repeated game minmax:
Player i ∈ N can be forced to xi ∈ R if, for each ε > 0, ∃ σ−i ∈ Σ−i and

T0 ∈ N such that, for each τi ∈ Σi and T ≥ T0, γT

i (τi , σ−i) ≤ xi + ε.

For each i ∈ N, v ∞

i

= inf{xi : i can be forced to xi}. IR∞ = {x ∈ R|C| : x ≥ v ∞} and FIR∞ = F ∩ IR∞.

Clearly, wi ≤ v ∞

i

≤ vi. With two communities, wi = v ∞

i

= vi.

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Introduction Model and Preliminaries Results Conclusion

Solution Concept

Strongly Uniform Equilibrium

Definition

A strategy profile σ is a strongly uniform equilibrium (SUE) with payoff x ∈ RN if limT→∞ γT(σ) = x and, for each ¯ t ∈ N, each ˆ h ∈ ˆ H¯

t, and each i ∈ N, the

following holds.

1

lim

T→∞

1 T

T+¯ t

t=¯

t

gi(at) = xi Pσ(· |ˆ h)-a.s.

2

For each τi ∈ Ai, lim sup

T→∞

1 T

T+¯ t

t=¯

t

gi(at) ≤ xi Pτi ,σ−i (· |ˆ h)-a.s.

3

For each ε > 0, there is T0 ∈ N such that, for each T ≥ T0 and each τi ∈ Ai, γT

i (τi, σ−i |ˆ

h) ≤ γT

i (σ |ˆ

h) + ε.

UE

SLIDE 33

Introduction Model and Preliminaries Results Conclusion

Solution Concept

Strongly Uniform Equilibrium

Uniformity of σ is independent of length of game.

SLIDE 34

Introduction Model and Preliminaries Results Conclusion

Solution Concept

Strongly Uniform Equilibrium

Uniformity of σ is independent of length of game.
Strong global stability
As T → ∞, x is not only the expected payoff, but also, almost surely, the

realized payoff.

Realized continuation payoff from any history on, is x a.s.

SLIDE 35

Introduction Model and Preliminaries Results Conclusion

Solution Concept

Strongly Uniform Equilibrium

Uniformity of σ is independent of length of game.
Strong global stability
As T → ∞, x is not only the expected payoff, but also, almost surely, the

realized payoff.

Realized continuation payoff from any history on, is x a.s.
After every history, all deviations are almost surely non-profitable (as T

goes to infinity).

SLIDE 36

Introduction Model and Preliminaries Results Conclusion

Solution Concept

Strongly Uniform Equilibrium

Uniformity of σ is independent of length of game.
Strong global stability
As T → ∞, x is not only the expected payoff, but also, almost surely, the

realized payoff.

Realized continuation payoff from any history on, is x a.s.
After every history, all deviations are almost surely non-profitable (as T

goes to infinity).

Strong notion of ε-sequential rationality: At any history h, for any ε > 0

and any deviation, if T is large enough, the expected profit is no larger than ε (regardless of beliefs).

SLIDE 37

Introduction Model and Preliminaries Results Conclusion

Solution Concept

Discounting and Moral Costs

Stronger than Uniform Equilibrium (UE)
Stronger than what we get by introducing ε-best replies in a discounted

repeated game.

If a strategy profile is an SUE, it is a sequential equilibrium in a setting

with discounting and moral costs.

Discounting

SLIDE 38

Introduction Model and Preliminaries Results Conclusion

Folk Theorem for Two Communities

SLIDE 39

Introduction Model and Preliminaries Results Conclusion

Folk Theorem for Two Communities

Theorem

Suppose |I| = 2. Then, E = F ∩ IR∞.

SLIDE 40

Introduction Model and Preliminaries Results Conclusion

Folk Theorem for Two Communities

Theorem

Suppose |I| = 2. Then, E = F ∩ IR∞.

We construct an SUE.
Equilibrium strategy profile simple.

SLIDE 41

Introduction Model and Preliminaries Results Conclusion

Folk Theorem for Two Communities

Theorem

Suppose |I| = 2. Then, E = F ∩ IR∞.

We construct an SUE.
Equilibrium strategy profile simple.

How do we sustain cooperation?

SLIDE 42

Introduction Model and Preliminaries Results Conclusion

Equilibrium Strategies

t = 0 · · · · · · ∞

SLIDE 43

Introduction Model and Preliminaries Results Conclusion

Equilibrium Strategies

t = 0 · · · · · · ∞ B3 B4 Bl Bl+1

l4 stages
(l+1)4 stages
Blocks of increasing length: For each l, block Bl has l4 stages.

SLIDE 44

Introduction Model and Preliminaries Results Conclusion

Equilibrium Strategies

t = 0 · · · · · · ∞ B3 B4 Bl Bl+1

l4 stages
(l+1)4 stages

“Grim Trigger” within a block “Restart” at start of next block

Let (¯

at)t be a seq. of pure action profiles approximating the target eq. payoff x, i.e., limT→∞ 1

T

t=1 ¯

at = x.

At start of each block, play according to sequence (¯

at)t.

Minmax opponent until the end of the block, if you observe a deviation.
In any block, a player can be in one of two states: On-path (uninfected)
r off-path (infected).

SLIDE 45

Introduction Model and Preliminaries Results Conclusion

Equilibrium Strategies

t = 0 · · · · · · ∞ B3 B4 Bl Bl+1

l4 stages
(l+1)4 stages

“Grim Trigger” within a block “Restart” at start of next block

Let (¯

at)t be a seq. of pure action profiles approximating the target eq. payoff x, i.e., limT→∞ 1

T

t=1 ¯

at = x.

At start of each block, play according to sequence (¯

at)t.

Minmax opponent until the end of the block, if you observe a deviation.
In any block, a player can be in one of two states: On-path (uninfected)
r off-path (infected).

Community enforcement in blocks

SLIDE 46

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

Global Stability

Assume (in this talk) that the minmax is in mixed actions, for both communities.

SLIDE 47

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

Global Stability

Assume (in this talk) that the minmax is in mixed actions, for both communities.

SLIDE 48

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

Global Stability

Assume (in this talk) that the minmax is in mixed actions, for both communities.

Easy to see that

lim

T→∞ γT(σ) = x.

SLIDE 49

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

Global Stability

Assume (in this talk) that the minmax is in mixed actions, for both communities.

Easy to see that

lim

T→∞ γT(σ) = x.

Now consider any period ¯ t ∈ N, history h ∈ H¯

t and player i.

SLIDE 50

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

Global Stability

Assume (in this talk) that the minmax is in mixed actions, for both communities.

Easy to see that

lim

T→∞ γT(σ) = x.

Now consider any period ¯ t ∈ N, history h ∈ H¯

t and player i.

Play “restarts” at the start of every block regardless of history. So, we will

have strong global stability. lim

T→∞

1 T

T+¯ t

t=¯

t

gi(at) = xi Pσ(· |ˆ h)-a.s

SLIDE 51

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

No profitable deviations

SLIDE 52

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

No profitable deviations

More subtle to check incentive to deviate after any history. Suppose

player 1 deviates.

SLIDE 53

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

No profitable deviations

More subtle to check incentive to deviate after any history. Suppose

player 1 deviates.

Player 2 observes a deviation, and reverts to the minmax action. This

starts contagion.

SLIDE 54

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

No profitable deviations

More subtle to check incentive to deviate after any history. Suppose

player 1 deviates.

Player 2 observes a deviation, and reverts to the minmax action. This

starts contagion.

α:= Max prob. a pure action is assigned in the minmax action.
If a player is infected, the probability that a new player gets infected in the

current period is, at least, 1−α

M , (unless all players in the other community

are already infected).

The probability that any player remains uninfected after l2 stages can be

bounded above using a binomial distribution (and applying Tchebychev’s inequality). P(infected players < 2M − 2) ≤

1 l

6 5 .

SLIDE 55

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

No profitable deviations

More subtle to check incentive to deviate after any history. Suppose

player 1 deviates.

Player 2 observes a deviation, and reverts to the minmax action. This

starts contagion.

α:= Max prob. a pure action is assigned in the minmax action.
If a player is infected, the probability that a new player gets infected in the

current period is, at least, 1−α

M , (unless all players in the other community

are already infected).

The probability that any player remains uninfected after l2 stages can be

bounded above using a binomial distribution (and applying Tchebychev’s inequality). P(infected players < 2M − 2) ≤

1 l

6 5 .

By definition of (¯

at)t, given any ε > 0, there is T0 such that, for all T ≥ T0, |T

t=1 g(¯ at ) t

− x| ≤ ε. Consider l ≥ T0

ε .

SLIDE 56

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

No profitable deviations t = 0 · · · · · · ∞

SLIDE 57

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

No profitable deviations t = 0 · · · · · · ∞ B3 B4 Bl Bl+1

SLIDE 58

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

No profitable deviations t = 0 · · · · · · ∞ B3 B4 Bl Bl+1

l4 stages
(l+1)4 stages

Bl,1 Bl,2 Bl,3 l3 l4 − 2l3 l3

Divide each block Bl in three sub-blocks: Bl,1, Bl,2 and Bl,3 with l3,

(l4 − 2l3) and l3 stages respectively.

We consider three cases: deviations in Bl,1, Bl,2 and Bl,3.

SLIDE 59

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

No profitable deviations t = 0 · · · · · · ∞ B3 B4 Bl Bl+1

l4 stages
(l+1)4 stages

Bl,1 Bl,2 Bl,3 l3 l4 − 2l3 l3

Divide each block Bl in three sub-blocks: Bl,1, Bl,2 and Bl,3 with l3,

(l4 − 2l3) and l3 stages respectively.

We consider three cases: deviations in Bl,1, Bl,2 and Bl,3.

Late (first) deviation in Bl,3:

The realized average payoff in the last (l4 − l3) periods is at most ε away

from x.

The player can make a gain of at most R (say) for at most l3 remaining

periods of the block.

Therefore late deviations cannot give more than (R + 1)ε gain.

SLIDE 60

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

No profitable deviations t = 0 · · · · · · ∞

SLIDE 61

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

No profitable deviations t = 0 · · · · · · ∞ B3 B4 Bl Bl+1

l4 stages
(l+1)4 stages

Bl,1 Bl,2 Bl,3 l3 l4 − 2l3 l3

Deviation in intermediate period ˜ t in Bl,2:

SLIDE 62

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

No profitable deviations t = 0 · · · · · · ∞ B3 B4 Bl Bl+1

l4 stages
(l+1)4 stages

Bl,1 Bl,2 Bl,3 l3 l4 − 2l3 l3

Deviation in intermediate period ˜ t in Bl,2:

In l2 stages, with prob. (1 −

1 l

6 5 ), every player will be infected.

Moreover, punishment lasts at least l3 − l2 stages: So, with high

probability (at least 1 −

R ε2(l3−l2)), the expected average payoff in those

stages will no more than ε away from v1.

Therefore, with prob. at most ( 1

l

6 5 +

R ε2(l3−l2)), a deviation can give no

more than (R + 1)ε gain.

SLIDE 63

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

No profitable deviations t = 0 · · · · · · ∞

SLIDE 64

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

No profitable deviations t = 0 · · · · · · ∞ B3 B4 Bl Bl+1

l4 stages
(l+1)4 stages

Bl,1 Bl,2 Bl,3 l3 l4 − 2l3 l3

Deviation in early period ˜ t in Bl,1:

SLIDE 65

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

No profitable deviations t = 0 · · · · · · ∞ B3 B4 Bl Bl+1

l4 stages
(l+1)4 stages

Bl,1 Bl,2 Bl,3 l3 l4 − 2l3 l3

Deviation in early period ˜ t in Bl,1:

In this case too, we have l2 stages of contagion, followed by more than l3

stages of punishment.

Therefore, we again have: With prob. at most ( 1

l

6 5 +

R ε2(l3−l2)), a deviation

can give no more than (R + 1)ε gain.

SLIDE 66

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

No profitable deviations

SLIDE 67

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

No profitable deviations

Player 1’s gain from deviating is more than (R + 1)ε, with probability at

most

1 l

6 5 +

R2 ε2(l3−l2).

There is l0 such that, with probability 1, for each l ≥ l0, the payoff to

player 1 in block l is smaller than xi + (R + 1)ε.

Since this is true for any ε, we must have limT→∞

T

t=1 g1(at) T

= xi Pτ1,σ2-a.s.

SLIDE 68

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

ε-Sequential Rationality

SLIDE 69

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

ε-Sequential Rationality

Finally, we need to establish that for each ε > 0, there is T0 ∈ N such

that, for each T ≥ T0 and each τi ∈ Ai, γT

i (τi, σ−i |h) ≤ γT i (σ |h) + ε.

SLIDE 70

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

ε-Sequential Rationality

Finally, we need to establish that for each ε > 0, there is T0 ∈ N such

that, for each T ≥ T0 and each τi ∈ Ai, γT

i (τi, σ−i |h) ≤ γT i (σ |h) + ε.

This follows quite easily from the previous analysis.

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Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

ε-Sequential Rationality

Finally, we need to establish that for each ε > 0, there is T0 ∈ N such

that, for each T ≥ T0 and each τi ∈ Ai, γT

i (τi, σ−i |h) ≤ γT i (σ |h) + ε.

This follows quite easily from the previous analysis.
For each block, in case of a late deviation, the payoff is essentially given by

the sequence (¯ at)t

The payoff in case of an intermediate deviation is given by (¯

at)t and the minmax phase

For an early deviation, the payoff is given by the minmax phase.

SLIDE 72

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof

ε-Sequential Rationality

Finally, we need to establish that for each ε > 0, there is T0 ∈ N such

that, for each T ≥ T0 and each τi ∈ Ai, γT

i (τi, σ−i |h) ≤ γT i (σ |h) + ε.

This follows quite easily from the previous analysis.
For each block, in case of a late deviation, the payoff is essentially given by

the sequence (¯ at)t

The payoff in case of an intermediate deviation is given by (¯

at)t and the minmax phase

For an early deviation, the payoff is given by the minmax phase.
If minmax is in pure actions...
If the sequence (¯

at)t plays the pure minmax action very often, then, after a deviation, contagion may spread too slowly.

Construction needs to be modified slightly.

SLIDE 73

Introduction Model and Preliminaries Results Conclusion

UE and SUE Payoffs

We characterized SUE payoff set for two communities. However, SUE strategies are delicate to construct, because of sequential rationality requirement at every history.

SLIDE 74

Introduction Model and Preliminaries Results Conclusion

UE and SUE Payoffs

We characterized SUE payoff set for two communities. However, SUE strategies are delicate to construct, because of sequential rationality requirement at every history.

SUE is a refinement of Uniform Equilibrium (UE).

SLIDE 75

Introduction Model and Preliminaries Results Conclusion

UE and SUE Payoffs

We characterized SUE payoff set for two communities. However, SUE strategies are delicate to construct, because of sequential rationality requirement at every history.

SUE is a refinement of Uniform Equilibrium (UE).
But it turns out that the refinement from UE to SUE is obtained “at no

cost”.

Theorem

For any infinitely repeated game with finite players, actions and signals, the set

f SUE payoffs is exactly the set of UE payoffs.

SLIDE 76

Introduction Model and Preliminaries Results Conclusion

UE and SUE Payoffs

We characterized SUE payoff set for two communities. However, SUE strategies are delicate to construct, because of sequential rationality requirement at every history.

SUE is a refinement of Uniform Equilibrium (UE).
But it turns out that the refinement from UE to SUE is obtained “at no

cost”.

Theorem

For any infinitely repeated game with finite players, actions and signals, the set

f SUE payoffs is exactly the set of UE payoffs.
Not just in RARMG setting
Implication in our setting: From now on, we can construct UE and

conclude that there exists an SUE with the same payoff.

SLIDE 77

Introduction Model and Preliminaries Results Conclusion

Folk Theorem for More than Two Communities

SLIDE 78

Introduction Model and Preliminaries Results Conclusion

Folk Theorem for More than Two Communities

Theorem

Suppose |I| > 2. Then, E = F ∩ IR∞.

SLIDE 79

Introduction Model and Preliminaries Results Conclusion

Folk Theorem for More than Two Communities

Theorem

Suppose |I| > 2. Then, E = F ∩ IR∞.

Proof by construction.
Construct a UE that achieves target payoff x. If x is a UE payoff, then it is

also an SUE payoff.

SLIDE 80

Introduction Model and Preliminaries Results Conclusion

Folk Theorem for More than Two Communities

Theorem

Suppose |I| > 2. Then, E = F ∩ IR∞.

Proof by construction.
Construct a UE that achieves target payoff x. If x is a UE payoff, then it is

also an SUE payoff.

Main Challenge: For punishments to be effective, need to detect deviations, and identify deviator.

Grim trigger - spreads information that a deviation has occurred. But need

a way to also spread the information about the identity of the deviator.

We do this with “Communication Blocks”

SLIDE 81

Introduction Model and Preliminaries Results Conclusion

Equilibrium Strategies

t = 0

SLIDE 82

Introduction Model and Preliminaries Results Conclusion

Equilibrium Strategies

t = 0 BL C L,1 C L,2

L2 stages

L|I| L|I|

“Payoff sub-block” “Communication”

Blocks of L2 + 2L|I| stages: One “payoff sub-block,” Bl and two

identical“communication sub-blocks,” C l,1, C l,2.

SLIDE 83

Introduction Model and Preliminaries Results Conclusion

Equilibrium Strategies

t = 0 BL C L,1 C L,2

L2 stages

L|I| L|I|

“Payoff sub-block” “Communication” · · · BL C L,1 C L,2 . . . L such blocks of L2 + 2L|I| stages each

Blocks of L2 + 2L|I| stages: One “payoff sub-block,” Bl and two

identical“communication sub-blocks,” C l,1, C l,2.

SLIDE 84

Introduction Model and Preliminaries Results Conclusion

Equilibrium Strategies

t = 0 BL C L,1 C L,2

L2 stages

L|I| L|I|

“Payoff sub-block” “Communication” · · · BL C L,1 C L,2 . . . L such blocks of L2 + 2L|I| stages each

Blocks of L2 + 2L|I| stages: One “payoff sub-block,” Bl and two

identical“communication sub-blocks,” C l,1, C l,2.

In first block, in pay-off sub-block, B1 (of length L2)
Play according to sequence (at)t that approximates x.
If L is large enough, then | 1

L

t∈B1 gi(at) − xi| ≤ ε, ∀i ∈ N.
At the end of B1, each player i is in state s(i, j) ∈ {0, 1}, for each

community j: s(i, j) = 1 ⇐ ⇒ i saw a deviation by anyone in community j during B1.

SLIDE 85

Introduction Model and Preliminaries Results Conclusion

Equilibrium Strategies

t = 0 BL C L,1 C L,2

L2 stages

L|I| L|I|

“Payoff sub-block” “Communication” · · · BL C L,1 C L,2 . . . L such blocks of L2 + 2L|I| stages each

SLIDE 86

Introduction Model and Preliminaries Results Conclusion

Equilibrium Strategies

t = 0 BL C L,1 C L,2

L2 stages

L|I| L|I|

“Payoff sub-block” “Communication” · · · BL C L,1 C L,2 . . . L such blocks of L2 + 2L|I| stages each

In first block, in communication sub-blocks C l,1, C l,2,
L stages targeting each community, one at a time.
When target is community j, each player i plays ai if s(i, j) = 0 and ˆ

ai if s(i, j) = 1, i.e. Playing ˆ ai means accusing someone in community j of having deviated.

SLIDE 87

Introduction Model and Preliminaries Results Conclusion

Equilibrium Strategies

t = 0 BL C L,1 C L,2

L2 stages

L|I| L|I|

“Payoff sub-block” “Communication” · · · BL C L,1 C L,2 . . . L such blocks of L2 + 2L|I| stages each

In first block, in communication sub-blocks C l,1, C l,2,
L stages targeting each community, one at a time.
When target is community j, each player i plays ai if s(i, j) = 0 and ˆ

ai if s(i, j) = 1, i.e. Playing ˆ ai means accusing someone in community j of having deviated.

At the end of the L stages targeting community j, each player i updates his

state s(i, j): Set s(i, j) = 1, if she started block Bl with s(i, j) = 0, and observed players of at least 2 different communities accusing community j. Otherwise, state s(i, j) is unchanged.

If s(i, j) = 1 and s(i, j′) = 0∀j′ = j, then we say, i is in “punish

community j phase.”

SLIDE 88

Introduction Model and Preliminaries Results Conclusion

Equilibrium Strategies

t = 0 BL C L,1 C L,2

L2 stages

L|I| L|I|

“Payoff sub-block” “Communication” · · · BL C L,1 C L,2 . . . L such blocks of L2 + 2L|I| stages each

SLIDE 89

Introduction Model and Preliminaries Results Conclusion

Equilibrium Strategies

t = 0 BL C L,1 C L,2

L2 stages

L|I| L|I|

“Payoff sub-block” “Communication” · · · BL C L,1 C L,2 . . . L such blocks of L2 + 2L|I| stages each

In any later payoff sub-block Bl, l ≥ 2,
A player i who starts in the “punish community j phase,” plays a strategy

¯ σj, that punishes community j in the L2-period game, i.e. γL2

i (τi, ¯

σi

−i) ≤ v L2 i

, for any τj.

Otherwise, players play as in B1.

SLIDE 90

Introduction Model and Preliminaries Results Conclusion

Equilibrium Strategies

t = 0 BL C L,1 C L,2

L2 stages

L|I| L|I|

“Payoff sub-block” “Communication” · · · BL C L,1 C L,2 . . . L such blocks of L2 + 2L|I| stages each

In any later payoff sub-block Bl, l ≥ 2,
A player i who starts in the “punish community j phase,” plays a strategy

¯ σj, that punishes community j in the L2-period game, i.e. γL2

i (τi, ¯

σi

−i) ≤ v L2 i

, for any τj.

Otherwise, players play as in B1.
In any later communication sub-block C l,1, C l,2, l ≥ 2,
A player i who starts in the “punish community j phase,” communicates her

state, but does not change her state. Other players communicate and update states.

At other histories, prescribe actions arbitrarily.

SLIDE 91

Introduction Model and Preliminaries Results Conclusion

Sketch of Equilibrium Proof

Lemma

Let sl(i, j) be the state of i regarding community j at the end of Bl. For each i ∈ N, and each strategy τi,

1

False accusations are ignored. Pτi ,σ−i (∀i′ = i, ∀j = ic, sL(i′, j) = 0) = 1 and Pτi ,σ−i (∀i′ = i, sl(i′, ic) = 0 |i has not deviated during B) = 1.

2

Deviations are detected. There is f (L, |M|) s.t. limL→∞ f (L, |M|) = 1 and Pτi ,σ−i (∀i′ = i, sl(i′, ic) = 1 |i has deviated during B) ≥ f (L, |M|).

SLIDE 92

Introduction Model and Preliminaries Results Conclusion

Sketch of Equilibrium Proof

Lemma

Let sl(i, j) be the state of i regarding community j at the end of Bl. For each i ∈ N, and each strategy τi,

1

False accusations are ignored. Pτi ,σ−i (∀i′ = i, ∀j = ic, sL(i′, j) = 0) = 1 and Pτi ,σ−i (∀i′ = i, sl(i′, ic) = 0 |i has not deviated during B) = 1.

2

Deviations are detected. There is f (L, |M|) s.t. limL→∞ f (L, |M|) = 1 and Pτi ,σ−i (∀i′ = i, sl(i′, ic) = 1 |i has deviated during B) ≥ f (L, |M|).

Accusations by only one community unsuccessful.

SLIDE 93

Introduction Model and Preliminaries Results Conclusion

Sketch of Equilibrium Proof

Lemma

Let sl(i, j) be the state of i regarding community j at the end of Bl. For each i ∈ N, and each strategy τi,

1

False accusations are ignored. Pτi ,σ−i (∀i′ = i, ∀j = ic, sL(i′, j) = 0) = 1 and Pτi ,σ−i (∀i′ = i, sl(i′, ic) = 0 |i has not deviated during B) = 1.

2

Deviations are detected. There is f (L, |M|) s.t. limL→∞ f (L, |M|) = 1 and Pτi ,σ−i (∀i′ = i, sl(i′, ic) = 1 |i has deviated during B) ≥ f (L, |M|).

Accusations by only one community unsuccessful.
If player i deviates in B, then, n − 1 ≥ 2 players from different communities
bserve this deviation and inform in block C l,1 and C l,2. Thus, given |M|,

the probability of the states sL(i′, ic) = 1 goes to 1 as L goes to infinity.

SLIDE 94

Introduction Model and Preliminaries Results Conclusion

Sketch of Equilibrium Proof

The strategy σ played for T = L(L2 + 2L|I|) stages.
Let L large enough so that
f (L, |M|) ≥ 1 − ε. (i.e. deviations are detected)
For each player i, v L2

i

≤ v ∞

i

+ ε.(i.e. punishment is harsh enough)

2L|I| ≤ εL2. (i.e. communication blocks are negligible)
R ≤ εL.

SLIDE 95

Introduction Model and Preliminaries Results Conclusion

Sketch of Equilibrium Proof

The strategy σ played for T = L(L2 + 2L|I|) stages.
Let L large enough so that
f (L, |M|) ≥ 1 − ε. (i.e. deviations are detected)
For each player i, v L2

i

≤ v ∞

i

+ ε.(i.e. punishment is harsh enough)

2L|I| ≤ εL2. (i.e. communication blocks are negligible)
R ≤ εL.
If players play the eq. strategy, easy to compute that payoffs are xi.

SLIDE 96

Introduction Model and Preliminaries Results Conclusion

Sketch of Equilibrium Proof

The strategy σ played for T = L(L2 + 2L|I|) stages.
Let L large enough so that
f (L, |M|) ≥ 1 − ε. (i.e. deviations are detected)
For each player i, v L2

i

≤ v ∞

i

+ ε.(i.e. punishment is harsh enough)

2L|I| ≤ εL2. (i.e. communication blocks are negligible)
R ≤ εL.
If players play the eq. strategy, easy to compute that payoffs are xi.
No incentive to deviate in communication blocks, since unilateral

accusations are ignored.

SLIDE 97

Introduction Model and Preliminaries Results Conclusion

Sketch of Equilibrium Proof

The strategy σ played for T = L(L2 + 2L|I|) stages.
Let L large enough so that
f (L, |M|) ≥ 1 − ε. (i.e. deviations are detected)
For each player i, v L2

i

≤ v ∞

i

+ ε.(i.e. punishment is harsh enough)

2L|I| ≤ εL2. (i.e. communication blocks are negligible)
R ≤ εL.
If players play the eq. strategy, easy to compute that payoffs are xi.
No incentive to deviate in communication blocks, since unilateral

accusations are ignored.

No incentive to deviate in payoff block: Obtain bound on deviation payoff.
With high probability (1 − ε), i’s deviation is detected at block C¯

l by all

ther players, so her continuation payoff falls to her minmax.
Three situations in which player i may have high payoffs - all low probability.

(i) In current payoff block, (ii) In communication blocks (proportion of periods ≤ ε) (iii) If her deviation is not detected (prob. ≤ ε)

Details

SLIDE 98

Introduction Model and Preliminaries Results Conclusion

Imperfect Monitoring within a Match

SLIDE 99

Introduction Model and Preliminaries Results Conclusion

Imperfect Monitoring within a Match

Are these results about cooperation robust to imperfections in monitoring?

SLIDE 100

Introduction Model and Preliminaries Results Conclusion

Imperfect Monitoring within a Match

Are these results about cooperation robust to imperfections in monitoring?
Consider two communities and allow imperfect monitoring.
Sets of signals Y1, Y2 for communities 1 and 2 respectively, and a function:

π : A → ∆(Y ), where Y = Y1 × Y2. Within a match, if the action profile is (a1, a2), then a joint signal (y1, y2) in Y is drawn according to π(a1, a2). The player from community 1 observes y1 and the player from community 2

bserves y2.

SLIDE 101

Introduction Model and Preliminaries Results Conclusion

Folk Theorem for Imperfect Monitoring

SLIDE 102

Introduction Model and Preliminaries Results Conclusion

Folk Theorem for Imperfect Monitoring

Condition C: For each community i = 1, 2, for each pure action ai in Ai

and mixed action si in ∆(Ai), if ai and si are different

SLIDE 103

Introduction Model and Preliminaries Results Conclusion

Folk Theorem for Imperfect Monitoring

Condition C: For each community i = 1, 2, for each pure action ai in Ai

and mixed action si in ∆(Ai), if ai and si are different then, there exists a pure action aj of the other community such that π(ai, aj) and π(si, aj) have distinct marginals on the set of signals Yj of community j.

SLIDE 104

Introduction Model and Preliminaries Results Conclusion

Folk Theorem for Imperfect Monitoring

Condition C: For each community i = 1, 2, for each pure action ai in Ai

and mixed action si in ∆(Ai), if ai and si are different then, there exists a pure action aj of the other community such that π(ai, aj) and π(si, aj) have distinct marginals on the set of signals Yj of community j.

Theorem

Suppose condition C is satisfied. Then, for all community sizes, E = F ∩ IR.

SLIDE 105

Introduction Model and Preliminaries Results Conclusion

Asymmetric Payoffs

Can some people get payoffs outside the feasible, IR set? Can we allow some free-riders?

SLIDE 106

Introduction Model and Preliminaries Results Conclusion

Asymmetric Payoffs

Can some people get payoffs outside the feasible, IR set? Can we allow some free-riders?

1, 1 −1, 2 2, −1 0, 0 Community 1 Cooperate Defect Community 2 Cooperate Defect

Theorem

Let M be the community size and let K ≤ M

2 . There is a non-symmetric SUE

in which

K players in community 2 get payoff of 2, while the rest get 1.
Players in community 1 get an expected payoff of 1 − 2K/M.

SLIDE 107

Introduction Model and Preliminaries Results Conclusion

Asymmetric Payoffs

How do we sustain asymmetric payoffs in an SUE?
Construct UE. Implies existence of SUE with same payoff.
Difficulties: How to prevent “impersonations” and ensure punishments
Result similar in spirit to Dal B´
(2007).

SLIDE 108

Introduction Model and Preliminaries Results Conclusion

Asymmetric Payoffs

How do we sustain asymmetric payoffs in an SUE?
Construct UE. Implies existence of SUE with same payoff.
Difficulties: How to prevent “impersonations” and ensure punishments
Result similar in spirit to Dal B´
(2007).

Equilibrium Strategies

Let D2 denote a fixed set of K players in community 2. Let C2 denote the

remaining M − K players. Denote community 1 by C1.

Two states for players: “Uninfected” or “infected.”

SLIDE 109

Introduction Model and Preliminaries Results Conclusion

Asymmetric Payoffs

t = 0 B1 C 1

L5 stages

L stages

· · · · · · Payoff sub-block Communication

SLIDE 110

Introduction Model and Preliminaries Results Conclusion

Asymmetric Payoffs

t = 0 B1 C 1

L5 stages

L stages

· · · · · · Payoff sub-block Communication . . . L blocks of L5 + L stages each · · · · · · BL C L

SLIDE 111

Introduction Model and Preliminaries Results Conclusion

Asymmetric Payoffs

t = 0 B1 C 1

L5 stages

L stages

· · · · · · BL C L C1 in Comm. 1: C2 in Comm. 2: D2 in Comm. 2: C C D C C C C C D C C C

SLIDE 112

Introduction Model and Preliminaries Results Conclusion

Asymmetric Payoffs

t = 0 B1 C 1

L5 stages

L stages

· · · · · · BL C L C1 in Comm. 1: C2 in Comm. 2: D2 in Comm. 2: C C D C C C C C D C C C

Transition from “uninfected” state to “infected” state
Community 1: If (i) Observes defection in block C l or (ii) Observes D in a

block Bl more than K

M + 1 L times.

Community 2: If and only if she faces a defection.

Proof Sketch

SLIDE 113

Introduction Model and Preliminaries Results Conclusion

Asymmetric Payoffs

t = 0 B1 C 1

L5 stages

L stages

· · · · · · BL C L C1 in Comm. 1: C2 in Comm. 2: D2 in Comm. 2: C C D C C C C C D C C C Play if Infected: D D D D

Transition from “uninfected” state to “infected” state
Community 1: If (i) Observes defection in block C l or (ii) Observes D in a

block Bl more than K

M + 1 L times.

Community 2: If and only if she faces a defection.

Proof Sketch

SLIDE 114

Introduction Model and Preliminaries Results Conclusion

Correlation in punishments: An Example

SLIDE 115

Introduction Model and Preliminaries Results Conclusion

Correlation in punishments: An Example

We characterize SUE payoffs, but we don’t characterize v ∞. How close

can we get to perfectly correlated punishments?

SLIDE 116

Introduction Model and Preliminaries Results Conclusion

Correlation in punishments: An Example

We characterize SUE payoffs, but we don’t characterize v ∞. How close

can we get to perfectly correlated punishments?

Don’t know! Example with eqm. payoff strictly below v, i.e. v ∞ < v.

SLIDE 117

Introduction Model and Preliminaries Results Conclusion

Correlation in punishments: An Example

We characterize SUE payoffs, but we don’t characterize v ∞. How close

can we get to perfectly correlated punishments?

Don’t know! Example with eqm. payoff strictly below v, i.e. v ∞ < v.

Consider the three-player game below: E2 W2 E1 W1 0, 0, −1 0, 0, 0 0, 0, 0 0, 0, 0

E3

E2 W2 0, 0, 0 0, 0, 0 0, 0, 0 0, 0, −1

W3
w = (0, 0, −1/2) and v = (0, 0, −1/4). Can we give players in

community 3 payoffs below 1

4?

SLIDE 118

Introduction Model and Preliminaries Results Conclusion

Correlation in punishments: An Example

We characterize SUE payoffs, but we don’t characterize v ∞. How close

can we get to perfectly correlated punishments?

Don’t know! Example with eqm. payoff strictly below v, i.e. v ∞ < v.

Consider the three-player game below: E2 W2 E1 W1 0, 0, −1 0, 0, 0 0, 0, 0 0, 0, 0

E3

E2 W2 0, 0, 0 0, 0, 0 0, 0, 0 0, 0, −1

W3
w = (0, 0, −1/2) and v = (0, 0, −1/4). Can we give players in

community 3 payoffs below 1

4?

Example with 5 players in each community. Can force one player in

community 3 to payoff below 1

4.

Players = i in Community 3 help players in community 1 and 2 correlate.

Example

SLIDE 119

Introduction Model and Preliminaries Results Conclusion

To conclude...

We consider games played in a repeated anonymous random matching

setting, and ask what payoffs are sustainable in equilibrium.

SLIDE 120

Introduction Model and Preliminaries Results Conclusion

To conclude...

We consider games played in a repeated anonymous random matching

setting, and ask what payoffs are sustainable in equilibrium.

We introduce a new solution concept: Strongly uniform equilibrium.
Folk Theorems for 2 communities
Folk Theorem for more than 2 communities.
Folk Theorem for 2 communities with imperfect monitoring.

SLIDE 121

Introduction Model and Preliminaries Results Conclusion

To conclude...

We consider games played in a repeated anonymous random matching

setting, and ask what payoffs are sustainable in equilibrium.

We introduce a new solution concept: Strongly uniform equilibrium.
Folk Theorems for 2 communities
Folk Theorem for more than 2 communities.
Folk Theorem for 2 communities with imperfect monitoring.
Contributions:
First possibility result that covers all games in the random matching

setting–Community enforcement works.

General result about uniform equilibrium and strongly uniform equilibrium

payoffs.

SLIDE 122

Uniform Folk Theorems in Repeated Anonymous Random Matching Games

Joyee Deb1 Julio Gonz´ alez-D´ ıaz2 J´ erˆ

me Renault3

1Stern School of Business, New York University

2Dept. of Statistics & Operations Research, University of Santiago de Compostela

3Toulouse School of Economics, Universit´

e Toulouse I

November 2013

SLIDE 123

Uniform Equilibrium

SLIDE 124

Uniform Equilibrium

σ is a uniform equilibrium (UE) with payoff x ∈ RN if
limT→∞ γT (σ) = x and,
For each ε > 0, there is T0 such that, for each T ≥ T0, σ is an ε-Nash eq.

in the finitely repeated game with T stages, i.e., for each i ∈ N and each τi ∈ Σi, γT

i (τi, σ−i) ≤ γT i (σ) + ε.

SLIDE 125

Uniform Equilibrium

σ is a uniform equilibrium (UE) with payoff x ∈ RN if
limT→∞ γT (σ) = x and,
For each ε > 0, there is T0 such that, for each T ≥ T0, σ is an ε-Nash eq.

in the finitely repeated game with T stages, i.e., for each i ∈ N and each τi ∈ Σi, γT

i (τi, σ−i) ≤ γT i (σ) + ε.

If σ is a SUE then it is also a UE. Converse not true.

Return

SLIDE 126

Discounting: DSUE and Moral Costs

Discounted SUE: “Moral Costs:”

SLIDE 127

Discounting: DSUE and Moral Costs

Discounted SUE:

σ is a discounted strongly uniform equilibrium (DSUE) with payoff

x ∈ Rn if limδ→1 γδ(σ) = x and, for each ¯ t ∈ N, each h ∈ H¯

t, and each

i ∈ N the following holds.

1

limδ→1(1 − δ) ∞

t=¯ t δt−¯ tgi(at) = xi

Pσ(· |ˆ h)-a.s.

2

For each τi ∈ Ai, lim supδ→1(1 − δ) ∞

t=¯ t δt−¯ tgi(at) ≤ xi

Pτi ,σ−i (· |ˆ h)-a.s.

3

For each ε > 0, there is δ0 ∈ (0, 1) such that, for each δ ∈ (δ0, 1) and each τi ∈ Ai, γδ

i (τi, σ−i |h) ≤ γδ i (σ ˆ

h) + ε.

If σ is a SUE then it is also a DSUE. Converse is not true.

“Moral Costs:”

SLIDE 128

Discounting: DSUE and Moral Costs

Discounted SUE:

σ is a discounted strongly uniform equilibrium (DSUE) with payoff

x ∈ Rn if limδ→1 γδ(σ) = x and, for each ¯ t ∈ N, each h ∈ H¯

t, and each

i ∈ N the following holds.

1

limδ→1(1 − δ) ∞

t=¯ t δt−¯ tgi(at) = xi

Pσ(· |ˆ h)-a.s.

2

For each τi ∈ Ai, lim supδ→1(1 − δ) ∞

t=¯ t δt−¯ tgi(at) ≤ xi

Pτi ,σ−i (· |ˆ h)-a.s.

3

For each ε > 0, there is δ0 ∈ (0, 1) such that, for each δ ∈ (δ0, 1) and each τi ∈ Ai, γδ

i (τi, σ−i |h) ≤ γδ i (σ ˆ

h) + ε.

If σ is a SUE then it is also a DSUE. Converse is not true.

“Moral Costs:”

Consider discounted RARMG, in which agents suffer a (small) moral cost

λ > 0, each time they deviate from a prescribed norm.

Lemma

If a strategy profile σ is an SUE then, for each λ > 0, there is δ0 such that, for each δ ∈ (δ0, 1), σ is a sequential equilibrium of the infinitely repeated game with discount δ and moral cost λ.

Return

SLIDE 129

Introduction Model and Preliminaries Results Conclusion

Sketch of Equilibrium Proof: |C| > 2

If players play the eq. strategy: All players are on-path.

|γT

i (σ) − xi| ≤ L(ε · L2 + 2R · 2L|I|)

L(L2 + 2L|I|) ≤ εL2 + 2RεL2 L2 + 2L|I| ≤ ε(1 + 2R).

SLIDE 130

Introduction Model and Preliminaries Results Conclusion

Sketch of Equilibrium Proof: |C| > 2

If players play the eq. strategy: All players are on-path.

|γT

i (σ) − xi| ≤ L(ε · L2 + 2R · 2L|I|)

L(L2 + 2L|I|) ≤ εL2 + 2RεL2 L2 + 2L|I| ≤ ε(1 + 2R).

No deviation in a communication block - Unilateral accusations ignored.

SLIDE 131

Introduction Model and Preliminaries Results Conclusion

Sketch of Equilibrium Proof: |C| > 2

If players play the eq. strategy: All players are on-path.

|γT

i (σ) − xi| ≤ L(ε · L2 + 2R · 2L|I|)

L(L2 + 2L|I|) ≤ εL2 + 2RεL2 L2 + 2L|I| ≤ ε(1 + 2R).

No deviation in a communication block - Unilateral accusations ignored.
If she deviates in a payoff-block:
Let ¯

l be the first block, such that i deviates in B¯

l.

With probability (1 − ε), i’s deviation is detected at block C¯

l by all other

players, so her payoff is below v ∞

i

+ ε in each future payoff block.

Three situations in which player i may have high payoffs.

(i) During payoff block B¯

l,

(ii) During communication blocks (proportion of periods ≤ ε) (ii) If her deviation is not detected (prob. ≤ ε) Her payoff is bounded by εR + (1 − ε)

εR + (1 − ε)(L − 1)(xi + ε) + R

L

≤ xi + 2ε(1 + R).

Return

SLIDE 132

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof: Asymmetric Payoffs

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Introduction Model and Preliminaries Results Conclusion

Sketch of Proof: Asymmetric Payoffs

Pick L large enough so that
If one player i deviates, and some player j = i is infected, then with

probability at least 1 − ε all players will be infected at end of the communication block.

Length of communication blocks small:

M L ≤ ε.

SLIDE 134

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof: Asymmetric Payoffs

Pick L large enough so that
If one player i deviates, and some player j = i is infected, then with

probability at least 1 − ε all players will be infected at end of the communication block.

Length of communication blocks small:

M L ≤ ε.

If no player deviates: We show the following:
With probability at least 1 − M

L2 , all players will be uninfected.

Since 1 − M

L2 ≥ 1 − ε, players in C1, C2 and D2 get payoffs close to (at most

5ε away from) M−2K

M

, 1 and 2 respectively.

SLIDE 135

Introduction Model and Preliminaries Results Conclusion

Sketch of Proof: Asymmetric Payoffs

Pick L large enough so that
If one player i deviates, and some player j = i is infected, then with

probability at least 1 − ε all players will be infected at end of the communication block.

Length of communication blocks small:

M L ≤ ε.

If no player deviates: We show the following:
With probability at least 1 − M

L2 , all players will be uninfected.

Since 1 − M

L2 ≥ 1 − ε, players in C1, C2 and D2 get payoffs close to (at most

5ε away from) M−2K

M

, 1 and 2 respectively.

If some player i deviates: We show the following:
Any deviation by a player in C1 is immediately detected.
Deviations by players in C2 detected with very high probability.
(No player in D2 wants to deviate.)
Payoff from deviating at most 8ε away from target payoffs.

Return

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Introduction Model and Preliminaries Results Conclusion

Correlation in punishments

SLIDE 137

Introduction Model and Preliminaries Results Conclusion

Correlation in punishments

3 communities with 5 players each. Fix i in community 3.

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Introduction Model and Preliminaries Results Conclusion

Correlation in punishments

3 communities with 5 players each. Fix i in community 3.
Play proceeds in blocks of (l + 1) stages.
Each player in community 3 other from player i plays W3, . . . , W3 or

E3, . . . , E3 with equal probabilities, in periods 1, . . . , l, and players in communities 1 and 2 minmax players in community 3 (randomize uniformly between their two actions).

In period l + 1 players in community 1 (2) play W1 (W2) if they have seen

W3 more often than E3 and, otherwise, play E1 (E2).

SLIDE 139

Introduction Model and Preliminaries Results Conclusion

Correlation in punishments

3 communities with 5 players each. Fix i in community 3.
Play proceeds in blocks of (l + 1) stages.
Each player in community 3 other from player i plays W3, . . . , W3 or

E3, . . . , E3 with equal probabilities, in periods 1, . . . , l, and players in communities 1 and 2 minmax players in community 3 (randomize uniformly between their two actions).

In period l + 1 players in community 1 (2) play W1 (W2) if they have seen

W3 more often than E3 and, otherwise, play E1 (E2).

In the first l stages of each block,

With prob. 5/16, 3 or 4 players in community 3 will be playing E3, and with the same prob. 3 or 4 will be playing W3 With prob. 6/16, 2 players will play E3 and 2 will play W3.

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Introduction Model and Preliminaries Results Conclusion

Example of correlation in punishments

SLIDE 141

Introduction Model and Preliminaries Results Conclusion

Example of correlation in punishments

The best reply of i is to stick to one action, say W3, in the first l stages;

(this increases the likelihood of communities 1 and 2 correlating at (W1, W2), and then switch to the opposite action, E3, at period l + 1.

SLIDE 142

Introduction Model and Preliminaries Results Conclusion

Example of correlation in punishments

The best reply of i is to stick to one action, say W3, in the first l stages;

(this increases the likelihood of communities 1 and 2 correlating at (W1, W2), and then switch to the opposite action, E3, at period l + 1.

However, i can only significantly influence play in period l + 1 if the other

players in his community were equally split between W3 and E3. Therefore, as l increases, the probability of successful correlation goes to 10/16, in which case player i would expect a payoff of −1/2 in period l + 1.

SLIDE 143

Introduction Model and Preliminaries Results Conclusion

Example of correlation in punishments

The best reply of i is to stick to one action, say W3, in the first l stages;

(this increases the likelihood of communities 1 and 2 correlating at (W1, W2), and then switch to the opposite action, E3, at period l + 1.

However, i can only significantly influence play in period l + 1 if the other

players in his community were equally split between W3 and E3. Therefore, as l increases, the probability of successful correlation goes to 10/16, in which case player i would expect a payoff of −1/2 in period l + 1.

Therefore, the expected payoff of player i in each block is bounded above

by H(l) = l · −1

4 + 1 · −1 2 10 16f (l)

l + 1 , where lim

l→∞ f (l) = 1.

Hence, there is l0 such that, for each l > l0, H(l) < − 1

4 = v3.

Return