1

G601, IO I Eric Rasmusen, erasmuse@indiana.edu22 September 2006

(Changed after the class was taught_)

Rafael Rob and Huanxing Yang, notes on "Long-Term Relationships as Safeguards," May 26, 2006.

2

Readings

21 September, Thursday. Reputation, Rob, and Perfect Bayesian Equilibrium

   Chapters 5 and 6, Games and Information. Rafael Rob and Huanxing Yang, notes on

"Long-Term Relationships as Safeguards," May 26, 2006.   Powerpoints on the Rob paper are available, as well as latex files on 5

and 6 and pdf files on 5 and 6.    Robert Axelrod and William Hamilton, ``The Evolution of

Cooperation,'' Science, 211: 1390-1396 (March 1981). Reader 24.    Harry Roberts and Roman Weil, ``The University of Chicago. Starting

Research Early,'' unpublished notes, University of Chicago Graduate School of Business (August 14, 1970). Reader 27.

   David Kreps, Paul Milgrom, John Roberts, and Robert Wilson, "Rational Cooperation in the Finitely Repeated Prisoners' Dilemma,"

Journal of Economic Theory, 27: 245-252 (1982). Reader 29.

3

Handouts

For next time: Hand in

Problem Set 3: 4.5, 5.4, 6.3, and 6.4.

4

ABSTRACT:We analyze a repeated prisoners’ dilemma game played in a community setting with heterogeneous types.

Some players are bad types, programmed to defect, others are good types, programmed to cooperate, and others yet choose actions to maximize their discounted payoffs.

Players are also able to strategically choose whether to continue interacting with the same partner - form a long term relationship - or separate and seek a new partner.

We show that the ability to form long term relationships facilitates the achievement of cooperative outcomes without information flows, without instability due to observational errors, and without a central coordinating device to synchronize players’ actions. We also show that the heterogeneity of types helps, rather than hinders, cooperative behavior by inducing players to avoid bad types that inflict low payoffs on them and seek good (or opportunistic) types that bestow high payoffs.

5

Players-Actions-Payoffs-Info

We consider a community of individuals (or players or agents), modeled as a continuum of measure 1.

Time is discrete and the horizon is infinite. Each individual is infinitely lived.

At the beginning of each period, the community is divided into partnerships (or relationships), and each pair of partners play a two-stage game.

In the first stage they play a prisoners’ dilemma game, and each partner chooses either C, which stands for “cooperate,” or D, which stands for “defect.” The payoff. matrix of this first-stage game is specified momentarily. [below]

6

After this stage, each partnership persists with probability p, and breaks with probability 1 –p.

If a partnership persists, the two partners go into a simultaneous-move second-stage game, in which each partner makes a stay-or-separate decision.

If both partners choose to stay, the current partnership continues into the next period. If at least one partner chooses to separate, or if the partnership (exogenously) breaks, both partners go into a pool of unmatched players.

Players in this pool are randomly matched at the beginning of the next period, forming new partnerships. Consequently, the dissolution and re-formation of partnerships are partly exogenous and partly endogenous.

No direct payoffs are associated with the second-stage game; its only role

is to endogenize the decision whether to interact with the same individual in the next period.

7

There are three types of players in the population.

There is a measure alpha of opportunistic types that we denote by O, a measure ß of bad types that we denote by B, and a measure gamma (= 1-alpha -ß) of good types that we denote by G.

G-type player always chooses C B-type player always chooses D O-type player chooses either C or D

8

9

INFORMATION

We assume that monitoring is perfect inside each partnership: a player observes his partner’s actions - beginning with the date at which this partnership is commenced. [unclear]

However, when a player is matched to a new partner he knows nothing about the partner’s past history of actions with other partners. That is, there are no information flows across matches. [delete]

Also, a player’s type is private information.

10

This is an infinitely repeated community game with incomplete information, so folk-theorem type arguments establish that there are many equilibria supported by a variety of repeated-game strategies.

For example, when ß = gamma = 0, Kandori’s (1992) “contagious equilibrium,” in which each player plays D forever if either he or one of his previous partners played D, is an equilibrium in our setting. [Grim Strategy]

11

Rather than prove folk theorems, this paper focuses on a certain class of equilibria. This class is defined by the following property.

Each player plays a stationary strategy against his current partner, which depends, possibly, on the history of actions within the current relationship, but is independent of calendar time and of history of actions in previous interactions (fresh start). [stationary? Wrong word, maybe. Will be precisely defined later]

12

Notice:

1. The Players-Actions-Payoffs-Info structure, laid out very clearly.

2. No talk of equilibrium till after that. 3. Just a few clarifying and motivating remarks

about the assumptions.4. Carefully chosen notation (gamma for good,

beta for bad)5. Bad choices: Cooperate/Defect, O-type and

time 0. 6. The first priority: Make sure your reader

understands the model. If that fails, the rest is useless.

13

Rasmusen Analysis

Important: They have a GOOD type, which always Cooperates, even if Defected against.

Since there are GOOD types, everybody wants to find them. GOOD ones too. So nobody wants to stay with a Defector.

14

(1) GOOD types. This is unusual. They do NOT have GRIM types or TitForTat types.

(2) Do the BAD types make any difference to the model? ---they do solve the technical problem of forming out-of-equilibrium beliefs, by making every action possible in any equilibrium.

(3) That there is a positive probability of exogenous switching is interesting. What effect does that have? Is it just like time preference? (No– see defect eq. below when gamma is big)

15

DEFECT, DEFECT equilibrium, unsatisfactory:

Good: Cooperate. Always switch.Opportunistic: DEFECT. Always switch.Bad: DEFECT. Always switch.

If the SWITCH decision was sequential, this would break down. Or, if there were a few tremblers. Or, if there were a few types that never switched.

That’s because SWITCH IF DEFECT weakly dominates ALWAYS SWITCH.

16

AN EQUILIBRIUM Strategy Profile?

Good: COOPERATE. Switch if DEFECT.Opportunistic: DEFECT. Switch if DEFECT.Bad: DEFECT. Switch if DEFECT.

Is this an equilibrium? For given gamma and low enough discount rates (and p=low enough), it is NOT (UNLESS: gamma is high enough—see next page). The Opportunistic type would deviate to Cooperate, and search for a GOOD partner with whom to get an infinite number of (Cooperate, Cooperate) payoffs.

They do not need their “stationarity” restriction to rule out the DEFECT equilibrium.

17

AN EQUILIBRIUM?--YES

Good: COOPERATE. Switch if DEFECT.Opportunistic: DEFECT. Switch if DEFECT.Bad: DEFECT. Switch if DEFECT.

Is this an equilibrium? Even for low discount rates (and p= not too low), it IS, if gamma is high enough. The reason: If an opportunistic type chooses DEFECT, he knows that most of the matching pool will be GOOD’s involutnarily sent there by P, so he will get to DEFECT again.

18

THE Rob-Yang EQUILIBRIUM

Good: Cooperate. Switch if Defect.Opportunistic: Cooperate. Switch if Defect.Bad: Defect. Switch if Defect.

This is an eq. for small enough discount rate, for given other parameters.

If there is more discounting, this works only if there are few enough BADs, or if almost everyone is BAD.

If there is lots of discounting, then we are back to the one-shot game, and this eq. fails, even with the Good types being in the model.

19

The Gang of Four Model

Kreps-Milgrom-Roberts-Weber (1982, JET).

If we add a tit-for-tat type to a FINITELY repeated game, we can get cooperation till near the end of it. If there are 8000 rounds, maybe there is cooperation till the last 1000 rounds.

But if there are 800,000 rounds, there will STILL be cooperation till the last 1000 rounds.

20

Back to Rob-Yang

Rob-Yang is an infinitely repeated game, but that doesn’t really matter, I think. The Gang of Four idea applies here. We could make the game have finite rounds, and cooperation would still be the only equilibrium if there is little discounting. The key is that it is worth searching a long time to find a GOOD type, and worth being nice to the GOOD type till near the end.

21

22

Course Website

A link to the course website

http://www.rasmusen.org/g601/0.g601.htm