Game-Theoretic Models for Effects of Social Embeddedness on Trust and

Cooperation

Werner Raub

Workshop on Social Theory, Trust, Social Networks, and Social Capital II

National Chengchi University – NCCUApril 2011

P1 T1 E1 P2

Problem of order

Game Theory

Implications for research

New research problem

Cooperation in Social Dilemmas

The problem of social order 1 Examples of the problem of social order: social dilemmas

• Trust• Hobbes, State of Nature• Collective goods, collective action (trade unions,

associations of common interests, protest campaigns)• Environmental pollution• Arms races• “Social Exchange” (e.g., help among friends)• Economic Relations

- transactions on stock markets (M. Weber)- cooperation between firms

2 General" The pursuit of self-interest by each leads to a poor outcome for all."

[Axelrod 1984:7]

The explanatory problem related tosocial dilemma situations

P T E

• Conditions for cooperation in social dilemma situations without external enforcement and/or internalized norms.

• Phenomena to be explained:

1) individual effect: choice of strategies

2) collective effect: Pareto (sub-)optimality

Prisoner’s Dilemma

R,R S,T

T,S P,P

C D

D

C

Player 2

Player 1

Assumptions:

• T>R>P>S

• Simultaneous moves

• No binding agreements

• Information: each player is informed on his or her own alternative actions and outcomes, as well as on alternative actions and outcomes for the partner

Refresher: basic concepts of game theory

• Best reply strategy: – A strategy that gives the highest payoff, given the strategy of the other

player

• Dominant strategy: – A strategy that is the best reply against every possible strategy of the

other player

• Nash equilibrium: – A combination of best reply strategies; no player has an incentive for

one-sided deviation

• Pareto-optimal outcome: – There is no other outcome that is an improvement for at least one of the

players without making someone else worse off

(Note: compare with the more formal definitions provided earlier)

Prisoner’s Dilemma

R,R S,T

T,S P,P

C D

D

C

Player 2

Player 1

Assumptions:

• T>R>P>S

• Simultaneous moves

• No binding agreements

• Information: each player is informed on his or her own alternative actions and outcomes, as well as on alternative actions and outcomes for the partner

*

**

Prisoner’s Dilemma:no cooperation in single encounters

A

C

B

D

Macro

Micro

One shot PD interaction

Pareto-suboptimal outcome

PD matrix PD matrix

T>R>P>S Players defect Dominant strategies and Nash equilibrium behavior

Conclusion for the one-shot Prisoner’s Dilemma

• Given goal-directed behavior, there will be no cooperation without external enforcement and without internalized norms in the one-shot PD.– Hence, PD as a social dilemma and problematic

social situation.• How to proceed?

– Does repeating the PD have an effect on behavior of goal-directed actors?

Robert Axelrod and “The Evolution of Cooperation” (1984)

Michael Taylor and “Anarchy and Cooperation” (1976; rev. ed.: The Possibility of Cooperation”)

The repeated Prisoner’s Dilemma

• The Prisoner’s Dilemma is played indefinitely often. After each round, each player is informed on the other player’s behavior (C or D) in that round.

• A player’s payoff for the repeated game is the discounted sum of his or her payoffs in each round, i.e.:

v = g1 + wg2 + w²g3 + ... + wt-1gt + ...

with: 0 < w < 1 for the discount parameter w

gt: payoff in round t = 1, 2, ....

• A player’s strategy for the repeated game is a rule specifying the player’s behavior (C or D) in each round as a function of what has happened in the game before that round.

Repeated interactions as a paradigmatic case of “social embeddedness”

• Dyadic embeddedness: repeated interactions between the same actors

• Network embeddedness: actors have (information) ties with partners of their partners

Intuition: why might cooperation be feasible for goal-directed actors in the repeated game?

• Basic idea: conditional cooperation– Behavior in the present round might affect the behavior of

the partner in future rounds and might thus affect one’s own future payoffs

– Thus, own defection in the present round will yield a higher payoff in the present round than own cooperation in the present round (T > R). However, own defection in the present round may induce the partner to defect himself in the future so that in future rounds one may get at most P < R. Hence, short-term incentives for defection and long-term incentives for cooperation. Question: what are conditions such that the long-term incentives become more important than the short-term incentives?

• Axelrod: shadow of the future

Types of strategies for the repeated game

Unconditional strategies (e.g.: ALL D, ALL C, Random)

Conditional strategies

Nice, Provocable (and Forgiving) Strategies (e.g.: TFT)

Others

A simple but important negative result for the repeated game

• Cooperation in the repeated game as a result of unconditional strategies would require that actors use ALL C

• Note: (ALL C, ALL C) cannot be a Nash equilibrium of the repeated game.

• Thus, playing ALLC is inconsistent with the idea of goal-directed behavior.

Cooperation in the repeated game as a result of goal-directed behavior can only be based on conditional strategies.

Two simple strategies for therepeated Prisoner’s Dilemma

ALL D: Play D in each round

Thus, ALL D is- Unconditional- Not Nice

TFT: 1 Play C in each round 1. 2 Imitate in each round (2,3,...,t,...) the other

player’s behavior in the previous round (1,2,...,t-1,...).

Thus, TFT is- Conditional- Nice - Provocable

Motivation for analyzing a simplified version of the repeated Prisoner’s Dilemma with only two

feasible strategies

• Repeated game can be analyzed as a simple 2x2-game.• Result for the simplified case is generalizable:

– Result applies also if strategy set for the repeated game is not restricted

– Result generalizes to many other game-theoretic models for social dilemmas such as the repeated Trust Game as well as n-person dilemmas

– Similar result for network embeddedness• Important feature of good model building: simplified

assumptions do not affect the main results. Main results are robust relative to modifications of simplified assumptions.

Repeated Prisoner’s Dilemma

TFT

Player 2

Player 1

ALL D

TFT ALL D

TFT vs. TFT

Player Round

1 2 3 … t t+1 …

1 (TFT) C C C … C C …

2 (TFT) C C C … C C …

Step 1: Moves per round

TFT vs. TFT

Player Round

1 2 3 … t t+1 …

1 (TFT) R R R … R R …

2 (TFT) R R R … R R …

Step 2: Payoffs per round

Step 3: Payoffs for the repeated game

V(TFT,TFT) = R + wR + w2R + … + wt+1R + …

1

1

t

t

w R

1

R

w

TFT vs. TFT

Player Round

1 2 3 … t t+1 …

1 (TFT) R R R … R R …

2 (TFT) R R R … R R …

Step 2: Payoffs per round

Step 3: Payoffs for the repeated game

V(TFT,TFT) = R + wR + w2R + … + wt+1R + …

1

1

t

t

w R

1

R

w

ALL D vs. ALL D

Player Round

1 2 3 … t t+1 …

1 (ALLD) D D D … D D …

2 (ALLD) D D D … D D …

Step 1: Moves per round

ALL D vs. ALL D

Player Round

1 2 3 … t t+1 …

1 (ALLD) P P P … P P …

2 (ALLD) P P P … P P …

Step 2: Payoffs per round

Step 3: Payoffs for the repeated game

V(ALLD, ALLD) = P + wP + w2P + … + wt+1P + …

1

1

t

t

w P

1

P

w

ALL D vs. TFT

Player Round

1 2 3 … t t+1 …

1 (ALLD) D D D … D D …

2 (TFT) C D D … D D …

Step 1: Moves per round

ALLD vs. TFT

Player Round

1 2 3 … t t+1 …

1 (ALLD) T P P … P P …

2 (TFT) S P P … P P …

Step 2: Payoffs per round

Step 3: Payoffs for the repeated game

Player 1: V(ALLD,TFT) = T + wP + w2P + … + wt+1P + …

1

wPT

w

Player 2: V(TFT,ALLD) = S + wP + w2P + … + wt+1P + …

1

wPS

w

Repeated Prisoner’s Dilemma

R R ------ ; ------1-w 1-w

wP wP S+ ------ ; T+ ----- 1-w 1-w

wP wP T+ ------ ; S+ ----- 1-w 1-w

P P ------ ; ------1-w 1-w

TFT

ALL D

TFT ALL D

Repeated Prisoner’s Dilemma

R R ------ ; ------1-w 1-w

wP wP S+ ------ ; T+ ----- 1-w 1-w

wP wP T+ ------ ; S+ ----- 1-w 1-w

P P ------ ; ------1-w 1-w

TFT

ALL D

TFT ALL D?

?

Equilibria

• (ALL D, ALL D) is always an equilibrium

1 1

P wPp

w w

• (ALL D, TFT) and (TFT, ALL D) are never equilibria• (TFT, TFT) is sometimes an equilibrium; namely if:

1 1

( )

R wPT

w wR T wT wP

w T P T R

T Rw

T P

Costs of cooperation

Costs of conflictStability of relation (“shadow of the future”)

w

wPS

w

Pnote

11:

Example

R=3; R=3 S=0; T=5

T=5; S=0 P=1; P=1

C

Player 2

Player 1

D

C D

Situation 1:

W=0.1

(Shadow of the future is small)

Situation 2:

W=0.9

(Shadow of the future is large)

Situation 1

3.3; 3.3 0.1; 5.1

5.1; 0.1 1.1; 1.1

TFT

Player 2

Player 1

ALL D

TFT ALL D

ALL D is dominant strategy

Situation 2

30; 30 9; 14

14; 9 10; 10

TFT

Player 2

Player 1

ALL D

TFT ALL D

TFT vs TFT results in a Nash equilibrium (but ALL D vs ALL D still is a NE too)

Cooperation in repeated social dilemmas: conclusions

• Goal-directed behavior can lead to cooperation without external enforcement and without internalized norm if the shadow of the future is large enough.

• Cooperation can be driven by enlightened self-interest.

Cooperation in the repeated Prisoner’s Dilemma

P T E

1 General Hypothesis(goal-directed behavior)• Strategies of actors

are in an equilibrium

2 Initial conditions and bridge-assumptions• Individual interactions:

PD-type• Repeated interactions with:

- stability w > T-R - cooperation costs T-P - perfect information on partner’s previous behavior

• two-sided expectation that partner plays TFT if (TFT,TFT) is equilibrium

3 Individual effects:Players use TFT Mutual cooperation

5 Collective effect:

Outcome is Pareto optimalNote: transformation rules and (some of the) conditions and bridge-assumptions are implicit in the PD matrix

4 Transformation rule• In problematic social

situations two-sided cooperation implies a Pareto-optimal outcome

Cooperation in repeated encounters

A

C

B

D

Macro

Micro

• Repeated PD interactions

• w > (T-R)/(T-P)Pareto-optimal outcome

PD matrix PD matrix

T>R>P>S

Coorientation

Players use TFT

Nash equilibrium behavior

Testable implications

P T E• Info on partner’s

behavior

• Stability of relation (shadow of the future)

• Costs of cooperation

• Coorientation

Cooperation

+

+

+

-

New Problems

P1 T E P2• Other strategies for the repeated game• Other games (other social dilemmas) - other payoff-matrix - more strategies than C and D in the “constituent game” - more actors• Network embeddedness: reputation effects• Partner selection selection and exit opportunities• Imperfect information on the behavior of the partner• Other mechanism of cooperation

- Voluntary commitments- Conditions for internalizing norms and values of cooperation

- Conditions for the emergence of external enforcement

Game theory and Axelrod’s analysis

• Nash equilibrium = +/- collective stability (see Axelrod, Propositions 2, 4, 5)

• Equilbrium analysis (collective stability): when is mutual cooperation stable?

Versus• Tournament approach and evolutionary analysis: (1) How

can cooperation emerge? (2) What are successful strategies in a variegated environment?