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

1. Game-Theoretic Models for Effects of Social Embeddedness on Trust and CooperationWerner Raub Workshop on Social Theory, Trust, Social Networks, and Social Capital II National Chengchi University – NCCUApril 2011

2. Cooperation in Social Dilemmas P1  T1  E1  P2 Implications for research New research problem Problem of order Game Theory

3. 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]

4. 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

5. Prisoner’s Dilemma Player 2 C D C Player 1 D • 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

6. 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)

7. Prisoner’s Dilemma Player 2 C D C * * Player 1 D * • 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

8. Prisoner’s Dilemma:no cooperation in single encounters Pareto-suboptimal outcome One shot PD interaction Macro C D PD matrix PD matrix A B T>R>P>S Players defect Dominant strategies and Nash equilibrium behavior Micro

9. 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?

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

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

12. 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.

13. 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

14. 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

15. Types of strategies for the repeated game Nice, Provocable (and Forgiving) Strategies (e.g.: TFT) Conditional strategies Others Unconditional strategies (e.g.: ALL D, ALL C, Random)

16. 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.

17. 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

18. 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.

19. Repeated Prisoner’s Dilemma Player 2 ALL D TFT TFT Player 1 ALL D

20. TFT vs. TFT Step 1: Moves per round

21. TFT vs. TFT Step 2: Payoffs per round Step 3: Payoffs for the repeated game V(TFT,TFT) = R + wR + w2R + … + wt+1R + …

23. TFT vs. TFT Step 2: Payoffs per round Step 3: Payoffs for the repeated game V(TFT,TFT) = R + wR + w2R + … + wt+1R + …

24. ALL D vs. ALL D Step 1: Moves per round

25. ALL D vs. ALL D Step 2: Payoffs per round Step 3: Payoffs for the repeated game V(ALLD, ALLD) = P + wP + w2P + … + wt+1P + …

26. ALL D vs. TFT Step 1: Moves per round

27. ALLD vs. TFT Step 2: Payoffs per round Step 3: Payoffs for the repeated game Player 1: V(ALLD,TFT) = T + wP + w2P + … + wt+1P + … Player 2: V(TFT,ALLD) = S + wP + w2P + … + wt+1P + …

28. Repeated Prisoner’s Dilemma ALL D TFT TFT ALL D

29. Repeated Prisoner’s Dilemma ? ALL D TFT TFT ? ALL D

30. Equilibria • (ALL D, ALL D) is always an equilibrium • (ALL D, TFT) and (TFT, ALL D) are never equilibria • (TFT, TFT) is sometimes an equilibrium; namely if: Costs of cooperation Stability of relation (“shadow of the future”) Costs of conflict

31. Example Player 2 D C C Player 1 D Situation 2: W=0.9 (Shadow of the future islarge) Situation 1: W=0.1 (Shadow of the future issmall)

32. Situation 1 Player 2 ALL D TFT TFT Player 1 ALL D ALL D is dominant strategy

33. Situation 2 Player 2 ALL D TFT TFT Player 1 ALL D TFT vs TFT results in a Nash equilibrium (but ALL D vs ALL D still is a NE too)

34. 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.

35. Cooperation in the repeated Prisoner’s Dilemma • Transformation rule • In problematic social situations two-sided cooperation implies a Pareto-optimal outcome 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 optimal Note: transformation rules and (some of the) conditions and bridge-assumptions are implicit in the PD matrix

36. Cooperation in repeated encounters Macro • Repeated PD interactions • w > (T-R)/(T-P) Pareto-optimal outcome C D PD matrix PD matrix A B Players use TFT Nash equilibrium behavior T>R>P>S Coorientation Micro

37. Testable implications • Info on partner’s behavior • Stability of relation (shadow of the future) • Costs of cooperation • Coorientation P  T E + + Cooperation - +

38. 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

39. 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?