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 – NCCU April 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 1Examples 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 2General " The pursuit of self-interest by each leads to a poor outcome for all." [Axelrod 1984:7]

The explanatory problem related to social 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,RS,T T,SP,P CD 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,SP,P CD 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 T>R>P>ST>R>P>SPlayers 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 = g 1 + wg 2 + w²g w t-1 g t +... with: 0 < w < 1 for the discount parameter w g t : 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 the repeated Prisoner’s Dilemma ALL D : Play D in each round Thus, ALL D is - Unconditional - Not Nice TFT : 1Play C in each round 1. 2Imitate 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 PlayerRound 123…tt+1… 1 (TFT)CCC…CC… 2 (TFT)CCC…CC… Step 1: Moves per round

TFT vs. TFT PlayerRound 123…tt+1… 1 (TFT)RRR…RR… 2 (TFT)RRR…RR… Step 2: Payoffs per round Step 3: Payoffs for the repeated game V(TFT,TFT) = R + wR + w 2 R + … + w t+1 R + …

TFT vs. TFT PlayerRound 123…tt+1… 1 (TFT)RRR…RR… 2 (TFT)RRR…RR… Step 2: Payoffs per round Step 3: Payoffs for the repeated game V(TFT,TFT) = R + wR + w 2 R + … + w t+1 R + …

ALL D vs. ALL D PlayerRound 123…tt+1… 1 (ALLD)DDD…DD… 2 (ALLD)DDD…DD… Step 1: Moves per round

ALL D vs. ALL D PlayerRound 123…tt+1… 1 (ALLD)PPP…PP… 2 (ALLD)PPP…PP… Step 2: Payoffs per round Step 3: Payoffs for the repeated game V(ALLD, ALLD) = P + wP + w 2 P + … + w t+1 P + …

ALL D vs. TFT PlayerRound 123…tt+1… 1 (ALLD)DDD…DD… 2 (TFT)CDD…DD… Step 1: Moves per round

ALLD vs. TFT PlayerRound 123…tt+1… 1 (ALLD)TPP…PP… 2 (TFT)SPP…PP… Step 2: Payoffs per round Step 3: Payoffs for the repeated game Player 1: V(ALLD,TFT) = T + wP + w 2 P + … + w t+1 P + … Player 2: V(TFT,ALLD) = S + wP + w 2 P + … + w t+1 P + …

Repeated Prisoner’s Dilemma R R ; w wP wP S ; T w 1-w wP wP T ; S w 1-w P P ; w TFT ALL D TFT ALL D

Repeated Prisoner’s Dilemma R R ; w wP wP S ; T w 1-w wP wP T ; S w 1-w P P ; w TFT ALL D TFT ALL D ? ?

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 Costs of conflict Stability of relation (“shadow of the future”)

Example R=3; R=3S=0; T=5 T=5; S=0P=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; ; ; ; 1.1 TFT Player 2 Player 1 ALL D TFT ALL D ALL D is dominant strategy

Situation 2 30; 309; 14 14; 910; 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 1General Hypothesis (goal-directed behavior) Strategies of actors are in an equilibrium 2Initial 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 3Individual effects: Players use TFTMutual cooperation 5Collective effect: Outcome is Pareto optimal Note: transformation rules and (some of the) conditions and bridge-assumptions are implicit in the PD matrix 4Transformation 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 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?