game-theoretic models for effects of social embeddedness on trust and cooperation werner raub
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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. Cooperation in Social Dilemmas. - PowerPoint PPT PresentationTRANSCRIPT
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?