Cooperation without Association
Transcript Of Cooperation without Association
Cooperation without Association
Christopher Ahern1, Robin Clark1, Steven Kimbrough2 1Department of Linguistics, 2Operations and Information Management
University of Pennsylvania
Contact Information: [email protected] [email protected] [email protected]
Introduction
How does association factor into the evolution of cooperation? The key to the evolution of cooperation, collective action, and social structure is [association]. [Association] of interactions allows the evolution of cooperative social structure that would otherwise be impossible. (Skyrms, 2004, pp. xii–xiii)
Association is central to: • Kin Selection (Hamilton, 1964) • Direct Reciprocity (Hamilton and Axelrod, 1981) • Indirect Reciprocity (Nowak and Sigmund, 1998) • Group Selection (Traulsen and Nowak, 2006)
Question: Can cooperation be sustained without association? Answer: Shadow of Society can loom as the Shadow of the Future.
Model
Prisoner’s Dilemma • Two agents choose between cooperating and defecting. • Cooperating provides a benefit to opponent, b, at a cost, c, where b > c > 0 • Cooperating is strictly dominated in one-shot game.
WIN-STAY, LOSE-SHIFT
• Strategy in an iterated PD (Nowak and Sigmund, 1993) • If agent WINS, continue with same action; if agent LOSES, shift to the other
action.
Cooperate Cooperate b − c, b − c
Defect b, −c
Defect −c, b 0, 0
Cooperate Defect Cooperate Stay, Stay Shift, Stay
Defect Stay, Shift Shift, Shift
Social Information • Agents have knowledge of others’ payoffs. • Winning is doing better than the average payoff of some subset of population.
Results
• Randomly-paired interactions in population with proportion of cooperators p. • Calculate the evolution and stability of p over time: p → p → p . . . • Closed circles represent stable equilibria, open circles represent unstable equi-
libria.
Insufficient Knowledge
p'
0.0 0.2 0.4 0.6 0.8 1.0
p'
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
p
0.0
0.2
0.4
0.6
0.8
1.0
p
Figure 1: Proportion of cooperators with knowledge of single opponent (left) and opponent and one more (right).
Cooperation cannot be sustained.
Sufficient Knowledge
p'
0.0 0.2 0.4 0.6 0.8 1.0
p'
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
p
0.0
0.2
0.4
0.6
0.8
1.0
p
Figure 2: Proportion of cooperators with knowledge opponent and two more where cb < 2 (left) and cb > 2 (right).
Two’s company, three is a crowd.
Limiting Analysis
p'
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
p
Figure 3: Proportion of cooperators with knowledge of population average
Maximum rate of cooperation.
Conclusions
• Cooperation is possible without association by WIN-STAY, LOSE-SHIFT with social information.
• Very modest epistemic requirements for positive amount of cooperation. • Maximum rate of cooperation is half (Palomino and Vega-Redondo, 1999).
Future Directions
• What does a little bit of association add? • What effect of punishment or reward? • What predictions for larger class of games?
References
Hamilton, W.D., and R. Axelrod. 1981. The evolution of cooperation. Science 211(27):1390–1396. Hamilton, William D. 1964. The genetical evolution of social behaviour. I and II. Journal of theoretical biology 7(1):1–16. Nowak, M.A., and K. Sigmund. 1998. Evolution of indirect reciprocity by image scoring. Nature 393(6685):573–577. Nowak, Martin, and Karl Sigmund. 1993. A strategy of win-stay, lose-shift that outperforms tit-for-tat in the Prisoner’s Dilemma game. Nature 364(6432):56–58. Palomino, Frederic, and Fernando Vega-Redondo. 1999. Convergence of aspirations and (partial) cooperation in the prisoner’s dilemma. International Journal of Game Theory 28(4):465–488. Skyrms, Brian. 2004. The Stag Hunt and the Evolution of Social Structure. , Cambridge, UK: Cambridge University Press. Traulsen, Arne, and Martin A Nowak. 2006. Evolution of cooperation by multilevel selection. Proceedings of the National Academy of Sciences 103(29):10952–10955.
Christopher Ahern1, Robin Clark1, Steven Kimbrough2 1Department of Linguistics, 2Operations and Information Management
University of Pennsylvania
Contact Information: [email protected] [email protected] [email protected]
Introduction
How does association factor into the evolution of cooperation? The key to the evolution of cooperation, collective action, and social structure is [association]. [Association] of interactions allows the evolution of cooperative social structure that would otherwise be impossible. (Skyrms, 2004, pp. xii–xiii)
Association is central to: • Kin Selection (Hamilton, 1964) • Direct Reciprocity (Hamilton and Axelrod, 1981) • Indirect Reciprocity (Nowak and Sigmund, 1998) • Group Selection (Traulsen and Nowak, 2006)
Question: Can cooperation be sustained without association? Answer: Shadow of Society can loom as the Shadow of the Future.
Model
Prisoner’s Dilemma • Two agents choose between cooperating and defecting. • Cooperating provides a benefit to opponent, b, at a cost, c, where b > c > 0 • Cooperating is strictly dominated in one-shot game.
WIN-STAY, LOSE-SHIFT
• Strategy in an iterated PD (Nowak and Sigmund, 1993) • If agent WINS, continue with same action; if agent LOSES, shift to the other
action.
Cooperate Cooperate b − c, b − c
Defect b, −c
Defect −c, b 0, 0
Cooperate Defect Cooperate Stay, Stay Shift, Stay
Defect Stay, Shift Shift, Shift
Social Information • Agents have knowledge of others’ payoffs. • Winning is doing better than the average payoff of some subset of population.
Results
• Randomly-paired interactions in population with proportion of cooperators p. • Calculate the evolution and stability of p over time: p → p → p . . . • Closed circles represent stable equilibria, open circles represent unstable equi-
libria.
Insufficient Knowledge
p'
0.0 0.2 0.4 0.6 0.8 1.0
p'
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
p
0.0
0.2
0.4
0.6
0.8
1.0
p
Figure 1: Proportion of cooperators with knowledge of single opponent (left) and opponent and one more (right).
Cooperation cannot be sustained.
Sufficient Knowledge
p'
0.0 0.2 0.4 0.6 0.8 1.0
p'
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
p
0.0
0.2
0.4
0.6
0.8
1.0
p
Figure 2: Proportion of cooperators with knowledge opponent and two more where cb < 2 (left) and cb > 2 (right).
Two’s company, three is a crowd.
Limiting Analysis
p'
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
p
Figure 3: Proportion of cooperators with knowledge of population average
Maximum rate of cooperation.
Conclusions
• Cooperation is possible without association by WIN-STAY, LOSE-SHIFT with social information.
• Very modest epistemic requirements for positive amount of cooperation. • Maximum rate of cooperation is half (Palomino and Vega-Redondo, 1999).
Future Directions
• What does a little bit of association add? • What effect of punishment or reward? • What predictions for larger class of games?
References
Hamilton, W.D., and R. Axelrod. 1981. The evolution of cooperation. Science 211(27):1390–1396. Hamilton, William D. 1964. The genetical evolution of social behaviour. I and II. Journal of theoretical biology 7(1):1–16. Nowak, M.A., and K. Sigmund. 1998. Evolution of indirect reciprocity by image scoring. Nature 393(6685):573–577. Nowak, Martin, and Karl Sigmund. 1993. A strategy of win-stay, lose-shift that outperforms tit-for-tat in the Prisoner’s Dilemma game. Nature 364(6432):56–58. Palomino, Frederic, and Fernando Vega-Redondo. 1999. Convergence of aspirations and (partial) cooperation in the prisoner’s dilemma. International Journal of Game Theory 28(4):465–488. Skyrms, Brian. 2004. The Stag Hunt and the Evolution of Social Structure. , Cambridge, UK: Cambridge University Press. Traulsen, Arne, and Martin A Nowak. 2006. Evolution of cooperation by multilevel selection. Proceedings of the National Academy of Sciences 103(29):10952–10955.