Revisiting “The Evolution of Reciprocity in Sizable Groups” Masanori Takezawa (Tilburg University, the Netherlands) Michael E. Price (Brunel University, UK)

Reciprocity is one mechanism sustaining the evolution of cooperation in two-person PD game but not the cooperation in n-person PD game.

Boyd & Richerson (1988) “The Evolution of Reciprocity in Sizable Groups” Argued that reciprocity per se cannot explain the evolution of cooperation in a repeated n-person PD game. Reciprocal strategy, Tn-1 : “Cooperate at the first round; cooperate if all the other group members (n - 1) cooperated at the last round; defect if one or more group members defected at the last round.” When common, Tn-1 does not allow invasion of rare All-D. When All-D is common, rare Tn-1 cannot invade into a population. In order for Tn-1 to evolve, almost entire population needs to be occupied by Tn-1.

If reciprocity cannot solve the problem of cooperation in n-person PD game, why is reciprocal behavior so frequently observed in the experiments of n-person PD game?

Evidence of Reciprocity in N-Person PD Game People seem to be conditional cooperators who cooperate only when they expect or observe that other people are cooperative. People match the level of contribution to the others’ contributions in the past; Fischbacher, Gaechter & Fehr (2001), Croson, Fatas & Neugebauser (2005), Croson (2006). Trust matters in n-person PD game; Yamagishi & Sato (1986), Yamagishi (1986).

The Difference Between Theories and Experiments Boyd & Richerson (1988; Joshi, 1987) modeled reciprocity as a trigger strategy that responds to the number of cooperators in the last round. Empirical evidence is weak that people use trigger strategy (Watabe, 1992; Watabe & Yamagishi, 1994; Ostrom, Gardner & Walker, 1994). Laboratory experiments indicate that people use matching strategies which determine an amount of contribution in response to average (or, minimum) contributions in the last round (see Cronson, 2005).

Fischbacher, Gaechter & Fehr (2001), p.400

A New Model: Replacing Tn-1with R Continuous reciprocal strategy: “Fully contribute at the first round; match an amount of contribution to the average contributions of the other n - 1 group members at the last round” All the rests are identical with Boyd & Richerson (1988); infinite population; groups of size n are randomly formed; interaction continues with a probability, w.

What happens if there is the third strategy, All-C What happens if there is the third strategy, All-C. All-C can invade into a population of R (or, T) but too many All-Cs feed All-D and lead the collapse of cooperation.

All-D Group size = 100 MPCR = 0.95 w = 0.7 All-C Tn-1

All-D Group size = 100 MPCR = 0.95 w = 0.7 R All-C

All-D Group size = 500 MPCR = 0.95 w = 0.7 R All-C

Continuous reciprocal strategy, R, looks much better than binary reciprocal strategy, Tn-1. What are the drawbacks of R? What are the implications of our study?

Problems of Continuous Reciprocal Strategy MPCR: A basin of attraction for R-C gets quickly smaller as MPCR (=efficiency of cooperation) gets worse. Mutation/Drift: In order for R to increase its share in a population, at least 5-6 % of a population needs to mutate into R from the other strategies. If mutation or drift is large enough to make it happen often, we need to consider in which of the two areas a system stays longer.

What Are the Implications of Our Study? Doubt a common knowledge! The difference between binary and continuous strategies looks very minor (e.g., both can explain gradual collapse of cooperation within a group) but produces significantly different result at evolutionary level. A warning to simple but tractable model? Binary reciprocity is a simple strategy which is rather easy to be analyzed but not a real model of human behaviors. Necessity of modeling empirically and psychologically realistic algorithms of human behaviors while keeping model tractability.

What Are the Implications of Our Study? Synergy effect of reciprocity + α? Reciprocity + punishment (e.g., Terai, Yamagishi & Watabe, 2003)? Reciprocity + assortative interaction (B&R, 1988)?

Thank you!