The Evolution of Fairness in the Ultimatum Game

Abstract

The article presents a model on the evolution of strategies in the Ultimatum game. In particular, the model considers the effect of role turnover and interaction structure. Past experimental studies showed that subjects behave more fairly than would be predicted by classical game theory. The goal of this model is to illuminate how fairness emerges in evolution when role turnover and preferential association are in effect. Simulation shows that evolution towards strategies of fairness is plausible. Some empirical implications are also discussed.

Keywords:

• evolution
• fairness
• network
• role turnover
• ultimatum game

Notes

¹ Put formally, suppose the strategy bundles of the actors are denoted by (x, y), where x ∊ [0, 1] represents his/her proposed offer, and y ∊ [0, 1] is the reservation value for being a responder. Perfect consistency, specified in this model, implies that the strategy space will be: x + y = 1.

²Although replicator dynamics were first proposed and discussed in biology, the idea is exchangeable with other similar mechanisms, such as imitation and reinforcement learning consistent with the socio-economic contexts we are interested here (Hopkins, 1999).

³Here k is a constant and is set to be 0.1 as it is not the main concern of my model.

⁴ Note that a zero entry in the payoff matrix P _ij (andR _ij ) will result in a zero entry in . This will bias in Equation (6) to be zero even though is greater than zero. To remedy this problem, we add all entries in P _ij (andR _ij ) by a tiny amount ε = 0.001 in advance such that the problem can be overcome while the ordering of payoffs in the new matrices P _ij and R _ij is not changed at all.