Analyzing the payoff of a heterogeneous population in the ultimatum game

Abstract

This paper aims at showing how analytical techniques can be employed to explain the global emerged behavior of a heterogeneous population of ultimatum game players, over different strategies, by calculating their payoff moments. The ultimatum game is a game, in which two players are offered a gift to be shared. One of the players (the proposer) suggests how to divide the offer while the other player (the responder) can either agree or reject the deal. Computer simulations were performed considering the concept of turns (in every turn each participant plays necessarily only once, which is equivalent to performing matching a graph) in the game. We reproduce by simulations the expected analytical results at the limit of high number of turns. From these results, we are capable of establishing diagrams to say where each strategy is the best (optimal strategy).Keywords: Evolutionary game theory; Ultimatum game; Payoff moments It is very hard to find realistic models to describe the complexity of economic behavior in real societies. The same can be said about biological populations with a large number of participants over different strategies competing to survive. However, some interesting answers can be provided using Game Theory [1], which should describe negotiations among different individuals via a payoff matrix that quantifies all possible actions that can be taken by players in the game.This theory, an accepted approach by theoretical economists due to contributions by John Nash [4] have also became accepted by biologists after the essential works of John Maynard Smith [2, 3] if we have in mind the different relations existing among species and among individuals of the same species because of a simple way different strategies can be tested and the results extrapolated to real situations in societies. In this evolutionary branch [8], this theory can bring about questions related to equilibrium, survival of strategies assuming not only one game play but several plays. Such theory is able to represent the dynamic behavior of games in average under different social contexts: in economy, such as public good games[9-11], minority game [12], in biology, prisoner dilemma [8] and others and it also can consider the different networks and their influence on the emerging collective behaviors.In the game defined in [5], the usual ultimatum game, two players must divide a quantity between them. One of them proposes a division (the proposer) and the other can accept or reject it (the responder). If the responder rejects the offer, both players receive nothing, otherwise the "money" will be shared accordingly. A good analogy of this game, in a situation involving human beings, is how a percentage paid to an agent in a sale can be negotiated.In human economic experiments, one could see a tendency to offer a division as near as possible to fifty-fifty and to reject values lower than 30% (see e.g. [5], [13]). However, this goes against results obtained by the classical game theory that claims participants play under Nash equilibr...

Defense Mechanisms of Empathetic Players in the Spatial Ultimatum Game

“…In fact, largely regardless of sex, age and the amount of money at stake, people refuse to accept offers they perceive as too small [3,4]. More precisely, offers below one third of the total amount are rejected as often as they are accepted, and not surprisingly, more than two thirds of all offers will be remarkably close to the fair 50:50 split.While explanations for the human fondness of fair division range from the psychologically-inspired definitions of utility functions [5] to the failure of "seizing the moment" (implying that after the game there will be no further interactions between the two players) [6], theoretical studies indicate that empathy [7,8], spatial structure [9-11], heterogeneity [12], and reputation [13] play a pivotal role. In particular, Page et al [9] have shown that in well-mixed populations natural selection favors the rational solution, while spatiality may lead to much fairer outcomes.…”

Random allocation of pies promotes the evolution of fairness in the Ultimatum Game

“…While others have shown that the preference of people towards fairness may be due to the repeated interactions in the Ultimatum Game 6 8 13 14 . In the context of evolutionary game theory 15 , theoretical studies indicate that small group size 16 , reputation 17 , empathy 18 , population structure 19 20 21 22 23 24 25 26 27 28 29 and heterogeneity 30 31 play a vital role in the evolution of fairness in the Ultimatum Game.…”

Randomness and arbitrary coordination in the reactive ultimatum game

“…We also can imagine it as parameters averaged by the different players in a large population, where the players interact at each time t (denoted by authors in refs. [11], and [12] as one 'turn') by pairs composing a perfect matching with N players (for the sake of simplicity N is an even number) randomly composed. In this pairing, no player is left out of the game, with each individual playing once by turn, by construction.…”