Randomness and arbitrary coordination in the reactive ultimatum game

Abstract

The ultimatum game explains and is a useful model in the analysis of several effects of bargaining in population dynamics. Darwin's theory of evolution -as introduced in game theory by Maynard Smith -is not the only important evolutionary aspect in a evolutionary dynamics, since complex interdependencies, competition, and growth should be modeled by, for example, reactive aspects. In biological models, computationally or analytically considered, several authors have been able to show the emergence of cooperation with stochastic or deterministic dynamics based on the mechanism of copying the best strategies. On the other hand, in the ultimatum game the reciprocity and the fifty-fifty partition seems to be a deviation from rational behavior of the players under the light of the Nash equilibrium concept. Such equilibrium emerges from the punishment of the responder who generally tends to refuse unfair proposals. In the iterated version of the game, the proposers are able to improve their proposals by adding an amount thus making fairer proposals. Such evolutionary aspects are not properly Darwinian-motivated, but they are endowed with a fundamental aspect: they reflect their actions according to value of the offers. Recently, a reactive version of the ultimatum game where the acceptance occurs with fixed probability was proposed. In this paper, we aim at exploring this reactive version of the ultimatum game where the acceptance by the players depends on the offer. In order to do so, we analyze two situations: (i) mean field and (ii) by considering the players inserted within the networks with arbitrary coordinations. In the proposed model we not only explore situations of occurrence of the fifty-fifty steady-state, in both homogeneous and heterogeneous populations, but also explore the fluctuations and payoff distribution characterized by the Gini coefficient of the population. We then show that the reactive aspect, here studied, thus far not analyzed in the evolutionary game theory literature can unveil an essential feature for the convergence to fifty-fifty split. Our approach concerns four different policies to be adopted by the players. In such policies the evolutionary aspects do not work through a Darwinian copying mechanism, but by following a policy that governs the increase or decrease of their offers according to the response of the result -i.e. acceptance or refusal. Moreover, we present results where the acceptance occurs with fixed probability. Our contribution is twofold: we present both analytical results and MC simulations which in turn are useful to design new controlled experiments in the ultimatum game in stochastic and deterministic scenarios.

A Reactive-Darwinian Model for the ultimatum game: On the dominance of moderation in high diffusion

“…Here, in a different way, we consider three different strategies and not four as in [6], since two of them are very similar. The strategies differ on the number of positive responses of the neighbours in regulating the offers.…”

“…In the first approach, we consider an analysis of the simple static combination of strategies, according to certain probabilities, studying the stationary offer of the players. In this case we consider that players i and j are uncorrelated and p( j|i;t) does not depend on j and only on the offer of the proposer, i.e., p( j|i;t) = O (i) t as considered in [6]. So we imagine an iterated game, where a player inserted in a population executes N iterations (plays or combats) per time step t. In each iteration, it makes offers to k neighbour players (its coordination).…”

“…Thus, when considering the effects of the ultimatum game on large populations with or without spatial structures, several perspectives on social, economic, and biological sense can be explored. Among the interesting ways to consider this game in large populations, we recently considered the reactive ultimatum game proposed in [5,6]. This is an interesting protocol based on direct responses the players can take given a bad or good result.…”

“…So the offers can incremented/decremented by a quantity ε > 0 if the responders reject/accept, as a kind of regulation. In [6] previous studies of a static version of the game (fixed strategies) were studied, considering that a player regulates their offer by following more conservative actions (in the financial sense) or being more greedy.…”

“…However other questions should be answered and explored under this approach, but first we suggest some reformulation and simplification of the model considered in [6], by reducing from 4 to 3 strategies 1…”