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.
We consider a version of the ultimatum game which simultaneously combines reactive and Darwinian aspects with offers in [0,1]. By reactive aspects, we consider the effects that lead the player to change their offer given the previous result. On the other hand, Darwinian aspects correspond to copying a better strategy according to best game payoff when the current player compares with one of their neighbours. Therefore, we consider three different strategies, which govern how the players change their offers: greedy, moderate, and conservative. First, we provide an analytic study of a static version of game, where Darwinian aspects are not considered. Then, by using numerical simulations of a detailed and complete multi-agent system on a two dimensional lattice, we add an extra feature, in which players probabilistically scape from extreme offers (those close to 0 or 1) for obvious reasons. The players are also endowed reciprocity on their gains as proposers, which is reflected on their gains as responders. We also analyse the influence of the player's mobility effects. An analysis of the emergence of coexistence of strategies and changes on the dominant strategies are observed, which in turn depends on the player's mobility rate.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.