Fairness emergence from zero-intelligence agents Wen-Qi Duan 1,2 and H. Eugene Stanley 2 1 School of Business Administration, Zhejiang Normal University, Jinhua 321004, People’s Republic of China 2 Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA Received 10 September 2009; revised manuscript received 11 November 2009; published 11 February 2010 Fairness plays a key role in explaining the emergence and maintenance of cooperation. Opponent-oriented social utility models were often proposed to explain the origins of fairness preferences in which agents take into account not only their own outcomes but are also concerned with the outcomes of their opponents. Here, we propose a payoff-oriented mechanism in which agents update their beliefs only based on the payoff signals of the previous ultimatum game, regardless of the behaviors and outcomes of the opponents themselves. Employing adaptive ultimatum game, we show that 1fairness behaviors can emerge out even under such minimalist assumptions, provided that agents are capable of responding to their payoff signals, 2the average game payoff per agent per round decreases with the increasing discrepancy rate between the average giving rate and the average asking rate, and 3the belief update process will lead to 50%-50% fair split provided that there is no mutation in the evolutionary dynamics. DOI: 10.1103/PhysRevE.81.026104 PACS numbers: 89.65.Ef, 87.23.Ge, 02.50.Le, 87.10.Rt Understanding the evolution of cooperative behavior is one of the greatest challenges in the modern biological and social sciences. Fairness plays a key role in explaining the emergence and maintenance of cooperation in evolving populations. In order to punish unfair behaviors, humans are even willing to forego material payoffs 1. Such fairness and other-regarding preferences have been widely studied by us- ing the ultimatum game 2. In the standard version of the ultimatum game, two players are given the opportunity to split a sum of money. They are assigned the role of either proposer or responder. The proposer makes an offer on how to divide this money and the responder can then accept or reject the proposer’s offer. If it is accepted, the money is split as proposed, whereas if it is rejected, then neither player receives anything. The canonical economic model of pure self-interest predicts that the proposer will offer the smallest share possible and that the responder will accept any positive offer because the alternative is a zero payoff. However, ulti- matum game literature indicates that, irrespective of the monetary sum, proposers typically make offers of 40% to 50% and responders routinely reject offers under 20% 3. Furthermore, this kind of inequity aversion may not be uniquely human quality. Once being put under conditions similar to those of humans, animals would also behave like humans, if they are able to grasp the fundamental aspects of these conditions. For example, brown capuchin monkey re- sponded negatively to unequal reward distribution in ex- changes with a human experimenter 4. There is a volumi- nous literature discussing the cultural, social, and genetically evolutionary origins of the observed fairness preferences, in which all players are capable of comparing their own efforts and payoffs with those of others 5–7. However, remember- ing opponents’ payoffs or reasoning out a better strategy is a difficult task for most nonhumans with extremely low intel- ligence. Therefore, we need to reduce the complexity of the rules, which an individual, animal, or human, must be able to grasp in order to show fairness behavior in the ultimatum game. This paper develops an adaptive ultimatum game model in which agents are only required to react adaptively to their own payoff signals of the previous ultimatum game. Unlike those opponent-oriented models, agents are not re- quired to have remembering or reasoning capability. Based on computer simulations, the results show that fairness be- haviors can always emerge from zero intelligence agents. The ultimatum game will even reach at 50%-50% fair split if there is no mutation in the evolutionary dynamics. To show how our payoff-oriented model can lead to the emergence of fairness behaviors, we introduced the adaptive ultimatum game and studied it from the perspective of evo- lutionary game theory. Different from the published litera- ture, we shift our focus on fairness behavior rather than on fairness sense. The opponent-oriented concept of fairness sense is related to player’s subjective understanding of the equity of the situation. Such subjective approaches might work in studying humans, but are impossible in studying nonhumans. Fairness behavior method relies on the informa- tion provided by players’ behavioral reactions to inequitable situations. It is a more general concept and can be applied to study both humans and nonhumans 8. Similar to the adap- tive dynamics framework to describe evolutionary ultimatum game in Refs. 9,10, fairness behaviors can be characterized by agents’ willingness to give to and anticipation to receive from their opponents. For a given monetary sum, the agent acting as proposer will offer its own giving rate p of the total money and the agent acting as responder will reject any offer smaller than its own asking rate q of the total money. With- out loss generality, we set the total money to 1. The values of p and q, called as a player’s strategy in Ref. 9, can repre- sent agents’ internal beliefs about how to respond to external payoff signals. Evidently, we can observe more fairness be- haviors when p and q are more close to 0.5. In our adaptive ultimatum game, we assume agents with little intelligence so that we can uncover the evolutionary mechanism of fairness from the view of population dynamics of zero intelligence agents 11. In the evolutionary process of our adaptive ulti- matum game, each agent makes offer or responds to other agent’s offer only based on its internal belief value p or q. Agents do not purposefully make any strategic offers that are less likely to be refused. They also do not intentionally pun- ish their opponents. For each ultimatum game, if the re- PHYSICAL REVIEW E 81, 026104 2010 1539-3755/2010/812/0261046©2010 The American Physical Society 026104-1