arXiv:0807.1823v3 [q-fin.GN] 4 Jan 2010 Cooperation Evolution in Random Multiplicative Environments. Gur Yaari The Racah Institute of Physics, The Hebrew University of Jerusalem, Edmond Safra Campus, Givat Ram, Jerusalem, 91905, Israel* Sorin Solomon The Racah Institute of Physics, The Hebrew University of Jerusalem, Edmond Safra Campus, Givat Ram, Jerusalem, 91905, Israel* and Institute for Scientific Interchange Viale Settimio Severo 65, 10113, Turin, Italy (Dated: January 4, 2010) 1

Abstract Most real life systems have a random component: the multitude of endogenous and exogenous factors influencing them result in stochastic fluctuations of the parameters determining their dy- namics. These empirical systems are in many cases subject to noise of multiplicative nature. The special properties of multiplicative noise as opposed to additive noise have been noticed for a long while. Even though apparently and formally the difference between free additive vs. multiplicative random walks consists in just a move from normal to log-normal distributions, in practice the im- plications are much more far reaching. While in an additive context the emergence and survival of cooperation requires special conditions (especially some level of reward, punishment, reciprocity), we find that in the multiplicative random context the emergence of cooperation is much more natural and effective. We study the various implications of this observation and its applications in various contexts. PACS numbers: 89.20.-a Interdisciplinary applications of physics , 89.65.-s Social and economic systems , 89.65.Gh Economics econophysics, financial markets, business and management , 89.75.Fb Structures and organization in complex systems Keywords: Multiplicative Random Process���High Risk��� Kelly strategy��� Altruism ���Cooperation *gyaari@gmail.com We acknowledge very instructive discussions with Nadav Shnerb, Ofer Biham, Itzhak Aharon, Damien Challet and Yi-Cheng Zhang. The present research was partially supported by the STREPs CO3 and DAPHNet of EC FP6. 2

I. BACKGROUND AND PREVIOUS KNOWLEDGE One of the puzzling facts in game theory is the recurring result, in a wide range of conditions that the general good is not reached by each of the individuals following its own self interest. Technically it means that in most of the games considered, the ���Nash Equilibrium��� is not an optimal situation for the ensemble of players taking part in the game [1]. Another name for this kind of situation is ���the tragedy of the commons��� [2]: if a set of players have access to a common good, and each of them will act only in his own interest, the common good is going to be over-exploited and ultimately lost/destroyed. The puzzle resides in the fact that the very existence of our society, human civilization and many animal societies is based on a much higher level of cooperation than suggested by the above models. The loopholes in the arguments leading to those pessimistic conclusions have been looked for at length and in certain cases some mechanisms to avoid this ���mean fate��� were found. For example, allowing for an infinitely long chain of iterated games might affect the non- cooperative result [1]. Five more mechanisms to ensure that cooperative traits will survive evolution in additive environments can be found here [3]. In the present paper we offer another alternative: we find that if the gains and losses of the participants are multiplicative, then the cooperative behavior becomes highly preferable and might in fact be realized in a very wide range of realistic examples. The fundamental feature of multiplicative process we exploit here is the fact that the expected gain of the players taking part in this iterative process depends in a crucial way on the number of players considered (number of independent realizations) and the number of time steps that the game is played. For long times (the number of time steps played in the game), the expected wealth of the players follows the geometric mean and not the arithmetic mean of the game (keep in mind that ���geometric mean ��� arithmetical mean���). In economics, one way to take into account this effect was to declare that what is to be maximized is not the wealth itself but rather the ���utility function��� [4]. The case where the ���utility function��� is the logarithm of the wealth U = ��� e p(e) ln(g(e)) reduces to considering the geometric mean rather than the arithmetic mean. Thus, the use of this utility function may be interpreted as a way to take into account the fact that in general a strategy is ap- plied repeatedly for long spans of time such that the frequency of the events approach their probability. Some of the behavioral anomalies studied over the years [5���7] can be related to 3