Evolutionary dynamics of group fairness
Introduction
Although evidence exists of collective action problems that other species engage on and successfully solve (Axelrod and Hamilton, 1981, Maynard-Smith and Price, 1973), humans are truly singular in the extent to which they resort to collective action (Barrett, 2003, Hardin, 1968, Henrich et al., 2001, Kollock, 1998, Levin, 2012, Ostrom, 1990). To this end they have developed highly sophisticated and unparalleled mechanisms (Apicella et al., 2012, Dietz et al., 2003, Fehr and Fischbacher, 2003, Fowler and Christakis, 2010, Gintis, 2000, Kollock, 1998, Nowak and Sigmund, 2005, Ohtsuki and Iwasa, 2004, Pacheco et al., 2009, Skyrms, 2004, Skyrms, 2010). Similar to what happens when two individuals interact, also when groups get together to make collective decisions, the concept of fairness is known to play a very important role (Fehr and Schmidt, 1999, Fehr and Gächter, 2000, Henrich et al., 2001, Rabin, 1993). However, and in sharp contrast to the two-person interaction cases, where fairness has been given considerable attention, both theoretically (Binmore, 1998, Gintis et al., 2003, Nowak et al., 2000, Page and Nowak, 2002) and experimentally (Fehr and Schmidt, 2006, Henrich et al., 2001, Henrich et al., 2010, Rand et al., 2013, Sanfey et al., 2003), the study of fairness in connection to group decisions has been limited to few investigations (Bornstein and Yaniv, 1998, Fischbacher et al., 2009, Messick et al., 1997, Robert and Carnevale, 1997).
The Ultimatum Game (UG) (Güth et al., 1982) has constituted the framework of choice (for an exception, see (Van Segbroeck et al., 2012)) with which to address the emergence and evolution of fairness in two-person interactions. In the two-person UG (and using common notation (Sigmund, 2010)), one individual plays the role of the Proposer whereas the other individual plays the role of the Responder. The Proposer is endowed with an amount (without loss of generality, we may assume it is 1 unit) and makes an offer, which is a fraction p of that amount. We may assume that the Responder has an expectation value q, and will accept the proposal if the condition p≥q is satisfied, in which case the Proposer earns 1−p and the Responder earns p. In such a scenario, a fair offer corresponds to an equal split, that is p=1/2.
Theoretical (Gale et al., 1995, Nowak et al., 2000, Page and Nowak, 2002, Rand et al., 2013) and experimental (Henrich et al., 2004, Henrich et al., 2001) investigations have shown that humans generally accept offers with p≥0.4, a feature which (with some variation) applies to both old and new societies, as a series of cross-culture studies have shown (Henrich et al., 2001). But what about group decisions? While there are no theoretical studies of the UG in a group context, there is one experiment (Messick et al., 1997) which naturally captures some limiting situations of the Multi-Player UG (MUG) that we develop below, and which puts in evidence a small part of the plethora of new possibilities enacted by group decisions, with direct consequences in what concerns the concept of fairness. Naturally, we shall take the opportunity to compare our independent theoretical predictions with available results from the aforementioned behavioral experiments.
Section snippets
Multiplayer ultimatum games
Let us assume a large population of individuals, who assemble into groups of size N. Each individual has a strategy encoded by 2 real numbers , whose significance is explained below. At any instance of the game, there will be 1 Proposer and N−1 Responders. Following the conventional notation of UG (Sigmund, 2010), the total amount initially given to the Proposer is equal to 1. The Proposer will offer an amount p∈[0,1] corresponding to her/his strategy, whereas each of the N−1
Results and discussion
The results of the numerical simulations are shown in Fig. 1, where we plot the average stationary values of p and q in the population (normalized by the corresponding values for N=2) as a function of the group size N, for two limiting group behaviors: M=1 (left panel) and M=N−1 (right panel). Thus, whereas the left panel shows how p and q evolve as a function of group size in situations in which offer acceptance by a single Responder is enough to ensure collective action, the right panel shows
Acknowledgments
This work was supported by national funds through Fundação para a Ciência e a Tecnologia (FCT) via the projects SFRH/BD/94736/2013 and EXPL/EEI-SII/2556/2013, and by multi-annual funding of INESC-ID and CBMA-UM (under the projects UID/CEC/50021/2013 and PEst-C/BIA/UI4050/2011).
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