The shared reward dilemma

Abstract

One of the most direct human mechanisms of promoting cooperation is rewarding it. We study the effect of sharing a reward among cooperators in the most stringent form of social dilemma, namely the Prisoner's Dilemma. Specif cally, for a group of players that collect payoffs by playing a pairwise Prisoner's Dilemma game with their partners, we consider an external entity that distributes a f xed reward equally among all cooperators. Thus, individuals confront a new dilemma: on the one hand, they may be inclined to choose the shared reward despite the possibility of being exploited by defectors; on the other hand, if too many players do that, cooperators will obtain a poor reward and defectors will outperform them. By appropriately tuning the amount to be shared a vast variety of scenarios arises, including traditional ones in the study of cooperation as well as more complex situations where unexpected behavior can occur. We provide a complete classif cation of the equilibria of the n-player game as well as of its evolutionary dynamics.

The shared reward dilemma on structured populations

“…2, analytically obtained for regular networks, show a phenomenology which is very similar to that reported for the population without structure in Cuesta et al (2008). In particular, when ζ < 1/2, we find coexistence of two mixed equilibria.…”

“…Of course, as we have already mentioned, even if it is a simple mechanism, it is not at all trivial in so far as larger rewards do not necessarily lead to larger levels of cooperation. To this knowledge, which was already hinted in Cuesta et al (2008) we can now add the main conclusion of this paper: the effectiveness of the reward depends on the structure of the population and on the size of the neighborhoods, i.e., on the diameter of the set of agents which a given one interacts with. Therefore, application of this proposal in actual situations, specifically, on social networks, would require a careful analysis to foresee the outcome of rewarding cooperation.…”

“…if n i = n for all 1 ≤ i ≤ N ), the conditions for full cooperation and full defection to be Nash equilibria become, respectively, δ ≥ 1 and δ ≤ 1/N ζ . Moreover, the proof of Theorem 1 in Cuesta et al (2008) can be adapted to show that for any regular network with degree n ≥ 3, the number of symmetric Nash equilibria in mixed strategies depends only on the values of δ and ζ . Specifically, the changes in the structure of equilibria when δ increases are described by one out of Figure 1 depicts the typical structure of equilibria for these three cases.…”

“…To this end, we study a simple mechanism for the emergence of cooperation, which we introduced in Cuesta et al (2008) and which we refer to as the shared reward. In this setting, players interact through a standard Prisoner's Dilemma, but in a second stage cooperators receive an additional payoff coming from an exogenous source available only to them and not to defectors.…”

Variability in group size and the evolution of collective action

“…As shown in the vast literature on nonlinear public goods games (e.g., Dugatkin, 1990;Motro, 1991;Bach et al, 2006;Hauert et al, 2006;Cuesta et al, 2008;Pacheco et al, 2009;Archetti and Scheuring, 2011) cooperative behavior may arise in the evolutionary solution of such games even when other mechanisms potentially promoting cooperation such as relatedness (Eshel and Motro, 1988;Archetti, 2009;Peña et al, 2015) and reciprocity in repeated interactions (Boyd and Richerson, 1988;Hilbe et al, 2014) are absent.…”