In this paper, we present a general framework for cooperation in evolutionary N-player games that encompasses and recovers traditional games such as the Prisoner’s Dilemma or Public Goods games as special cases. The basis of our framework is formed by the concept of discounting and synergy, which simply takes into account that the actual value of the benefits provided by cooperators may depend on the total number of cooperators in the group. Thus, with discounting the value of the benefit provided by the first cooperator in a group is b, but the value of the benefits provided by each additional cooperator is discounted by a factor w < 1 as compared to the previous cooperator. All cooperators pay a cost c. For example, this leads to the definition of an N-player Snowdrift game if the discounting factor w and the cost-benefit ratio cN/b are sufficiently small (). It is important to recall that discounting does not refer to potential future benefits but rather to the process of accumulating benefits provided by multiple cooperators. In the case of synergy, the value of the benefit provided by each additional cooperator is synergistically enhanced by a factor w > 1.
Viewing the traditional games from the perspective of this general framework emphasizes that the various scenarios - Prisoner’s Dilemma or Public Goods games, Snowdrift games, by-product mutualism, and bistability - are interconnected through variations of the continuous parameters w, c/b
and the group size N
, which seamlessly relates seemingly disparate biological situations. For example, the discomfort with the Prisoner’s Dilemma as the sole model for cooperation is increasing (Clutton-Brock, 2002
, Heinsohn & Parker, 1995
, West et al., 2002
), but viewing cooperation in the framework of discounting opens up natural connections to related scenarios, such as the Snowdrift game. In this way, our framework could prove to be helpful in bridging the gap between theoretical advances and experimental evidence.
In experimental settings it is notoriously difficult to quantitatively assess the fitness of strategic/behavioral patterns. For example, sticklebacks inspect their predators preferably in pairs and are believed to be trapped in a Prisoner’s Dilemma (Milinski, 1987
). However, despite tremendous efforts, only the payoff ranking T
has been experimentally confirmed (Milinski et al., 1997
). Consequently, it remains unresolved whether the fish indeed engage in a Prisoner’s Dilemma (requiring R
) or rather in a Snowdrift game (R
). In another example, a Prisoner’s Dilemma interaction has been shown to occur between RNA phages within host cells (Turner & Chao, 1999
), but selection alters the payoff structure such that cooperative and defective phage strains coexist in a Snowdrift game (Turner & Chao, 2003
Similarly, it has been argued that Prisoner’s Dilemma interactions occur in the aforementioned case of enzyme production in foraging yeast cells (Greig & Travisano, 2004
). Despite the apparent connection to the Prisoner’s Dilemma game given by the possibility of cheating, such frequency dependent benefits may be better captured by the Snowdrift game: if cooperators abound, defection is dominant and selfish individuals exploit the accrued benefits but as cooperators become rare, the costly enzyme production, may provide sufficient advantage to the producing individual despite by-product benefits to others, such that cooperation becomes dominant. Indeed, Greig & Travisano (2004)
report that cheating was beneficial only if a substantial fraction of the yeast population was cooperating, i.e., producing the enzyme.
Outside of biology, the study of social dilemmas has received particular attention by experimental economists and anthropologists (Fehr & Gächter, 2002
, Henrich et al., 2001
, Panchanathan & Boyd, 2004
). Humans display an apparently irrational, high readiness to cooperate in Public Goods and Ultimatum games (Güth et al., 1982
, Nowak et al., 2000
), which confounds the basic rationality assumptions of homo oeconomicus
. In both games, defection is dominant but the Ultimatum game adds aspects of punishment because it can be interpreted as a Prisoner’s Dilemma interaction followed by a round of (costly) punishment (Sigmund et al., 2001
). Punishment and reputation have been identified as very potent promoters of human cooperation in social dilemmas (Fehr & Gächter, 2002
, Milinski et al., 2002
, Wedekind & Milinski, 2000
). Such additional mechanisms can be easily incorporated into our framework. However, already in Public Goods interactions, where only the multiplication factor of the common good depends on the total amount invested, qualitatively different outcomes can be generated, which allow e.g. for co-existence of cooperators and defectors in a generalized Snowdrift game. In experimental settings, variations of the multiplication factor could test the sensitivity of human behavior to quantitative and qualitative changes of the interaction characteristics.
In summary, Snowdrift games can be considered as social dilemmas that are intermediate between Prisoner’s Dilemma games (or Public Goods games in larger groups) and by-product mutualism, which occur whenever ordinary selfish behavior benefits others (Brown, 1983
, West-Eberhard, 1975
). By-product mutualism has also been put forth to challenge the Prisoner’s Dilemma for explaining patterns of cooperation in natural populations (Connor, 1995
, Dugatkin, 1996
, Milinski, 1996
). Our general theoretical framework for cooperation in social dilemmas seems capable of reconciling the different viewpoints and emphasizes that the different dynamical domains of social dilemmas are related by continuous changes in biologically meaningful parameters.