Snowdrift games, cooperation, and "tragedy of the commune"

It’s the second day of June, which means it’s a good time to consider snowdrifts. OK, maybe not – but at least we’re far enough from winter now that the thought of snowdrifts out the window isn’t enough to give me a chill.

The Snowdrift Game is a theoretical model of cooperation within the context of game theory. I gave a short introduction to game theory a couple of years ago, focusing on the games of Chicken and the Prisoner’s Dilemma. There are really only two formal varieties of two-player games involving cooperation or defection in the absence of information transfer. When defection is always the optimal strategy, it’s the Prisoner’s Dilemma. When a mixed strategy of cooperation and defection is optimal, it’s Chicken.

But there are other names for this game. I’m not sure why, exactly—I suppose it’s because teenage boys in dragsters don’t appeal to everybody. One familiar name is the Hawk-Dove game. An individual can adopt two strategies: either attack and fight for a resource, or share equally and retreat when attacked. In the game, fighting carries a high cost (like wrecking your car into somebody) so a mixed strategy is optimal. When hawks are common, it’s better to be a dove and avoid fighting. When doves are common, it’s better to be a hawk because you always win.

A third name for this game is Snowdrift. Imagine you’re riding in a car that becomes stuck in a snowdrift. You and a fellow passenger share the same interest: you both want the snowdrift to be removed. But who’s going to get out and shovel? It might seem fair just to get out and shovel the snow together—in other words, to cooperate. But what if the other passenger just sits there and refuses to help? If the cost of shoveling is low compared to the benefit of getting out of the drift, it will be in your interest to shovel by yourself. Sure, the other passenger is a freeloader who shares the benefit undeservedly, but so what? If the cost of shoveling was too high for you to bear, you’d have refused to do it, letting both of you freeze there. That would be the Prisoner’s Dilemma. But if the cost of shoveling is low compared to the costs of doing nothing, then a mixed strategy will be optimal. As long as freeloaders aren’t too common, that strategy will pay off. So a population engaged in the Snowdrift game will come to a mixed proportion of shovelers and freeloaders.

Doebeli et al. (2004) considered the Snowdrift game as a model for the evolution of cooperation. A mixed strategy of cooperation and defection can emerge under a Snowdrift game system of payoffs, which makes it very different from the Prisoner’s Dilemma. Remember that in the Prisoner’s Dilemma, defection always generates a higher payoff than cooperation, regardless of the opponent’s strategy. So stable cooperation can only evolve under a Prisoner’s Dilemma system of payoffs if some kind of information transfer is possible. One example is the Iterated Prisoner’s Dilemma, in which two players encounter each other repeatedly. In this circumstance, one player can punish defection, leading to conditional strategies — the most famous of which is “tit for tat” — that yield a positive payoff for cooperation. It is worth pointing out that the cumulative payoffs under “tit for tat” or other conditional strategies come to approximate the payoffs of the Snowdrift game. The transfer of information changes one payoff structure into another.

Here, we have unveiled a different paradox of cooperation, which could be termed the ”tragedy of the commune”: In a cooperative system, in which every individual contributes to a common good and benefits from its own investment, selection does not always generate the evolution of uniform and intermediate investment levels but may instead lead to an asymmetric stable state, in which some individuals make high levels of cooperative investment and others invest little or nothing.

In practice, it is often difficult to determine the payoffs in social interactions and hence to distinguish prisoner’s dilemma and snowdrift interactions [a phage system marks a rare exception, but interestingly, selection turns the prisoner’s dilemma into a snowdrift game (24)]. Nevertheless, the mere existence of high- and low-investing individuals has often been taken as prima facie evidence that the interaction is governed by a prisoner’s dilemma, with some additional mechanism, such as reciprocity, responsible for the co-existence of altruists and nonaltruists. The tragedy of the commune, however, provides a quite different and, in many ways, simpler explanation for the coexistence of high- and low-investing individuals, which potentially applies to a wide range of cooperative and communal enterprises in biological systems (Doebeli et al. 2004:861–862).

How is this relevant to paleoanthropology? The last paragraph of the paper suggests one way:

In behavioral ecology, classical examples of cooperation include collective hunting and territory defense in lions (28), predator inspection in sticklebacks (29), and alarm calls in meerkats (30). In theoretical discussions of these examples, the existence of cooperators providing a common good and defectors exploiting it has been assumed a priori. The tragedy of the commune, however, suggests an evolutionary mechanism for the emergence of distinct behavioral patterns with differing degrees of provisions to the common good. This mechanism may also apply to cultural evolution in human societies, in which large differences in cooperative contributions to communal enterprises could give rise to conflicts on the basis of accepted notions of fairness (Doebeli et al. 2004:862).

Food sharing in human hunter-gatherers includes many asymmetries. For example, hunters differ greatly in their hunting returns and expenditure of effort. Yet good hunters tolerate the presence of poor hunters and share food with them. As with hunting but extended to both men and women, people invest greatly varying degrees of effort into gathering plant foods, with resulting variation in caloric returns. Some of the variation in investment and success is age-related, some is likely directly environmentally induced, and some may reflect frequency-dependent strategies.

Over the next few days, I’ll be considering human hunting from the perspective of the Snowdrift game. I’ll start with some very simple deterministic models and then try to make them a bit more relevant by considering the effects of stochastic payoffs and asymmetries among players.

Next: Defining the Snowdrift game for hunting


   Doebeli M, Hauert C, Killingback T. 2004. The evolutionary origin of cooperators and defectors. Science 306:859–862. doi:10.1126/science.1101456.