A Snowdrift game version of hunting

5 minute read

I want to run through some examples of how we can apply game theory to consider hunting decisions in human groups. First, I describe a simple Snowdrift model applied to hunting. This is part 2 of a series, part 1 introduces the topic of the Snowdrift game. A reader sent along a story after reading the first post:

In reading your snowdrift blog post, I was reminded of an experiment that does not require game theory to understand. You may have heard of it. Two pigs are in a pen. One is dominant. To get food one of them presses a bar, but the food is dispensed at the other side of the pen. If the subdominant pig presses the bar, it gets no reward, as the dominant pig hogs the food, eating it all. The result is that the dominant pig presses the bar while the subdominant pig waits at the food trough. Then the dominant pig rushes over to the trough to push the subdominant pig aside. Both pigs get fed, but the dominant pig does all the work
It's a great example of asymmetrical rewards. I'll get to those in the next few posts on this topic, because the asymmetries are very important to understanding dynamics in hunter-gatherers. But first, we have to describe the simple symmetrical case, including the algebra defining the evolutionarily stable equilibrium between the two simple strategies.

Suppose we have two hunters, who will share whatever game either of them kills. A man may choose on a given day to hunt. By hunting, he suffers a cost c and brings back a large fixed benefit b for each man. The two men may both choose to hunt on the same day, resulting in the same benefit b but a lowered cost c∕2 for each man. The two men decide whether to hunt simultaneously and without conferring — that is, there is no information transfer between them that would affect their decisions.

Here is the payoff matrix of the game for player 1 (choices on left) given the strategy selected by player 2 (on top):

hunt no hunt
hunt b - c∕2 b - c
no hunt b 0

Given the existence of the two strategies, “hunt” and “no hunt,” the ESS is the ratio at which the two strategies have equal expected returns. If individuals select a strategy and do not vary, the ESS represents the frequency of these variants in the population. If in contrast, individuals can choose to adopt either strategy, then the ESS also will be the optimal proportion of the two strategies in one individual’s repertoire. The two strategies will yield equal payoffs when the ESS satisfies the following equation, where p represents the proportion of “hunt” and 1 - p the proportion of “no hunt”:

p(b- c∕2)+ (1- p)(b - c) = pb

…which simplifies to p = 2(b - c)(2b - c). That expression is positive where b > c, and approaches unity where c is very small relative to b. If in contrast b c then the scenario is the Prisoner’s Dilemma, where the only ESS is a pure “no hunt” strategy.

Let’s also look at a slightly different case. As above, each man’s return from hunting will be b regardless of whether one man or both choose to hunt. But in the payoff matrix below, the cost of hunting is also the same whether one man or both choose to hunt. So there is no reduction in the cost of hunting if both men do it.

hunt no hunt
hunt b - c b - c
no hunt b 0

Now, in this case, the ESS satisfies the equation:

p(b- c)+ (1- p)(b - c) = pb

Again, p is the frequency of the “hunt” strategy. This simplifies to p = (b-c)∕b, which again yields the Prisoner’s Dilemma when b < c.

OK, that’s the simple Snowdrift game model, described in the language of hunting instead of winter car accidents. It is quite simplistic in many ways. We might expect real hunters to have successes and costs that vary as stochastic functions of the environment. A real hunter must decide whether to hunt based not only on the odds his companion will hunt, but also upon some appraisal of the companion’s likelihood of success. Men in hunting societies are not paired up by the buddy system, but instead make their decisions about hunting in the context of a larger group’s activities.

Maybe most confusing, there are two possible kinds of currency in which benefits and costs may be expressed. A benefit from hunting may be most naturally measured in calories. If we average hunting returns across many episodes, then our result would be mean calories per day, or per hour of effort. Likewise, it might seem natural to discuss costs in terms of calories, as we might consider the cost of locomotion or cost of transport associated with foraging.

But the only currency that matters to evolution is fitness. We cannot assume that maximizing caloric returns will maximize fitness. Transport and locomotor costs may be minor compared to the mortality risk from predation when foraging far from camp. The caloric benefits from hunting matter more to a starving child than to a satiated adult.

So the measures of costs and benefits that define the ESS should be expressed in terms of fitness. That’s a problem, because fitness outcomes are a lot harder to measure than caloric returns. To figure out caloric expenditure and returns, you can measure oxygen consumption, work out distances, and weigh meat. To measure fitness, you have to record lifetime reproduction. To assess the relationship between caloric returns and fitness, you need a lifetime of caloric returns.

So far, hunter-gatherer demographic data and hunting returns are both known from a small number of transverse studies. Longitudinal data on hunter-gatherer demography are limited, and mostly known by retrospective methods — that is, informants share their knowledge about the history of their groups. The fitness effects of a single individual’s hunting effort over time are not known.

If fitness outcomes are hard for the scientist to measure, they are equally hard for a social actor to predict. Even intelligent actors like humans know little about the effects of their actions upon their future reproduction. Men sometimes do poorly with information directly relevant to fitness, like “Is the child mine?” That’s not to say that men may not follow highly sophisticated strategies to allocate hunting effort. But we should develop explanations that do not assume that a man knows the fitness benefits and costs of his choices. Next: Life history and asymmetrical strategies