Imagine you and some friends really really want ice cream. But to get it, one of you is going to have queue for hours. Someone has to take the costly plunge. Who will do it? This kind of scenario  the 'volunteer's dilemma'  is the sort of thing that arises when only one individual (or, at least, a set number) is needed to generate a public good. In this case, it is ice cream for everyone. If nobody volunteers, nobody gets ice cream. If somebody volunteers, they can do the queuing. It pays everyone else to remain silent. Nobody wants to queue for hours. The volunteer's dilemma has recently been under a bit of scrutiny in the context of punishing cheats. When you've got a group with more than two actors (an nplayer group), and a cheat arises, that cheat can be deterred from cheating in the future if they are punished. The positive payoff from preventing cheating is a public good, the benefits of which are shared across the group. But who will step up and pay the personal cost of punishing the cheat? Exotically, we end up with what we call a 'secondorder' cheating problem: it pays to freeride on the goodwill of someone else who'll punish the first freerider (1). To keep with the ice cream example, it's like somebody has stolen everyone else's delicious chocolate 99flakes, but nobody particularly wants to step up to punish them because it's a costly behaviour that will be injurious to them (let's say it involves a bit of a punchup), even if it benefits everyone by deterring 99flake thieving in the future. There are two main ways the payoffs in this secondorder problem can work. It could be that more punishers means a proportionally better result. Or it could be that only one punisher (or, in more complicated setups of the volunteer's dilemma, a set number of punishers) are needed. The first is an nplayer prisoner's dilemma scenario; the second is an nplayer volunteer's dilemma. Another way of describing this is to say that the former is based on a linear payoff function, whilst the latter relies upon a nonlinear, binary payoff function, which makes a sudden change at a certain level of punishment. There are, of course, a number of intermediate nonlinear functions. Let's imagine the nplayer volunteer's dilemma in its simplest form, in which only one volunteer need step up in order to achieve the step change in payoff to the group by getting on with the punishment. The evolutionary outcome of such a game is that both secondorder freeriders and secondorder volunteers persist in the population at equilibrium, and perturbations from this equilibrium result in a return to it (i.e. it is evolutionarily stable). Why? Because fitness in this game changes in a negatively frequencydependent fashion. To see this point, have a think about this: the fitness payoff of being a volunteer in a population where everyone is freeriding is higher than being a freerider, but the fitness payoff of being a freerider in a population where everyone is volunteering is higher than volunteering. There's a school of thought that says the volunteer's dilemma is a much more realistic vision of punishment than the prisoner's dilemma. Nichola Raihani and Redouan Bshary, for instance, think that we should expect the real world to involve stepchanges: a set number of individuals should step forward, and any more would be silly, increasing the cost to the group for no gain (1). In the prisoner's dilemma (i.e. when the payoff function is linear), we end up with freeriding winning unless individuals are able to assort in some way (such as by grouping with relatives, or remembering reputations). Nonlinearity creates a stable mixed equilibrium through negative frequencydependence (unless the cost of punishing is simply too big), so does away with the need for these additional assumptions (2). The next step is demonstrating that such nonlinearity of payoffs is common in real punishment situations. Recognising this nonlinearity may help transform our understanding of public goods problems, so watch this space! To find out more! (1) Raihani, N. & Bshary, R. 2011. The evolution of punishment in nplayer public goods games: a volunteer's dilemma. Evolution, 65: 27252728 (2) Archetti, M. & Scheuring, I. 2010. Coexistence of cooperation and defection in public goods games. Evolution, 65: 11401148
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June 2020

Adventures of a
trainee zoologist
trainee zoologist
Dr Patrick Kennedy, Radford Lab, University of Bristol  Zoology