Abstract
Public goods games often assume that the effect of the public good is a linear function of the number of contributions. In many cases, however, especially in biology, public goods have nonlinear effects, and nonlinear games are known to have dynamics and equilibria that can differ dramatically from linear games. Here I explain how to analyze nonlinear public goods games using the properties of Bernstein polynomials, and how to approximate the equilibria. I use mainly examples from the evolutionary game theory of cancer, but the approach can be used for a wide range of nonlinear public goods games.
Highlights
Examples can be found in almost all areas of human knowledge, from biology to economics, from selfish genetic elements [2], microbes secreting diffusible molecules [3] and cancer cells secreting growth factors [4,5] to cooperative hunting in mammals [6] and, the exploitation of shared natural resources described by Hardin [1]
The social dilemma described by Hardin [1] is essentially a multiplayer version of the PD (NPD [12,13]): individuals can be cooperators or defectors; only cooperators pay a contribution; all contributions are summed, the sum is multiplied by a reward factor and redistributed to all individuals
As we have seen here, nonlinearities have profound effects; the main difference is that pairwise and linear games usually lack the internal equilibria that are often observed in nonlinear games
Summary
Fifty years after Garret Hardin’s “Tragedy of the Commons” [1], the problem of collective action remains one of the most influential concepts in science: free-riding on the contributions of others enables free-riders to thrive at the expense of cooperators. The major transitions in evolution are considered solutions to social dilemmas of this kind [7]. The problem of cooperation can be modelled by games with at least one Pareto inefficient equilibrium: an alternative outcome exists in which at least one player could have a higher payoff without reducing any other player’s payoff (a Pareto improvement is possible; the inefficiency); no one, has an incentive to change their behavior ( the equilibrium). Dilemma (PD) [8] is the most famous among such games, and it has been used extensively to describe the problem. The game of Chicken [9] ( known as the Hawk–Dove game [10] or the Snowdrift game [11]) has been used to study cooperation
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