‘Raise the stakes’ evolves into a defector

Abstract

To understand how cooperation can evolve by reciprocal altruism when individuals can make variable investments, Roberts and Sherratt¹ have introduced a new strategy, ‘raise the stakes’ (RTS), for a continuous version of the iterated ‘prisoner's dilemma’. An individual investing I bears a cost I, while the recipient gets a benefit kI. For k>1, this generalizes the standard prisoner's dilemma²,⁵. Over R alternating encounters⁶,⁷, RTS is defined as follows: on the first move, invest a, subsequently raise your investment by 2b (or b) if your partner's previous investment bettered (or equalled) your last move, otherwise match your partner's last move. This strategy is denoted by σ=(a,b). Roberts and Sherratt¹ reported that the strategy σ=(1,1) performs well in computer simulations against various alternative strategies but did not consider how a population of RTS strategies with different a and b values evolves. We find that selection within RTS populations always acts to lower the values of a and b, hence RTS cooperation is not a robust phenomenon.

Main

Assuming that mutations are small and rare, evolution in a population of RTS strategies can be understood analytically by using adaptive dynamics⁸. If the population consists of individuals using the strategy ς̂=(â, b̂), then the vector field determines the direction that optimizes the increase in payoff of a mutant strategy σ=(a,b) (ref. 8), where S is the payoff from an iterated interaction. S, and hence ξ, can be calculated analytically and it can be shown that evolution acts to lower the (a,b) parameters of the population. This yields the general prediction that the (a,b) parameters in a population of RTS strategies evolve to zero under natural selection.

This prediction is verified by evolutionary simulations. Consider a population of RTS strategies, with new mutants introduced at a certain rate. In every generation, each strategy plays against all the others and their frequencies in the next generation are calculated using standard game dynamics⁸. Any strategy whose frequency falls below a given threshold is eliminated. A typical simulation (for parameter values used in ref. 1) is shown in Fig. 1. As predicted, the (a,b) parameters evolve to zero. Extensive simulation has confirmed the analytical result for all parameter values studied (including extreme cases, such as k=100, R=1,000).

Figure 1: Simulation of the evolution of RTS strategies in the game studied in ref. 1.

a, Changes in the population mean values of the RTS parameters a and b (starting values, a=1 and b=1). b, Change in the mean payoff. In this simulation, k=2, R=20 (the same as in all figures in ref. 1).

Thus, in general, RTS evolves under natural selection into an unconditional defector (a=0, b=0). The lack of robustness of RTS arises because, although it is essential from an evolutionary perspective to allow the strategies σ=(a,b) to varycontinuously (as mutations can, in principle, result in arbitrary changes in a and b), the definition of RTS is discontinuous. From a biological viewpoint, the discontinuous nature of RTS is unrealistic as it is implausible that two strategies that are arbitrarily close would have qualitatively different behaviour.

Although reciprocal altruism with variable investments is an important approach to understanding the evolution of cooperation, our results indicate that new strategies are required to give a satisfactory theoretical account of this process. We have found, both analytically and by simulation, that investment strategies based on an individual's payoff in the previous round (see those used to study mutualism in ref. 9), rather than on the partner's investment, are evolutionarily robust and show how intraspecific cooperation can emerge with variable investments. We believe that these payoff-based strategies represent a more fertile area for future research than RTS strategies.

Reply — Sherratt et al.