Part III. Spatial interactions on a grid
III-3. Extension to any number of strategies
1. Goal
Our goal here is to extend the model we have created in the previous chapter –which accepted games with 2 strategies only– to model (2-player symmetric) games with any number of strategies.
2. Motivation. Spatial Hawk-Dove-Retaliator
The model we are going to develop in this chapter will allow us to explore games with any number of strategies. Thus, we will be able to model games like the classical Hawk-Dove-Retaliator (Maynard Smith, 1982, pp. 17-18), which is an extension of the Hawk-Dove game, with the additional strategy Retaliator. Retaliators are just like Doves, except in contests against Hawks. When playing against Hawks, Retaliators behave like Hawks. A possible payoff matrix for this symmetric game is the following:
| Hawk (H) | Dove (D) | Retaliator (R) | |
| Hawk (H) | -1 | 2 | -1 |
| Dove (D) | 0 | 1 | 1 |
| Retaliator (R) | -1 | 1 | 1 |
Let us consider the population game where agents are matched to play the normal form game with payoffs as above.[1] The only Evolutionarily Stable State (ESS; see Thomas (1984) and Sandholm (2010a, section 8.3)) of this population game is the state (½H + ½D), with half the population playing Hawk and the other half playing Dove (Maynard Smith, 1982, appendix E; Binmore, 2013). Also, note that Retaliators are weakly dominated by Doves: they get a strictly lower expected payoff than Doves in any situation, except in those population states with no Hawks whatsoever (at which retaliators get exactly the same payoff as Doves).
Figure 1 below shows the best response correspondence of this game. Population states are represented in a simplex, and the color at any population state indicates the strategy that provides the highest expected payoff at that state: orange for Hawk, green for Dove, and blue for Retaliator. As an example, the population state where the three strategies are equally present, i.e. (⅓H + ⅓D +⅓R), which lies at the barycenter of the simplex, is colored in green, denoting that the strategy that provides the highest expected payoff at that state is Dove.
We would like to study the dynamic stability of the unique ESS (½H + ½D) in spatial contexts. In unstructured populations, ESSs are asymptotically stable under a wide range of revision protocols (see e.g. Sandholm (2010a, theorem 8.4.7)), and in particular under the best response rule. Therefore, one might be tempted to think that in our spatial model with the imitate the best neighbor rule (including some noise to allow for the occasional entry of any strategy), simulations will tend to spend most of the time around the unique (½H + ½D) and Retaliators would hardly be observed. This hypothesis may be further supported by the fact that the area around the unique ESS where Retaliators are suboptimal is quite sizable. In no situation can Retaliators obtain a higher expected payoff than Doves, and departing from the unique ESS, at least one half of t