Effective games and the confusion over spatial structure

A typical study of space in evolutionary game theory starts with a specification of how local interactions impact fitness, and then simulates that interaction over a model of space to show a surprising difference in dynamics between the spatial model and its nonspatial counterpart. Usually, this difference is revealed at the population level. Exceptional works like that of Nanda and Durrett (1) and Ohtsuki and Nowak (2) provide a general method for combining local interactions and spatial structure into dynamically equivalent nonspatial models. We call this microscopic local interaction “reductive game” and its macroscopic population-level summary “effective game” (3). Thus, Nanda and Durrett (1) or Ohtsuki and Nowak (2) provide a transform from reductive to effective game. That effective game then has the same nonspatial dynamics as the reductive game played out on the spatial structure. We can think of this typical approach as bottom-up: Start with the reductive game, find the corresponding effective game, and use this as a prediction to compare against observed phenomena.

However, when we apply this pipeline, how do we know that the local interactions of the reductive game are the right ones to start with? For macroscopic systems like humans or other large animals, we might be able to directly design this game. In microscopic systems, however, we tend to guess these games from intuitions acquired by looking at population-level experiments. Unfortunately, these experiments seldom explicitly account for the effect of their spatial structure. Hence, they are actually intuitions about the effective game that we then feed into our models as the reductive game. This is the confusion over spatial structure. We are taking a game from a top-level view, feeding it into the bottom level, getting a different result at the top level, and publishing that surprising conclusion.

This is backward. At best, it is just telling us that our intuitions about the game were wrong—since correct intuitions about the reductive game should yield the observed effective game. At worst, this is a type error and thus incoherent: We are feeding in an effective game where we should be putting a reductive game.

To make the theory developed by Nanda and Durrett (1) useful to microscopic systems like cancer, we have to invert the typical pipeline. We must start at the top with a carefully designed game assay to measure the effective game played by the population (4, 5) or design new assays that measure both the game and space together (3). In some systems, this might be difficult due to time-dependent effects of space over long timescales. This measured effective game encodes the combined effect of the reductive game, spatial structure, and other aspects of the experimental system. We can then push down by measuring spatial structure and inverting the transforms of Nanda and Durrett (1) or Ohtsuki and Nowak (2). This removes the contribution of space, allowing us to arrive at the reductive game. When this reductive game ends up contrary to our intuitions, then we know that we have a surprising conclusion.