Game landscapes: from fitness scalars to fitness functions

My biology writing focuses heavily on fitness landscapes and evolutionary games. On the surface, these might seem fundamentally different from each other, with their only common feature being that they are both about evolution. But there are many ways that we can interconnect these two approaches.

The most popular connection is to view these models as two different extremes in terms of time-scale.

When we are looking at evolution on short time-scales, we are primarily interested which of a limited number of extant variants will take over the population or how they’ll co-exist. We can take the effort to model the interactions of the different types with each other, and we summarize these interactions as games.

But when we zoom out to longer and longer timescales, the importance of these short term dynamics diminish. And we start to worry about how new types arise and take over the population. At this timescale, the details of the type interactions are not as important and we can just focus on the first-order: fitness. What starts to matter is how fitness of nearby mutants compares to each other, so that we can reason about long-term evolutionary trajectories. We summarize this as fitness landscapes.

From this perspective, the fitness landscapes are the more foundational concept. Games are the details that only matter in the short term.

But this isn’t the only perspective we can take. In my recent contribution with Peter Jeavons to Russell Rockne’s 2019 Mathematical Oncology Roadmap, I wanted to sketch a different perspective. In this post I want to sketch this alternative perspective and discuss how ‘game landscapes’ generalize the traditional view of fitness landscapes. In this way, the post can be viewed as my third entry on progressively more general views of fitness landscapes. The previous two were on generalizing the NK-model, and replacing scalar fitness by a probability distribution.

In this post, I will take this exploration of fitness landscapes a little further and finally connect to games. Nothing profound will be said, but maybe it will give another look at a well-known object.

Our contribution to the mathematical oncology roadmap discussed both fitness landscapes and games. In particular, we focused on the need for both theoretical and empirical abstraction in mathematical oncology. I’ve already written about the evolutionary games side of this abstraction.

But I avoided discussing fitness landscapes.

This was mostly because I feel that fitness landscapes have had less impact than games as concrete models in oncology.

Fitness landscapes conceptualize fitness as a single scalar value — a number. A scalar can only express cell-autonomous effects, where fitness is inherent to the properties of a single. But cancer displays important non-cell-autonomous effects that allow fitness to depend on a cell’s micro-environmental context, including the frequency of other cell types. And this is certainly not limited to cancer. Microenvironmental context and the abundance of other organisms almost always matters to the fitness of a particular type.

To accommodate this non-cell-autonomous (or more generally, non-type-autonomous) fitness, EGT views the types as strategies and models fitness as a function which depends on the abundance of strategies in the population.

This is the other way that we can connect fitness landscapes and games:

Fitness landscapes map types to a fitness scalar. Games map types to a fitness function.

Since any scalar can be represented as a constant function, this perspective makes games the more general and foundational perspective. At least in appearance, although often not in practice.

As is often the case, greater expressiveness comes at a price. In the case of the greater expressiveness of games, this price is a loss of analysis techniques. For example, when dealing with fitness landscapes, we can often consider the strong-selection weak-mutation limit. In this regime, we image that mutations are so rare that the population remains monomorphic except for a brief time during a selective sweep. This allows us as modellers to replace a population by a single point in the landscape.

In the case of evolutionary games, such an approximation is unreasonable since it would eliminate the very ecological interactions that EGT aims to study. This means that the strategy space that can be analysed in an evolutionary game is usually much smaller than the genotype/phenotype space considered in a fitness landscape. Typical EGT studies consider just a handful of strategies (most often just two, or three), while fitness landscapes start at dozens of genotypes and go up to tens of thousands (or even hyper-astronomical numbers of genotypes in theoretical work).

But suppose that we did want to work with a huge combinatorially structured space of strategies with frequency dependent fitness. A game landscape. Could we get started?

In the case of scalar fitness, it becomes useful to think about a fitness graph: with each edge between nearby mutants oriented from the lower fitness to the higher fitness type. This results in just two kinds of edges: a direction from one to the other type, or a neutral edge with no direction (for equal fitness). More importantly, these edges have a nice global property: their directed graph is acyclic.

Can we get something with game landscapes? The obvious first step is to limit to linear games and consider what weak mutation dynamics might look like. Now our edge count increases to four: we can still have the two old types plus two new ones. But even with the old directed edge, the acyclic property disappears: just consider the rock-paper-scissors game. The new types get us even more. Let’s look at three and four.

The third edge type arises in games like Stag-Hunt where a repulsive fixed point exists: this results in a fitness graph edge that points against either direction. This stops a population from moving across the edge, even in the random drift limit.

The fourth edge type is the most interesting. It arises in games like Hawk-Dove where an attractive fixed point exists. This results in a fitness graph edge that points inwards. It moves the population to a fixed point where both types co-exist and thus even in the strong-selection weak-mutation limit creates a polymorphic population. This can be difficult to deal with.

But the most non-obvious part of game landscapes is how to represent them. Traditional fitness landscapes are exponentially large objects, so we don’t simply store a long list of scalar fitness values. Instead, we use a compact representation like the NK-model that tells us how to quickly compute a fitness from a genotype. We would need something similar with game landscapes: a mapping from genotype to fitness function. Here two new difficulties arise. First, we need to output a function, not a scalar, so doing something simple like adding together fitness components (as the NK-model does) doesn’t work anumore. Second, what is the domain of the resulting fitness function? The naive answer is that it is all the genotypes: so an exponential domain. This means that our compact mapping from genotype to fitness function has to output a compact mapping of its own (and not just an array of linear coefficients for each potential interaction partner)!

Do you have any suggestions on how to handle these difficulties, dear reader?

I think that these challenges of game landscapes can be addressed in interesting ways. In ways that differ from the stand use of fitness landscapes or the replicator-mutator equation. But I’ll tackle thoughts on this in a future post.