Conceived and designed the experiments: JG AT. Performed the experiments: JG AT. Analyzed the data: JG AT. Wrote the paper: JG AT.
The authors have declared that no competing interests exist.
Evolutionary game dynamics in finite populations assumes that all mutations are equally likely, i.e., if there are
Evolutionary game dynamics can be used to study the evolution of phenotypes. It usually considers the fate of a population of strategies playing a game, subject to selection and mutation. In this framework one of the most studied formalisms is the Moran process, it allows for studying the interplay between selection and mutation under demographic noise. The Moran process considers a finite population of constant size. At every time step one strategy is chosen for reproduction in proportion to its performance in the current population. A copy of this strategy is added to the population after removing a random strategy. With a small probability, the strategy that is copied changes its type to any of the other available strategies. This process results in an ergodic Markov chain. The effect of selection and mutation can be assessed by inspecting the average composition of the population in the long run.
The Moran process is often studied in the limit of small mutation probability
Evolutionary processes have been traditionally given two possible interpretations. In cultural evolution, the process of selection is taken to represent a situation in which successful strategies spread by imitation. Here, mutations are generally interpreted as mistakes in the process of imitating others, or intended exploration undertaken by individuals
Some previous studies have already considered nonuniform mutation rates. The idea of local mutations is central in adaptive dynamics, but here the literature is strictly concerned with infinite populations and continuous strategies in metric spaces
We analyze mutation structures in two examples dealing with the evolution of cooperation. We restrict ourselves to mutation structures that are nonfrequency dependent and do not vary with time. Our examples resemble the concept of a “protein space”, as envisioned by Maynard Smith
Consider the simple case of competition between two strategies
The evolutionary dynamics is considerably simplified for small mutation rates. If mutations are small enough
For
These results hold for any finite intensity of selection and small (positive) mutations. Even for small mutation rates, the specific model of how mutations arise can dramatically change the fate of an evolving population, as shown previously in
Usually, the mutation rate is one single parameter. In larger systems, studying nonuniform mutation rates requires us to specify how likely it is that any given strategy
Let us now look at how such kernels may be specified for particular examples, and how known results do change when departing from uniform mutations.
We start by studying the evolution of direct reciprocity
The one shot game we are interested in is a prisoner's dilemma with the payoff matrix
A finite population will spend most of the time in the cooperative strategy
The complete set of strategies for the repeated prisoner's dilemma is infinite. Therefore, studying a particular dynamics implies some restriction in the strategy set. In this section we study the
Strategy  Behavior  Binary code  
0 

Always cooperate  000 
1 

Tit for tat  001 
2 

Cooperate on the first move  010 
then reverse the opponent's last move  
3 

Cooperate once and then always defect  011 
4 

Defect once and then always cooperate  100 
5 

Defect once and then copy the opponent's last move  101 
6 

Defect once and then reverse the opponent's last move  110 
7 

Always defect  111 
The derivation of the payoff matrix for all the strategies in
For the uniform mutation structure, a mutation occurs with probability
Panel A shows uniform mutations, and Panel B shows the results for the bitwise kernel. Continuous lines represent the theoretical approximation. Dots represent simulation results averaged over 500 repetitions of
We now assume that strategies are represented by the binary code, as described in
Arrows indicate the direction of selection, and dashed lines indicate neutral paths. Blue strategies are completely cooperative and red strategies are completely uncooperative when paired with themselves. The kernel structure shuts down paths that would normally be available with the standard assumption that all mutation paths are possible.
The results are shown in Panel B of
The results of the bitwise kernel are of course invariant to changing the meaning of each position (e.g., the last bit, instead of the first, determines what to do in the first round), or changing the meaning of each bit (e.g., cooperation is coded by
Note that arbitrary kernels can produce results that differ in more radical ways from the standard result with uniform mutations. For instance, changing the labels by swapping strategies
We now turn to cooperation without repetition. The evolution of strategies in the optional public goods game has been studied extensively since proposed by Fowler
Here, we will follow the version of the game presented in
For uniform mutations the systems spends most of the time in a population completely made up of cooperators that punish defectors. This can be seen in
Panel A shows abundance in stationarity as a function of the intensity of selection. Continuous lines represent the theoretical approximation. Dots represent simulation results averaged over 500 repetitions of
Analytical results greatly simplify in the limit of strong selection (i.e.,
We can depart from the standard assumption of uniform mutations, for instance, assuming that mutations from loners towards the other strategies are rarer by a factor
We can provide a straightforward interpretation for this mutational structure, biologically as well as from a cultural perspective. Biologically, mutations towards strategies that do actually play the game (
In the limit of strong selection the calculations are again greatly simplified. The stationary distribution is given by:
Thus, in the limit of strong selection, playing the game is more popular than abstaining whenever
Panel A shows abundance in stationarity as a function of the intensity of selection (
Accordingly, we show that in finite populations some risk aversion may deter cooperation, as most individuals prefer not to play the game. This is radically different from what happens in the case of uniform mutations, where strong selection always leads to total predominance of altruistic punishers. Other mutation structures are of course possible, but once again, it is difficult not to be completely arbitrary. In the next section we inspect a bitwise mutation kernel for this game.
A larger strategy set for the optional public good games with punishment has been recently studied by Rand and Nowak
The game has the same parameters and structure as the game considered above; the only difference comes in the specification of payoffs for each one of the
We compare the uniform mutation structure, where all strategies can be reached from each other with the same weight, with a bitwiselike kernel that has the following structure. Each strategies is a chain of four positions. The first position has base 3: D stands for defection, C stands for cooperation, and L stands for loner. The second, third and fourth positions are binary. P in the second position means punish cooperators, whereas N means do not punish cooperators. The third position takes care of punishing defectors, and the fourth position codes for punishing loners. In the bitwiselike kernel, mutations can only take you to a strategy that differs in one position. This means that each strategy has
For example, defectors who do not punish (DNNN) can only mutate into defectors that punish loners (DNNP), defectors that punish other defectors (DNPN), defectors that punish cooperators (DPNN), cooperators that do not punish (CNNN) and loners that do not punish (LNNN). Each one of these events happens with probability
Abundance in stationarity as a function of the intensity of selection (
Introducing the bitwiselike mutation structure we also find that no strategy is overwhelmingly prevalent. In particular, the stationary distribution has more variation and other strategies become abundant. The most popular strategies are now altruistic punishers, loners that punish cooperators, and individuals that refrain from taking any action whatsoever. It is noteworthy that introducing this kernel considerably favors the autarkic option of individuals that abstain from the game, and forgo any punishment.
We have formalized a Moran process with nonuniform mutations. We show that mutation structure plays an important role, even if mutations are assumed to be small. In three examples we have come up with specific reasonable kernels that overturn known results. Our mutation kernels are akin to MaynardSmiths's concept of protein spaces, where phenotypes are connected by unit mutational steps
We first study the evolution of direct reciprocity in a set of
Next we turn to a model of cooperation without repetition. We study the evolution of altruistic punishment in optional public good games. Assuming a reasonable kernel with a clear biological and behavioural interpretation leads to the possibility of abstention being more successful than playing the game. A specific condition is specifically worked out for the case of strong selection. Finally, we study optional public good games with punishment in a much larger strategy space. The structure of the space also lends itself to an interpretation that makes it easy to come up with a reasonable mutation kernel. This kernel changes the results in a significant manner, particularly showing that allowing for so many strategies can actually result in no play being a very successful alternative.
Even though we have focused our analysis on systems in the limit of small mutation rates and without population structure, there is no reason to suspect that the effects we have highlighted will not be salient as well in systems with larger mutation rates
Our results call into question the “modelless” approach to mutations in evolutionary dynamics, where given a strategy set, all mutations are available an equally likely. Even in the limit of rare mutations, the mutation structure can make a substantial difference on what gets selected. It is important to observe that all models that follow the methodology studied here, rest on a specific assumption of mutation structure
The evolutionary dynamics is studied based on the Moran process
We asses the effect of selection and mutation by inspecting the average composition of the population in the long run. The stationary distribution can be computed exactly, if mutations are sufficiently small
Supporting information.
(PDF)
We thank C.S. Gokhale for help.