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The replicator-mutator dynamics is a set of differential equations frequently used in biological and socioeconomic contexts to model evolutionary processes subject to mutation, error or experimentation. The replicator-mutator dynamics generalizes the widely used replicator dynamics, which appears in this framework as the extreme case where replication is perfectly precise. This paper studies the influence of strictly dominated strategies on the location of the rest points of the replicator-mutator dynamics, at the limit where the mutation terms become arbitrarily small. It can be proved that such limit rest points for small mutation are Nash equilibria, so strictly dominated strategies do not occur at limit stationary points. However, we show through a simple case how strictly dominated strategies can have an influence on the location of the limit rest points for small mutation. Consequently, the characterization of the limit rest points of the replicator-mutator dynamics cannot in general proceed safely by readily eliminating strictly dominated strategies.

The continuous time selection mutation equation [

Here _{i}_{ij}_{j}_{ij}_{ij}_{ij}_{i}_{j}_{j}_{ij}_{i}_{i}f_{i}

In the replicator-mutator dynamics, the unit simplex Δ_{n}_{1},…,_{n}^{n}_{i}_{i}_{i}_{1},…,_{n}_{ij}

For the particular case of symmetric payoff matrices (which corresponds to the standard population-genetic model of natural selection on a large diploid population) and mutation rates _{ji}_{i}

A well-known particular instance of the Replicator-Mutator Dynamics is the widely-used replicator dynamics. The Replicator Dynamics (RD) appears in this framework as the extreme case where replication is perfectly precise, _{ij}_{ij}

In this paper we focus on the influence of strictly dominated strategies on the location of the rest points of the RMD in the limit of small mutation. Following Samuelson [

It is well known that weakly dominated strategies in the RD may remain present forever [

In the RMD with _{ij}

Let us now turn to strictly dominated strategies. In the RD, starting from any interior point, strictly dominated strategies are asymptotically wiped out [

For the RMD with _{ij}_{ii}

A biological implication of this result is that, in an evolutionary system in which the flow of mutations between behaviors, species or varieties is small, a very poorly fit and very rarely observed behavior in an ecosystem can be the main force explaining, predicting or controlling the observed proportions of the other behaviors or species. The impact of the poorly fit behavior can be more profound than the mere selection of a particular point within a set of neutrally stable equilibria subject to random drift (like in the RD): in the RMD with arbitrarily rare mutations, the poorly fit behavior may modify the location of a unique asymptotically stable point. Below, we use an example to show that equilibria that, in the RD, would be subject to random drift within a large range of values, can actually be stabilized at a precise particular level which is influenced by the scarce appearance of strictly dominated strategies or entrants.

The dependence of limit stationary states of the RMD on strictly dominated strategies is not as disheartening as it may seem: there are some cases where one can safely calculate the limit stationary states of the RMD with relation to the stationary states of the RD [

We study a replicator-mutator system with three strategies, one of which is strictly dominated, and we provide an analytical formula for the limit of its stationary state as the mutation rate goes to zero. The limit stationary state is shown to depend on the parameters that correspond to the strictly dominated strategy.

Let _{i}_{ji}_{i}_{i}_{i} ·m_{i}_{i}_{i}_{i}_{ii}_{i}

Assuming _{ii}_{i}_{i}_{i}

Let us now consider the payoff matrix

Let

Consequently, for small _{1}, 1 − _{1}, 0]which satisfies

Note that _{1}) is a quadratic function. Given that _{1}(1 − _{32}) < 0 and _{2}(1 − _{31}) > 0, there is a unique solution to the equation _{1}) = 0 in _{1} ∈[0,1], which proves that there is a unique limit stationary state

Note also that the limit stationary state does not depend on the payoff _{33}, and the analysis is locally valid dropping the constraint _{33} < min(_{13}, _{23}), as there would still be an _{3} < _{2} and _{3} < _{1}, for _{3} <

Finally, notice that, without mutation, the 3-strategy case we have selected presents a connected component of critical points along the edge where the strictly dominated strategy is null (_{3} = 0). In this neutral component the other two strategies are payoff-equivalent (_{1} = _{2}). Thus, the selection of one single limit stationary point within this neutral component is necessarily due to the second-order forces induced by mutations [

Consider the case
_{ji}_{i}

For small mutation the limit stationary state _{1}, 1 − _{1}, 0] can be calculated according to _{1} =_{2} = _{3} = 1/3, this leads to

_{1} as a function of _{1}, 1 − _{1}, 0] can vary from one extreme to the other with the parameters

It might be thought that which particular point is selected out of a continua of RD equilibria, such as the line _{1} + _{2} = 1 in this example, is not very relevant, given that strategy 1 and strategy 2 are equivalent in the absence of other strategies. Note, however, that the response of this system to shocks or perturbations can be very different depending on the proportions of strategy 1 and strategy 2. Suppose, for instance, that the model corresponds to the coexistence of three species in an ecosystem, and the system is resting at the stationary point where _{1} + _{2} ≈ 1. If a small fraction _{4} of individuals of a fourth species arrives at this ecosystem, and this fourth species rates poorly against species 1, but well against species 2 and against itself, needing a minimal value of _{2} in order to proliferate (

The limit stationary points of the RMD constitute a subset of the stationary points of the RD. Bomze and Bürger [

Proportion _{1} of strategy 1 in the limit stationary state of the RMD for matrix _{1} = m_{2} = _{3}.

The red circle shows the unique stationary state of the RMD for matrix _{1} = _{2} = _{3}, and mutation rate ^{−4}, for three different combinations of values of the payoffs [

The authors would like to thank Fernando Vega-Redondo and two anonymous reviewers for comments. This work has received financial support from the Spanish JCyL (GREX251-2009 and VA006B09), Ministry of Science and Innovation (TIN2008-06464-C03-02, DPI2010-16920 and CSD2010-00034) and Ministry of Education (grant JC2009-00263).

From _{3}_{3} = 1), we have

From

So,

Besides,

So, from

From the conditions of stationary state

Considering that

And considering that

As a consequence of