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I have a problem in evolutionary optimization that I think resembles a question that arises in game theory. Suppose a population lives in a fluctuating environment. The environment $G_t$ takes on one of $N_g$ different different possibilities at each generation $t$. (Generations are assumed to be nonoverlapping, and $G_t$ is a stationary ergodic discrete-time Markov process, though I don't think those details matter for this question.) Each organism in the population chooses a strategy $X_t$ from one of $N_x$ different possibilities. In the next generation, it will produce $f(X_t,G_t)\ge0$ progeny. $f$ is the fitness matrix.

Obviously, if the elements of any row $x$ of $f$ are strictly less than the corresponding elements of some other row $y$, strategy $x$ is dominated by strategy $y$ and can be eliminated. That's easy to detect. However, the population can also employ mixed strategies, where individuals choose strategy $x$ with probability $p_x$. ($p$ must have the usual characteristics of a probability distribution: each $p_x\ge0$ and $\sum_x{p_x}=1$.) The fitness of mixed strategy $p$ in environment $g$ is

$$\gamma_g(p)=\sum_y{p_y f_{yg}}$$

If for some row $x$, there exists a mixed strategy $p$ with $p_x=0$ such that, for all $g$, $\gamma_g(p)>f(x,g)$, then strategy $x$ is dominated by the mixed strategy and can be eliminated from consideration.

My question is this: is there a straightforward way, given the fitness matrix $F=(f_{xg})$, to discover whether any of its strategies is dominated by some convex combination of the other strategies? Would it help if $F$ is lower triangular?

Thanks for any insight.

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    In a two-player finite strategic form games, a strategy that is never a best-reply is strictly dominated. This result extends to $N$ players if you allow correlation in opposing strategies. Not sure this is terribly helpful to your circumstance but perhaps you can see a way to implement that check. – 2017-02-26

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For what it's worth, I have worked out a practical solution for numerical matrices. Let $F$ be an $n\times m$ fitness matrix. To find out if $m$-vector $v$ is dominated by a convex combination of the rows of matrix $F$, find the positive $n$-vector $p$ that minimizes $p\cdot \vec{1}$ subject to $x^T F\ge v$. (Here $\vec{1}$ represents the $n$ vector all of whose elements are 1.) Then, if $p\cdot \vec{1}\le 1$, $v$ is dominated by a convex combination of the rows of $F$. In practice, to find out if a matrix has a dominated row, you simply let $v$ be each row in turn and let $F$ be the fitness matrix with that row deleted.

This is not a particularly elegant solution, but since the numerical solution of linear programming problems is computationally easy, it works OK in practice.