[Paper Review] Deterministic monotone dynamics and dominated strategies

Yannick Viossat|arXiv (Cornell University)|Oct 28, 2011

Game Theory and Applications33 references3 citations

TL;DR

This paper unifies and extends results on deterministic monotonic dynamics in evolutionary game theory, proving that under concave monotonic dynamics—dual to Hofbauer and Weibull's convex monotonic dynamics—strictly dominated strategies are eliminated over time. The key contribution is establishing conditions under which dominated strategies vanish, even in non-autonomous or time-varying environments, using growth rate ordering and Lyapunov-like arguments.

ABSTRACT

We survey and unify results on elimination of dominated strategies by monotonic dynamics and prove some new results that may be seen as dual to those of Hofbauer and Weibull (J. Econ. Theory, 1996, 558-573) on convex monotonic dynamics.

Motivation & Objective

• To unify and generalize existing results on the elimination of strictly dominated strategies under monotonic evolutionary dynamics.
• To establish a duality between Hofbauer and Weibull’s (1996) results on convex monotonic dynamics and new results on concave monotonic dynamics.
• To analyze the survival or elimination of mixed strategies under continuous- and discrete-time dynamics, particularly when payoffs and growth rates vary over time.
• To examine conditions under which iteratively strictly dominated strategies are eliminated, especially in non-autonomous settings.
• To clarify the role of aggregate monotonicity and growth rate ordering in ensuring long-run elimination of dominated strategies.

Proposed method

• Uses continuous-time dynamics of the form $\dot{x}_i = x_i(g_i(\mathbf{x},\mathbf{y}) - \sum_k x_k g_k(\mathbf{x},\mathbf{y}))$, where $g_i$ represents the unnormalized growth rate of strategy $i$.
• Defines aggregate monotonicity as preserving payoff order in growth rates: if $U_\mathbf{p}(\mathbf{y}) > U_\mathbf{q}(\mathbf{y})$, then $g_\mathbf{p}(\mathbf{x},\mathbf{y}) > g_\mathbf{q}(\mathbf{x},\mathbf{y})$ for all $\mathbf{x}, \mathbf{y}$.
• Introduces concave monotonic dynamics as a dual to Hofbauer and Weibull’s convex monotonic dynamics, where the condition holds when $\mathbf{p}$ is pure.
• Applies Lyapunov function techniques and time-averaged growth rate analysis to show that dominated strategies vanish when the dynamics satisfy concave monotonicity.
• Considers discrete-time dynamics of the form $x_i(n+1) = x_i(n) \frac{C + g_i(\mathbf{x},\mathbf{y})}{C + \sum_k x_k g_k(\mathbf{x},\mathbf{y})}$, with $C$ large or growing over time.
• Uses asymptotic approximation $\ln(1 + g_i/C) \approx g_i/C$ for large $C$, showing that discrete dynamics converge to continuous dynamics in the limit.

Experimental results

Research questions

• RQ1 Under what conditions on the growth rate functions $g_i$ do monotonic dynamics eliminate strictly dominated strategies?
• RQ2 Can the duality between convex and concave monotonic dynamics be formally established, and what are the implications for strategy elimination?
• RQ3 Do discrete-time dynamics with large or time-varying constants $C$ still eliminate dominated strategies, and what conditions ensure this?
• RQ4 Can dominated strategies survive under non-autonomous or time-varying opponent strategies, even when the dynamics are monotonic?
• RQ5 How does the elimination of mixed strategies depend on the structure of the payoff functions and the monotonicity class of the dynamics?

Key findings

• For concave monotonic dynamics, strictly dominated mixed strategies are eliminated as $t \to \infty$, provided the dynamics satisfy the growth rate ordering condition.
• In non-autonomous settings with time-varying opponent strategies $\mathbf{y}(t)$, dominated strategies can still be eliminated if the dynamics are aggregate monotonic and the time-averaged growth rate favors non-dominated strategies.
• Discrete-time dynamics with $C \to \infty$ eliminate dominated strategies if $\sum_n 1/C_n = \infty$, ensuring sufficient long-run growth differential.
• When $C$ is large but fixed, discrete dynamics eliminate dominated strategies if the dynamics are aggregate monotonic or convex monotonic with pure $\mathbf{q}$, for sufficiently large $\bar{C}$.
• The survival of a strictly dominated mixed strategy $\mathbf{q}$ is possible under time-varying dynamics if the growth rate differences cancel over cycles, as shown via piecewise linear $\mathbf{y}(t)$ and balanced integral conditions.
• The constant $\bar{C}$ required for elimination in discrete dynamics depends on the payoff gap $\varepsilon$ and the growth rate bounds, making it sensitive to the specific strategies $\mathbf{p}$ and $\mathbf{q}$.