Completely Mixed Nets of Strategies

• Completely mixed nets of strategies are a generalization of completely mixed profiles to infinite or measure-theoretic game settings, ensuring all possible deviations are eventually covered.
• They are constructed as nets where each strategy component assigns positive probability to every measurable carrier under suitable topological and convexity assumptions.
• Their role in perfect equilibrium is pivotal, as they guarantee non-emptiness and compactness of equilibria even in games with discontinuous or finitely additive payoffs.

A completely mixed net of strategies is a fundamental concept in the theory of equilibrium refinements for both finite and infinite games. These objects generalize the notion of "completely mixed" strategy profiles to broad topological and measure-theoretic settings where the classical finite-dimensional simplex structure is absent. By systematically realizing all possible "carriers"—collections of measurable sets assigned positive measure by a strategy—in a directed net, completely mixed nets underlie the modern definition and existence theory for perfect equilibrium, capturing the intuition of handling all conceivable rare deviations or "trembles" in infinite and discontinuous games (Flesch et al., 20 Nov 2025).

1. Formal Definition and the Carrier Concept

Let $\Gamma=(N,A,\mathcal F,\Sigma,U)$ denote a strategic-form game with $N$ players, where each player $i$ possesses an action set $A_i$, a measurable field $\mathcal F_i$, a mixed strategy space $\Sigma_i$, and a payoff function $U_i$. For any $\sigma_i\in\Sigma_i$, the carrier of $\sigma_i$ is the collection

$$\mathcal C_i(\sigma_i) = \{ E_i \in \mathcal F_i : \sigma_i(E_i) > 0 \}$$

which maintains a record of every measurable set with strictly positive probability under $\sigma_i$. This notion refines the classical concept of the support of a probability measure, as it tracks the strictly positive sets rather than just the minimal nonnull sets.

A collection $\mathcal C_i \subseteq \mathcal F_i$ is called a carrier if there exists $\sigma_i\in\Sigma_i$ such that $\mathcal C_i(\sigma_i) = \mathcal C_i$.

2. Completely Mixed Nets: Definition and Rationale

In finite games, a completely mixed strategy is a profile where each pure action receives strictly positive probability. In infinite or measure-theoretic settings, such a maximally diffuse strategy may not exist within the space of mixed strategies, as no single probability measure can assign positive measure to all nonempty measurable subsets in non-atomic spaces. The appropriate generalization is therefore a net of strategies.

A completely mixed net is a net $(\sigma_\alpha)_{\alpha\in I} \subseteq \Sigma$ such that for every possible carrier $\mathcal C=\prod_i \mathcal C_i$, there exists $\alpha_0\in I$ so that for all $\alpha\geq\alpha_0$,

$$\mathcal C_i \subseteq \mathcal C_i(\sigma_{\alpha,i}) \quad \forall\, i\in N,$$

i.e., each player's component strategy eventually assigns positive probability to every set in the carrier $\mathcal C_i$. This property ensures the net "eventually covers" all conceivable small mistakes—a linchpin for equilibrium robustness.

3. Topological and Structural Assumptions for Existence

The existence and properties of completely mixed nets hinge upon topological structure in the strategy spaces. The general definition of perfect equilibrium requires only a Hausdorff topology on each $\Sigma_i$. For existence (Theorem 6.2), the following conditions are imposed:

• The product strategy space $\Sigma = \prod_i \Sigma_i$ must be compact and convex within a locally convex topological vector space.
• Each payoff $U_i$ is required to be multilinear and continuous over $\Sigma$.

Canonical classes that satisfy these conditions include (a) finitely additive strategies under the Tychonov topology and (b) countably additive strategies with compact metric $A_i$ equipped with the Prokhorov topology. These settings encompass games with infinite action spaces and those with discontinuous or finitely additive payoffs (Flesch et al., 20 Nov 2025).

4. Existential Construction of Completely Mixed Nets

The proof of existence and construction of completely mixed nets relies on the finite intersection property among the family of closed sets $\mathrm{PS}(\mathcal C)$, comprising all limit points of nets that witness carrier $\mathcal C$. The directed set of indices is

$$I = \{ (\mathcal C, U) \mid \mathcal C\ \text{a carrier},\ U\subseteq\Sigma\ \text{open},\ \sigma\in U \}$$

ordered by $$(\mathcal C_1,U_1)\le(\mathcal C_2,U_2) \iff \mathcal C_1\supseteq\mathcal C_2,\ U_1\subseteq U_2$$. Given the local property [2] of Theorem 3.3, for each $(\mathcal C, U)$, one can select $\sigma_{(\mathcal C, U)} \in U$ such that $$\mathcal C \subseteq \mathcal C(\sigma_{(\mathcal C, U)})$$, producing a net that systematically realizes every carrier as the net evolves. The limit point of this net serves as a candidate for perfect equilibrium (Flesch et al., 20 Nov 2025).

5. Role in Perfect Equilibrium: Existence, Non-emptiness, and Compactness

Completely mixed nets are integral to the definition and existence theory for perfect equilibrium in infinite and discontinuous games. Their key roles are:

• Non-emptiness: By perturbing each player’s strategy space (reminiscent of Selten’s construction), one ensures strict convexity and utilizes Kakutani’s theorem to obtain equilibria in perturbed games. A subnet-convergence and index-net argument then yields a completely mixed net converging to a perfect equilibrium.
• Compactness: Perfect equilibria are characterized as the intersection over all carriers $\mathcal C$ of the closed sets $\mathrm{PS}(\mathcal C)$ within $\Sigma$, and by the finite intersection property and compactness of $\Sigma$, their intersection is nonempty and compact (Theorem 3.3).

This framework generalizes the classical finite case and aligns with Selten (1975) for finite games, with Simon–Stinchcombe (1995) weak perfect equilibria for compact-continuous games, and Marinacci (1997) perfection under finitely additive strategies (Flesch et al., 20 Nov 2025).

6. Illustrative Examples

Completely mixed nets capture crucial phenomena in a variety of settings:

• Discontinuous coordination on $[0,1]$: Payoff 1 occurs only at $(a_1,a_2) = (1,1)$. Any completely mixed net must eventually allocate positive mass to $\{1\}\times\{1\}$. Therefore, only $(\delta_1, \delta_1)$ survives as perfect.
• Cantor-set coordination: Payoff is 1 only if both actions belong to a Cantor set $\mathbb D$. Carriers must include $\mathbb D$, so perfect equilibria assign all mass to $\mathbb D$.
• Finitely additive "variant Wald" game: With payoffs $\frac{1}{\max\{i,j\}}$, there is no countably additive Nash equilibrium, but in the full finitely additive space, every diffuse equilibrium is perfect. A completely mixed net must eventually include every singleton, achievable only with finitely additive mixtures.

In each instance, the net realizes all small possible `mistakes' (perturbations), demonstrating the robustness properties imparted by the completely mixedness criterion (Flesch et al., 20 Nov 2025).

7. Relation to Finite Games and Half-Space Covering Criteria

In finite games, completely mixed strategies coincide with interior points of the simplex. For two-player finite games, the emergence of completely mixed profiles is characterized using Farkas’ Lemma and the half-space covering condition: the "payoff-difference matrix" $D$ for a player leads to column vectors $d_j$, each inducing a half-space $$HS(d_j) = \{ w\in\mathbb R^{n-1} : w^\top d_j \leq 0 \}$$. The player can be made indifferent among all pure strategies if and only if $\bigcup_j HS(d_j) = \mathbb R^{n-1}$. This provides both necessary and, in many cases, sufficient conditions for the existence of a completely mixed Nash equilibrium (Herold et al., 3 Apr 2024).

This finite-dimensional theory underpins concrete constructions of nets of mixed strategies converging towards complete indifference, forming the bridge to the infinite, measure-theoretic generality captured by completely mixed nets in perfect equilibrium theory.

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References

• "A general definition of perfect equilibrium" (Flesch et al., 20 Nov 2025)
• "Farkas’ Lemma and Complete Indifference" (Herold et al., 3 Apr 2024)