Pure-Strategy Perfect Equilibrium

• Pure-strategy perfect equilibrium is a game theory concept defined by deterministic strategies that are optimal in every subgame and under slight perturbations.
• The equilibrium employs value recursion and dynamic programming to exclude non-credible mixed strategies, particularly in stochastic and extensive-form games.
• Practical applications include solving large extensive-form games and verifying trembling-hand perfection, although NP-hard issues arise in multi-player settings.

A pure-strategy perfect equilibrium is an equilibrium refinement in game theory characterized by profiles in which each player's strategy is deterministic (pure) and satisfies optimality within every conceivable subgame or in response to all vanishing perturbations of opponents' strategies. It excludes reliance on randomization except where warranted by the structure of the game, and enforces robust credibility of threats and choices throughout the entire game tree or strategic form.

1. Formal Definitions and Conceptual Frameworks

Three principal notions arise in the literature:

• Markov Perfect Equilibrium (MPE) in Continuous-Time Games In dynamic settings (e.g., war-of-attrition), a Markov pure strategy specifies closed exit regions $E_i \subseteq \mathcal{X}$, with exit occurring only upon hitting $E_i$, and a hazard function $\lambda_i : \mathcal{X} \to [0, \infty)$ vanishes identically, i.e., $\lambda_i \equiv 0$ (Georgiadis et al., 2021). The equilibrium requirement is

$V_i(x; a_i^*, a_{-i}^*) \geq V_i(x; a_i, a_{-i}^*)$

for every alternative $a_i$ and state $x$, with $V_i$ the expected discounted payoff given the Markov strategies.

• Trembling-Hand Perfect Equilibrium (THPE) in Strategic Form Games Given pure strategy profile $p = (p_1, \ldots, p_n)$ in $G = (N, \{S_i\}, \{u_i\})$, $p$ is trembling-hand perfect if there exists a sequence of fully mixed strategy profiles $\{\sigma^k\}$ converging to $p$ such that $p_i$ is a best reply to $\sigma^k_{-i}$ for all $i$ and $k$ (0812.0492). The defining property is:

$$u_i(p_i, \sigma^k_{-i}) = \max_{s_i \in S_i} u_i(s_i, \sigma^k_{-i})$$

• Subgame-Perfect Equilibrium (SPE) via Value Recursion in Extensive-Form Games A pure-strategy profile is subgame-perfect if it induces Nash equilibrium play in every subgame. In pentaform (extensive-form) representation, equilibrium is characterized by value recursion over subroots, formalized as

$$v(t) = \begin{cases} v(\max O_t(s)), & \text{if } \max O_t(s) \in T \ u(R(O_t(s))), & \text{otherwise} \end{cases}$$

where each restriction $s|_t$ is Nash on piece-form $Q_t$ (Streufert, 2023).

2. Existence and Structure in Dynamic and Strategic Games

2.1 Dynamic War-of-Attrition (Stochastic Payoff Diffusion)

Pure-strategy MPEs emerge in stochastic war-of-attrition with heterogeneous exit payoffs (i.e., $l_1 < l_2$) and irreducible diffusion $dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dB_t$, $\sigma(x) > 0$. Each firm's optimal response is characterized by threshold rules:

• Define single-player stopping thresholds $\theta_i^*$ as solutions to the optimal stopping problem (Lemma 1), yielding $\theta_1^* < \theta_2^*$.
• Two MPE profiles:
  • Profile A: Firm $2$ exits once $X_t \leq \theta_2^*$; Firm $1$ never exits.
  • Profile B (when $|l_2 - l_1|$ is small): Firm $1$ exits once $X_t \leq \theta_1^*$; Firm $2$ never exits.
• The equilibrium region is sharply delineated: for $X_0 \geq \max(\theta_1^*, \theta_2^*)$, only A or B can occur (Georgiadis et al., 2021).

2.2 Elimination of Mixed Strategies

Under stochastic payoffs ($\sigma(x) > 0$) and heterogeneous exit values, all mixed-strategy Markov equilibria are excluded. Lemma 2 demonstrates that common support and the need for indifference cannot be achieved when $\theta_1^* \neq \theta_2^*$ (Theorem 1). If payoffs are deterministic ($\sigma \equiv 0$) or exit payoffs are homogeneous ($l_1 = l_2$), classic mixed-strategy equilibria remain feasible.

3. Computation and Verification Complexity

3.1 Trembling-Hand Perfection: NP-Hardness

For strategic-form games with $n \geq 3$ players and integer payoffs, deciding whether a given pure-strategy profile is trembling-hand perfect is shown to be NP-hard (0812.0492). The reduction originates from the three-player MINMAX problem and constructs an augmented game with an added absorbing strategy $L$:

• If the minmax value for Player 1 in the base game $G$ is $< r$, then $(L, L, L)$ is trembling-hand perfect.
• If the minmax value is $> r$, trembling-hand perfection fails for $(L, L, L)$. No polynomial algorithm can certify perfection unless P=NP. For two-player games, trembling-hand perfection coincides with dominance and is computable via LP.

3.2 Dynamic Programming Approach for Extensive Games

The value-recursion characterization for pure-strategy subgame-perfect equilibria allows recursive computation over subroots ("piece-forms"). Provided upper- and lower-convergence of the utility function $u$, admissibility and persistence of value functions ensure authenticity—meaning that the continuation value at each subroot matches the realized utility along the induced outcome (Streufert, 2023).

4. Mathematical Characterizations and Criteria

4.1 Hamilton–Jacobi–Bellman and Variational Inequality

In continuous-time dynamic games, candidate value functions $V_i$ must solve the variational inequality: $$0 = \max\left\{ (\mathcal{L}V_i)(x) + \pi(x),\, l_i - V_i(x) \right\}$$ with boundary conditions ("smooth-pasting") at thresholds $x = \theta_i^*$: $$V_i(\theta_i^*) = l_i, \quad (\mathcal{L}V_i)(\theta_i^*) + \pi(\theta_i^*) = 0$$ These characterize optimal exit policies (and equilibrium regions).

4.2 One-Piece Unimprovability

Pure-strategy subgame-perfect equilibrium in pentaform games is equivalent to "one-piece unimprovability": for each subroot $t$, player $i$'s realized utility $u_i(R(O_t(s)))$ weakly dominates the utility attainable by any single-piece deviation at $t$ (Streufert, 2023).

5. Comparative Cases and Limitations

• When exit options are homogeneous ($l_1 = l_2$) and/or payoffs are deterministic ($\sigma \equiv 0$), symmetric mixed-strategy equilibria can be constructed (hazard rates $\lambda(x)$ explicitly specified).
• The jump from $n=2$ to $n=3$ players in THPE induces a sharp complexity transition.
• For proper equilibrium (Myerson), NP-hardness also applies in three-player settings (0812.0492). The two-player computational status remains unresolved.

6. Interpretive Remarks and Theoretical Significance

• Pure-strategy perfect equilibrium refines Nash equilibrium by demanding optimality in every subgame and excluding non-credible threats.
• In dynamic programming treatments, equilibrium existence and computation proceed via Bellman-like recursion over subgame decomposition (subroots/piece-forms), unifying dynamic programming and equilibrium theory.
• Trembling-hand perfection and subgame perfection are robust against vanishing perturbations and payoff changes at future nodes, provided technical conditions (upper/lower convergence).
• The absence of mixed-strategy equilibria in perturbed dynamic games with heterogeneous payoffs highlights the fragility of mixed-strategy constructions to stochasticity and payoff asymmetry (Georgiadis et al., 2021).

7. Practical Applications and Computational Implications

• Large extensive-form games can be solved by recursive value-function methods over subroots, extending backwards-induction to general imperfect-information and infinite-horizon settings.
• For verifying trembling-hand perfection in three-player games, only exponential-time algorithms exist subject to current complexity bounds.
• Existence proofs and computational schemes directly inherit methods from dynamic programming and optimal control theory, relating Bellman equations to equilibrium recursion.

A plausible implication is that refinements such as pure-strategy perfect equilibrium may be preferable in environments featuring payoff stochasticity and heterogeneity, as they yield credibly stable and computationally tractable solutions in settings where mixed strategies become degenerate or untestable.