LLM-Nash Equilibria: A Prompt-Space Analysis

• LLM-Nash Equilibria are a game-theoretic framework where strategic agents select natural language prompts to guide LLM decisions, highlighting bounded rationality.
• The model translates prompt selection into a constrained optimization problem, using LLM inference to map prompts to action distributions and compute equilibrium strategies.
• Divergences from classical Nash outcomes arise due to cognitive constraints and limited prompt expressiveness, impacting equilibrium computation and learning dynamics.

LLM Nash Equilibria (LLM-Nash Equilibria) are a game-theoretic construct in which strategic agents select natural language prompts—representing their reasoning processes—to guide LLMs in decision-making. Unlike classical Nash equilibrium, which assumes directly optimized choice over mixed or pure strategies in the action space, the LLM-Nash framework models bounded rationality by formalizing strategic interaction in the prompt space. Agent actions are generated as the stochastic output of an LLM conditioned on chosen prompts, information sets, and internal worldview parameters. This approach enables explicit analysis of cognitive constraints, reasoning expressiveness, and learning dynamics in LLM-mediated systems, and reveals divergences from classical equilibrium predictions (Zhu, 10 Jul 2025).

1. Formal Framework

The LLM-Nash model considers two agents, typically an Attacker (A) and a Defender (D), with finite action sets $\mathcal{A}$ (for A) and $\mathcal{D}$ (for D). Each agent receives information sets $I_A \in \mathcal{I}_A$ and $I_D \in \mathcal{I}_D$. The strategy space is recast as prompt spaces $x \in \mathcal{X}$ for A and $y \in \mathcal{Y}$ for D, each prompt encoding a natural language reasoning process (chain-of-thought, framing, heuristics). Agents also possess internal parameters $\theta, \delta \in \Xi$ reflecting worldview or knowledge priors. The LLM induces a policy over actions: $$\mu_A(a) = \tilde{\gamma}_A(a | I_A, x, \theta), \quad \mu_D(d) = \tilde{\gamma}_D(d | I_D, y, \delta)$$ Payoff functions $u_A:\mathcal{A}\times\mathcal{D}\to\mathbb{R}$ and $u_D$ (possibly $\mathcal{D}$0 for zero-sum) are used to evaluate agents' expected utilities.

LLM-Nash Reasoning Equilibrium is defined as a prompt pair $\mathcal{D}$1 such that

$\mathcal{D}$2

$\mathcal{D}$3

with $\mathcal{D}$4.

2. Strategy Space: From Prompts to Actions

Classical strategies $\mathcal{D}$5 are replaced by selecting prompts $\mathcal{D}$6, each inducing an action distribution via LLM inference. The set of feasible behavioral policies under a mindset $\mathcal{D}$7 is

$\mathcal{D}$8

Prompt selection thus constitutes a constrained optimization in prompt space, reflecting the agent’s expressive and inferential resources.

3. Best Responses and Equilibrium Computation

A best response in prompt space is defined by maximizing expected utility over prompt choices, given the other agent's prompt. The best-response mapping is: $\mathcal{D}$9

$I_A \in \mathcal{I}_A$0

A fixed-point in prompt space corresponds to an LLM-Nash equilibrium. In the case of finite prompt sets, this extends to mixed-prompt strategies $I_A \in \mathcal{I}_A$1, $I_A \in \mathcal{I}_A$2, leading to classical mixed equilibrium inequalities over prompt distributions.

4. Divergence from Classical Nash Equilibria

LLM-Nash equilibria generally do not coincide with classical Nash equilibria due to the restricted image of the LLM mapping from prompts to action distributions. For instance, in a Rock–Paper–Scissors scenario, classical Nash equilibrium prescribes uniform randomization $I_A \in \mathcal{I}_A$3. Under an LLM-Nash model, prompt sets such as $I_A \in \mathcal{I}_A$4 and $I_A \in \mathcal{I}_A$5 may induce policies

$I_A \in \mathcal{I}_A$6

$I_A \in \mathcal{I}_A$7

Payoffs $I_A \in \mathcal{I}_A$8 computed for these policy pairs may yield equilibria outside the classical support. In the stated case, $I_A \in \mathcal{I}_A$9 emerges as the reasoning equilibrium, differing from the classical Nash outcome (Zhu, 10 Jul 2025).

5. Existence, Uniqueness, and Multiplicity

Existence of pure-prompt equilibrium is not guaranteed when the prompt spaces $I_D \in \mathcal{I}_D$0 are discrete or non-convex. However, if $I_D \in \mathcal{I}_D$1 are finite, Nash’s theorem ensures the existence of at least one mixed‐prompt equilibrium $I_D \in \mathcal{I}_D$2 in the space of distributions over prompts. Uniqueness does not generally hold; multiple equilibrium prompt profiles or supports may satisfy the equilibrium conditions. Assumptions include fixed LLM inference mappings and bounded rationality—i.e., optimization is over prompts, not directly over arbitrary mixed action strategies.

6. Mindset Expressiveness and Epistemic Learning

Mindsets are ordered by expressiveness: $I_D \in \mathcal{I}_D$3 is more expressive than $I_D \in \mathcal{I}_D$4 if every LLM-induced policy from $I_D \in \mathcal{I}_D$5 is realizable within $I_D \in \mathcal{I}_D$6. In single-agent contexts, greater expressiveness leads to a weakly higher behavioral utility. However, in interactive settings, increased prompt expressiveness can be exploitable by opponents and does not guarantee higher equilibrium payoffs. Epistemic learning allows agents to enhance $I_D \in \mathcal{I}_D$7 or $I_D \in \mathcal{I}_D$8 through neurosymbolic methods or fine-tuning, effectively increasing cognitive scaffolding and potentially shifting reasoning equilibria towards or away from classical Nash.

7. Computational Considerations

For finite prompt spaces, the prompt game reduces to a normal-form game with utilities $I_D \in \mathcal{I}_D$9, $x \in \mathcal{X}$0 indexed by prompt pairs $x \in \mathcal{X}$1. Standard algorithms—support enumeration, best-response dynamics, or (for zero-sum) linear programming—are applicable to compute mixed-strategy equilibria. In large or continuous prompt spaces, heuristic procedures are necessary: prompt candidates may be sampled and evaluated via Monte Carlo rollouts, with iterated best-response and optimization in prompt embedding space (e.g., Bayesian optimization) employed to search for equilibria.

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The LLM-Nash paradigm fundamentally shifts strategic analysis from the action space to the reasoning prompt space, making equilibrium characterization sensitive to both the LLM’s inferential structure and agents’ cognitive affordances. The observable divergence between LLM-Nash and classical Nash equilibria underscores the practical implications of bounded rationality and model constraints in LLM-mediated strategic systems (Zhu, 10 Jul 2025).