Neuromorphic Realization of Best Response in Finite-Action Games

Abstract

We develop a mechanistic dynamical-systems formulation of best response in finite-action games with relational structure on the action set. The proposed neuromorphic decision dynamics realize bestresponse as the stable outcome of an internal state-space process, rather than as an externally imposed choice rule. This provides a deterministic account of commitment formation, symmetry resolution through basins of attraction, and hysteresis and decision persistence under perturbations. For action spaces with circulant coupling, we prove using Lyapunov-Schmidt reduction that the action-coupling operator determines which components of evidence govern decision formation. We further show that the dynamics implicitly compute a geometry-aware utility, converge exponentially to the corresponding best response with rate independent of the number of actions, and switch only when evidence is sufficiently strong. In contrast, supplying the same geometry-aware utility directly to logit dynamics does not recover these properties, showing that relational structure must be embedded in the decision mechanism itself. We illustrate the framework in a repeated coverage game, prove that the induced game is an exact potential game, and show that its Nash equilibria are reached by the neuromorphic dynamics.

Figures (3)

• Figure C1: Bifurcation geometry and action-selection in the critical eigenspace $E_c = \mathrm{span}\{\boldsymbol{\phi}_{k^\star}, \boldsymbol{\psi}_{k^\star}\}$.(A) At the onset $\alpha=\alpha_c$, a supercritical pitchfork generates a continuum of equilibria parameterized by $(r^\star,\theta^\star)$, forming a ring in $E_c$ (blue) of phase-indeterminate committed states along the stable branch $r=r^\star$. (B) In the presence of an input $\boldsymbol{b}$ (purple), its projection $\Psi_A(\boldsymbol{b})$ onto $E_c$ (magenta) selects a unique equilibrium on the ring. The intersection determines the phase $\theta^b_{k^\star}$ and amplitude $r^\star$, yielding the committed state $(r^\star,\theta^b_{k^\star})$.
• Figure D1: (A) Raw event density $V(k,0)$ on the action ring. (B) Mexican-hat action-coupling matrix $A$ with dominant eigenmode $k_\star=1$.
• Figure D2: Subcritical NDD vs. logit dynamics on a coverage task. Top: Heatmap shows event density $V(k,t)$; white circles are agents scaled by population fraction at each action. The strip below each heatmap shows the modal action (red, largest population fraction) and effective utility (black). Neuromorphic agents commit to dominant peaks and track them stably; logit agents scatter without committing. Middle: Cumulative switches over 700 steps: $\mathcal{O}(10)$ for NDD vs. $\mathcal{O}$(5000) for logit. Bottom: NDD maintains BR fraction = 1 throughout; logit fluctuates erratically.

Theorems & Definitions (12)

• Lemma 1: Reduced critical equation
• proof
• Proposition 1: Local polar reduced equations
• proof
• Lemma 2: Suppression of noncritical evidence
• proof
• Lemma 3: Exact potential for the projected game
• proof
• Proposition 2: Local convergence to projected BR
• proof