Parametric Open Source Games¶

🕒 Published (v1): 2026-06-25 14:14 UTC · Source: Arxiv · Venue: ICML · link

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ICML; continuous parametric extension of open-source games; rigorous theory for inter-agent dependencies

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TL;DR¶

This paper introduces parametric open-source games, a continuous analogue of classical program equilibria in which players choose parameter vectors and a semantics map converts the full parameter profile into mixed actions. For symmetric $2\times2$ games under sigmoid semantics, the authors derive an exact coupling threshold $\gamma^\star$ at which selfish gradient ascent switches from defection to cooperation, and extend this to a neural semantics class governed by the ratio $\beta/\alpha$ of cross-player to self-player sensitivity.

Problem¶

Classical open-source game theory (program equilibria) relies on discrete, symbolic, or proof-theoretic representations incompatible with modern differentiable learning systems parameterized by high-dimensional vectors. Standard differentiable multi-agent optimization, conversely, assumes each player's policy depends only on its own parameters, discarding the strategic effects of mutual parameter inspection. No existing framework unifies open-source reasoning with continuous gradient-based optimization.

Method¶

A parametric open-source game augments a finite normal-form game $G = (N, (S_i), (u_i))$ with compact convex parameter spaces $\Theta_i \subset \mathbb{R}^{d_i}$ and continuous semantics maps $\varphi_i : \Theta \to \Delta(S_i)$ that map the full parameter profile $\theta = (\theta_i)_{i\in N}$ to mixed actions. The induced payoff is $U_i(\theta) = u_i(\varphi_1(\theta), \ldots, \varphi_n(\theta))$, and a parametric program Nash equilibrium (PPNE) is a fixed point of mutual best responses in $\Theta$.

Two concrete semantics families are analyzed:

Sigmoid semantics: $p_i(\theta) = \sigma(\theta_i + \gamma\theta_{-i})$ with scalar coupling $\gamma \geq 0$. Setting $\gamma = 0$ recovers closed-source. The critical coupling is: $$\gamma^\star = \frac{T + P - R - S}{R + T - P - S}$$ derived by evaluating the gradient $\partial U_1/\partial\theta_1$ at the symmetric midpoint $\theta = (0,0)$.
Neural semantics: $p_i(\theta) = \sigma\!\bigl(\alpha\theta_i + \beta\theta_{-i} + h(\theta_i^2, \theta_{-i}^2, \theta_i\theta_{-i}; W)\bigr)$ where $h$ is a two-layer tanh network taking only quadratic features (so its first-order derivative vanishes at $\theta=(0,0)$). The cooperation threshold is then $\beta/\alpha > \gamma^\star$.

Equilibrium verification uses a one-dimensional boundary PPNE test: freeze the opponent at boundary value $B$ and check that $\frac{d}{d\theta_i}U_i(\theta_i, B) \geq 0$ on $(-B, B)$.

Key Contributions¶

Formal definition of parametric open-source games and proof of PPNE existence via Glicksberg's and Kakutani's theorems.
Exact phase-transition threshold $\gamma^\star$ for sigmoid semantics in symmetric $2\times2$ games, with empirical verification across four canonical games (Prisoner's Dilemma, Stag Hunt, Chicken, Harsh PD).
One-dimensional boundary PPNE characterization reducing equilibrium verification from a 2D landscape to independent per-player sweeps.
Neural semantics class that preserves the same local cooperation condition through the ratio $\beta/\alpha$, with experimental evidence that warm initialization above threshold enables cooperative learning while cold starts do not.

Results¶

Empirical phase transition in mean terminal cooperation probability matches the analytical $\gamma^\star$ closely across all four canonical games (20 seeds each; Figure 2).
Under sigmoid open-source semantics in the Prisoner's Dilemma: equilibrium cooperation $p_1 \approx p_2 \approx 1.00$, individual payoff $\approx 3.00$, social welfare $= 6.00$ vs. closed-source $p_1 \approx p_2 \approx 0.01$, social welfare $\approx 2.07$.
Fixed neural semantics with matching $\beta/\alpha$ reproduce sigmoid baselines exactly; neural-fixed-closed tracks sigmoid-closed and neural-fixed-open tracks sigmoid-open.
Warm-started learned neural semantics ($\alpha, \beta) = (1, 1.5)$, above threshold) reach high-welfare regimes comparable to fixed open-source models; cold-started ($\alpha, \beta) = (0, 0)$) do not reliably discover cooperative coupling.

Limitations¶

Assumes full, noiseless access to opponent parameters — unrealistic in deployed multi-agent systems.
Restricted to symmetric two-player settings with one-shot base games; results do not immediately extend to asymmetric, $n$-player, or sequential (Markov game) settings.
Sigmoid semantics hard-codes linear cross-player dependence; the neural extension addresses expressivity but introduces an optimization barrier (initialization sensitivity).
Theorem 3 is a local result near $\theta = (0,0)$; global convergence and basin-of-attraction characterization are not fully analyzed.
The framework is intentionally idealized and does not address certifiable or partial transparency, which are more realistic notions of "open-source" in practice.

Relevance to Agentic AI / LLM Agents¶

As LLM-based agents are increasingly deployed in multi-agent settings where they may share weights, activations, or chain-of-thought, the question of how mutual parameter visibility reshapes incentives becomes practically urgent. This paper provides a formal, gradient-compatible framework showing that parameter transparency can fundamentally alter equilibrium structure — specifically enabling cooperation in otherwise-defective games — which directly informs the design of multi-agent LLM systems where agents observe each other's internal states (e.g., shared scratchpads, model weights, or embeddings). The neural semantics extension and its initialization-sensitivity findings are particularly relevant to training pipelines where cooperative behavior must be discoverable via gradient descent. The boundary PPNE test also offers a tractable verification method applicable to continuous-parameter agent policies.