Game-to-Real Gap: Quantifying the Effect of Model Misspecification in Network Games

TL;DR

This paper studies robustness of game-theoretic planning under model misspecification in network games. It defines the game-to-real gap as the difference between realized costs under heterogeneous predictions and the costs predicted by each player's own model, with network games having $u_i$ actions and costs $J_i(u)=\tfrac{1}{2} u_i^T u_i - u_i^T( P_{i,-} u + \epsilon_i)$ and Nash equilibria under $(P+P^T)/2 \prec I$. The key results show that misspecifications in external shocks or in the interaction matrix $P$ can yield arbitrarily large gaps, even with small parameter differences, revealing fragility in predictions. To quantify this sensitivity, the authors introduce two centrality-like metrics: the Shock Misspecification Centrality $\mathcal{B}_{i,j}$ and the Interaction-Graph Misspecification Centrality $\mathcal{C}_{i,j}$, enabling exact bounds on the gap via closed-form expressions and line-integral arguments. Numerical experiments, including misaligned shocks and Monte Carlo simulations, demonstrate that conventional centrality measures can mislead about a node's impact and underscore the utility of the proposed metrics for diagnosing and mitigating model misspecification in networked systems.

Abstract

Game-theoretic models and solution concepts provide rigorous tools for predicting collective behavior in multi-agent systems. In practice, however, different agents may rely on different game-theoretic models to design their strategies. As a result, when these heterogeneous models interact, the realized outcome can deviate substantially from the outcome each agent expects based on its own local model. In this work, we introduce the game-to-real gap, a new metric that quantifies the impact of such model misspecification in multi-agent environments. The game-to-real gap is defined as the difference between the utility an agent actually obtains in the multi-agent environment (where other agents may have misspecified models) and the utility it expects under its own game model. Focusing on quadratic network games, we show that misspecifications in either (i) the external shock or (ii) the player interaction network can lead to arbitrarily large game-to-real gaps. We further develop novel network centrality measures that allow exact evaluation of this gap in quadratic network games. Our analysis reveals that standard network centrality measures fail to capture the effects of model misspecification, underscoring the need for new structural metrics that account for this limitation. Finally, through illustrative numerical experiments, we show that existing centrality measures in network games may provide a counterintuitive understanding of the impact of model misspecification.

Figures (3)

• Figure 1: Illustration of misaligned game theoretic planning. Each player $i \in N$ conjectures a game $G^{(i)}$ and devises their action from a solution concept $\texttt{Soln}$, e.g., Nash equilibrium. If players' conjectures are misaligned, the realized joint action need not be an equilibrium and can offer degraded performance relative to each player's prediction.
• Figure 2: Example where performance gap is bigger between two players with similar network effects but smaller Bonacich centrality (i.e., 5 has the highest centrality, but 4 has worse loss if misaligned with 3)
• Figure 3: Bean plots of the Monte-Carlo simulation of network aggregative games with misconjectures. In each case, the type of possible misconjecture and thus misalignment between players varies. Along each horizontal axis is the bound on the level of respective misalignment. As the misalignment increases, the chance spread of the relative difference in predicted and realized cost grows.

Theorems & Definitions (13)

• Definition 1
• Definition 2
• Definition 3
• Proposition 1
• Proposition 2
• Definition 4
• Theorem 1
• proof
• Definition 5
• Proposition 3