game theorists

Abstract: We study the computational complexity of fundamental algorithmic problems -- membership testing, separation, valid-inequality testing, and linear optimization -- over polytopes and cones arising from cooperative games (also known as pseudo-Boolean functions). A central obstacle in the study of such problems is that a general cooperative game on $n$ players requires $2^n$ values, so the input size is $2^n$ for a game with $n$ players, making these computational tasks theoretically trivial. Restricting to $k$-additive games reduces the input size to $O(n^k)$, making such games a natural target for meaningful questions about the existence of efficient algorithms. On the positive side, we give an explicit extended formulation of size $O(n^k)$ for the core of $k$-additive $k$-monotone games, allowing all four problems to be solved by a single polynomial-size linear program -- in particular, circumventing the ellipsoid method that is needed when building from earlier tractability results of Deng and Papadimitriou, or of Edmonds. For the cone of $k$-additive $(k{-}1)$-monotone games, we give a complete characterization of its extreme rays and derive the same $O(n^k)$ bound on extension complexity, yielding a geometry-based proof and generalization of a result of Billionnet and Minoux. On the negative side, we show that for $l \leq k-2$ the cone of $k$-additive $l$-monotone games is computationally intractable: membership testing is not in NP (unless NP\,=\,coNP), valid-inequality testing is NP-complete, and extension complexity is at least $1.5^n$. Our hardness results yield, as a special case, a result of Crama and of Gallo and Simone. Furthermore, our hardness results also explain the lack of any good characterization of the extreme rays of the cone of $k$-additive $(k{-}2)$-monotone games.

Abstract: We study feasible stable equilibria of Lotka-Volterra systems and their higher-order extensions. We complete the classification of impossible ecological interaction networks with at most four species and extend several of these impossibility results to families with arbitrarily many species. We then show that these sign-pattern obstructions are specific to the pairwise Lotka-Volterra model: arbitrary prescribed growth rates and pairwise coefficients can be supplemented by higher-order interactions so as to admit a feasible asymptotically stable equilibrium. Through the correspondence with replicator dynamics, we interpret feasible equilibria of higher-order Lotka-Volterra systems as totally mixed symmetric Nash equilibria of symmetric multiplayer games, derive bounds on their number, and study their robustness under perturbations of the payoff tensors. We conclude by showing that every impossible ecology determines a nonempty open class of symmetric two-player games with no totally mixed evolutionarily stable strategy.

Abstract: We study risk-sensitive evolutionary learning dynamics and their long-run equilibrium selection behaviors in coordination games. Agents' risk attitudes enter through the classical entropic risk measure, which evaluates opponent-induced payoff uncertainty and feeds into noisy best responses under two standard revision protocols: best response with mutations and logit choice. We first analyze $2\times 2$ coordination games in both single-population symmetric and two-population asymmetric settings. In the single-population setting, unlike the risk-neutral case where the dynamics are known to favor the risk-dominant equilibrium, we show that risk sensitivity can change the stochastically stable outcome: a greater risk-seeking attitude favors the payoff-dominant equilibrium, while a greater risk-averse attitude favors the maximin equilibrium. Thus, the population's risk attitude may act as a control knob for long-run equilibrium selection. In both population settings, we also identify a robust regime: any super-dominant equilibrium is stochastically stable for all risk attitudes, under both protocols, and across populations. We further extend the single-population analysis to symmetric $k$-action games, which include symmetric $k$-action coordination games as a special case, under risk-sensitive best response with mutations. In this setting, we show that, for sufficiently large populations, sufficiently risk-seeking agents uniquely select the strongly payoff-dominant equilibrium when it exists, whereas sufficiently risk-averse agents uniquely select the strongly maximin equilibrium when it exists. These results show that entropic risk sensitivity may serve as a systematic mechanism for steering equilibrium selection in evolutionary games, beyond the classical risk-neutral benchmark.

Abstract: Offline multi-agent payoff models are estimated under a logging distribution but used on distributions induced by learned solutions and unilateral deviations. Standard held-out loss can therefore favor an interaction class that predicts logged play well while distorting strategic incentives. We introduce Selective Interaction-Rank Validation (SIRV) for finite games with known logging distributions. A training split fits nested payoff models and constructs a common union of all candidate deployment and unilateral-replacement distributions; an independent calibration split evaluates every candidate on this same union. SIRV returns the smallest rank whose simultaneous upper worst-target risk is within tolerance of the best upper score, and abstains when a declared target is unsupported or too imprecisely estimated. A common coverage event yields a finite-candidate target-risk bound and a candidate-specific coarse correlated equilibrium (CCE) gap certificate. We also isolate an exact two-point off-support non-identifiability result. In a controlled factorial study with 2,048 independent games per family, empirical-Bernstein bounds reduce the median CCE-gap certificate by 42.5% relative to Hoeffding bounds on common returns, with a 1.36-point reduction in supported return. Under paired rank misspecification and in a separately generated congestion family, the SIRV-EB fallback rule lowers mean true candidate-selection CCE regret relative to ID-Mean, while retaining game-level losses. Across 384 games at $N=3,5,8$, ID-Mean-relative mean CCE-regret effects stay positive while certified return falls sharply under weak coverage. These results separate certifiable model selection from universal strategic improvement.