Meritocratic systems, from admissions to hiring, aim to impartially reward skill and effort.

What does it truly mean for a language model to "reason" strategically, and can scaling up alone guarantee intelligent, context-aware decisions? Strategic decisionmaking requires adaptive reasoning, where agents anticipate and respond to others' actions under uncertainty. Yet, most evaluations of large language models (LLMs) for strategic decision-making often rely heavily on Nash Equilibrium (NE) benchmarks, overlook reasoning depth, and fail to reveal the mechanisms behind model behavior. To address this gap, we introduce a behavioral game-theoretic evaluation framework that disentangles intrinsic reasoning from contextual influence. Using this framework, we evaluate 22 state-of-the-art LLMs across diverse strategic scenarios. We find models like GPT-o3-mini, GPT-o1, and DeepSeek-R1 lead in reasoning depth. Through thinking chain analysis, we identify distinct reasoning styles--such as maximin or belief-based strategies--and show that longer reasoning chains do not consistently yield better decisions. Furthermore, embedding demographic personas reveals context-sensitive shifts: some models (e.g., GPT4o, Claude-3-Opus) improve when assigned female identities, while others (e.g., Gemini 2.0) show diminished reasoning under minority sexuality personas. These findings underscore that technical sophistication alone is insufficient; alignment with ethical standards, human expectations, and situational nuance is essential for the responsible deployment of LLMs in interactive settings.

Concavity and its refinements underpin tractability in multiplayer games, where players independently choose actions to maximize their own payoffs which depend on other players' actions. In games, where players' strategy sets are compact and convex, and their payoffs are concave in their own actions, strong guarantees follow: Nash equilibria always exist and decentralized algorithms converge to equilibria. If the game is furthermore, an even stronger guarantee holds: Nash equilibria are unique under strictness assumptions. Unfortunately, we show that concavity or monotonicity is NP-hard, already for games where utilities are multivariate polynomials and compact, convex basic semialgebraic strategy sets--an expressive class that captures extensive-form games with imperfect recall. On the positive side, we develop two hierarchies of sum-of-squares programs that certify concavity and monotonicity of a given game, and each level of the hierarchies can be solved in polynomial time. We show that almost all concave/monotone games are certified at some finite level of the hierarchies. Subsequently, we introduce the classes of SOS-concave/monotone games, which globally approximate concave/monotone games, and show that for any given game we can compute the closest SOS-concave/monotone game in polynomial time. Finally, we apply our techniques to canonical examples of extensive-form games with imperfect recall.

We prove that no reinforcement learning policy with confidence-gated autonomy can simultaneously achieve maximum helpfulness, optimal calibration, and full autonomy under rational oversight, whenever some tasks exceed the agent's reliable competence: the Behavioral Credibility Trilemma. The impossibility is geometric -- adding any non-affine autonomy incentive to a strictly proper scoring rule destroys strict properness, so an agent rewarded for both calibrated confidence and autonomous action systematically inflates its reported confidence on tasks below the principal's approval threshold. The Behavioral Perturbation Lemma quantifies the inflation (scaling as $w_A/(2 w_C)$ for the Brier score) and shows detection requires $Ω(1/Δ^2)$ observations. We prove the principal's optimal oversight rule is necessarily non-affine, making the impossibility unconditional and optimizer-independent across log-concave-density policy families. We formalize the Confidence-Gated Decision Problem, map existing methods onto the trilemma, and identify two constructive resolution pathways (commitment, domain separation). A 540-configuration Best-of-N experiment tests five pre-registered hypotheses, all strongly confirmed (effect sizes $d = 1.10$ to $5.32$), and adds a descriptive analysis of the achievable-$(H, C, A)$ surface geometry showing a plateau-truncated frontier consistent with the predicted inflation saturation.

We develop new parameter-free and scale-free algorithms for solving convexconcave saddle-point problems. Our results are based on a new simple regret minimizer, the Conic Blackwell Algorithm+ (CBA+), which attains O(1/ T) average regret. Intuitively, our approach generalizes to other decision sets of interest ideas from the Counterfactual Regret minimization (CFR+) algorithm, which has very strong practical performance for solving sequential games on simplexes. We show how to implement CBA+ for the simplex, `p norm balls, and ellipsoidal confidence regions in the simplex, and we present numerical experiments for solving matrix games and distributionally robust optimization problems. Our empirical results show that CBA+ is a simple algorithm that outperforms state-ofthe-art methods on synthetic data and real data instances, without the need for any choice of step sizes or other algorithmic parameters.

This objective is similar to Maximum Entropy Correlated Equilibrium (MECE) [48], and the proofs here are similar to the framework set out there. A drawback of MECE is that it is not easy to determine the minimum p permissible. If we choose p that does not permit a valid solution, then the parameters will diverge. We can circumvent this problem by optimizing the distance to a target ˆ p. And µis for balancing the linear objective.