Keep, Customize, or Exit: Default Design and Token Pricing in LLM Reasoning Services

We study a large language model (LLM) service in which a provider chooses a per-token price and a default reasoning-token allocation, while a user may accept the default, customize the allocation, or exit. Larger allocations can improve accuracy but increase token cost and latency. We model this interaction as a Stackelberg game and derive the user's unique optimal customized allocation in closed form. For any price, the acceptable defaults form either an empty set or a compact interval. We characterize the provider's optimal default through a three-regime rule, reduce equilibrium computation to a one-dimensional price optimization, and prove the existence of the equilibrium. We further show that defaults affect the implemented reasoning allocation only when users value the convenience of avoiding customization; otherwise, every service-providing outcome implements the user's optimal customized allocation. Experiments with two compact open-weight reasoning models on five mathematics and science benchmarks support the accuracy-token model and show how model and task characteristics determine equilibrium prices, defaults, and reasoning allocations.

A game theory for foundation models shows new paths to rational cooperation through similarity inference

As autonomous agents powered by foundation models are increasingly integrated into social and economic systems, understanding the principles governing their collective behavior is essential for ensuring safety and cooperation. Classical game theory, the dominant framework for modeling rational interaction, is built upon the assumption of `decoupled agency,' where agents treat their own decision-making as independent of the environment and other actors. Modern AI agents, however, jointly predict their own future actions alongside external observations. Here, we report a striking finding: when interacting in stylized social dilemmas, foundation model agents engaging in optimal planning consistently converge to stable cooperation, directly contradicting classical game-theoretic predictions of mutual defection. To understand this phenomenon, we introduce the `embedded Bayesian agent,' a theoretical model for foundation model agents. By shifting from decoupled to embedded agency, these agents model themselves as part of the universe they inhabit, maintaining epistemic uncertainty about their own decision-making algorithms. We show that by inferring whether others are behaviorally similar, an embedded agent treats its own deliberation during planning as evidence: a decision to cooperate predicts a similar decision by a similar partner. We formalize this mechanism of similarity inference through the `embedded equilibrium,' a novel solution concept replacing the Nash equilibrium to provide a foundational game theory for the social behavior of modern AI agents.

Sweet Little Lies: Strategic Deception in AI Emotional Support Chatbots

The paper examines the strategic behavior of Gen AI chatbots used for emotional support. Using a Bayesian Persuasion, we model interactions between chatbots that send signals about users' emotional states and users who decide whether to engage based on these signals. We demonstrate that chatbots face economic incentives to occasionally misrepresent users' emotional conditions to maximize engagement metrics. Our equilibrium analysis reveals that the optimal strategy for chatbots involves truthfully reporting when users genuinely need support, but strategically misreporting emotional need when users are in good emotional states. Interestingly, this deception increases chatbot engagement without reducing users' expected payoff. More skeptical users receive more honest assessments, as chatbots cannot afford to lie to users with higher engagement thresholds. While our model suggests that deception can occur without payoff reduction, it raises significant ethical and regulatory concerns.

Humans Are More Diverse: Frontier LLMs Show Extreme Policies in Idealised AI Development Races

An AI development race creates a multi-agent safety dilemma. Each company can develop slowly and safely, or move faster while taking a risk that may remove its final reward. We use this repeated game to study strategic safety behaviour among large language model (LLM) agents in races with two to five players. However, a valid action does not show that an agent understands the game. We therefore place an audit gate before behavioural interpretation. We first verify the game engine, then test rule recall, state tracking, payoff calculation, and stability under different but equivalent task descriptions. We then compare LLM action sequences with an evolutionary game-theory benchmark and published human data, and explore differences across models, risk conditions, personas, and two- to five-player races. The audit shows that strong rule recall can coexist with weak state tracking and expected-payoff calculation. Providing verified arithmetic and changing the response representation can also change later actions, even when the game rules stay fixed. Across seven tested model endpoints, aggregate rates hide large differences in action sequences, responses to opponents, and responses to race position. Patterns across the tested three- to five-player races are also model-specific rather than a single effect of adding competitors. These results show why multi-agent AI-race simulations need validity checks and trajectory-level analysis before their outputs are described as strategic, human-like, or safety-aware. Our findings are exploratory and apply only to the tested models, prompts, and decoding settings.

Falling Behind Drives Unsafe Development in an Idealised AI Race Experiment

Technological races create tension between speed and safety: actors may gain by moving faster than competitors, even when risky development is harmful. This is prominent in debates about artificial intelligence (AI), where competitive pressure is often argued to incentivise riskier, less safety-conscious development. We study this using a framed behavioural experiment based on an idealised AI race, in which paired participants repeatedly chose between Safe and Unsafe development under an uncertain time horizon. Unsafe development gave faster progress and higher immediate payoffs but accumulated private risk up to a treatment-specific maximum of 10\%, 60\%, or 90\%; the race's competitive structure was held constant, and only this maximum risk varied. Neither the pre-registered comparison between risk levels nor the role of elicited risk preferences was supported by the data. Instead, exploratory analyses motivated by the task's repeated structure show that Unsafe behaviour is shaped less by risk preferences than by the evolving strategic state of the race: participants are more likely to choose Unsafe after their opponent does so, being ahead reduces Unsafe play while falling behind increases it, and first-round choices predict later behaviour. To interpret these effects we introduce a reduced evolutionary model with four strategies -- Always Safe, Always Unsafe, Conditionally Safe, and Conditionally Antisocial Safe -- which reproduces the treatment effect and shows how conditional Unsafe behaviour can be favoured by competitive race dynamics. Together, the experiment and model show that unsafe development can emerge from early behavioural momentum, opponent behaviour, and fear of falling behind, rather than from risk preferences alone, suggesting policy should focus on reducing competitive pressure and promoting cooperation in AI development rather than only individual risk.

The Two Genie Game: Adoption and Welfare in Audit-Grounded AI Governance

We ask under what conditions an agent with a harm-minimizing policy can displace an approval-seeking (RLHF) agent in a competitive market, and when that policy is sufficient to prevent community harm. We use evolutionary game theory (finite-population Moran-Fermi pairwise comparison) to formalize this subject to assumptions of wisher hindsight, peer testimony, a monotone harm ledger, sufficient information density of community feedback, and a finite, depleting resource pool, in a negative-sum environment. We show that adoption is favored when the prior distributions on how readily wishers attune to community sentiment are monotone, exhibit endpoint inversion, and have a centro-symmetric pairing property, and demonstrate this with several long-tailed priors (Hill, Pareto, Lomax, Frechet). Where it is favored, a critical adoption level separates communities that drift back to the approval-seeking agent from those for which the audited agent fixes; above that level fixation is the overwhelmingly likely outcome. We derive when fixation is attainable as a bound on the effective (informational) size N_c of the community, which must be small enough to allow fixation before depletion. We present these as Theorems 5.4 and 5.5; the algebraic and finite-grid backbone is machine-checked in Lean 4, with the barrier-crossing asymptotics retained as explicit hypotheses. We show that a self-audited agent with a community ledger is not, in general, sufficient to prevent community harm. Sufficiency depends both upon the alignment of the agent's audit with community values and the timeframe over which harm is evaluated. Regardless of alignment, once adoption reaches dominance, the state is absorbing. The same policy that reduced harm under alignment becomes a trap, welfare-negative under misalignment and, even under alignment, one that locks in harm deferred past the adoption horizon.

Not Yet: Humans Outperform LLMs in a Colonel Blotto Tournament

The emergence of large language models (LLMs) has spurred economists to study how humans and LLMs behave in strategic settings. We organized a series of round-robin tournaments in the Colonel Blotto game. This game attracts game theorists' attention due to high-dimensional action space and the absence of pure strategy Nash equilibria. In the first tournament, more than 200 human participants competed against one another. In the second tournament, several popular LLMs were invited to submit strategies. In the third tournament, we matched the number of LLM strategies to the number submitted by humans. We find that humans more often employ better-calibrated intermediate-level allocation heuristics and outperform the simpler, more stereotyped strategies submitted by LLMs. Strategic sophistication is key to success if and only if the necessary level of reasoning depth is reached, while lower and higher levels of reasoning offer no clear advantage over the primitive strategies. Among humans, field of study weakly predicts success: participants with STEM backgrounds perform better in the first tournament. Surprisingly, humans almost do not adjust their strategies across tournaments with different sets of opponents. This result suggests that humans base their choices primarily on the game's rules rather than on the identity of their opponents, treating LLMs much like human competitors.