Papers

A common problem in social choice is to determine whether there is a social choice procedure, such as a voting method, satisfying some desired criteria. Computer-aided methods such as SAT solving can sometimes answer these questions. However, under typical encodings, a SAT solver may only synthesize a voting method on a finite domain, while we may want one on an infinite domain, such as the domain of all preference profiles for a fixed number of candidates but any finite number of voters. In this paper, we use an approach based on reasoning with constrained Horn clauses and computation with polyhedra to synthesize a voting method on an infinite domain. We then use SMT and Lean to verify its properties. Our main result is a possibility theorem about four well-known criteria from voting theory: the Condorcet winner and loser criteria, positive involvement, and resolvability. Previous work has shown that for five or more candidates, there is no voting method satisfying these axioms, and that for four candidates, there is no method satisfying these core axioms plus one more invariance axiom. Here we show that for four candidates, there does exist a method satisfying the core axioms and more.

Generative AI is shifting digital commerce from browsing toward agentic search, in which consumers delegate product discovery to AI agents. We compare manual search, which accurately evaluates a limited product set, with agentic search, which screens a broad catalog through noisy representations of preferences and products. Preference complexity is the number of satisfaction-relevant dimensions that are difficult to articulate before search but readily evaluated upon inspection. Consumers have finite attention and choose search intensity: products inspected manually or preference-refinement depth with an agent. We obtain three findings. First, manual search collapses beyond a finite complexity threshold: inspection ceases, mismatch reaches the no-search benchmark, and platform revenue falls to zero. Agentic search avoids this collapse. Once refinement becomes worthwhile, it remains worthwhile as complexity rises; mismatch stays below the no-search benchmark and revenue remains positive, although articulation effort and mismatch may increase. Second, platforms rank the regimes by conversion revenue, whereas consumers also bear search expenditure. When manual inspection is sufficiently inexpensive, agentic search becomes revenue-superior before consumers voluntarily adopt it, creating an adoption lag in which consumers rationally continue manual search. Third, conditional on agentic participation, platforms may assign lower fidelity to consumers with larger attention budgets because they can offset noisier representations through additional refinement, yielding an inverted fidelity allocation. Agentic commerce thus shifts scarcity from product inspection to preference articulation, making consumers' willingness and ability to interact central to voluntary use and platform fidelity design.

We define a category of strategic games in which a morphism is a pair of a map between sets of players and a map between sets of strategy profiles with specific properties and show that this category is well-defined. Three cases are considered for the maps between sets of strategy profiles: they may be order-preserving, order-reflecting or order-embedding with respect to each player's preference relation. We define a presheaf on the category of games valued in a category of sets that sends each strategic game to a set of strategy profiles and present conditions for this presheaf to be well-defined. Two cases are considered for the morphisms in the category of sets: they may be relations or maps. We define a presheaf that sends each strategic game to the set of Nash equilibria (resp. Pareto efficient strategy profiles) and show that this presheaf is well-defined if and only if the maps between sets of strategy profiles are order-reflecting or order-embedding (resp. order-embedding), and the morphisms in the category of sets are relations.

As firms increasingly deploy machine learning for strategic decision-making, understanding algorithmic interactions has become central to operations research and economics. This paper studies learning in infinite-horizon, nonzero-sum linear-quadratic stochastic games under a radically uncoupled information structure, where players are either unaware of opponents or strategically oblivious, observing only a common state and their own action history. Under this minimal information, we analyze an asynchronous decentralized learning process in which each player independently runs a single-agent $\epsilon$-greedy iterated least-squares algorithm. We prove that, despite being unable to identify the system parameters, players' learning dynamics converge almost surely to the complete-information Nash equilibrium and characterize the convergence rate. We then apply the framework to a dynamic Cournot competition with sticky prices. Numerical experiments validate the theoretical results and show that learning under limited information reduces firm profits under both low and high price stickiness, while total surplus declines and market concentration increases when price stickiness is high. Publicly revealing aggregate market output substantially accelerates convergence and mitigates these welfare losses.

This paper explores the idea of promoting well-being and safety in human-AI interactions by forcing AI agents explicitly to empower humans and to manage the power balance between humans and AI agents in a desirable way. Using a principled, partially axiomatic approach based on desirable properties, we design a parametrizable and decomposable objective function for AI systems that represents an inequality- and risk-averse long-term aggregate of human power. It can take into account models of human bounded rationality and social norms, and crucially, considers a wide variety of possible human goals. We prove how certain desiderata enforce particular functional forms and restrict parameter ranges. We exemplify the consequences of softly maximizing this metric in several paradigmatic situations and describe what instrumental sub-goals it will likely imply.

Merit-based scholarships are widely used to attract high-achieving students, but do they guarantee an improvement in the merit of a university's student pool? We study this question in centralized college admissions using a matching with contracts framework. We first introduce the scholarship choice rule, which processes scholarship levels from highest to lowest and selects the highest-merit students at each level. We show that scholarship feasibility, scholarship maximality, and no justified envy uniquely characterize this rule. We further characterize the student-proposing deferred-acceptance mechanism based on this rule as the unique mechanism, up to outcome equivalence, satisfying the extensions of the three axioms, together with individual rationality, and strategy-proofness. We then study the welfare effects of scholarship provision. In general matching markets, a university may be matched with lower-merit students after introducing a scholarship. Finally, when all schools use the scholarship choice rule and every student is eligible for every scholarship level a school offers, we show that a common merit ranking is necessary and sufficient for scholarship provision to guarantee, for every student-preference profile satisfying within-school monotonicity, a weak improvement in the scholarship-providing school's student pool.

In this note, we show that equilibrium profit is zero in Bertrand competition with a finite number of firms and consumers whose willingness to pay are bounded, under any market segmentation profile.

The paper places the Samuelson–Merton “util-prob” construction and preference-based state-price interpretation inside a continuous-time equivalent-martingale-measure framework. The contribution is not the existence of a marginal-utility pricing kernel, but a compensated stochastic-exponential representation of the “util-prob” density. Behavioral weighting by marginal utility becomes an equivalent martingale measure only after the drift generated by risk aversion and prudence is compensated. Under positivity, integrability, normalization, and martingale conditions, the behavioral density is generated by the Brownian kernel $-A_t\sigma_t$, where $A_t$ is Arrow–Pratt risk attitude and $\sigma_t$ is state volatility. The required compensator identifies the finite-variation term that must be removed before marginal-utility weighting defines a true martingale density. The framework preserves put–call parity under a common admissible kernel and yields a Cox–Ross risk-premium identity linking option and underlying premia through volatility exposures. Empirically, the paper maps option-implied and physical densities into a signed Arrow–Pratt decomposition, with positive curvature interpreted as risk aversion and negative curvature as risk-seeking or speculative tilt. In a panel of liquid index ETF and single-name options, arbitrage-penalized EBMM has lower pooled holdout squared-error loss than arbitrage-penalized SVI when both models receive the same total-variance positivity and butterfly-density penalties, although slice-level results remain mixed. The exercise is best read as a density-ratio admissibility test rather than a replacement for production volatility-surface estimation.

Empirical and experimental studies suggest that a decision maker can remain with their default option even when preferred alternatives are available, yet standard choice data may not record which option is the default. Existing default-based models often assume that the default option is observed. In contrast, I study a decision maker subject to status quo bias and choice overload and illustrate how one can identify both the decision maker's default and preference from the observed behavior. This paper shows that anomalous choice behavior may arise from a latent default, a possibility largely overlooked in studies of standard choice data.

Detecting algorithmic collusion is challenging because regulators often have limited access to firms' algorithms, training data, and market information. We study an intermediate-information regime in which an auditor can query firms' frozen pricing policies and construct the induced strategy graph. Using a complete characterization of Nash equilibria in a repeated pricing game, we identify graph-theoretic features of strategy graphs that are associated with collusive reward-and-punishment schemes, including maximum betweenness, attractor in-degree, and average path length. We then test these metrics on policies learned by decentralized Q-learning and the Q-learning algorithm of Calvano et al. (2020). We find that especially the maximum betweenness and attractor in-degree are strongly correlated with the standard profit-based Collusion Index. Importantly, the proposed metrics rely only on the unlabeled topology of strategy graphs and require neither price histories, demand estimates, nor competitive and monopoly benchmarks. Our results suggest that the structure of frozen pricing policies contains robust signals of collusion among reinforcement learning algorithms and provides a promising basis for auditing algorithmic pricing systems under limited information.

A choice rule is $q$-acceptant if it chooses $\min\{q,|X|\}$ alternatives from each set $X$. We show that a path-independent rule of maximum cardinality at most $q$ need not have a $q$-acceptant path-independent expansion, refuting Chambers and Yenmez (2017, Theorem 4). We construct a one-school matching market whose unique stable matching leaves a seat vacant that no path-independent expansion of the school's rule fills. Every path-independent rule satisfying the law of aggregate demand has such an expansion. We characterize the choice rules admitting an acceptant expansion by monotone selections of rejected alternatives.

Consider a revenue-maximizing seller who can access a binary signal about two bidders` joint values. We explore what kind of information is most valuable to the seller by studying three classes of signals, each capturing a distinct dimension of bidders` values: their overall level (demand), their relative strength (ranking), and their dispersion while preserving bidder anonymity (competitiveness). We characterize the optimal signal and corresponding auction mechanism within each class, and find that competitiveness signals are particularly effective. Under certain regularity conditions, the optimal competitiveness signal yields at least as much revenue as any ranking signal or demand signal. Moreover, for signals that induce a monotone allocation, the optimal competitiveness signal yields at least as much revenue as any other binary signal.

I argue that there is a sense in which universal equilibrium (defined loosely) existence in games is incompatible with eschewing strictly dominated strategies and a sense in which it isn't.

We develop a continuous-time stochastic choice theory with endogenous preference evolution. Unlike dynamic random utility, observed behavior affects future preferences through the conditional distribution of latent preference states, generating endogenous distributional feedback. We show that this feedback has observable behavioral implications and characterize stochastic choice by a behavioral representation consisting of contemporaneous choice and continuation behavior. This representation is identified from stochastic choice, yields a rigidity result linking structural preference dynamics to observable behavior, and characterizes exactly when distribution dependent utility is behaviorally reducible to dynamic random utility. We further prove a behavioral impossibility theorem: stochastic choice arrays exhibiting behavioral distributional feedback admit no dynamic random utility representation. On the probabilistic side, we establish existence and weak uniqueness for the underlying conditional McKean-Vlasov system with conditional law feedback. The structure unifies endogenous information, latent preference dynamics, behavioral identification, and stochastic choice within a single continuous-time model.

This paper investigates the strategic and welfare properties of endogenous population partitioning (secession) within large-population anonymous games featuring strategic heterogeneity. We consider a continuum-player framework with a binary action space where players are categorized either as fol- lowers, who experience positive network externalities from conformity, or as contrarians, who seek distinctiveness via anti-conformism. We fully characterize the set of Nash equilibria and establish con- ditions under which costless secession yields structural Pareto improvements. We demonstrate that in any strategically mixed society, every mixed-strategy Nash equilibrium admits a Pareto-improving se- cession. With finitely many types, secession systematically mitigates coordination frictions, enhancing both individual payoffs and aggregate utility. Furthermore, we characterize social planner configura- tions optimizing weighted aggregate utility, establishing a formal mathematical isomorphism between optimal jurisdictional design and the theory of Bayesian persuasion solved via concavification. Finally, we derive the structural conditions governing migration stability when subgroups can unilaterally re- locate across distinct societies.

Generative AI makes expert output uninformative about the human fallback capability its provider keeps in reserve. A liability commitment can signal that capability: a provider whose staff rescue more of the cases the AI fails pays damages less often, so the commitment certifies an asset built by keeping people on cases and lost by taking them off. This paper asks where contract law leaves room for such a signal. Four legal primitives map a posted cap into retained exposure and close the message space from both sides: litigation viability, the penalty doctrine, displacement of liability to the state or an indemnity pool, and mandatory control of standard terms. Where the law leaves an option of zero exposure, low types pool at zero, intermediate types separate on a schedule anchored at the mandatory floor, and high types pool at the ceiling; where it does not, separation starts at the bottom type. A statutory floor removes the lower pool but pushes the schedule toward the ceiling, so its net effect is not monotone. The penalty doctrine decides whether recovery adequate for deterrence can be restored by contract or only by investing in verifiability. Holding fallback skill constant demands more human engagement the less often the AI fails. A calibration to five occupations and twelve legal configurations in Germany, Austria, Switzerland, the United Kingdom and the United States locates the binding margin of each cell and supports no ranking by legal family or contracting channel.

We study the expected average rank achieved by the Efficiency-Adjusted Deferred Acceptance (EADA) mechanism in i.i.d.\ matching markets. While student-proposing Deferred Acceptance gives students an expected average rank of logarithmic order, we prove that EADA's expected average rank is at most $4\log\log n+O(1)$. Therefore, EADA improves the asymptotic order of students' assignments. At the cost of a weaker bound, $O((\log\log n)^2)$, we extend this conclusion to a much larger class of mechanisms. Namely, every Pareto-efficient mechanism that weakly Pareto-dominates DA breaks DA's logarithmic barrier. These are the first asymptotic guarantees for the expected average rank of EADA and of the broader class of Pareto-efficient improvements of DA. The conclusions extend to many-to-one markets with bounded quotas and random markets with correlated preferences.

We study a binary choice problem in which an agent chooses between two actions whose payoff depends on a continuous state. The agent chooses how much effort to invest in learning about the state. Equivalently, we can think of the state as the strength of a stimulus, with the agent exerting costly effort to be more responsive to it. Taking as given the Fisher information cost introduced by Hebert and Woodford (2021), we analyze the optimal state-dependent choice rule using a variational approach. The main result is that agents' optimal response is an S-shaped function of the state under mild conditions. This prediction is aligned with the widely documented psychometric curve response profile observed in the experimental literature in psychology and economics.

Which social norms are self-correcting under rational learning? We show that conduct sustained by false beliefs cannot persist if a single departure from prevailing behavior generates evidence against those beliefs. We study the overlapping-generations learning model of Fudenberg and Levine (1993), in which finitely lived Bayesian agents are repeatedly and randomly matched with agents in other player roles, observe only their own matches, and learn from experience. In simple extensive-form games with nodewise-independent, nondegenerate priors, as agents live increasingly long lives and become sufficiently patient, every limiting game outcome is path-equivalent to a subgame-confirmed equilibrium. This establishes the converse of Fudenberg and Levine (2006). The mechanism is endogenous experimentation: uncertainty about the consequences of a potentially profitable departure gives patient agents an incentive to test it, generating the observations that correct beliefs and discipline continuation play.

This paper provides an axiomatic foundation for catastrophic thinking, a form of pessimism in which an agent evaluates uncertain alternatives by attending only to a subset of adverse outcomes. We introduce Catastrophic Attention Preferences (CAP), under which an act is evaluated by its subjective expected utility conditional on the worst outcomes, up to a subjectively determined probability threshold $q$. The resulting functional is a subjective counterpart of Expected Shortfall: both the agent's belief $\mu$ and her threshold $q$ are derived from preferences rather than assumed, without a probability distribution given as a primitive. Our main result is a complete behavioral characterization: six axioms, one of which, Catastrophic Complementarity, carries the behavioral content of catastrophic thinking, together with two standard richness conditions, are equivalent to the existence of a CAP representation, and the parameters $(\mu, q)$ are unique. The parameters are fully identified from probability equivalents of events, simple binary bets that can be elicited experimentally. We characterize comparative ambiguity aversion within the class: with common beliefs, ambiguity aversion is completely ordered by $q$; with different beliefs, we provide a necessary and sufficient condition on the two belief-threshold pairs. The model admits an equivalent multiple priors representation with a closed-form set of priors, nests subjective expected utility at $q = 1$, and converges to maxmin expected utility as $q \rightarrow 0$.

A foundational question in General Equilibrium theory—when is an equilibrium stable?—has remained largely unresolved for over 150 years despite important contributions. This paper proposes a resolution.

Choice alternatives are often multidimensional and risky. We introduce and axiomatize the structured multidimensional expected utility representation, a unified framework that generalizes existing approaches to evaluating such alternatives. The representation uses a rooted clustered tree to organize the joint, separate, and conditional evaluation of risk across dimensions within a common structure. We analyze the uniqueness of the representation and characterize useful special cases. We apply the representation to inequality across individuals, groups, and generations and to multisource income, characterizing the implications of bracketing for stochastic dominance and the avoidance of multidimensional risk.

Representation theorems in decision theory establish that behavior satisfies certain axioms if and only if it can be rationalized by a well-defined objective. I argue that this “if and only if” structure provides a potentially useful foundation for label-free evaluation and regularization of LLMs and other AI systems. Axiom compliance can be checked from the model's own responses to synthetic choice problems, with no external labels or human feedback, and the penalties are readily computable. Because the axioms are necessary and sufficient, the resulting checks exhaust the implications of the relevant rationality standard for the elicited data: a model that passes cannot be rejected on rationality grounds by any further test of the same data. I discuss three instantiations: probabilistic coherence via a theorem of de Finetti, preference rationality via Afriat's theorem, and subjective expected utility via a theorem of Echenique and Saito (2015), each yielding a continuous penalty that is zero whenever behavior can be rationalized. Since coherence does not restrict which objective rationalizes behavior, these penalties complement rather than replace other evaluation and training signals.

Bergemann and Morris (2016) show that one information structure is more informative than another exactly when it induces a smaller set of Bayes correlated equilibrium outcomes in every game. We build the quantum analogue. An information structure becomes a family of density operators indexed by the payoff state, which the mediator observes. We show that obedience is equivalent to a Loewner domination between operators on one player's subsystem. The equilibrium set is then a nonempty compact spectrahedron computable by semidefinite programming, classical structures embed exactly, and under quantum individual sufficiency more information shrinks the equilibrium set in every game.

A common monetary standard can expand credit while weakening the alternative practices needed to recover from its failure and identify its cause. I study a growing credit economy in which safeguarded real settlements supply recovery capacity, matched observations for a reform decision, and apprenticeships in an alternative practice. The efficient reserve grows in absolute size but becomes a vanishing share of the economy. Protection based on a fixed population share therefore eventually fails, even along a path with expanding capacity and evidence. Voluntary cooperation provides only bounded readiness, while cheaper drills can preserve recovery without producing evidence. Apprenticeship and expected political survival reinforce one another, allowing exposed and protected continuations. I characterize protection indexed to the required service and the least-cost funding of replacement if the original institution is cancelled. The resulting arrangement combines verified procurement, training support when needed, and safe real collateral held by an outside custodian.

We study auctions of k identical objects to n bidders, each of whom wants at most one. The objects have a common but unknown value and the bidders receive private signals about this value. The discriminatory price auction and the uniform-price auction are compared in terms of how informative the resulting auction prices (not bids) are in conveying the true value to an outside observer/investor. Since both auctions have symmetric, monotone equilibria, the problem reduces to comparing the informativeness of the highest order statistic of a sample to the (k+1)st highest. We find sufficient conditions under which the highest order statistic is superior—in the sense of Lehmann—in this regard. The sufficient conditions involve the informativeness of high versus low signals and the ratio k/n of objects to bidders. These conditions are also qualitatively necessary.

Providers deploying imperfect AI must decide how much to keep humans engaged: engagement costs output but builds the fallback skill needed when it fails. A liability pledge whose expected cost falls in that skill restores single crossing on a type that is neither drawn nor embodied once in a good, but built, decaying and poachable. One engagement decision thus produces both the client-facing signal and the trajectory on which mobile workers sort. That conjunction is what is new. It yields a two-sided band of served work, a stakes threshold above which building beats free-riding, and a signal that never dies.

In the standard Downsian model, two office-seeking parties converge to the median voter. However alienated voters may abstain and turn out only for a party within their tolerance radius. For single-peaked voter distributions, convergence survives but relocates to a central voter, the median of the electorate that participates at the convergent platform. However single-peakedness of the voter distribution is an empirically contested assumption. So we first characterize pure-strategy equilibrium for any continuous voter distribution. For general distributions, pure-strategy equilibrium can fail to exist or be non-unique, and existence of equilibrium need not persist as the tolerance radius of the voters increases. In order to resolve these issues, we propose a fundamental object: centripetal structure for which there is a single anchor platform toward which competition always pulls. We show this structure produces convergence at equilibrium under alienation based abstention. Our main result concerns the emergence and persistence of this new structure as the tolerance radius increases. Even though equilibria for office-seeking parties themselves can vanish and reappear as the radius grows, once centripetal structure emerges, it persists as long as the midpoint voter is not alienated. Moreover, the centripetal structure always emerges, and this structure classifies equilibrium completely when parties are policy motivated.

Success under a demanding method does not show that its additional demands are worthwhile. I study this distinction in a model of intergenerational learning with hidden effort. Moderate effort is informative about the value of a less demanding method; high effort produces more successes but no information about that question. I first characterize efficient experimentation with a complete record. I then show that parents who underweight descendants' effort costs can induce earlier retirement from experimentation. On an explicit region of priors, a paternalistic equilibrium stops after one failed experiment, whereas an aligned equilibrium allows two. Both transmit the relevant record truthfully and withstand communication deviations: every sender already obtains its preferred full-record continuation. Nevertheless, paternalism lowers discounted material welfare, raises long-run effort and success, and leaves less informative public evidence. Under stronger paternalism, parents may instead conceal knowledge to induce experimentation. I characterize a pooling continuation, establish its local communication stability, and show that it can occur in a full equilibrium. The results distinguish the efficient limits to learning from the additional loss of evidence caused by disagreement over effort costs.

We consider a variant of the Assignment Game of Shapley and Shubik (1971), where agents do not observe the assignment or the surplus division of other matched pairs. We propose a set-valued solution concept (Self-Stabilizing Set) and characterize the largest such set. This leads to the formulation of the Stable Payoff Guarantee (SPG) set, which assumes that agents receive at least their payoff guarantees and that this is common knowledge. Our main result is that, for generic surplus matrices, the SPG set consists only of the efficient assignment. The associated payoff vectors are given by the smallest interval that contains the worker-optimal and firm-optimal stable payoffs.