Turnbull (Econ Lett 13: 253–258, 1983) describes unique symmetric Nash equilibria for supply function competition under demand uncertainty which consist of affine strategies that are nonlinear. For the same economy Klemperer and Meyer (Econometrica 79: 1919–1966, 1989) derive unique Nash equilibria that are linear. This note shows that the friction between both results comes from a mathematical error in Turnbull (Econ Lett 13: 253–258, 1983). Correcting this error establishes the same linear Nash equilibria for both models.
Whether taking an optional pass-fail test is optimal depends on pass probabilities, which increase in productivity and are each individual's private information, and on the wages earned when passing or failing the test, which are conditional expectations formed about productivity by employers who know its distribution in the population subset that chooses to take the test. This paper models the relevant rational-expectations equilibrium and shows that plausible pass-probability functions imply that as a difficult test becomes more difficult, taking it becomes more appealing for low-productivity individuals who are likely to fail. Along with other features of real-life tests, this rationalizes the very low success rates of some optional admission and publication processes.
Waiting lists are used to allocate scarce resources when demand exceeds supply and pricing mechanisms are insufficient or inappropriate. We examine the interplay of strategic behavior of agents in waiting lists and supply-side interventions, such as improvements in resource quality and arrival rates. Using a dynamic assignment model and fixing the service discipline to First Come, First Served (FCFS), we show that such interventions can unintentionally increase waste and waiting time by making agents more selective. These are the first known formal analytical results on the effect of supply shocks on overall system performance in the presence of agent choice.
We consider an evidence game in which a principal seeks to award a high-quality agent. Agents, competing for the award, may hold evidence about their own and peers’ qualities (i.i.d. good or bad) as determined by a network. They can conceal or disclose each piece of evidence to the principal. Baumann and Dutta (2026) analyze the sequential equilibria where the principal has no commitment power and moves last. We show that with commitment power, the principal cannot achieve her first-best outcome. However, she can improve on the outcome of the adversarial disclosure equilibrium identified by Baumann and Dutta (2026), which is a sequentially stable outcome that is principal-preferred to the outcome of any equilibrium in iteratively interim undominated strategies.
In expected utility theory (EUT) risk preferences are entirely captured by the curvature of (Bernoulli) utility function. This note presents a minor one-parameter generalization of EUT that disentangles the two. The preference foundation (axiomatization) of the new model rests on completeness, transitivity, continuity, Thomsen condition, dominance, and one new axiom (a weaker version of the independence axiom of EUT).
We study global uniqueness of competitive equilibrium in two-good pure-exchange economies with heterogeneous patience types and a common HARA Bernoulli utility. The HARA utility is posed on its natural domain; after the affine shift X_i=x_i+h , Y_i=y_i+h , the consumer problem becomes a CRRA problem with shifted endowments, so the first-order demand used in the proof is genuinely interior. The main result is a global coefficient-ratio criterion: after a change of variables, market clearing can be written as r^θ =A(r)/B(r) , where A and B have positive coefficients built from elementary symmetric polynomials in the patience parameters. If the associated coefficient ratios form a monotone chain, then A(r)/B(r) is nonincreasing while r^θ is strictly increasing, and the equilibrium price is unique. Sorting of patience and endowment composition—more patient agents holding weakly more of the first good and weakly less of the second—is an economically interpretable sufficient condition for this chain. In the sorted case, the result recovers the ordered-endowment CRRA benchmark of Geanakoplos and Walsh (2018) for every b≥ 0 . The autonomous contribution is the coefficient-ratio criterion itself: it is strictly more general than sorting, as shown by a non-sorted, non-collinear example satisfying the chain.
This paper investigates the compatibility of the properties of rectangularity and central symmetry for the set of probabilities in the alpha-Maxmin Expected Utility model. In this framework, rectangularity is the condition that ensures dynamic consistency. We show that when a set of probabilities is centrally symmetric, this requirement forces a strict trade-off: ambiguity, defined as non-singleton beliefs, must vanish either for the marginal probabilities over partitions or for the conditional probabilities given partition elements at each stage of the filtration.
This paper characterizes three new rules for allocating resources in bilateral conflicting claims problems. These rules incorporate lower and upper exclusion thresholds that determine whether an individual is excluded from initial gains or losses. After prioritizing individuals according to these thresholds, the first rule assigns awards proportionally to them. The other two rules use weighted proportional divisions to ensure equal treatment of gains and losses once an exclusion threshold is reached. By incorporating exclusion thresholds into bilateral claims problems, our framework extends the class of division rules for resource allocation in contexts where symmetry is not appropriate due to legal and policy considerations.
Lovo and Spaenjers (Econ Lett 156:142–144, 2017) proved that no trade occurs in all equilibria of a second-price auction with a secret reserve price and an entry cost, since no buyer types enter the auction. They assume buyers bid truthfully, which is the unique equilibrium bidding strategy with a positive reserve and at least three buyers. We show that this result does not encompass the case of exactly two buyers, where equilibria with entry and trade may exist when buyers follow non-truthful bidding strategies.
We study a multilateral bargaining game in which players can attempt short-lived commitments before bargaining. Existing work shows that simultaneous commitments under unanimity lead to inefficient delayed agreements due to players’ strategic over-commitments. We show that this inefficiency is not inherent in unanimity itself, but arises from a coordination failure caused by the timing of simultaneous commitments. We introduce a sequential commitment protocol: Players are randomly ordered to make their commitment attempts after observing prior stochastic commitments, modeling real-time credibility updating during negotiations. Sequential commitment yields efficient and immediate agreements in the unique stationary Markov-perfect equilibrium: The first successful committer effectively becomes the proposer, while subsequent players do not attempt to demand more than their continuation payoffs. The result is robust to heterogeneity in players’ time preferences, commitment powers, and their likelihood to be the proposer. Our findings highlight how inefficiency under unanimity arises from timing frictions—not the rule itself—and demonstrate that structured commitment can promote efficient cooperation.
Click-fraud is commonly seen as a key vulnerability of pro-rata revenue sharing rule on music streaming platforms, whereas user-centric is largely immune. This paper develops a tractable non-cooperative model in which artists can purchase fraud activity that generates undetectable fake streams up to a technological limit. We defend pro-rata by showing that it is fraud-robust: when fraud technology is weak, honesty is a strictly dominant strategy, and an efficient fraud-free equilibrium obtains; when fraud technology is strong, a unique fraud equilibrium arises, yet aggregate fake streams remain bounded. Although fraud is inefficient, the resulting redistribution may improve fairness in some cases. To mitigate fraud without abandoning pro-rata, we introduce a parametric weighted rule that interpolates between pro-rata and user-centric, and characterize parameter ranges that restore a fraud-free equilibrium under technology constraint. We also discuss implications of Spotify’s modernized royalty system for fraud incentives.
This paper develops a prior-free model of data-driven decision making in which the decision maker observes the entire distribution of signals generated by a known experiment under an unknown distribution of the state variable and evaluates actions according to their worst-case payoff over the set of state distributions consistent with that observation. We propose a ranking of experiments in which E is robustly more informative than E' if the value of the decision maker’s problem after observing E is always at least as high as the value of the decision maker’s problem after observing E'. This comparison, which is strictly weaker than Blackwell’s classical order, holds if and only if the null space of E is contained in the null space of E'.
This paper studies sequential second-price auctions with heterogeneous goods and both global and local bidders. We show that selling order affects not only revenue and efficiency, but also the incidence of bidder losses. As the variance of one good vanishes, selling it first yields (weakly) higher revenue with probability approaching one, while selling it second yields (weakly) higher efficiency. Specifically, as the variance shrinks, the only order-dependent configuration that generates a revenue–welfare trade-off and survives with positive probability is the one in which the global bidder wins the low-variance good first but then loses the high-variance good, eliminating synergy and generating an ex-post loss. This exposure raises revenue but lowers welfare relative to the alternative order. By contrast, selling the high-variance good first makes this loss-making configuration unlikely and yields efficient allocations with (weakly) lower revenue. Simulations show that such losses occur with non-negligible probability, reinforcing the practical relevance. The results uncover a novel trade-off and identify selling order as a design instrument in sequential auctions.
If a complete-markets economy is composed of expected utility maximizers with heterogeneous beliefs and logarithmic utility, then it may be well represented via a single expected utility-maximizing agent with a “consensus belief”. For more general preferences, an “aggregation bias” necessitates alteration of the economy’s fundamentals if such a representation is to be constructed. But no such aggregation bias arises if the representative agent is allowed to be ambiguity averse, in the sense that he maximizes his Choquet expected utility under a convex “consensus capacity”. The economy thus becomes ambiguity averse in the aggregate as a consequence of heterogeneity in beliefs.
The paper analyzes unbeatable strategies in a Cournot duopoly model with differentiated goods. We conduct an extensive comparative analysis of Cournot-Nash equilibrium strategies and unbeatable ones in this framework. This article focuses on an asymmetric model, extending the results previously obtained in the symmetric case.
This paper revisits the n-player rent-seeking contest with homogeneous valuations and increasing returns. Our main result says that, for any m∈{2,… ,n-1} , there are threshold values 1
The cooperative-competitive value (the coco value) of Kalai and Kalai (Q J Econ 128(2):917–966, 2013) coincides with the payoffs of every Nash equilibrium in 2-person zero-sum games, and with the payoffs of every efficient equilibrium in 2-person common-interest games. This note presents several generalizations of these facts.
This paper explores market distortions in a monopolistic competition framework with multiple heterogeneous sectors and endogenous labor supply. We show that the aggregate labor supply in the market equilibrium is lower than its socially optimal level. We also find that, if the labor elasticity is relatively high, then all the sectors in the economy are underemployed in the market equilibrium compared to the social optimum. For sufficiently low values of the labor elasticity, some sectors with a relatively high elasticity of substitution between varieties appear to be overemployed, while the other sectors are underemployed. We then provide a quantitative illustration for the welfare losses caused by the inefficient labor supply and find that these losses can be substantial.
In this note we extend the result of von Stengel and Koller (Games Econ Behav 21:309–321, 1997) to infinite games. Specifically, we show that every infinite zero-sum game between a team and an adversary admits a pure-strategy team-maxmin equilibrium when (i) the strategy sets are compact, convex subsets of (possibly infinite-dimensional) topological vector spaces, and (ii) the payoff function is bounded, upper-semicontinuous on the team’s strategy-profile set and concave in each team member’s strategy, while being lower-semicontinuous and convex in the adversary’s strategy. Because the vector spaces may have arbitrary dimension, we obtain the following corollary: a mixed-strategy team-maxmin equilibrium exists whenever (i) the strategy sets are compact subsets of metric spaces, and (ii) the payoff function is bounded, measurable, upper-semicontinuous on the team’s strategy-profile set, and lower-semicontinuous on the adversary’s strategy set. The proof employs Sion’s minimax theorem.
