Osaka’s public high school admissions system uses a common score-based priority order and schools’ preferences, while restricting each student to reporting a single school. We show that the Osaka mechanism is neither priority-stable nor preference-stable, is not strategy-proof for students, and can conceal substantial instability in the admission records. Allowing students to report more schools weakly reduces underlying instability and the number of manipulable students, while making more of the remaining blocking pairs visible. Simulations show that tight reporting compresses quantitative differences across mechanisms. In the unstructured environment, the Osaka mechanism occupies an intermediate position between serial dictatorship and deferred acceptance, but this position disappears in the structured environment. The results show the benefits of broader reporting and the limits of Osaka’s compromise between score-based priority and school discretion.
Cooperative games with transferable utility are a standard framework for modeling collaboration, fair division, cost sharing, and coalition formation. Computing central solution concepts such as the Shapley value, the core, the least core, and the nucleolus is often computationally difficult because a general game contains exponentially many coalitions. Recent work on extremal dividend structures, such as sunflower-dividend games, shows that restricting the Harsanyi dividend support to a structured set family can restore substantial algorithmic tractability. This paper introduces Frankl–Wilson dividend games, also called L-intersecting dividend games, a class of cooperative games in which the nonzero non-singleton Harsanyi dividends are supported on a structured intersection family. We distinguish two precise cases. In the uniform ordinary case, the dividend support is an r-uniform L-intersecting family: every two distinct dividend coalitions have an intersection size belonging to a fixed set L. In the modular Frankl–Wilson case, the dividend support satisfies prescribed intersection congruences modulo a prime p, together with the admissibility condition that coalition sizes avoid the allowed intersection residues modulo p. The algorithmic tools used in this paper are classical: the Shapley value is computed from Harsanyi dividends, and coalition-excess separation for non-negative dividends is performed via the standard maximum-weight closure construction. The paper does not propose a new Shapley algorithm, a new maximum-flow algorithm, or a new linear-programming method. Its main contribution is structural: it identifies Frankl–Wilson admissible dividend supports as a cooperative-game condition under which these classical procedures receive a polynomially compact input and therefore become polynomial-time algorithms in the number of players when the intersection complexity is fixed. We prove several results. First, for a modular Frankl–Wilson dividend game with residue set A and s = ∣A∣, the non-singleton dividend support has size at most ∑i=0sni, which is polynomial in n for fixed s. In the uniform ordinary case with ∣L∣ = s, the support size is at most ns. Second, when the dividend support is explicitly given, the Shapley value is computable in time polynomial in n, the support size, and the input bit length. Third, for non-negative dividends, core and least-core separation can be performed through a maximum-weight closure construction whose graph size is polynomial in the dividend support. Hence, for fixed intersection complexity s, the least core is polynomial-time computable under explicit support. Fourth, we state a conditional nucleolus result. If the fixed constraints in the lexicographic scheme have a polynomial-size explicit representation and a polynomial-time constrained K-best closure oracle is available, then the nucleolus is polynomial-time computable. Without these additional assumptions, nucleolus computation for general Frankl–Wilson dividend games remains open. Fifth, if a game’s Harsanyi dividends can be approximated in dividend mass by an L-intersecting or Frankl–Wilson-admissible subfamily, then its Shapley value can be approximated with additive error bounded by the omitted dividend mass. We also show that total dividend approximation error gives explicit approximate least-core and approximate-core stability bounds. Frankl–Wilson dividend games are closely related to several extremal dividend structures. Sunflower active coalitions with common kernel size κ form a {κ}-intersecting family; when the active coalitions are uniform, they fit the Type I case, and under additional modular admissibility conditions they may also fit Type II. Pairwise disjoint dividend coalitions correspond to L = {0}. Finite-geometric designs, block designs, and modular intersection families provide natural examples. The paper therefore extends the research program of using extremal set theory to create cooperative-game classes with exact or certified approximate tractability.
Managerial belief bias and risk preference shape strategic decisions through different channels: beliefs affect perceived consumer valuations, whereas risk preference affects the valuation of profit volatility. We develop a differentiated-duopoly model in which firms choose quality before competing in prices. One manager may hold an optimistic or pessimistic assessment of consumers’ willingness to pay and exhibit risk-seeking, risk-neutral, or risk-averse preferences; the rival holds unbiased beliefs and is risk-neutral. Under simultaneous pricing, the biased firm’s price increases with its willingness-to-pay assessment, while the rival’s price may respond non-monotonically. Relative to the unbiased benchmark, either direction of nonzero belief bias raises the biased firm’s unit margin but reduces its actual expected demand. The rival benefits from every nonzero admissible bias. Both firms gain when the biased manager is risk-averse and the distortion is sufficiently small relative to the degree of risk aversion. Under risk neutrality or risk seeking, bias reduces the focal firm’s actual expected profit. Sequential-pricing extensions show that managerial characteristics can reverse leader–follower profit rankings. The price leader earns more than the follower over an intermediate range of risk aversion, while sufficiently large admissible belief distortions reverse the biased follower’s profit advantage under risk neutrality. These findings identify risk preference as a condition governing the profitability of biased beliefs and explain how managerial heterogeneity shapes price–quality decisions and competitive spillovers.
In this paper, we compare behavior in the dictator, ultimatum, and public goods games before and during the Gezi Park protests in laboratory experiments conducted in Türkiye. In May 2013, protests began against the proposed destruction of Gezi Park—one of the few remaining green spaces in Istanbul’s city center—for the construction of a shopping mall. Quickly spreading across Türkiye, many ordinary citizens participated in the protests in the following weeks. The collective action demonstrated in these protests is similar to the types of behavior economists study in games where individuals accept personal costs for the benefit of others. We find that participants in the protests were just as likely to choose the equal payoff option as it increased in value in the modified dictator game, punish uncooperative behavior in the ultimatum game, and reward cooperative behavior in the public goods game as those who participated in the experiments before the protests. This was true of the behavior of the nonparticipants in the protests in the ultimatum and public goods games. In contrast, nonparticipants in the protests were less willing to forgo the selfish payoff option for their coplayer’s benefit in the modified dictator game.
Bayesian games model interactions where players possess private information and have incomplete information about each other. Computing Bayesian Nash equilibria (BNE) in continuous Bayesian games remains challenging due to infinite-dimensional strategy spaces, multi-objective conditions, and intractable expected utilities. To address these computational bottlenecks, this paper proposes a novel learning framework. First, we reformulate the game’s equilibrium problem into a tractable single-objective minimization task by leveraging the Fischer–Burmeister function to combine all players’ Karush–Kuhn–Tucker (KKT) conditions into a global merit function. Second, we introduce a dual neural network architecture per player—coupling strategy and Lagrange networks—to effectively approximate continuous strategy spaces. Third, we integrate Monte Carlo sampling to tractably evaluate expected utilities during training. We theoretically establish the asymptotic convergence of the cost function under stochastic gradient descent (SGD) and show that, conditional on the trained networks achieving a sufficiently small empirical merit loss, the resulting strategy profiles constitute an ϵ-Bayesian Nash equilibrium (ϵ-BNE) under sufficiently large sample sizes. Experiments and theoretical analysis demonstrate the effectiveness of the proposed approach for continuous incomplete-information games with strongly concave utilities that are thrice continuously differentiable and have bounded first and second derivatives.
