We study a Hotelling-type model with income constrained consumers, deriving closed-form subgame perfect equilibria (SPE) locations and prices as functions of income levels and transportation costs. For most income levels, a unique symmetric equilibrium exists. When reservation prices are binding, SPE yields intermediate differentiation and lower prices, boosting consumer surplus. However, under severe income constraints, firms leave part of the market uncovered and behave like local monopolists. Further income reductions reduce consumer surplus. Thus, firms improve the welfare of lower- income consumers under moderate reservation price constraints but worsen it under extreme income restrictions.
In cooperative games with transferable utility (TU-games), a player’s contribution may vary depending on the coalition they join, reflecting different levels of synergy with other members. In this paper, we introduce a family of axioms referred to as sub-game order preservation axioms, which formalize the intuition that a player’s payoff should increase with the degree of synergy they exhibit within a coalition. We propose four distinct axioms, each of which captures a different interpretation of what constitutes synergy in the context of TU-games. We demonstrate that one of these variants is incompatible with the classical Efficiency axiom, thereby giving rise to an impossibility result. The remaining three axioms, when combined with Efficiency, lead to unique characterizations of three well-known solution concepts: the Shapley value, the Center of the imputation set (CIS), and the Equal allocation of non-separable contributions (ENSC) value, respectively.
In this paper, we develop a dynamic model of moral preferences in repeated public goods games on networks. We incorporate informational nudges and derive conditions for convergence of behavior and optimal targeting of policies. Importantly, the long-term impact of moral nudges on preferences and behavior depends on how preferences evolve. When preferences include a component of persistent intrinsic morality, the effect of a moral nudge dissipates over time. Conversely, if social preferences continuously evolve, influenced by recent periods and present experiences, the potential for long-lasting effects of behavioral interventions becomes more promising. We also discuss heterogeneous adaptation, which improves the model’s predictive power for laboratory public-goods experiments. With the framework constructed in this paper, outcomes in experimental, field and natural experiments can be evaluated to further understand what drives individual decision making.
We consider two-stage parallel contests with a finite set of agents, as well as a finite set of heterogeneous contests with varying winning prizes. In the first stage, each agent simultaneously chooses in which contest he wants to compete, and in the second stage, the agents who chose the same contest compete against each other in a contest for a single prize. We first assume a general Tullock contest success function with nonlinear cost functions, and then we present an algorithm that converges to equilibrium, namely, the algorithm organizes the allocation of homogeneous or heterogeneous agents of two types, each with a different cost of effort, among the contests until no agent wishes to change his current contest. Later, we assume the Tullock contest success function with linear cost functions, and we show that for homogeneous agents, our model and a potential game are equivalent, allowing us to explicitly characterize the subgame perfect equilibrium with pure strategies.
Many real-world negotiations are persistently delayed, yet imposing deadlines is costly because it induces inefficient separations. Do all deadlines in one-on-one market transactions need to be perfectly credible? To improve trade efficiency, I propose a mechanism that introduces an intermediate, imperfectly credible deadline to facilitate agreement. Using a canonical seller–buyer dynamic bargaining model with a credible deadline, I analytically characterize the optimal degree of credibility of such a deadline that maximizes trade efficiency, leading to early agreements without triggering separations. Under a non-zero risk of separation, the seller is tempted to discount the price to secure a payoff, while the buyer is more likely to accept at the intermediate deadline because pricing resembles ultimatum offers. A laboratory experiment provides direct evidence of the mechanism’s effects that extend beyond the theoretical benchmark.
This paper presents an evolutionary model of patriotism. The state interacts with a population of citizens who may differ in their levels of patriotism, reflecting how much they care about the state. Citizens’ patriotism is subject to evolutionary pressure in the long run. Under complete information, if the state’s payoff function exhibits strategic complementarity, patriotism can emerge through evolution. In the presence of incomplete information, however, patriotism fails to thrive. Despite this, if citizens can costlessly disclose their levels of patriotism to the state in a truthful but potentially imprecise manner, full information revelation can be achieved under certain monotonicity conditions—one of which can be derived from the supermodularity of the interaction between the state and citizens—thus allowing patriotism to evolve in a manner similar to the complete information case.
We conducted an online Dictator Game experiment (N = 1195) to test three hypotheses about the role of monetary incentives in prosocial behavior. First, we examined whether real incentives of £1 reduce the dispersion of responses compared to hypothetical ones. Surprisingly, we found the opposite: hypothetical responses were less dispersed, with choices clustering around the egalitarian split. This pattern held in a replication (N = 308) with higher stakes (£5), offering no support for the first hypothesis. Second, we tested whether real incentives—by involving actual monetary consequences—lead to more selfish decisions, as they are expected to reduce socially desirable responses. With £1 stakes, no significant differences emerged across conditions. However, when the stake was increased to £5, participants became more selfish under real incentives, supporting the second hypothesis only when the amount at stake is substantial. Third, we explored whether probabilistic payments trigger differential behavior. At low stakes, probabilistic incentives resembled real ones. But with higher stakes, real and probabilistic outcomes diverged, suggesting participants respond to expected value only when it is meaningful. Finally, in a separate study (N = 299), we found that many participants facing standard hypothetical-payment instructions still expected real payments. Only explicit phrasing stating that “unfortunately, the money is not real” alleviated this confusion. This result underscores the importance of precise wording in experimental design and potentially explains why hypothetical treatments do not yield dramatically different results compared to real-money treatments.
Processes of bargaining are studied in which the players reach interim agreements that serve as status quo points for further bargaining. This is modeled in Nash’s setup of bargaining problems, where the solution is a time parameterized path of interim agreements rather than a single point. We characterize path solutions for linear problems that satisfy the axioms of restarting and covariance, and show that if a Pareto efficient agreement is not reached immediately, then it is never reached in finite time. Adding the axioms of individual rationality, relevance, and monotonicity, we characterize the family of continuous Raiffa solutions and show that these solutions converge to a Pareto efficient agreement but never reach it in finite time. Finally, if a deadline is added to the bargaining problem, and the speed of bargaining is proportionally inverse to the deadline, then a Pareto efficient agreement is reached exactly at the deadline.
Dilmé (Internat J Game Theory 53:673 2024) shows that an assessment is Kreps-Wilson (1982) consistent if and only if it is the limit of a particularly simple kind of sequence of strictly mixed strategies—which Dilmé calls a “power sequence”—and its associated sequence of Bayes-rule induced beliefs. We use Farkas’ lemma to provide a brief and simple proof of Dilmé’s result.
We investigate implications of the consistency principle for the reordering problem, also known as the queueing problem with an initial queue. The consistency principle specifies how an allocation rule should respond when an agent leaves the problem. We introduce four different consistency properties for the reordering problem and characterize three allocations rules, the pairwise equal-splitting rule (Curiel et al. Eur J Operat Res 40:344-351, 1989), the maximum price rule and the minimum price rule. Balanced consistency requires that for each pair of agents i and j, the impact on agent i’s net utility when agent j leaves the initial queue and the agents behind her move forward by one position, should be equal to the impact on agent j’s net utility when agent i leaves the initial queue and the agents behind her move forward by one position. Balanced cost reduction requires that if an agent leaves the initial queue and the agents behind her move forward by one position, then the total net utilities of the remaining agents should be reduced by the amount equal to the net utility of the departing agent. Smallest-cost consistency (respectively, largest-cost consistency) requires that if an agent with the smallest (respectively, largest) unit waiting cost leaves the initial queue and the agents behind her move forward by one position, then the net utilities of the remaining agents should not be affected. We show that either balanced consistency or balanced cost reduction, together with the three basic properties of queue-efficiency, budget-balance and Pareto indifference, characterizes the pairwise equal-splitting rule. On the other hand, together with the three basic properties, smallest-cost consistency characterizes the maximum price rule and largest-cost consistency the minimum price rule.
Makino defined 2-pile divisor nim as a variation of 2-pile nim. In this paper, we generalize it to m-pile divisor nim and give a winning strategy. Moreover, we also show that the nim-value of a position of m-pile divisor nim is periodic.
In n-person games with compact strategy spaces and continuous payoff functions, the expected payoff functions of the mixed extension are continuous with respect to the product of the weak* topologies. We provide an elementary proof of this fact. The analysis reveals that the Hausdorff separation axiom, commonly imposed on the topology of players’ strategy spaces, is not needed. We also show that the definition of expected payoffs as an iterated integral does not depend on the ordering of the players.
This paper studies optimal communication under language constraints. The model features multiple Senders and a single Receiver, in which the only barrier to communication is a limitation imposed by language: Senders are identically informed, all players have aligned incentives, the state space is infinite, but the Senders’ message spaces are each finite. Thus, while players prefer full information revelation, their language does not allow for it. Efficient communication can always be sustained with a language in which the meaning of message profiles is convex, but the same does not hold for the meaning of individual messages.
This paper examines a Bertrand duopoly with homogeneous products and one-sided cost uncertainty. We characterize the undominated equilibria which yield the unique equilibrium payoffs. This uniqueness result could be applied to many important market situations. In particular, we apply our results to examine cost-reducing R D by an entrant firm that enters a monopolistic market. While significant cost-reducing investments can result in a low marginal cost of production, we show that it may be optimal for the entrant to induce some technological uncertainty by restraining its investment level within certain limits as a means to ease any potential intense competition resulting from such investments.
Characterizing and explicitly computing equilibria of undiscounted dynamic games have been a challenge for many years. In this paper, we study quitting games, which are stopping games where the terminal payoff does not depend on the stage of termination. We adapt the recursive approach of Abreu et al. (Econometrica 58(5):1041–1063, 1990) to characterize a certain subset of the set of subgame-perfect ε -equilibrium payoffs. Our approach is based on the novel representation of strategy profiles through absorption paths, which was developed in Ashkenazi-Golan et al. (Math Program 203(1–2):735–762, 2024), and our characterization focuses on absorption paths in which exactly one player randomizes between quitting and continuing at any point in time. Since quitting games form a special case of both stopping games and stochastic games, our approach may be useful in studying more general classes of these games.
The lower and upper exact core bounds describe the minimum and maximum individual payoffs within the core of a balanced game, respectively. Based on these bounds, we introduce lower and upper core bound reduced games and study the relation with other reduced games. We axiomatically characterize new solutions for balanced games as unique core bound consistent extensions of the classical two-player principles of unanimity, standardness, and constrained egalitarianism. We name these solutions core covers, equal gap values, and constrained equality values, respectively.
Perspective invariance in Nash bargaining means that it should not matter for the final agreement whether gains with respect to the disagreement payoffs or sacrifices with respect to the maximal payoffs are considered. In a recent paper Nalebuff (Manag Sci 67:577–593, 2021) discusses a version of Nash’s (Econometrica 18: 155– 162, 1950) contraction independence condition that should allow for perspective invariance but then shows that the Nash bargaining solution is not perspective invariant. In contrast, we propose an alternative version of Nash’s Contraction Independence condition that does allow for an extension of the Nash bargaining solution which is perspective invariant. We also study a condition of restricted monotonicity, resulting in, among other solutions, a perspective invariant version of the Kalai-Smorodinsky solution.
Whether every quasi-convex game has a population monotonic allocation scheme (PMAS) has been an open question for more than 30 years (Sprumont in Games Econ Behav 2:378–394, 1990). In this note, we provide an answer to this question negatively: namely, we exhibit a quasi-convex game that has no PMAS.
We extend the Agreement Theorem of Aumann (Ann Stat 4(6):1236–1239, 1976) in two key directions. First, we introduce Knightian uncertainty by modeling beliefs as sets of probability measures, allowing for ambiguity in agents’ posterior beliefs. Second, we relax the assumption that agents observe perfectly each other’s posterior probabilities, replacing it with the assumption that they perceive only certain properties of their posterior probability sets. Our main result establishes that if agents share at least one common prior, they can only have common knowledge that their posterior probabilities satisfy a given property if these properties are mutually compatible. Furthermore, we explore economic implications in the context of trade under asymmetric information, deriving a No Trade result under ambiguity and highlighting conditions under which trade may still occur.
In this paper, we consider a class of oligopolistic markets with interval-valued inverse demand function and cost functions. Under the noncooperative hypothesis, we first prove the existence of interval Cournot-Nash equilibria. Furthermore, we define the NTU interval cooperative equilibrium and TU interval cooperative equilibrium, and prove their existence theorems. Under the noncooperative and cooperative hypothesis, we define the interval hybrid equilibrium with a partition of coalitions, and prove the existence theorem. Finally, we provide some examples to illustrate the models and equilibrium concepts.
