Perfect Equilibrium Research Articles

Cooperation in temporary partnerships

The literature on cooperation in infinitely repeated Prisoner's Dilemmas covers the extreme opposites of the matching spectrum: partners, a player's opponent never changes, and strangers, a player's opponent randomly changes in every period. Here, we extend the analysis to settings where the opponent changes, but not in every period. In these temporary partnerships, players can deter some deviations by directly sanctioning their partner. Hence, relaxing the extreme assumption of one-period matchings can support some cooperation also off equilibrium because a class of strategies emerges that are less extreme than the typical “grim” strategy. We establish conditions supporting full cooperation as a subgame perfect equilibrium under a social norm that complements direct sanctions with a cyclical community sanction. Though this strategy less effectively incentivizes cooperation, it more effectively incentivizes punishment after a deviation, hence, can be preferable to the grim strategy under certain conditions.

Stationary Markov Equilibrium Strategies in Asynchronous Stochastic Games: Existence and Computation

We study Asynchronous Dynamic games and show that in games with a finite state space and finite action sets, one can obtain the pure strategy Markov perfect equilibrium by using a simple backward induction method when the time period for the game is finite. The equilibrium strategies for games with an infinite horizon are then obtained as the point-wise limit of the equilibrium strategies of a sequence of finite horizon games, where the finite horizon games are truncated versions of the original game with successively longer time periods. We also show that if the game has a fixed K-period cycle, then there is a stationary Markov equilibrium. Using these results, we derive an algorithm to compute the equilibrium strategies. We test the algorithm in three experiments. The first is a two-player asynchronous game with three states and three actions. In the second experiment, we compute the equilibrium of a cybersecurity game in which there are two players, an attacker and a defender. In the third experiment, we compute the stationary equilibrium of a duopoly game with two firms that choose an output in alternate periods.

There is no known nonlinear Markov perfect equilibrium strategies for the infinite horizon linear quadratic differential game

We discuss published accounts of nonlinear equilibrium strategies in linear quadratic differential games. The main reference: Tsutsui-Mino 1990, is shown to be misleading.

Simulation of the Stackelberg–Hotelling Game

This work studies the Hotelling game with sequential choice of prices, that is, the Stackelberg–Hotelling (SHOT) game. The game is studied through numerical simulation, which provides the subgame perfect equilibrium solution not only in the unrestricted game but also in the game with reservation cost and with elastic demand. The simulation technique is tested first in the unconstrained game, where the analytical subgame perfect equilibrium solution was already known. Then, the numerical procedure is generalized to cope with the SHOT game with reservation cost and with elastic demand. These enriched formulations of the SHOT game have not been studied so far, so this article provides an exploratory study of them.

A sequentially variant Blotto game with one-sided and incomplete information

We develop a sequentially variant Blotto game with one-sided and incomplete information to investigate strategic interactions between a defender and an attacker whose target site values are unknown. The defender first allocates defensive resources before the attacker decides a probability distribution over which site to attack between the target sites. The attacker perfectly observes the defender’s resource allocations. The attacker’s type is continuous, following the attacker’s private values of victoriously attacking each site. We find the game’s essentially unique subgame perfect equilibrium. In equilibrium, the site the attacker attacks with a higher probability is the site with a lower expected loss for the defender when the defender defends both sites. We present numerical examples to examine (1) the impacts of the informational uncertainty concerning the attacker’s site values, (2) the impacts of the site values of the defender, (3) the impacts of the site values of the attacker, and (4) the impacts of the defender’s defense efficiency on the equilibrium behavior.

Single Crystalline GeSe Van Der Waals Ribbons With Uniform Layer Stacking, High Carrier Mobility, and Adjustable Edge Morphology.

Performance of the group IV monochalcogenide GeSe in solar cells, electronic, and optoelectronic devices is expected to improve when high-quality single crystalline material is used rather than polycrystalline films. Crystalline flakes represent an attractive alternative to bulk single crystals as their synthesis may be developed to be scalable, faster, and with higher overall yield. However, large - and especially large and thin - single crystal flakes are notoriously hard to synthesize. Here it is demonstrated that vapor-liquid-solid growth combined with direct lateral vapor-solid incorporation produces high-quality single crystalline GeSe ribbons with tens of micrometers size and controllable thickness. Electron microscopy shows that the ribbons exhibit perfect equilibrium (AB) van der Waals stacking order without extended defects across the entire thickness, in contrast to the conventional case of substrate-supported flakes where material is added via layer-by-layer nucleation and growth on the basal plane. Electrical measurements show anisotropic transport and a high Hall mobility of 85 cm2V-1s-1, on par with the best single crystals to date. Growth from mixed GeSe and SnSe vapors, finally, yields ribbons with unchanged structure and composition but with jagged edges, promising for applications that rely on ample chemically active edge sites, such as catalysis or photocatalysis.

The epistemic spirit of divinity

We study strategic reasoning in a signaling game where players initially have common belief in an outcome distribution and in the event that the sender's beliefs are independent of her type. We characterize the behavioral implications of these epistemic hypotheses by means of a rationalizability procedure with belief restrictions. Our solution concept is related to, but weaker than Divine Equilibrium (Banks and Sobel, 1987). First, we do not obtain a sequential equilibrium, but just a perfect Bayesian equilibrium with heterogeneous off-path beliefs (Fudenberg and He, 2018). Second, when we model how the receiver may rationalize a particular deviation, we take into account that some types could have preferred a different deviation, and we show this is natural and relevant via an economic example.

Service Competition in Two-Tier Healthcare Markets with Heterogeneous Patients Balking

Recently, public and private providers within the two-tier healthcare market engage in a fierce competition for patients, which is significantly influenced by patients’ multifaceted sensitivities to delays and payments, particularly when patients’ potential balking is considered. However, existing literature studying the two-tier healthcare market often presupposes patients as homogeneous or one-dimensionally heterogeneous, and neglects their balking behavior. In stark contrast, this paper delves into the combined effects of patients’ delay- and payment-sensitive heterogeneities, alongside their joining and balking decisions, on the service competition dynamics within the two-tier healthcare market. Leveraging a game-theoretic framework, we initially derive subgame equilibria, both inclusive and exclusive of patient balking, uncovering that patient balking can lead to a subgame perfect Nash equilibrium in the presence of two-dimensional patient heterogeneity. Intriguingly, our findings further suggest that the non-profit public provider may, counterintuitively, curtail its service capacity despite budgetary adequacy, contradicting the conventional assumption that such providers would fully allocate funds towards patient care. Ultimately, from a patient welfare standpoint, we find heightened payment heterogeneity discrimination proves detrimental to patients, whereas increased differentiation in delay heterogeneity benefits them. This revelation underscores a pressing social issue inherent in the two-tier healthcare market model: It has the potential to exacerbate healthcare disparities rooted in economic inequalities within the population.

Green Promotion Service Allocation and Information Sharing Strategy in a Dual-Channel Circumstance

Credit purchase enables the manufacturers in the e-commerce environment to provide pre-sales service that consumers can experience first and pay later. This paper considers demand associated with price and green promotion service level and builds four decentralized game models to study two green promotion service allocation strategies and demand forecasting information sharing strategies in a dual-channel environment. The effects of the degree of dual-channel competition and free-riding on the perfect Bayesian Nash equilibrium are studied. The results show that the retailer should actively cooperate with the manufacturer and share private forecasting information if the coefficient of channel substitution is relatively high. Sharing information will aggravate double marginalization and hurt the retailer. In addition, the retailer’s profit is positively influenced by the forecasting accuracy in four models. When the manufacturer invests in the green promotion service, the prediction accuracy hurts the manufacturer’s profit without information sharing and there is a positive impact with information sharing. In particular, when a retailer provides service, we take the consumer’s free-riding behavior into account, and we find that free-riding hurts both parties and the whole supply chain. In addition, the manufacturer’s profit is irrelevant to the prediction accuracy without information sharing and positively influenced by the accuracy with information sharing.

On fraud and certification of green production

AbstractWe present a novel model of fraud and certification in green production. We focus on settings where firms decide between a green or a standard version of a product together with an advertising strategy that can include fraud. In addition, green firms can choose to certify their production to guarantee the truthfulness of their claims. This results in four production‐advertising possibilities (standard, genuine green, fraudulent green, and certified green), by which we provide new insights about the prevalence of fraud and certification. We characterize the perfect Bayesian equilibrium of the resulting game for given green production costs, certification costs, and consumers' willingness to pay for standard production, and we perform comparative statics for the main parameters of the resulting game. We find that changes in certification and green production costs affect consumers' beliefs differently, whereby increases in certification and decreases in green production costs can broaden the likelihood of fraud. These novel results are robust to different market structures and question the general desirability of public subsidies for promoting green production without accompanying certification.

Equilibrium analysis in majority-based coalitional bargaining games

This paper introduces majority rule into coalitional bargaining games, adapting traditional models that rely on unanimous consent to more accurately mirror decision-making processes in real-world scenarios. We introduce a majority-based coalitional bargaining game (MBCBG), wherein coalitions pass proposals via majority votes. Our analysis of the stationary subgame perfect equilibrium (SSPE) not only establishes the necessary and sufficient conditions for SSPE strategy profiles but also confirms the existence of no-delay SSPEs in MBCBGs. We further delve into symmetric MBCBGs, delineating conditions that ensure equitable outcomes for homogeneous players. Furthermore, we provide a necessary and sufficient condition for the formation of the grand coalition under SSPEs. Additionally, we briefly explore how asymmetries in coalition values, proposal probabilities, and voting weights may influence both the dynamics of coalition formation and the expected equilibrium payoffs.

Intelligent Decision-Making Algorithm for UAV Swarm Confrontation Jamming: An M2AC-Based Approach

Unmanned aerial vehicle (UAV) swarm confrontation jamming offers a cost-effective and long-range countermeasure against hostile swarms. Intelligent decision-making is a key factor in ensuring its effectiveness. In response to the low-timeliness problem caused by linear programming in current algorithms, this paper proposes an intelligent decision-making algorithm for UAV swarm confrontation jamming based on the multi-agent actor–critic (M2AC) model. First, based on Markov games, an intelligent mathematical decision-making model is constructed to transform the confrontation jamming scenario into a symbolized mathematical problem. Second, the indicator function under this learning paradigm is designed by combining the actor–critic algorithm with Markov games. Finally, by employing a reinforcement learning algorithm with multithreaded parallel training–contrastive execution for solving the model, a Markov perfect equilibrium solution is obtained. The experimental results indicate that the algorithm based on M2AC can achieve faster training and decision-making speeds, while effectively obtaining a Markov perfect equilibrium solution. The training time is reduced to less than 50% compared to the baseline algorithm, with decision times maintained below 0.05 s across all simulation conditions. This helps alleviate the low-timeliness problem of UAV swarm confrontation jamming intelligent decision-making algorithms under highly dynamic real-time conditions, leading to more effective and efficient UAV swarm operations in various jamming and electronic warfare scenarios.

Interacting mechanisms: A perspective on generalized principal–agent problems

Myerson (1982) formalizes general principal–agent problems, in which agents have private information and choose actions. His contribution is best known for a version of the revelation principle in the case of a single principal but he also introduces a model of interacting principals. We push the latter forward by studying the perfect Bayesian equilibrium outcomes of the corporations’ game in which every principal proposes a mechanism to his agents. We show that several versions of the revelation principle hold in our framework and that, under certain conditions, every principals’ equilibrium, as defined in Myerson (1982), is a perfect Bayesian equilibrium outcome of the corporations’ game.

High-stakes failures of backward induction

We examine high-stakes strategic choice using more than 40 years of data from the American TV game show The Price Is Right. In every episode, contestants play the Showcase Showdown, a sequential game of perfect information for which the optimal strategy can be found through backward induction. We find that contestants systematically deviate from the subgame perfect Nash equilibrium. These departures from optimality are well explained by a modified agent quantal response model that allows for limited foresight. The results suggest that many contestants simplify the decision problem by adopting a myopic representation, and optimize their chances of beating the next contestant only. In line with learning, contestants' choices improve over the course of our sample period.

Fomenting conflict

I study information disclosure as a means to create conflict. A sender aims to keep two parties engaged in a war of attrition and reveals information about their relative strength. In the unique Markov Perfect Equilibrium, the sender employs “shifting rhetoric”: she alternates pipetting good and bad news about each party, so that neither appears too strong. Information designed to induce one party to continue fighting weakens the other party's incentives. This spillover effect may lead to early resolution and leave the sender worse off. With commitment, the sender provides delayed noisy disclosures. A partisan sender, who favors one party, instead provides information which leads to immediate resolution.

Returns from herding as a function of the social network of agents: a simulation study

PurposeThis study combines different perspectives on herding, viewing it as a social network heuristic in comparison to other heuristics. The purpose is to use the heuristic view of herding as found in early literature and test it on grounds of efficiency and payoff, in essence, combining the heuristic and rational agent view of herding. The simulated double auction setting includes agents embedded in a social network, allowing for an examination of herding alongside rational behaviour and imperfect signals.Design/methodology/approachIn each round of the simulation, levels of homophily, density and fractions of types of agents is set and agents are allowed to follow their respective heuristics under those conditions. Characteristics of the social network, such as the size, levels of different homophilies, density and fractions of different types of agents are varied randomly to gauge their effect on the performance of herders vis-à-vis others and the overall market efficiency through simulation based approach. The data used for the study has been developed in Python and linear models are estimated using R.FindingsHerding decreases total surplus in private value double auctions, but herders are not worse off than other agents and perform equally in common value auctions. Further, herders and random offerers reduce payoffs of other agents as well, and herding effects the surplus per transaction and not the quantum.Research limitations/implicationsThis study explores herding as a strategic behaviour coexisting with rationality and other strategies in specific circumstances. It presents intriguing findings on the impact of herding on individual outcomes and market efficiency, raising new avenues for future research. Implication to research includes a dent on the “sieve” argument of markets rooting out irrationality and from it, a policy implication that follows is the need for corrective measures as markets cannot self-correct this, given herders do not perform worse than others.Originality/valueThe study links the phenomenon of herding to the dynamics of social networks and heuristic-based learning mechanisms that sets apart this research from the majority of existing literature, which predominantly conceptualizes herding as an outcome derived from a perfect Bayesian Equilibrium and a rational learning process.

Voluntary Disclosure in Asymmetric Contests

Abstract This article studies the incentives for interim voluntary disclosure of verifiable information in probabilistic all-pay contests with two-sided incomplete information. Private information may concern marginal cost, valuations, and ability. Our main result says that, if the contest is uniformly asymmetric, then full revelation is the unique perfect Bayesian equilibrium outcome. This is so because the weakest type of the underdog reveals her type in an attempt to moderate the favourite, while the strongest type of the favourite tries to discourage the underdog—so that the contest unravels. This strong-form disclosure principle is robust with respect to correlation, partitional evidence, randomized disclosures, sequential moves, and continuous type spaces. Moreover, the assumption of uniform asymmetry is not needed when incomplete information is one-sided. However, the principle may break down when type distributions are too similar, contestants possess commitment power, or information is unverifiable. In fact, cheap talk will always be ignored, even if mediated by a trustworthy third party.

The Rise of the Superstars: Uncovering the Composition Effect of International Trade That Cements the Dominant Position of Big Businesses

International markets are extremely polarised, with a few big superstar businesses operating alongside numerous small competitors, and globalisation has been highlighted as a powerful force behind the superstars’ increasingly dominant presence. The empirical literature has established that superstars are more efficient compared to their smaller counterparts, and, unlike them, they exhibit strategic behaviour. Building on this evidence, we develop a model to examine how an initial productivity advantage allows a select few firms to expand, via innovation, to the extent that it becomes optimal to adopt strategic behaviour, and show how polarised markets emerge endogenously as the unique subgame perfect equilibrium in pure strategies. We then introduce international trade and show that, in polarised markets, trade liberalisation puts into motion a novel composition effect, reallocating market share from smaller to larger rivals and raising large firms’ profits. This effect suppresses the pro-competitive welfare gains from trade and cements the dominant position of big businesses, who come out as the big winners of globalisation. We find that, although trade increases welfare, by reducing average markup and markup heterogeneity, in the presence of a handful of large powerful firms, welfare gains are severely diminished, and subsidising smaller enterprises may turn out to be welfare-enhancing.

Efficient and Convergent Sequential Pseudo-Likelihood Estimation of Dynamic Discrete Games

Abstract We propose a new sequential Efficient Pseudo-Likelihood (k-EPL) estimator for dynamic discrete choice games of incomplete information. k-EPL considers the joint behaviour of multiple players simultaneously, as opposed to individual responses to other agents’ equilibrium play. This, in addition to reframing the problem from conditional choice probability (CCP) space to value function space, yields a computationally tractable, stable, and efficient estimator. We show that each iteration in the k-EPL sequence is consistent and asymptotically efficient, so the first-order asymptotic properties do not vary across iterations. Furthermore, we show the sequence achieves higher-order equivalence to the finite-sample maximum-likelihood estimator with iteration and that the sequence of estimators converges almost surely to the maximum-likelihood estimator at a nearly superlinear rate when the data are generated by any regular Markov perfect equilibrium, including equilibria that lead to inconsistency of other sequential estimators. When utility is linear in parameters, k-EPL iterations are computationally simple, only requiring that the researcher solve linear systems of equations to generate pseudo-regressors which are used in a static logit/probit regression. Monte Carlo simulations demonstrate the theoretical results and show k-EPL’s good performance in finite samples in both small- and large-scale games, even when the game admits spurious equilibria in addition to one that generated the data. We apply the estimator to analyse competition in the U.S. wholesale club industry.

Game theoretical analysis in two-echelon sustainable supply chains to manage and coordinate strategic decisions

Business firms and consumers are increasingly and rapidly realizing the benefits of adopting environmental responsibility. The business firms stress including it in their supply chain strategy and/or operations. This paper explores the environmental consciousness and quality attentiveness of consumers in a supply chain. Demand for the product depends on its price, eco-friendly level, and quality of the product. Three decentralized models are developed under three different game theoretical settings. A comparative study of the three game-theoretical models reveals several characteristics. It determines conditions under which channel members can decide the policy for the best possible benefit. The present work explores the effect of customer environmental awareness on the optimal decision. Production of eco-friendly items is also the prime consideration of our paper. It reveals that the manufacturer gets the most profit in the manufacturer Stackelberg game model, but the retailer gets the maximum profit from the retailer Stackelberg game model. Although channel members prefer their leading position to make the game theoretical decision-making process, the most decentralized channel profit is obtained from the Nash game model. Thus, consumers will also benefit, and the best channel profit can be obtained if the channel members agree to make decentralized decisions using the Nash game model. The centralized channel profit outperforms all three decentralized channel profits. The paper applies the concepts of sub-game perfect equilibrium and alternating offer bargaining to coordinate the channel and distribute surplus profit. Finally, analytical results are verified using numerical examples.