Perfect Equilibrium Research Articles

A Variant of the Logistic Quantal Response Equilibrium to Select a Perfect Equilibrium

A Variant of the Logistic Quantal Response Equilibrium to Select a Perfect Equilibrium

Dynamic Price Dispersion of Seasonal Goods in Bertrand–Edgeworth Competition

The primary distinguishing characteristics of various revenue management systems include fixed capacities, uniform products, and customers' sensitivity to pricing when making purchase decisions. Even with only two or three competitors, the competition remains intense. This study examines the sub-game perfect Nash equilibrium of price competition within an oligopoly market where goods are perishable assets. Each seller possesses a single unit of a non-replenishable good and competes by setting prices to sell their inventory over a finite sales period. Each period, buyers seek to purchase one unit of the good, and the number of buyers entering is random. All sellers' prices are visible to buyers, and there is no cost associated with searching for goods. Through stochastic dynamic programming methods, sellers' optimal responses can be determined by treating the competition as a one-time price-setting game, considering the remaining sales periods and the current demand pattern. Assuming a binary demand model, it is demonstrated that the duopoly model exhibits a unique Nash equilibrium. In addition, it is shown that the oligopoly model does not result in price dispersion based on a discussed metric. Moreover, when a generalized demand model is considered, the duopoly model features a unique Nash equilibrium with mixed strategies, whereas the oligopoly model exhibits a unique symmetric Nash equilibrium with mixed strategies.

Study on the Phase Behavior Simulation Method of High-Salinity Reservoirs.

The presence of salinity affects the accuracy of existing correlations used in the equation of state. Moreover, the variation of salinity is often ignored in the systematic analysis of the phase diagram, resulting in a large error in the final calculation result. It is obvious that the conventional phase equilibrium calculation is not applicable in a high-salinity reservoir. By introducing the hydrocarbon-brine binary interaction coefficient and α-function, combined with the definition of salinity, and considering the variation of salinity under different pressure and temperature conditions, a more perfect phase equilibrium calculation model was established. The complete phase diagram was drawn, and the calculation results of salinity distribution are obtained. The effect of the mole percentage of water and salt content on the phase behavior was simulated. Finally, the phase distribution simulation is carried out based on the measured data. The phase state and salinity variation law of a high-salinity reservoir are obtained. According to the fluid composition of different periods, the real phase state of the high-salinity reservoir can be monitored in real time. It can provide a theoretical basis for the gas reservoir development and the dynamic evaluation of gas storage injection and production with a hydrocarbon-brine two-phase system.

A Multi-Step Model for Pie Cutting with Random Offers

The problem of dividing a pie between two persons is considered. An arbitration procedure for dividing the pie is proposed, in which the arbitrator is a random number generator. In this procedure, the arbitrator makes an offer to the players at each step, and the players can either accept or reject the arbitrator’s offer. If there is no consensus, negotiations move on to the next step. At the same time, the arbitrator punishes the rejecting player by reducing the amount of the resource in favor of the consenting player. A subgame perfect equilibrium is found in the process.

Computing perfect stationary equilibria in stochastic games

The notion of stationary equilibrium is one of the most crucial solution concepts in stochastic games. However, a stochastic game can have multiple stationary equilibria, some of which may be unstable or counterintuitive. As a refinement of stationary equilibrium, we extend the concept of perfect equilibrium in strategic games to stochastic games and formulate the notion of perfect stationary equilibrium (PeSE). To further promote its applications, we develop a differentiable homotopy method to compute such an equilibrium. We incorporate vanishing logarithmic barrier terms into the payoff functions, thereby constituting a logarithmic-barrier stochastic game. As a result of this barrier game, we attain a continuously differentiable homotopy system. To reduce the number of variables in the homotopy system, we eliminate the Bellman equations through a replacement of variables and derive an equivalent system. We use the equivalent system to establish the existence of a smooth path, which starts from an arbitrary total mixed strategy profile and ends at a PeSE. Extensive numerical experiments, including relevant applications like dynamic oligopoly models and dynamic legislative voting, further affirm the effectiveness and efficiency of the method.

Green quality choice in a duopoly

AbstractThis article considers a quantity‐setting duopoly (Cournot rivalry) in which firms adopt an abatement technology as a device to improve the quality of products. Consumer preferences capture vertical product differentiation (quality) towards “green” products. This introduces a trade‐off on the production side, as firms that do not abate, in turn, do not sustain any abatement cost but the demand for their product is low. On the contrary, firms that choose to abate incur abatement costs, but the demand for their product is high. The article aims to study and understand whether this kind of preference may lead firms to strategically invest in green technology and introduces a new, private‐based (that contrasts the well‐known public‐based) mechanism through which pollution abatement can emerge as a sub‐game perfect Nash equilibrium (SPNE) of a non‐cooperative abatement decision game with product quality and complete information. The model is developed in a parsimonious way to pinpoint the main determinants of the endogenous market outcomes ranging from an anti‐prisoner's dilemma in which self‐interest and mutual benefit of non‐abatement do not conflict to an anti‐prisoner's dilemma in which self‐interest and mutual benefit of abatement do not conflict, passing through to an anti‐coordination scenario. Additionally, the welfare analysis reveals the existence of a win‐win solution from a societal perspective. The article shows that the results obtained in the Cournot setting also hold considering a Bertrand duopoly.

Uniformly strict equilibrium for repeated games with private monitoring and communication

AbstractCooperation through repetition is an important theme in game theory. In this regard, various celebrated “folk theorems” have been proposed for repeated games in increasingly more complex environments. There has, however, been insufficient attention paid to the robustness of a large set of equilibria that is needed for such folk theorems. Starting with perfect public equilibrium as our starting point, we study uniformly strict equilibria in repeated games with private monitoring and direct communication (cheap talk). We characterize the limit equilibrium payoff set and identify the conditions for the folk theorem to hold with uniformly strict equilibrium.

Competition among Pairwise Lottery Contests

We investigate a two-stage competitive model involving multiple contests. In this model, each contest designer chooses two participants from a pool of candidate contestants and determines the biases. Contestants strategically distribute their efforts across various contests within their budget. We first show the existence of a pure strategy Nash equilibrium (PNE) for the contestants, and propose a fully polynomial-time approximation scheme to compute an approximate PNE. In the scenario where designers simultaneously decide the participants and biases, the subgame perfect equilibrium (SPE) may not exist. Nonetheless, when designers' decisions are made in two substages, the existence of SPE is established. In the scenario where designers can hold multiple contests, we show that the SPE always exists under mild conditions and can be computed efficiently.

Pore-scale mushy layer modelling

Equations describing mushy systems, in which solid and liquid are described as a single continuum, have been extensively studied. Most studies of mushy layers have assumed them to be ‘ideal’, such that the liquid and solid were in perfect thermodynamic equilibrium. It has become possible to simulate flows of passive porous media at the pore scale, where liquid and solid are treated as separate continua. In this contribution, we study the simplest possible mushy layers at the pore scale, modelling a single straight cylindrical pore surrounded by a cylindrical annulus representing the solid matrix. Heat and solute can be exchanged at the solid–liquid boundary. We consider harmonic temperature and concentration perturbations and examine their transport rates due to advection and diffusion and the melting and solidification driven by this transport. We compare the results of this numerical model with those of a one-dimensional ideal mushy layer and with analytical solutions valid for ideal mushy layers for small temperature variations. We demonstrate that for small values of an appropriately defined Péclet number, the results of the pore-scale model agree well with ideal mushy layer theory for both transport rates and melting. As this Péclet number increases, the temperature and concentration profiles with radius within the pore differ significantly from constant, and the behaviour of the pore-scale model differs significantly from that of an ideal mushy layer. Some effects of mechanical dispersion arise naturally in our pore-scale model and are shown to be important at high Péclet number.

A Subsidization Scheme for Maximizing Social Welfare in Mobile Communications Markets

In contemporary mobile communications markets, various agents or players interact to pursue welfare. Regulatory policies enacted by governments in certain markets aim to maximize social welfare. However, some countries, both least developed and developing, often adopt successful models from developed nations without local market validation. Therefore, network economics serves as a pertinent framework for analyzing such policies. This paper introduces a novel scheme based on constrained optimization problems, where the constraints represent multilevel economic game equilibria within a system model involving three agents: the central planner, the mobile network operator, and the mobile data users. These agents strategically optimize their payoff functions by considering benefit factors and decision variables such as the subsidization factor, pricing, and data consumption. To this end, a three-stage dynamic game is proposed to model the players’ interactions, employing the backward induction method to ascertain the subgame perfect equilibrium from the Nash equilibrium. A case study is presented, demonstrating a 31.16% increase in social welfare between scenarios involving no adoption of the subsidization factor and its adoption at the optimal value when the central planner enacts it to other players in the game, even if they do not necessarily attain maximum payoff values. In countries aligning with this proposed model, social welfare is maximized through a subsidization scheme.

A paradox of coalition building in public good provision

Does partial coalition among players help mitigate the issue of underprovision of public goods? This study considers endogenous coalition formation and endogenous technology choices in a model of the private provision of global public goods. We show that the possibility of future interstate (partial) coordination may hinder the current adoption of better technology by a country outside the cooperation, which may exacerbate an existing underprovision problem. Specifically, in the subgame perfect equilibrium of a three-stage game, we find two paradoxical results: the inability to form future partial coalitions encourages a country outside the cooperation to adopt better technology, which could lead to an increase in the total public good supply and an improvement of global welfare. These results have important policy implications. For example, in the context of the Paris Agreement, if a major country announces reduced nationally determined contributions in a strategic move to adopt less advanced technology to motivate other countries to form coalitions, it may ultimately lead to a decrease in the overall supply of public goods and global welfare.

Strategic Partial Inattention in Oligopoly

Abstract This article examines firms’ choices on partial (in)attention prior to quantity competition. Being partially (in)attentive, a firm accounts for a part of its market impact. We find that regardless of the number of firms, there is always a unique subgame perfect Nash equilibrium where all firms choose to be partially attentive. The optimal attention level decreases in the number of firms, increases in product differentiation, and converges to zero, i.e. firms tend to be fully inattentive, as either the number of firms goes to infinity or product differentiation goes to zero.

Emergence of trust in the trust game under best experienced payoff dynamics

The subgame perfect equilibrium (SPE) of the trust game is Pareto inferior. In that equilibrium, player 2 will not honor player 1’s trust and hence, player 1 will not trust player 2. We consider a large population version of the trust game. We apply a population game dynamic called the ‘best experienced payoff’ dynamic to this game. We establish conditions under which the SPE outcome in the trust game is unstable under this dynamic. We also show that an outcome where a substantial fraction of agents in population 1 displaying trust and a substantial fraction of agents in population 2 honoring that trust can be globally stable.

The strategic jump-the order effect on winning “The Final Three” in long jump competitions

Abstract The tournament rules for long jump competitions have changed in recent years. Today, only the three athletes with the best jumps from the five initial attempts are qualified to make an additional sixth jump – a format called The Final Three. In the first implemented version of The Final Three, the top athletes sequentially make one final jump, starting with the athlete ranked third place from the initial attempts. The athlete with the longest jump in this sixth attempt wins the competition, irrespective of achieved results in previous attempts. In this study, we analyze the effect of the athletes’ jump order on the probability of winning the competition within this first implemented version of The Final Three. We derive the final’s symmetric subgame perfect equilibrium and compute the corresponding equilibrium winning probabilities, given estimated distributional parameters from the Olympic long jumping final in Tokyo 2021. The modeling of the game is preceded by a development of a stochastic model for the outcome in long jumping. Our results indicate a last mover advantage, albeit small. Our model also reveals the importance of having a low variation in the approach run length and thinking strategically in this tournament format.

Optimal investor life cycle decisions with time-inconsistent preferences

We examine the lifetime decisions of an investor with time-inconsistent preferences, obtaining optimal time-consistent strategies for investment, consumption, and life insurance premiums explicitly under a stationary Markov perfect equilibrium framework. We find that the investor increases consumption to seek immediate gratification, and simultaneously increases life insurance purchasing to fulfill a legacy need. However, at a later stage in the life cycle, the investor confronts slower wealth accumulation and reduces consumption and life insurance purchasing accordingly. We also investigate the effects of naivete, degree of time inconsistency, risk preferences, legacy weight, insurance cost, death duty, and labor income on the investor's decision making. Our study reveals the balance of consumption, bequest, and terminal wealth motives in life cycle decisions featuring microfounded bequest utility and stochastic labor income.

The co-determination decision game with consumption externalities

This work aims to investigate the effects of co-determination in a game-theoretic setting by considering network externalities in consumption. The received theoretical literature, so far focused only on standard (non-network) industries, showed that co-determination might emerge as the sub-game perfect Nash equilibrium (SPNE) of a non-cooperative Cournot duopoly, but this outcome generates a prisoner’s dilemma. The present research shows that (positive) network externalities may solve the prisoner’s dilemma and let co-determination become a Pareto-efficient SPNE with homogeneous or heterogeneous products. In a network industry, co-determination becomes a Pareto-superior institution for society, as firms, consumers and workers are better off than under profit maximisation. These results hold for exogenous (homogeneous and heterogeneous) co-determination and endogenous co-determination.

Fighting terrorism: How to position rapid response teams?

AbstractIn light of recent terrorist attacks, we introduce and study a Stackelberg game between a government and a terrorist. In this game, the government positions a number of heavily‐armed rapid response teams on a line segment (e.g., a long boulevard or shopping avenue) and then the terrorist attacks a location with the highest potential impact of an attack. This potential impact, which we call damage, is the product of the time it takes the closest rapid response team to react and the damage caused per time unit, which is modeled via a damage rate function. We prove that there exists a subgame perfect Nash equilibrium that balances the possible damage on all intervals of the line segment that result from positioning the rapid response teams. We discuss the implications for various types of damage rate functions including one mimicking a busy boulevard with various hotspot locations.

Playing Extensive Games with Learning of Opponent's Cognition.

Decision-making is a basic component of agents' (e.g., intelligent sensors) behaviors, in which one's cognition plays a crucial role in the process and outcome. Extensive games, a class of interactive decision-making scenarios, have been studied in diverse fields. Recently, a model of extensive games was proposed in which agent cognition of the structure of the underlying game and the quality of the game situations are encoded by artificial neural networks. This model refines the classic model of extensive games, and the corresponding equilibrium concept-cognitive perfect equilibrium (CPE)-differs from the classic subgame perfect equilibrium, since CPE takes agent cognition into consideration. However, this model neglects the consideration that game-playing processes are greatly affected by agents' cognition of their opponents. To this end, in this work, we go one step further by proposing a framework in which agents' cognition of their opponents is incorporated. A method is presented for evaluating opponents' cognition about the game being played, and thus, an algorithm designed for playing such games is analyzed. The resulting equilibrium concept is defined as adversarial cognition equilibrium (ACE). By means of a running example, we demonstrate that the ACE is more realistic than the CPE, since it involves learning about opponents' cognition. Further results are presented regarding the computational complexity, soundness, and completeness of the game-solving algorithm and the existence of the equilibrium solution. This model suggests the possibility of enhancing an agent's strategic ability by evaluating opponents' cognition.

Endogenous price discrimination with asymmetric firms

We explore a third-degree spatial price discrimination model within the context of asymmetric duopolies. We find that if the degree of asymmetry is small, both firms charge uniform price is the unique sub-game perfect Nash equilibrium. As the degree of asymmetry reaches intermediate, there exists no pure strategy sub-game perfect Nash equilibrium. In the mixed strategy equilibrium, both firms price discriminate with certain probability. When the degree of asymmetry is large, there exists one or two asymmetric equilibria: the advantaged firm price discriminating and the disadvantaged firm uniformly pricing IN and vice versa in the other NI. In both equilibria, the price-discriminating firm consistently benefits, and the uniform-pricing firm also experiences higher profits when the degree of asymmetry is large, despite being driven out of the smaller market. Moreover, when the asymmetry is large enough, social welfare can also be higher.

Compromise, Don’t Optimize: Generalizing Perfect Bayesian Equilibrium to Allow for Ambiguity

Compromise, Don’t Optimize: Generalizing Perfect Bayesian Equilibrium to Allow for Ambiguity