Subgame Perfect Nash Equilibrium Research Articles

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.

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.

Order or Collaborate? Manufacturers Utilize 3D-Printed Parts to Sustainably Facilitate Increased Product Variety

3D printing (3DP) has garnered significant attention from industries, prompting traditional manufacturers to adopt 3DP to sustainably facilitate increased product variety. Observing manufacturers’ two adoption strategies, ordering parts and collaboratively printing 3DP parts, in a real-world setting, we utilize a wholesale price contract and a Nash Bargaining contract to describe these two strategies and then develop a supply-chain model including a 3DP supplier (Supplier) and a traditional manufacturer (Manufacturer). Further, we employ backward induction to solve the subgame-perfect Nash equilibrium for the model to reveal differences between these two strategies and the impact of 3DP’s improved resource efficiency. According to equilibrium outcomes, analytical results show that first, as long as the unit cost of each 3DP part is not overly high and 3DP’s resource efficiency is not extremely low, the Manufacturer is willing to implement 3DP to increase product variety. Second, a rise in the resource efficiency can create a “win-win” scenario for the Manufacturer and the Supplier. Third, supply-chain collaboration can be achieved when the Manufacturer’s and the Supplier’s bargaining powers approach equality. Interestingly, a Nash bargaining contract can incentivize the manufacturer to substitute a base product with a variety of products, a change facilitated by an increase in the retail price of this base product. The managerial implication of this research is that enhanced resource efficiency can lead to less environmental pollution in the collaboration model by resulting in the sale of lower quantities of the base product, which would otherwise consume more resources and generate greater environmental pollution.

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.

Folk theorem under bankruptcy

In this paper, we examine Subgame Perfect Nash Equilibria (SPE) within an infinite horizon oligopolistic market, accounting for the possibility of firm bankruptcy. Central to our dynamic analysis is the concept of Bankruptcy-Free (BF) allocations. The consideration of bankruptcy introduces several distinctive features not encompassed by standard models of infinitely repeated games. Notably, when firms are sufficiently patient towards future outcomes, achieving certain collusive outcomes as equilibria may not be possible. We discern qualitatively different results depending on the number of firms in the market; in a duopoly, only specific BF allocations can be sustained as SPE, while in scenarios with more than two firms, allocations outside the BF set can also be sustained as SPE. In both instances, it is crucial that the payoffs are above the minimax BF payoff, a condition more restrictive than the standard minimax payoff advocated in Folk Theorems.

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.

On Construction of Subgame Perfect Nash Equilibria in Stackelberg Games

Identifying a Subgame Perfect Nash Equilibrium (SPNE) of a two-player Stackelberg game could be not a manageable task, especially when the players have a continuum of actions and the follower’s best reply correspondence is not single-valued. Aim of the paper is to investigate the issue of construction of SPNEs in Stackelberg games by exploiting perturbations of both the action sets and the payoff functions of the leader and the follower. To achieve the goal, since the limit of SPNEs of perturbed games is not necessarily an SPNE of the original game even for classical perturbations, we prove under nonrestrictive convergence conditions how to produce an SPNE starting from a sequence of SPNEs of general perturbed games. This result allows to describe a procedure to find SPNEs that can accommodate various types of perturbations. More precisely, under mild assumptions on the data of the original game, we show that a large class of perturbed games (including, for example, perturbation approaches relying on the Tikhonov and entropic regularizations or motivated by altruistic and antagonistic behaviors) satisfies the convergence conditions for constructing an SPNE. The specific SPNE selections associated to such a class, together with their possible behavioral interpretations, are discussed and an illustrative example is provided.

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.

Last Time Buys during Product Rollovers: Manufacturer & Supplier Equilibria

Durable goods customers expect long-term availability of spare parts to maintain their product. This puts a strain on supply chains. Part suppliers want to obsolete older generations of parts with declining demand and reassign resources to new, growing generations. Meanwhile, durable goods manufacturers that purchase and sell spare parts to customers want suppliers to keep producing old parts. When suppliers offer a “last time buy” opportunity for legacy parts in anticipation of their retirement, manufacturers push back. We study the strategic interaction and equilibria of a supplier and a manufacturer during the rollover between a legacy part and its successor in a durable good supply chain. Evidence from our interviews shows that manufacturers attempt to use the future business of the new part to delay a supplier’s last time buy of the old part. However, this tactic is often ineffective in practice. We propose a two-stage noncooperative game that reproduces the behaviors observed in interviews. Our analysis of this game distills the conditions under which a manufacturer can leverage the future business to delay a last time buy. There exist only six strategies that achieve this delay. We present a necessary and sufficient condition for these six strategies to be subgame perfect Nash equilibria. The condition reduces to a closed-form threshold on the new part volume. We conclude with practical insights that can guide managers during a rollover discussion.

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.

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.

Sequential licensing with several competing technologies

We assume a multistage oligopoly wherein a given number of innovators compete by selling their substitutive technologies. Each innovator sequentially and independently chooses how many licenses to sell, and subsequently, all licensees compete à la Cournot in the product market. We show that, in equilibrium, the total number of licensees grows exponentially with the number of innovators. In addition, this sequential outcome is also obtained as a subgame perfect Nash equilibrium in pure strategies of a game with endogenous timing. Interestingly, by extending the duopoly model of Badia et al. (Math Soc Sci 108:8–13, 2020) to the case of more than two innovators and exploring pure strategy equilibria instead of mixed strategy equilibria, we derive drastically different policy implications, in terms of patent regulations. Our results suggest that more competition in the upstream market (e.g., by relaxing patent protection against the appearance of similar technologies) tends to increase downstream competition and welfare instead of discouraging or delaying technology adoption. In addition, our analysis is extended to explore the strategic role of public investment in basic R &D.

Innovation Race, Politics, and Resource Misallocation

Purpose - This paper aims to explain how politics can affect the innovation race in the private sector. Ideally, politics can help the economy develop or grow faster. However, it can slow the economy by obstructing the innovative activities of firms. The main objective of this paper is to examine how politics generates inefficiency and cost through Nash competition for political gain.
 Design/Methodology/Approach - This paper launches a game-theoretic approach to find the impact that political competition can leave upon an innovation race and R&D investments. It is modelled that private firms are involved non-cooperatively in a dynamic game for innovation. The strategic choices of R&D investment represent a subgame-perfect Nash equilibrium.
 Findings - Before an election, political parties tend to announce a commitment to supporting innovative activities for the private sector. It is ideal that the commitment helps firms establish R&D investment schedules as early as possible, and their innovation contributes to improving social welfare. However, political over-competition raises the problems of inconsistency or inefficiency.
 Research Implications - Public supportive programs can be designed to inspire innovative activities. However, public supportive programs can obstruct innovative activities and hurt social welfare through the misallocation of public resources. This paper can contribute to the research area of R&D and trade.

Product Quality and Product Compatibility in Network Industries

Abstract Using an appropriate game-theoretic approach, this article develops a non-cooperative two-stage game in a Cournot duopolistic network industry in which firms strategically choose whether to produce compatible goods in the first decision-making stage. Quality differentiation affects the sub-game perfect Nash equilibrium (SPNE): (i) the network effect acts differently between low- and high-quality firms, depending on their compatibility choice; (ii) if the network externality is positive (resp. negative), to produce compatible (resp. incompatible) goods is the unique SPNE; however, this equilibrium configuration leads the high-quality firm to be worse off; (iii) there is room for a side payment from the high- to the low-quality firm to deviate toward incompatibility (resp. compatibility) under positive (resp. negative) network externality. This payment represents a Pareto improvement on the firm side but not from a societal perspective, as consumers would be worse off. The article also pinpoints the social welfare outcomes corresponding to the SPNE.

Corporate social responsibility and network externalities: a game-theoretic approach

This research revisits the pioneering work by Katz and Shapiro (Am Econom Rev 75:424–440, 1985) with network (consumption) externalities in a twofold way: first, it considers Corporate Socially Responsible (CSR), instead of profit-maximising, firms; second, it uses a game-theoretic approach and analyses the commitment decision game in which firms face the binary choice to credibly commit (C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C$$\\end{document}) or not to commit (NC\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$NC$$\\end{document}) themselves to an announced output level in the first decision-making stage. Competition at the market stage occurs à la Cournot. Results show a rich spectrum of sub-game perfect Nash equilibrium (SPNE) outcomes, ranging from the prisoner’s dilemma (self-interest and mutual benefit of output commitment conflict) to the anti-prisoner’s dilemma or deadlock (self-interest and mutual benefit of output commitment do not conflict), passing from the coordination to the anti-coordination game. These outcomes depend on the intensity of the social concern in the firm’s objective and the network size. The article also pinpoints the welfare outcomes corresponding to the SPNE and extends the analysis to a Stackelberg rivalry setting.

Multi‐narrow type obstacle avoidance algorithm for UAV swarm based on game theory

AbstractA Flocking obstacle avoidance algorithm based on the extensive game with perfect information is proposed for the blockage problem of UAV swarm in front of multi‐narrow type obstacles. The two UAVs closest to the target are selected as participants of the game, and the game tree is defined to determine the combination of the motion strategies of the two UAVs to obtain the payoff matrix. Determine the subgame perfect Nash equilibrium to get the optimal strategy, and give the UAVs different motion states respectively so as to ensure that the UAVs can successfully pass multi‐narrow type obstacles. Simulation results demonstrate that the proposed algorithm has a higher over‐hole rate in the case of the multi‐narrow type obstacle compared to the static game‐based Flocking obstacle avoidance algorithm.

Delay to Deal: Bargaining with Indivisibility and Round-Dependent Transfer

We examine a bargaining game in which players cannot make arbitrary offers. Instead, players alternately decide whether to accept or delay, and are rewarded with an indivisible portion and a perishable transfer that depends on the round. Our analysis demonstrates that when the initial transfer is large enough, the subgame perfect Nash equilibrium consists of a finite number of rounds of delay before an agreement is reached. The equilibrium delay is longer when the players are more patient, and when the transfer is initially higher and depreciates slower. Nevertheless, the game’s chaotic characteristic makes it arduous to forecast the exact number of delayed rounds or which player will make the ultimate decision. This game can be applied to many social scenarios, particularly those with exogenous costs.

Strategic product compatibility in network industries

This article considers a product’s compatibility as a strategic variable in a Cournot duopoly with network consumption externalities. It develops a non-cooperative compatibility decision game (CDG) in which firms choose whether to let products be (in)compatible. With costless compatibility, the unique (Pareto-efficient) sub-game perfect Nash equilibrium (SPNE) of the CDG is universal compatibility. With quasi-fixed compatibility costs, the SPNE depends on whether product compatibility is an endogenous (i.e., a profit-maximising) or exogenous (i.e., given by technical constraints) variable. In the case of endogenous compatibility, the SPNE can vary from one unique to multiple regimes of (in)compatibility, such as the anti-prisoner’s dilemma (deadlock), prisoner’s dilemma, and coordination game. In the case of exogenous compatibility, the SPNE can become an anti-coordination game as well. The article also discusses the welfare outcomes corresponding to the SPNE.

Equilibria and efficiency in a reinsurance market

We study equilibria in a reinsurance market with multiple reinsurers that are endowed with heterogeneous beliefs, where preferences are given by distortion risk measures, and pricing is done via Choquet integrals. We construct a model in the form of a sequential economic game, where the reinsurers have the first-mover advantage over the insurer, as in the Stackelberg setting. However, unlike the Stackelberg setting, which assumes a single monopolistic reinsurer on the supply side, our model accounts for strategic price competition between reinsurers. We argue that the notion of a Subgame Perfect Nash Equilibrium (SPNE) is the appropriate solution concept for analyzing equilibria in the reinsurance market, and we characterize SPNEs using a set of sufficient conditions. We then examine efficiency properties of the contracts induced by an SPNE, and show that these contracts result in Pareto-efficient allocations. Additionally, we show that under mild conditions, the insurer realizes a strict welfare gain, which addresses the concerns of Boonen and Ghossoub (2022) with the Stackelberg model and thereby ultimately reflects the benefit to the insurer of competition on the supply side. We illustrate this point with a numerical example.