Subgame Perfect Nash Equilibrium Research Articles

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.

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.

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

AbstractThis 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

AbstractWe 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

AbstractThis 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 ($$C$$ C ) or not to commit ($$NC$$ NC ) 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.

Too-big-to-fail in federations?

We consider jurisdictions of different population size that provide local public goods with positive spillovers. Matching grants can induce optimal expenditure levels, but the regions can exploit the rationale behind this system to induce bailouts. We formalize the too-big-to-fail result of Wildasin (1997) by proving that it exists in a subgame-perfect Nash equilibrium, in which the central government’s decisions are taken by regional representatives. Furthermore, our model contains the too-big-to-fail and too-small-to-fail outcomes as special cases, and we are the first to derive the conditions under which each result emerges.

Strategic rationing in Stackelberg games

This paper examines Stackelberg price-quantity competition with imperfectly substitutable products. Under general cost and demand conditions, we establish existence of a subgame-perfect Nash equilibrium and provide a full characterization of the set of subgame-perfect Nash equilibria. In each equilibrium, the leader rations part of its clientele. Assuming linear demand and cost functions, first- and second-mover advantages are shown to critically depend on the degree of spillover demand, i.e., how many of the unserved customers visit the follower. We identify the presence of a spillover demand paradox. The leader may be the bigger firm and make more profit, but only when a sufficiently large part of its unmet demand shifts to the follower.

Collective bargaining in a shrinking group game: The role of information and communication

Majority decisions often resolve situations that involve strategic considerations of several individuals. Although these individuals often have identical voting power, their strategic position may be different. We utilize the pirate game in which a pre-existing minimum winning coalition emerges from the subgame perfect Nash equilibrium. This opens the possibility of investigating how equilibrium information about being pivotal and communication in different group constellations affect the size of proposals of the different players. Our results provide evidence that equilibrium information considerably increases the pivotal players' position at the expense of the non-pivotal players. This effect goes far beyond mere coordination of a pivotal triplet if this group has no information about the equilibrium. Communication leads more towards an equal distribution amongst the communication participants in both the subgroup and the large group. Whole group communication, directly following communication within informed subgroups, diminishes the pivotal players’ advantage, although not completely.

Me, myself and I: A general theory of non-Markovian time-inconsistent stochastic control for sophisticated agents

We develop a theory for continuous-time non-Markovian stochastic control problems which are inherently time-inconsistent. Their distinguishing feature is that the classical Bellman optimality principle no longer holds. Our formulation is cast within the framework of a controlled non-Markovian forward stochastic differential equation, and a general objective functional setting. We adopt a game-theoretic approach to study such problems, meaning that we seek for sub-game perfect Nash equilibrium points. As a first novelty of this work, we introduce and motivate a refinement of the definition of equilibrium that allows us to establish a direct and rigorous proof of an extended dynamic programming principle, in the same spirit as in the classical theory. This in turn allows us to introduce a system consisting of an infinite family of backward stochastic differential equations analogous to the classical HJB equation. We prove that this system is fundamental, in the sense that its well-posedness is both necessary and sufficient to characterise the value function and equilibria. As a final step we provide an existence and uniqueness result. Some examples and extensions of our results are also presented.

Network Externalities and Downstream Collusion under Asymmetric Costs: A Note

This paper considers the collusive stability of downstream competition in a vertical market with network externalities and cost asymmetry. A dynamic collusion game is constructed, and backward induction is employed to solve the subgame perfect Nash equilibrium. We show that larger network externalities lead to less collusive incentive for an inefficient firm, while for an efficient firm, this depends on the efficiency gap. An increase in network externalities will destabilize the downstream collusion when the cost asymmetry is large and network externalities are relatively weak.