Subgame Perfect Equilibrium In Pure Strategies Research Articles

Competition with exclusive contracts in vertically related markets: An equilibrium non-existence result

I study a model in which two upstream firms compete to supply a homogeneous input to two downstream firms selling differentiated products. Upstream firms offer exclusive, discriminatory, public, two-part tariff contracts to the downstream firms. I show that, under very general conditions, this game does not have a pure-strategy subgame-perfect equilibrium. The intuition is that variable parts in such an equilibrium would have to be pairwise-stable; however, with pairwise-stable variable parts, downstream competitive externalities are not internalized, implying that upstream firms can profitably deviate. I contrast this non-existence result with earlier papers that found equilibria in related models.

Simple collective equilibria in stopping games

At each moment in time, an alternative from a finite set is selected by a stochastic process. Players observe the selected alternative and sequentially cast a yes or a no vote. If the set of players casting a yes vote is decisive for the selected alternative, it is accepted and the game ends. Otherwise the next period begins. We refer to this class of problems as stopping games. Collective choice games, quitting games, and coalition formation games are particular examples. When the core of a stopping game is non-empty, a subgame perfect equilibrium in pure stationary strategies is shown to exist. But in general, even subgame perfect equilibria in mixed stationary strategies may not exist. We show that aggregate voting behavior can be summarized by a collective strategy. We insist on pure strategies, allow for simple forms of punishment, and provide a constructive proof to show that so-called two-step simple collective equilibria always exist. This implies the existence of a pure strategy subgame perfect equilibrium. We apply our approach to the case with three alternatives exhibiting a Condorcet cycle and to a model of redistributive politics.

Computing Equilibria of Dynamic Games

We develop a numerical method for computing all pure strategy subgame-perfect equilibrium values of dynamic strategic games with discrete states and actions. We define a monotone mapping that eliminates dominated strategies, and when applied iteratively, delivers an accurate approximation to the true equilibrium payoffs of the underlying game. Our algorithm has three parts. The first provides an outer approximation to equilibrium values, constructed so that any value outside of this approximation is not an equilibrium value. The second provides an inner approximation; any value contained within this approximation is an equilibrium value. Together, the two approximations deliver a practical check of approximation accuracy. The third part of our algorithm delivers sample equilibrium paths. To illustrate our method, we apply it to a dynamic oligopoly competition with endogenous production capacity.

Spatial equilibrium in a duopoly under asymmetric network externalities

Network externalities are present in a variety of applications of spatial and network economics. For example, in energy networks the consumers often are assumed to benefit from a large number of other households using the same type of energy. In general, different firms can be asymmetric in terms of their network externality effect. This paper addresses the role of asymmetric network externalities to the existence and nature of spatial equilibrium in a duopoly with a quadratic transport cost. The study is based on a discrete choice Hotelling-type model where non-cooperative suppliers decide on prices and locations. The model has a pure strategy subgame-perfect equilibrium with maximal differentiation for a wide range of the parameter values of the asymmetric (or symmetric) network externalities. The model is applied to study equilibrium in an energy market consisting of a non-renewable energy producer and a renewable energy producer.

Existence of free entry equilibrium in aggregative games with asymmetric agents

In this paper, we examine a free entry aggregative game where agents can be asymmetric. We show the existence of a pure strategy subgame perfect equilibrium of this game. The proof is a constructive one and therefore we provide a method to derive a subgame perfect equilibrium within a reasonable time.

FRACTAL GEOMETRY OF EQUILIBRIUM PAYOFFS IN DISCOUNTED SUPERGAMES

This paper examines the pure-strategy subgame-perfect equilibrium payoffs in discounted supergames with perfect monitoring. It is shown that the equilibrium payoffs can be identified as sub-self-affine sets or graph-directed iterated function systems. We propose a method to estimate the Hausdorff dimension of the equilibrium payoffs and relate it to the equilibrium paths and their graph presentation.

Choice of strategic variables under relative profit maximization in asymmetric oligopoly: non-equivalence of price strategy and quantity strategy

We consider choice of strategic variables under relative profit maximization in an asymmetric oligopoly with differentiated goods such that there are three firms, Firm 2 and 3 have the same cost function, but Firm 1 has a different cost function. We show that there are two pure strategy sub-game perfect equilibria. 1) All firms choose the outputs as their strategic variables, 2)Firm 2 and 3 choose the outputs as their strategic variables, and Firm 1 chooses the price as its strategic variable.

Innovation in a Generalized Timing Game

We examine innovation as a timing game with complete information and observable actions in which firms decide when to enter a market. We characterize all pure strategy subgame perfect equilibria for the two-player symmetric game. In particular, we describe all subgame perfect equilibria when both the leader's and the followers' payoff functions are multi-peaked, non-monotonic and discontinuous. We find that there are potentially multiple equilibria, which could involve: joint adoption by both firms, with and without rent equalization; and, alternatively, single-firm adoption with a second-mover advantage. Economic applications are discussed including process and product innovation and the timing of the sale of an asset.

Bargaining with revoking costs

A simple two stage bilateral bargaining game is analyzed. The players simultaneously demand shares of a unit size pie. If the demands add up to more than one, the players simultaneously choose whether to stick to their demand or accept the otherʼs offer. While both parties sticking to their offers leads to an impasse, accepting a lower share than the original demand is costly for each party. The set of pure strategy subgame perfect equilibria of the game is characterized for continuously differentiable payoff and cost functions, strictly increasing in the pie share and the amount conceded, respectively. Higher cost functions are shown to improve bargaining power. The limit equilibrium prediction of the model, as the cost functions are made arbitrarily high, selects a unique equilibrium in the Nash Demand Game that corresponds to a Proportional Bargaining Solution of Kalai (1977).

An Algorithm for Two Player Repeated Games with Perfect Monitoring

Consider repeated two-player games with perfect information and discounting. We provide an algorithm that computes the set of payoff pairs V* of all pure strategy subgame perfect equilibria with public randomization. The algorithm provides significant efficiency gains over the existing implementations of the algorithm from Abreu, Pearce and Stacchetti (1990). These efficiency gains arise from a better understanding of the manner in which extreme points of the equilibrium payoff set are generated. An important theoretical implication of our algorithm is that the set of extreme points E of V* is finite. Indeed, |E| ≤ 3|A|, where A is the set of action profiles of the stage game.

A strategic analysis of the war against transnational terrorism

We study a two stage game in which a transnational terrorist organization interacts with an arbitrary number of countries that may differ in their political or economic power, their military effectiveness, the benefit from cooperating against terrorism and the value they assign to damage. Only a subset of countries that emerges endogenously takes proactive measures to fight the terrorist, while all countries incur defensive expenditures to protect their soil. We characterize analytically the pure strategy subgame perfect equilibrium of the game and show how the equilibrium strategies depend on the key model parameters. We provide an algorithm to find the endogenous set of cooperating countries based on their benefit from cooperation and their political/economic power.

Global dynamics in repeated games with additively separable payoffs

This paper studies the global dynamics of a class of infinitely repeated two-player games in which the action space of each player is an interval, and the one-shot payoff of each player is additively separable in actions. We define an immediately reactive equilibrium (IRE) as a pure-strategy subgame perfect equilibrium such that each player's action is a stationary function of the opponent's last action. We completely characterize IREs and their dynamics in terms of certain indifference curves. Our results are used to show that in a prisoners' dilemma game with mixed strategies, gradual cooperation occurs when the players are sufficiently patient, and that in a certain duopoly game, kinked demand curves emerge naturally.

The Impact of Quick Response in Inventory-Based Competition

We propose an extension of the competitive newsvendor model to investigate the impact of quick response under competition. For this purpose, we consider two retailers that compete in terms of inventory: customers that face a stockout at their first-choice store will look for the product at the other store. Consequently, the total demand that each retailer faces depends on the competitor's inventory level. We allow for asymmetric reordering capabilities, and we are particularly interested in the case when one of the firms has a lower ordering cost but can only produce at the beginning of the selling season, whereas the second firm has higher costs but can replenish stock in a quick response manner, taking advantage of any incremental knowledge about demand (if it is available). We visualize this problem as the competition between a traditional make-to-stock retailer that builds up inventory before the season starts versus a retailer with a responsive supply chain that can react to early demand information. We provide conditions for this game to have a unique pure-strategy subgame-perfect equilibrium, which then allows us to perform numerical comparative statics. We confirm that quick response is more beneficial when demand uncertainty is higher or exhibits a higher correlation over time. We also find that the competitive advantage from quick response is larger when facing a slow response competitor, and interestingly, asymmetric competition can be desirable to both competitors.

Pure subgame-perfect equilibria in free transition games

We consider a class of stochastic games, where each state is identified with a player. At any moment during play, one of the players is called active. The active player can terminate the game, or he can announce any player, who then becomes the active player. There is a non-negative payoff for each player upon termination of the game, which depends only on the player who decided to terminate. We give a combinatorial proof of the existence of subgame-perfect equilibria in pure strategies for the games in our class.

Sequential, nonzero-sum “Blotto”: Allocating defensive resources prior to attack

The strategic allocation of resources across multiple fronts has long been studied in the context of Blotto games in which two players simultaneously select their allocations. However many allocation problems are sequential. For example, a state trying to defend against a terrorist attack generally allocates some or all of its resources before the attacker decides where to strike. This paper studies the allocation problem confronting a defender who must decide how to distribute limited resources across multiple sites before an attacker chooses where to strike. Unlike many Blotto games which only have very complicated mixed-strategy equilibria, the sequential, nonzero-sum “Blotto” game always has a very simple pure-strategy subgame perfect equilibrium. Further, the defender always plays the same pure strategy in any equilibrium, and the attacker's equilibrium response is generically unique and entails no mixing. The defender minmaxes the attacker in equilibrium even though the game is nonzero-sum, and the attacker strikes the site among its best replies that minimizes the defender's expected losses.

Price competition in markets with customer testing: the captive customer effect

We introduce product differentiation into the analysis of price competi- tion in markets where suppliers test customers in order to assess whether they will pay for received goods or services. We find that, if the degree of differentiation is suffi- ciently high, suppliers may improve the average probability that their clientele will pay by charging higher prices. This helps suppliers to sustain high prices in equilibrium. Moreover, endogenizing locations in product space, we demonstrate that the high price level can be implemented in a pure-strategy subgame-perfect equilibrium with a high degree of differentiation. This is in contrast to the original Hotelling model with linear travel costs where a pure-strategy subgame-perfect equilibrium fails to exist.

Price and Capacity Competition

We study the efficiency of oligopoly equilibria in a model where firms compete over capacities and prices. The motivating example is a communication network where service providers invest in capacities and then compete in prices. Our model economy corresponds to a two-stage game. First, firms (service providers) independently choose their capacity levels. Second, after the capacity levels are observed, they set prices. Given the capacities and prices, users (consumers) allocate their demands across the firms. We first establish the existence of pure strategy subgame perfect equilibria (oligopoly equilibria) and characterize the set of equilibria. These equilibria feature pure strategies along the equilibrium path, but off-the-equilibrium path they are supported by mixed strategies. We then investigate the efficiency properties of these equilibria, where efficiency is defined as the ratio of surplus in equilibrium relative to the first best. We show that efficiency in the worst oligopoly equilibria of this game can be arbitrarily low. However, if the best oligopoly equilibrium is selected (among multiple equilibria), the worst-case efficiency loss has a tight bound, approximately equal to 5/6 with 2 firms. This bound monotonically decreases towards zero when the number of firms increases. We also suggest a simple way of implementing the best oligopoly equilibrium. With two firms, this involves the lower-cost firm acting as a Stackelberg leader and choosing its capacity first. We show that in this Stackelberg game form, there exists a unique equilibrium corresponding to the best oligopoly equilibrium. We also show that an alternative game form where capacities and prices are chosen simultaneously always fails to have a pure strategy equilibrium. These results suggest that the timing of capacity and price choices in oligopolistic environments is important both for the existence of equilibrium and for the extent of efficiency losses in equilibrium.

Endogenous Voting Agendas

Existence of a "simple" pure strategy subgame perfect equilibrium is established in a model of endogenous agenda formation and sophisticated vot- ing; upper hemicontinuity of simple equilibrium outcomes is demonstrated; and connections to the set of undominated, or "core," alternatives are examined. In one dimension with single-peaked preferences, the simple equilibrium outcome is essentially unique and lies in the core, providing a game-theoretic foundation for the median voter theorem in terms of endogenous agenda setting. Existence of equilibrium relies on a general characterization of sophisticated voting outcomes in the presence of "majority-ties," rather than the standard tie-breaking convention in voting subgames in favor of the alternative proposed later. The model is illustrated in a three-agent distributive politics setting, and it is shown there that the standard tie-breaking convention leads to non-existence of equilibrium.

Capacity precommitment and price competition yield the Cournot outcome

We introduce a simple model of oligopolistic competition where firms first build capacity, and then, after observing the capacity decisions, choose a reservation price at which they are willing to supply their capacities. This model describes many markets more realistically than the model of Kreps and Scheinkman [Kreps, D., Scheinkman, J., 1983. Quantity precommitment and Bertrand competition yield Cournot outcomes. Bell J. Econ. 14, 326–337]. We show that in this new model every pure strategy equilibrium yields the Cournot outcome, and that the Cournot outcome can be sustained by a pure strategy subgame perfect equilibrium.

THE EXISTENCE OF EQUILIBRIUM IN A DIFFERENTIATED DUOPOLY WITH NETWORK EXTERNALITIES*

The existence of a pure-strategy subgame-perfect equilibrium in qualities and prices is investigated in a duopoly model of vertical differentiation where quality improvements require a quadratic variable cost and network externalities operate. We show that there exists a parameter region where the incentive to predate at the quality stage prevents firms from reaching a pure-strategy non-cooperative equilibrium with prices above marginal costs. If network externalities are sufficiently large, a Bertrand equilibrium with zero profits may arise, although the amount of product differentiation is strictly positive.