Markov Perfect Equilibrium Research Articles

Social capital and the status externality

AbstractThis paper investigates how the presence of social capital affects the externality arising from status‐seeking preference as a parable for inefficient antagonistic behavior. It is assumed that the stock of social capital accumulates through the strategic interaction among a finite number of rational, infinitely‐lived individuals. Using a differential game, we show that there are two types of Markov perfect equilibrium strategy, of which one leads a society to zero social capital, while the other leads to the satiation level of social capital. When there is an unstable interior steady state, there is a threshold: with any initial stock of social capital above (below) that, society is able to build social capital (get stuck in a poverty trap of null social capital). In the latter case, the intervention of governments is called upon, because social welfare in the poverty trap is less than that in the social capital‐rich society.

Dynamic oligopoly pricing with reference-price effects

We analyze the strategic implications of consumers’ reference-price effects, either symmetric (for loss-neutral consumers) or asymmetric (for loss-averse consumers), in a differentiated oligopoly model where firms compete either in prices (à la Bertrand) or in quantities (à la Cournot) over an infinite time horizon. The dynamic game is specified in continuous time. The solution concept is Markov Perfect Equilibrium. We show how price dynamics in the presence of reference-price effects crucially depends on the nature of market competition. One of the main results of our analysis is that, with loss-averse consumers, there exists an interval of initial reference prices such that firms adopt the same constant-pricing strategy in both the Bertrand and the Cournot games, implying that the distinction between price and quantity competition has no impact on market conduct and performance.

Mergers and acquisitions with conditional and unconditional offers

This paper proposes a dynamic model for the process of industry consolidation by sequences of mergers and acquisitions that create synergy gains to merging firms and may impose positive or negative externalities on the remaining firms in the industry. We allow firms to make acquisition offers that are conditional and unconditional on acceptances of target firms, instead of the usual conditional offers. We show that this expansion of the offer set results in faster and more economically efficient industry consolidations, as acquirers do not have to trade-off surplus extraction and efficiency. The acquirer can make surplus maximizing merger offers conditional to acceptances, and simultaneously capture efficiency gains by making unconditional offers to the remaining firms. We characterize the Markov perfect equilibrium and show that equilibrium always exists and is Pareto efficient. Finally, to illustrate the model and our main findings, we show that the equilibrium is unique in industries with three firms, and we derive the closed-form solution for the equilibrium value and the merger dynamics.

Eternal Peace in the Tug-of-War?

The tug-of-war between single players is known to have a non-cooperative Markov-perfect equilibrium in which both players expend zero efforts and neither player drives the Markov process to one of the terminal states. We show that these peaceful outcomes vanish if the single players are replaced by teams with team members permanently assigned to the different Markov states and interacting pairwise in an all-pay auction. The reason for this phenomenon is that the members of the teams can externalize future effort costs while the single players cannot. Our analysis also highlights the impact of the discount factor on the expected trajectory of the tug-of-war, the dynamics of the expected efforts, and the degree of rent dissipation.

Optimal and equilibrium execution strategies with generalized price impact

This paper examines the execution problems of large traders with a generalized price impact. Constructing two related models in a discrete-time setting, we solve these problems by applying the backward induction method of dynamic programming. In the first problem, we formulate the expected utility maximization problem of a single large trader as a Markov decision process and derive an optimal execution strategy. Then, in the second model, we formulate the expected utility maximization problem of two large traders as a Markov game and derive an equilibrium execution strategy at a Markov perfect equilibrium. Both of these two models enable us to investigate how the execution strategies and trade performances of a large trader are affected by the existence of other traders. Moreover, we find that these optimal and equilibrium execution strategies become deterministic when the total execution volumes of non-large traders are deterministic. We also show, by some numerical examples, the comparative statics results with respect to several problem parameters.

Dynamic Durable Goods Monopoly and Market Power

We analyze a vertically differentiated market for an imperfectly durable good served by a monopolist in an infinite-horizon, discrete-time game. Our goal is to identify the Markov perfect stationary equilibria where the seller can maintain his monopoly power. We establish that the set of parameters supporting a monopoly outcome is larger when the seller offers different quality versions of the same product. Hence, our results suggest that, when the innate durability of a product is high, the seller should offer different quality versions of the product.

Commitment and (in) Efficiency: A Bargaining Experiment

ABSTRACT In many matching markets, bargaining determines who matches with whom and on what terms. We experimentally investigate allocative efficiency and how subjects’ payoffs depend on their matching opportunities in such markets. We consider three simple markets. There are no information asymmetries, subjects are patient, and a perfectly equitable outcome is both feasible and efficient. Efficient perfect equilibria of the corresponding bargaining game exist, but are increasingly complicated to sustain across the three markets. Consistent with the predictions of simple (Markov perfect) equilibria, we find considerable mismatch in two of the markets. Mismatch is reduced but remains substantial when we change the nature of bargaining by moving from a structured experimental protocol to permitting free-form negotiations, and when we allow players to renege on their agreements. Our results suggest that mismatch is driven by players correctly anticipating that they might lose their strong bargaining positions, and by players in weak bargaining positions demanding equitable payoffs.

Vertical differentiation, product innovation, and dynamic competition

AbstractThis paper presents an infinite horizon dynamic model in which two firms compete in a market vertically differentiated by the qualities of their products and consumers have heterogeneous preferences for quality. Given the product qualities offered, the firms engage in price competition that segments the market. In each period each firm can spend on product innovation that if successful increases the quality of its product. Three types of Markov perfect equilibria are identified. A running–coasting equilibrium exhibits increasing quality dominance with one firm undertaking innovation and the other coasting to free ride on the innovation by the first firm. The firm that coasts can have the larger dynamic payoff, so quality dominance does not imply payoff dominance. A second is a leap‐frog equilibrium in which the trailing firm undertakes innovation to leap into the lead. The trailing firm never innovates solely to narrow the gap with the leader, so catch up strategies are never used. In the third both firms undertake innovation, but if both have innovation successes, product differentiation remains the same and profits are reduced by the cost of innovation. The rivalry between Intel and AMD in microprocessors for personal computers provides a motivating example.

Quality and price personalization under customer recognition: A dynamic monopoly model with contrasting equilibria

We present a model of market hyper-segmentation, where a monopolist acquires within a short time all information about the preferences of consumers who purchase its vertically differentiated products. The firm offers a new price/quality schedule after each commitment period. Lower consumer types may have an incentive to delay their purchases until next period to obtain a better introductory offer. The monopolist counters this incentive by offering higher informational rents. Considering the dynamic game played by the monopolist and its customers, we find that there is always a Markov perfect equilibrium (MPE) in which the firm immediately sells the good to all customers, offering the Mussa–Rosen static equilibrium schedule to first time customers (and getting full commitment profits). However, if the commitment period between two offers is long enough, there is another MPE with gradual market expansion. Contrary to the Coasian result for a durable-good monopoly, we find that in both equilibria the profit of the monopolist increases (and the aggregate consumers surplus decreases) as the interval of commitment shrinks. The model yields policy implications for regulations on collection and storage of customers information.

Oligopolistic equilibrium and financial constraints

AbstractWe model a dynamic duopoly in which firms can potentially drive their rivals from the market. For some parameter values, the Cournot equilibrium outcome cannot be sustained in an infinitely repeated setting. In those cases, there is a Markov perfect equilibrium in mixed strategies in which one firm, eventually, will exit the market with probability one. Producer surplus in the maximum collusive outcome is greater under bankruptcy consideration, because the outcome that maximizes joint profits is skewed in favor of the more efficient firm. Consumer surplus and social welfare also increase in many cases, although those effects are generally ambiguous.

Spatial dynamic models with intertemporal optimization: Specification and estimation

In this paper, we introduce a dynamic spatial interaction econometric model. There are n forward-looking agents of whom each has a parametric linear-quadratic payoff, and they interact with their neighbors through a spatial network. Considering a Markov perfect equilibrium (MPE), we derive a unique Nash equilibrium equation and construct a new spatial dynamic panel data (SDPD) model. For estimation, we suggest mainly the quasi-maximum likelihood (QML) method. Asymptotic properties of the QML estimator are investigated. In a Monte Carlo study, we estimate the model’s parameters and compare the results with those from traditional SDPD models. The model is applied to an empirical study on counties’ public safety spending in North Carolina.

POLITICAL CONFLICT AND BARGAINING IN A NEW KEYNESIAN MODEL OF FISCAL STABILIZATION

Recent work on optimal monetary and fiscal policy in New Keynesian models has tended to focus on policy set by an infinitely lived benevolent policy maker, often with access to a commitment technology. In this paper, we explore deviations from this ideal, by allowing (time-consistent) policy to be set by a process of bargaining between two political players with different weights on elements of the social welfare function. We characterize the (linear) Markov perfect equilibrium and, in a series of numerical examples, we explore the resultant policy response to shocks which cannot be perfectly offset with the available instruments due to their fiscal consequences. We find that, even although the players, on average, have the socially desirable objective function, the process of bargaining implies an outcome which deviates from the time-consistent policy chosen by the benevolent policy maker. Moreover, the range of instruments available mean that policy makers will bargain across the entire policy mix, sometimes implying outcomes which are quite different from those that would be chosen by a single policy maker. These policy outcomes depend crucially on the nature of the conflict and also the level of government debt.

Tullock Brings Perseverance and Suspense to Tug-of-War

We model the dynamic contest between two players as a game of tug-of-war with a Tullock contest success function (CSF). We show that (pure strategy) Markov perfect equilibrium of this game exists, and it is unique. In this equilibrium - in stark contrast to a model of tug-of-war with an all pay auction CSF - players exert positive efforts until the very last battle. Since the outcome of an individual battle is determined stochastically, even disadvantaged players who fell behind will occasionally win battles and hence the advantage likely change hands. We deliver a set of empirically appealing results on effort dynamics.

Vertically Differentiated Durable Goods Monopoly: Coexistence of Contrasting Markov-Perfect Equilibria

Vertically Differentiated Durable Goods Monopoly: Coexistence of Contrasting Markov-Perfect Equilibria

Equilibrium payoffs and proposal ratios in bargaining models

We analyze a bargaining model which is a generalization of the model of Rubinstein (Econometrica 50(1):97–109, 1982) from the viewpoint of the process of how a proposer is decided in each period. In our model, a player’s probability to be a proposer depends on the history of proposers and players divide a pie of size 1. We derive a subgame perfect equilibrium (SPE) and analyze how its SPE payoffs are related to the process. In the bilateral model, there is a unique SPE. In the n-player model, although SPE may not be unique, a Markov perfect equilibrium (MPE) similar to the SPE in the bilateral model exists. In the case where the discount factor is sufficiently large, if the ratio of opportunities to be a proposer converges to some value, players divide the pie according to the ratio of this convergent value under these equilibria. This result implies that although our process has less regularity than a Markov process, the same result as in the model that uses a Markov process holds. In addition to these results, we show that the limit of the SPE (or the MPE) payoffs coincides with the asymmetric Nash bargaining solution weighted by the convergent values of the ratio of the opportunities to be a proposer.

Competition in Pricing Algorithms

We document new facts about pricing technology using high-frequency data, and we examine the implications for competition. Some online retailers employ technology that allows for more frequent price changes and automated responses to price changes by rivals. Motivated by these facts, we consider a model in which firms can differ in pricing frequency and choose pricing algorithms that are a function of rivals' prices. In competitive (Markov perfect) equilibrium, the introduction of simple pricing algorithms can generate price dispersion, increase price levels, and exacerbate the price effects of mergers.

Valuations and dynamics of negotiations

AbstractThis paper analyzes three‐party negotiations in the presence of externalities. We obtain a closed‐form solution for the Markov perfect equilibrium of a multilateral non‐cooperative bargaining model, yielding an equilibrium value and dynamics of negotiations that are supported by experimental studies. Players’ values are monotonically increasing (or decreasing) in the amount of negative (or positive) externalities that they impose on others. Moreover, players’ values are continuous and piecewise linear on the worth of bilateral coalitions, and are inextricably related to their negotiation strategies: the equilibrium value is the Nash bargaining solution when no bilateral coalitions form; the Shapley value when all bilateral coalitions form; or the nucleolus, when either one bilateral coalition among “natural partners” or two bilateral coalitions including a “pivotal player” form.

Globalized robust Markov perfect equilibrium for discounted stochastic games and its application on intrusion detection in wireless sensor networks: Part I—theory

In this article, we study a discounted stochastic game to model resource optimal intrusion detection in wireless sensor networks. To address the problem of uncertainties in various network parameters, we propose a globalized robust game-theoretic framework for discounted robust stochastic games. A robust solution to the considered problem is an optimal point that is feasible for all realizations of data from a given uncertainty set. To allow a controlled violation of the constraints when the parameters move out of the uncertainty set, the concept of globalized robust framework comes into view. In this article, we formulate a globalized robust counterpart for the discounted stochastic game under consideration. With the help of globalized robust optimization, a concept of globalized robust Markov perfect equilibrium is introduced. The existence of such an equilibrium is shown for a discounted stochastic game when the number of actions of the players is finite. The contraction mapping theorem, Kakutani fixed point theorem and the concept of equicontinuity are used to prove the existence result. To compute a globalized robust Markov perfect equilibrium for the considered discounted stochastic game, a tractable representation of the proposed globalized robust counterpart is also provided. Using the derived tractable representation, we formulate a globalized robust intrusion detection system for wireless sensor networks.

Cheating as a dynamic marketing strategy in monopoly, cartel and duopoly

In 2015 it was discovered that Volkswagen had manipulated the exhaust emissions of its (diesel) cars. Since then, numerous other automotive car manufacturers were strongly suspected to violate against the same emission standards. This paper investigates how and why firms (monopoly, cartel and duopoly) engage in cheating, more precisely, promising attributes that are actually not part of the product. Firms make claims in order to better market their product but risk damaging their future reputation. The upshot of the paper is the stark difference between open loop and Markov perfect oligopolistic equilibrium outcomes. More precisely, the latter mitigates cheating substantially even below the levels attained by monopolies and cartels (unless consumers have a very short memory), which is contrary to the outcome in the limiting static version of the game. Therefore, revealing the true state (e.g., by mandating strict inspections) could force firms to use this information and play in Markov instead of open loop strategies.

Quality and Price Personalization under Customer Recognition: a Dynamic Monopoly Model

We present a model of market hyper-segmentation, where a monopolist acquires within a short time all information about the preferences of consumers who purchase its vertically differentiated products within a given period. The firrm offers a new price/quality schedule after each commitment period. Lower consumer types may have an incentive to delay their purchases until next period to obtain a better introductory offer to new customers. The monopolist counters this incentive by offering higher informational rents. Considering the dynamic game played by the monopolist and its customers, we find that there is always a Markov perfect equilibrium (MPE) in which the firm immediately sells the good to all its customers, offering the Mussa-Rosen static equilibrium schedule to first time customers (and getting full commitment profits). However, if the commitment period between two offers is long enough, there is another MPE, where the number of customers that are served expands only gradually. Contrary to the Coasian result for a durable-good monopoly, we find that in both equilibria the profit of the monopolist increases (and the consumers' expected surplus decreases) as the interval of commitment shrinks. The model yields policy implications for regulation on information collection and commitment period.