Symmetric Markov Perfect Equilibrium Research Articles

Uncertainty-driven symmetry-breaking and stochastic stability in a generic differential game of lobbying

We study a 2-player stochastic differential game of lobbying. Players invest in lobbying activities to alter the legislation in her own benefit. The payoffs are quadratic and uncertainty is driven by a Wiener process. We consider the Nash symmetric game where players face the same cost and extract symmetric payoffs, and we solve for Markov Perfect Equilibria (MPE) in the class of affine functions. First, we prove a general sufficient (catching up) optimality condition for two-player stochastic games with uncertainty driven by Wiener processes. Second, we prove that the number and nature of MPE depend on the extent of uncertainty (i.e. the variance of the Wiener processes). In particular, we prove that while a symmetric MPE always exists, two asymmetric MPE emerge if and only if uncertainty is large enough. Third, we study the stochastic stability of all the equilibria. We notably find, that the state converges to a stationary invariant distribution under asymmetric MPE. Fourth, we study the implications for rent dissipation asymptotically and compare the outcomes of symmetric vs asymmetric MPE in this respect, ultimately enhancing again the role of uncertainty.

Perseverance and suspense in tug-of-war

We study a tug-of-war game between two players using the lottery contest success function (CSF) and a quadratic cost (of effort) function. We construct a pure strategy symmetric Markov perfect equilibrium of this game, show that it is unique, and provide closed-form solutions for equilibrium strategies and values. 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 in this equilibrium. We deliver a set of empirically appealing results on effort dynamics.

Avoiding Fields on Fire: Information Dissemination Policies for Environmentally Safe Crop-Residue Management

Agricultural open burning, i.e., the practice of burning crop residue in harvested fields to prepare land for sowing a new crop, is well-recognized as a significant contributor to CO2 and black-carbon emissions, and long-term climate change. Low-soil-tillage practices using a specific agricultural machine called Happy Seeder, which can sow the new seed without removing the previous crop residue, have emerged as the most effective and profitable alternative to open burning. However, given the limited number of Happy Seeders that the government can supply, and the fact that farmers incur a significant yield loss if they delay sowing the new crop, farmers are often unwilling to wait to be processed by the Happy Seeder and, instead, decide to burn their crop residue. We study how the government can use effective information-disclosure policies in the operation of Happy Seeders to minimize agricultural open burning. A Happy Seeder is assigned to process a group of farms in an arbitrary order. The government knows, but does not necessarily disclose, the schedule for the Happy Seeder at the start of the sowing season. Farmers incur a disutility per unit of time while waiting for the Happy Seeder due to the yield loss as a result of late sowing of the new crop. If the Happy Seeder processes a farm, then the farmer gains a positive utility. At the beginning of each period, each farmer decides whether to burn her crop residue or to wait, given the information provided by the government about the Happy Seeder's schedule. We propose a class of information-disclosure policies, which we refer to as dilatory policies, that provide no information to the farmers about the schedule until a pre-specified period and then reveal the entire schedule. By obtaining the unique symmetric Markov perfect equilibrium under any dilatory policy, we show that the use of an optimal dilatory policy can significantly lower the number of farms burnt compared to that under the full-disclosure and the no-disclosure policies. Using data from the rice-wheat crop system in northwestern India -- an area of the world with the highest prevalence of open burning -- we conduct a comprehensive case study and demonstrate that the optimal dilatory policy can reduce CO2 and black-carbon emissions by at least 14%.

Discretion rather than rules: Equilibrium uniqueness and forward guidance with inconsistent optimal plans

New Keynesian economies with active interest rate rules gain equilibrium determinacy from the central bank's incredible off-equilibrium-path promises (Cochrane, 2011). We suppose instead that the central bank sets interest rate paths and occasionally has the discretion to change them. With empirically-reasonable frequencies of central bank reoptimization, the monetary policy game has a unique symmetric Markov-perfect equilibrium wherein forward guidance influences current outcomes without displaying a forward guidance puzzle.

Undiscounted bandit games

We analyze undiscounted continuous-time games of strategic experimentation with two-armed bandits. The risky arm generates payoffs according to a Lévy process with an unknown average payoff per unit of time which nature draws from an arbitrary finite set. Observing all actions and realized payoffs, plus a free background signal, players use Markov strategies with the common posterior belief about the unknown parameter as the state variable. We show that the unique symmetric Markov perfect equilibrium can be computed in a simple closed form involving only the payoff of the safe arm, the expected current payoff of the risky arm, and the expected full-information payoff, given the current belief. In particular, the equilibrium does not depend on the precise specification of the payoff-generating processes.

Multi-Battle Contests, Finite Automata, and the Tug-of-War

This paper examines multi-battle contests whose extensive form can be represented in terms of a finite state machine. We start by showing that any contest that satisfies our assumptions decomposes into two phases, a principal phase (in which states cannot be revisited) and a concluding tie-breaking phase (in which all non-terminal states can be revisited). Degenerate cases are the finite-horizon contests on the one hand (e.g., the match race), and the tug-of-war on the other. Next, assuming a probabilistic technology in each battle, we show that any contest satisfying our assumptions, with either finite or infinite horizon, admits a unique symmetric and interior Markov perfect equilibrium. This entails, in particular, a complete characterization of the equilibrium in the tug-of-war. Finally, we explore, both analytically and numerically, the intricate problem of a contest designer that maximizes expected total effort. In the absence of a complexity constraint, the revenue-maximizing contest is always a match race, where the optimal length of the race increases as the technology of the component contest becomes more noisy. If, however, the complexity constraint is binding, then the optimal contest is typically (but not always) a tug-of-war.

Very Simple Markov-Perfect Industry Dynamics: Theory

This paper develops a simple model of firm entry, competition, and exit in oligopolistic markets. It features toughness of competition, sunk entry costs, and market-level demand and cost shocks, but assumes that firms' expected payoffs are identical when entry and survival decisions are made. We prove that this model has an essentially unique symmetric Markov-perfect equilibrium, and we provide an algorithm for its computation. Because this algorithm only requires finding the fixed points of a finite sequence of contraction mappings, it is guaranteed to converge quickly.

Very Simple Markov-Perfect Industry Dynamics: Theory

This paper develops a simple model of firm entry, competition, and exit in oligopolistic markets. It features toughness of competition, sunk entry costs, and market-level demand and cost shocks, but assumes that firms' expected payoffs are identical when entry and survival decisions are made. We prove that this model has an essentially unique symmetric Markov-perfect equilibrium, and we provide an algorithm for its computation. Because this algorithm only requires finding the fixed points of a finite sequence of contraction mappings, it is guaranteed to converge quickly.

Designing Fair Tiebreak Mechanisms: The Case of FIFA Penalty Shootouts

In the current FIFA penalty shootout mechanism, a coin toss decides which team will kick first. Empirical evidence suggests that the team taking the first kick has a higher probability to win a shootout. We design sequentially fair shootout mechanisms such that in all symmetric Markov-perfect equilibria each of the skill-balanced teams has exactly 50% chance to win whenever the score is tied at any round. Consistent with empirical evidence, we show that the current mechanism is not sequentially fair and characterize all sequentially fair mechanisms. Taking additional desirable properties into consideration, we propose and uniquely characterize a practical mechanism.

Shipwrecks and Treasure Hunters

We examine dynamic search as a game in which two rivals explore (an island) for a hidden prize of known value. In every period until its discovery, the players decide how much of the unsearched area to comb. If a player finds the prize alone he wins it and the game ends. Players have a per-period discount factor and costs proportional to the area they search. First, as a benchmark for efficiency, we solve the monopoly problem. Second, in the duopoly setting we show that if players are sufficiently impatient they can inefficiently over-search in the unique symmetric Markov perfect equilibrium (SMPE) – a result akin to the tragedy of the commons. On the other hand, if players are patient: the SMPE is unique except for possibly one point; and either over- or under-search can result. Finally, with patient players, several counter-intuitive results can arise: for example, players might be better off searching a larger island or looking for a less valuable prize.

Very Simple Markov-Perfect Industry Dynamics: Empirics

This paper develops an econometric model of firm entry, competition, and exit in oligopolistic markets. The model has an essentially unique symmetric Markov-perfect equilibrium, which can be computed very quickly. We show that its primitives are identified from market-level data on the number of active firms and demand shifters, and we implement a nested fixed point procedure for its estimation. Estimates from County Business Patterns data on U.S. local cinema markets point to tough local competition. Sunk costs make the industry's transition following a permanent demand shock last 10 to 15 years.

Penny auctions

This paper studies penny auctions, a novel auction format in which every bid increases the price by a small amount, but placing a bid is costly. Outcomes of real-life penny auctions are often surprising. Even when selling cash, the seller may obtain revenue that is much higher or lower than its nominal value, and losers in an auction sometimes pay much more than the winner. This paper characterizes all symmetric Markov-perfect equilibria of penny auctions and studies penny auctions’ properties. The results show that a high variance of outcomes is a natural property of the penny auction format and high revenues are inconsistent with rational risk-neutral participants.

Dynamic price dispersion in Bertrand–Edgeworth competition

This paper studies a dynamic oligopoly model of price competition under demand uncertainty. Sellers are endowed with one unit of the good and compete by posting prices in every period. Buyers each demand one unit of the good and have a common reservation price. They have full information regarding the prices posted by each firm in the market; hence, search is costless. The number of buyers coming to the market in each period is random. Demand uncertainty is said to be high if there are at least two non-zero demand states that give a seller different option values of waiting to sell. Our model features a unique symmetric Markov perfect equilibrium in which price dispersion prevails if and only if the degree of demand uncertainty is high. Several testable theoretical implications on the distribution of market prices are derived.

Very Simple Markov-Perfect Industry Dynamics

Abstract: This paper develops an econometric model of industry dynamics for concentrated markets that can be estimated very quickly from market-level data on demand shifters and the number of producers. We show that the model has an essentially unique symmetric Markov-perfect equilibrium that can be calculated from the xed points of low-dimensional contraction mappings. We characterize the model's identi cation and extend Rust's (1987) nested xed point estimator to account for the observable implications of mixed strategies on survival. We illustrate the model's application with ten years of County Business Patterns data from Motion Picture Theaters in 573 Micropolitan Statistical Areas.

On Asynchronicity of Moves and Coordination

This paper shows that asynchronicity of moves can lead to a unique prediction in coordination games, in an infinite-horizon setting, under certain conditions on off-equilibrium payoffs. In two-player games we derive necessary and sufficient conditions for play ultimately being absorbed in the Pareto dominant Nash equilibrium of the stage game, for every Markov perfect equilibrium. For players patient enough, the condition is that the Pareto dominant Nash equilibrium is also risk dominant, but for lower levels of patience the condition departs from simple risk-dominance. For general n-player symmetric games with patient players, we show that a necessary and sufficient condition for the Pareto dominant Nash equilibrium to be the unique limit outcome in all symmetric Markov perfect equilibrium is a particular generalization of risk-dominance for more than two players. We provide extensions to the unique selection results to all subgame perfect Nash equilibria, and to coordination games in which different players prefer different Nash equilibria of the stage game.

Simple Markov-Perfect Industry Dynamics

This paper develops a tractable model for the computational and empirical analysis of infinite-horizon oligopoly dynamics. It features aggregate demand uncertainty, sunk entry costs, stochastic idiosyncratic technological progress, and irreversible exit. We develop an algorithm for computing a symmetric Markov-perfect equilibrium quickly by finding the fixed points to a finite sequence of low-dimensional contraction mappings. If at most two heterogenous firms serve the industry, the resuilt is the unique equilibrium in which a high protability firm never exits leaving behind a low protability competitor. With more than two firms, the algorithm always finds a natural equilibrium. We present a simple rule for checking ex post whether the calculated equilibrium is unique, and we illustrate the model's application by assessing the robustness of Fershtman and Pakes' (2000) finding that collusive pricing can increase consumer surplus by stimulating product development. The hundreds of equilibrium calculations this requires take only a few minutes on an off-the-shelf laptop computer. We also present a feasible algorithm for the model's estimation using the generalized method of moments.

Very Simple Markov-Perfect Industry Dynamics

This paper develops an econometric model of industry dynamics for concentrated markets that can be estimated very quickly from market-level panel data on the number of producers and consumers using a nested fixed-point algorithm. We show that the model has an essentially unique symmetric Markov-perfect equilibrium that can be calculated from the fixed points of a finite sequence of low-dimensional contraction mappings. Our nested fixed point procedure extends Rust's (1987) to account for the observable implications of mixed strategies on survival. We illustrate the model's empirical application with ten years of County Business Patterns data from the Motion Picture Theaters industry in 573 Micropolitan Statistical Areas. The results are suggestive of fierce competition between theaters in the market for film exhibition rights.

Very Simple Markov-Perfect Industry Dynamics

This paper develops an econometric model of firm entry, competition, and exit in dynamic oligopolistic markets. The model entertains market-level demand and cost shocks, sunk entry costs, and parameters that capture economic barriers to entry and the toughness of price competition. Nevertheless its analysis is very simple, because it takes firms to be homogenous. We show that the model has an essentially unique symmetric Markov-perfect equilibrium that can be computed quickly by solving a finite- sequence of low-dimensional contraction mappings. We develop a nested fixed-point procedure for the model's maximum-likelihood estimation from market-level panel data and compare the procedure's performance to that of a mathematical programming with equilibrium constraints approach. The framework is rich enough for a range of applications, such as the welfare analysis of licensing requirements, start-up subsidies, and environmental laws. Moreover, its analysis provides a starting point for the solution of more general models.

Climate Contracts: A Game of Emissions, Investments, Negotiations, and Renegotiations

The paper presents a dynamic game where players can contribute to a public bad, invest in technologies, and write incomplete contracts. Despite the n + 1 stocks in the model, the analysis is tractable and the symmetric Markov perfect equilibrium unique. The framework helps to analyze emissions, investments, and international environmental agreements. If only the emission levels are contractible, then investments are suboptimally small if the contract is short-term or close to its expiration date. To encourage investments, the optimal contract is more ambitious if it is short-term, and it is tougher to satisfy close to its expiration date and for countries with small investment costs. If renegotiation is possible, such an incomplete contract implements the …rst-best. The results have important lessons for how to design a climate treaty.

Simple Markov-Perfect Industry Dynamics

This paper develops a tractable model for the computational and empirical analysis of innite-horizon oligopoly dynamics. It features aggregate demand uncertainty, sunk entry costs, stochastic idiosyncratic technological progress, and irreversible exit. We develop an algorithm for computing a symmetric Markov-perfect equilibrium quickly