Markov perfect equilibrium

Markov perfect equilibrium
A solution concept in game theory
Relationships
Subset of	Subgame perfect equilibrium
Significance
Proposed by	Eric Maskin, Jean Tirole
Used for	tacit collusion; price wars; oligopolistic competition

A Markov perfect equilibrium is an equilibrium concept in game theory. It is the refinement of the concept of subgame perfect equilibrium to extensive form games for which a pay-off relevant state space can be readily identified. The term appeared in publications starting about 1988 in the work of economists Jean Tirole and Eric Maskin.^[1] It has since been used, among else, in the analysis of industrial organization, macroeconomics and political economy.

Definition

In extensive form games, and specifically in stochastic games, a Markov perfect equilibrium is a set of mixed strategies for each of the players which satisfy the following criteria:

The strategies have the Markov property of memorylessness, meaning that each player's mixed strategy can be conditioned only on the state of the game. These strategies are called Markov reaction functions.
The state can only encode payoff-relevant information. This rules out strategies that depend on non-substantive moves by the opponent. It excludes strategies that depend on signals, negotiation, or cooperation between the players (e.g. cheap talk or contracts).
The strategies form a subgame perfect equilibrium of the game.^[2]

Focus on symmetric equilibria

In symmetric games, when the players have strategy and action sets which are mirror images of one another, often the analysis focuses on symmetric equilibria, where all players play the same mixed strategy. As in the rest of game theory, this is done both because these are easier to find analytically and because they are perceived to be stronger focal points than asymmetric equilibria.

Lack of robustness

Markov perfect equilibria are not stable with respect to small changes in the game itself. A small change in payoffs can cause a large change in the set of Markov perfect equilibria. This is because a state with a tiny effect on payoffs can be used to carry signals, but if its payoff difference from any other state drops to zero, it must be merged with it, eliminating the possibility of using it to carry signals.

Industrial organization extended example

As an example of the use of this equilibrium concept we consider the competition between firms which had invested heavily into fixed costs and are dominant producers in an industry, forming an oligopoly. The players are taken to be committed to levels of production capacity in the short run, and the strategies describe their decisions in setting prices. The firms' objectives are modeled as maximizing the present discounted value of profits.^[3]

Airfare game

Often an airplane ticket for a certain route has the same price on either airline A or airline B. Presumably, the two airlines do not have exactly the same costs, nor do they face the same demand function given their varying frequent-flyer programs, the different connections their passengers will make, and so forth. Thus, a realistic general equilibrium model would be unlikely to result in nearly identical prices.

Both airlines have made sunk investments into the equipment, personnel, and legal framework. In the near term we may think of them as committed to offering service. We therefore see that they are engaged, or trapped, in a strategic game with one another when setting prices.

Equilibrium

Consider the following strategy of an airline for setting the ticket price for a certain route. At every price-setting opportunity:

if the other airline is charging $300 or more, or is not selling tickets on that flight, charge $300
if the other airline is charging between $200 and $300, charge the same price
if the other airline is charging $200 or less, choose randomly between the following three options with equal probability: matching that price, charging $300, or exiting the game by ceasing indefinitely to offer service on this route.

This is a Markov strategy because it does not depend on a history of past observations. It satisfies also the Markov reaction function definition because it does not depend on other information which is irrelevant to revenues and profits.

Assume now that both airlines follow this strategy exactly. Assume further that passengers always choose the cheapest flight and so if the airlines charge different prices, the one charging the higher price gets zero passengers. Then if each airline assumes that the other airline will follow this strategy, there is no higher-payoff alternative strategy for itself, i.e. it is playing a best response to the other airline strategy. If both airlines followed this strategy, it would form a Nash equilibrium in every proper subgame, thus a subgame-perfect Nash equilibrium.^{[note 1]}

Discussion

The purpose of studying this model in the context of the airline industry is not to claim that airlines follow exactly these strategies. Rather, it is used to explain the observation that airlines often charge exactly the same price, even though a general equilibrium model specifying non-perfect substitutability would generally not provide such a result. The equilibrium concept of a Markov perfect equilibrium helps to shed light on what may be the cause of tacit collusion in an oligopoly setting.

One strength of an explicit game-theoretical framework is that it allows us to make predictions about the behaviors of the airlines if and when the equal-price outcome breaks down, and interpreting and examining these price wars in light of different equilibrium concepts.^[4] In contrasting to another equilibrium concept, Maskin and Tirole identify an empirical attribute of such price wars: in a Markov strategy price war, "a firm cuts its price not to punish its competitor, [rather only to] regain market share" whereas in a general repeated game framework a price cut may be a punishment to the other player. The authors claim that the market share justification is closer to the empirical account than the punishment justification, and so the Markov perfect equilibrium concept proves more informative, in this case.^[5]

Notes

↑ This kind of extreme simplification is necessary to get through the example but could be relaxed in a more thorough study. A more complete specification of the game, including payoffs, would be necessary to show that these strategies can form a subgame-perfect Nash equilibrium. For illustration let us suppose however that the strategies do form such an equilibrium and therefore that they also constitute a Markov perfect equilibrium.

References

↑ Tirole (1988) and Maskin and Tirole (1988)
↑ We shall define a Markov Perfect Equilibrium (MPE) to be a subgame perfect equilibrium in which all players use Markov strategies. Eric Maskin and Jean Tirole. 2001. Markov Perfect Equilibrium. Journal of Economic Theory 100, 191-219. doi:10.1006/jeth.2000.2785, available online at http://www.idealibrary.com
↑ Tirole (1988), p. 254
↑ See for example Maskin and Tirole, p.571
↑ Maskin and Tirole, 1988, p.592

Bibliography

Fudenberg, Drew, and Jean Tirole. 1991/1993. Game Theory, pp 501–2
Tirole, Jean. 1988. The Theory of Industrial Organization. Cambridge, MA: The MIT Press.
Maskin, Eric, and Jean Tirole. 1988. "A Theory of Dynamic Oligopoly: I & II" Econometrica 56:3, 549-600.

Topics in game theory

Definitions	Normal-form game Extensive-form game Escalation of commitment Graphical game Cooperative game Succinct game Information set Hierarchy of beliefs Preference

Equilibrium concepts	Nash equilibrium Subgame perfection Mertens-stable equilibrium Bayesian-Nash Perfect Bayesian Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance Core Shapley value Pareto efficiency Quantal response equilibrium Self-confirming equilibrium Strong Nash equilibrium Markov perfect equilibrium

Strategies	Dominant strategies Pure strategy Mixed strategy Tit for tat Grim trigger Collusion Backward induction Forward induction Markov strategy

Classes of games	Symmetric game Perfect information Simultaneous game Sequential game Repeated game Signaling game Cheap talk Zero-sum game Mechanism design Bargaining problem Stochastic game n-player game Large Poisson game Nontransitive game Global games Strictly determined game

Games	Prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede game Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Rock-paper-scissors Pirate game Dictator game Public goods game Blotto games War of attrition El Farol Bar problem Fair division Fair cake-cutting Cournot game Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Screening game Prisoners and hats puzzle Trust game Princess and monster game Monty Hall problem Rendezvous problem

Theorems	Minimax theorem Nash's theorem Purification theorem Folk theorem Revelation principle Arrow's impossibility theorem

Key figures	Kenneth Arrow Robert Aumann Kenneth Binmore Samuel Bowles Melvin Dresher Merrill M. Flood Drew Fudenberg Donald B. Gillies John Harsanyi Leonid Hurwicz David K. Levine Daniel Kahneman David M. Kreps Harold W. Kuhn Eric Maskin Jean-François Mertens Paul Milgrom Oskar Morgenstern Hervé Moulin Roger Myerson John Nash John von Neumann Ariel Rubinstein Thomas Schelling Reinhard Selten Herbert Simon Lloyd Shapley John Maynard Smith Jean Tirole Albert W. Tucker Amos Tversky William Vickrey Robert B. Wilson Peyton Young

See also	Tragedy of the commons Tyranny of small decisions All-pay auction List of games in game theory Confrontation analysis List of game theorists Combinatorial game theory Alpha–beta pruning Bertrand paradox Bounded rationality No-win situation Coopetition

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