Markov perfect equilibrium

A Markov perfect equilibrium is an equilibrium concept in game theory. It has been used in analyses of industrial organization, macroeconomics, and political economy. It is a refinement of the concept of subgame perfect equilibrium to extensive form games for which a pay-off relevant state space can be identified. The term appeared in publications starting about 1988 in the work of economists Jean Tirole and Eric Maskin.[1][2][3][4]

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.[5]

Focus on symmetric equilibria

In symmetric games, when the players have a 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.
Examples

For examples of this equilibrium concept, consider the competition between firms which have 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 modelled as maximizing the present discounted value of profits.[6]
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, thus committing to offering service. They are engaged or trapped, in a strategic game with one another when setting prices.

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]

A Markov-perfect equilibrium concept has also been used to model aircraft production, as different companies evaluate their future profits and how much they will learn from production experience in light of demand and what others firms might supply.[7]
Discussion

Airlines do not literally or exactly follow these strategies, but the model helps 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 Markov perfect equilibrium model helps shed light on tacit collusion in an oligopoly setting, and make predictions for cases not observed.

One strength of an explicit game-theoretical framework is that it allows us to make predictions about the behaviours 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.[8] 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.[9]
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

Maskin E, Tirole J. A Theory of Dynamic Oligopoly, I: Overview and Quantity Competition with Large Fixed Costs. Econometrica 1988;56:549.
Maskin and Maskin E, Tirole J. A Theory of Dynamic Oligopoly, II: Price Competition, Kinked Demand Curves, and Edgeworth Cycles. Econometrica 1988;56:571
Maskin E, Tirole J. Markov Perfect Equilibrium. J Econ Theory 2001;100:191–219.
Fudenberg D, Tirole J. Game Theory. 1991:603.
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 Archived 2011-10-05 at the Wayback Machine. Journal of Economic Theory 100, 191-219. doi:10.1006/jeth.2000.2785, available online at http://www.idealibrary.com
Tirole (1988), p. 254
C. Lanier Benkard. 2000. Learning and forgetting: The dynamics of aircraft production. American Economic Review 90:4, 1034–1054. (jstor)
See for example Maskin and Tirole, p.571

Maskin and Tirole, 1988, p.592

Bibliography

Fudenberg, Drew; Tirole, Jean (1991). Game theory. Cambridge, Massachusetts: MIT Press. pp. 501–502. ISBN 9780262061414. Book preview.
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.

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Topics in game theory
Definitions

Cooperative game Determinacy Escalation of commitment Extensive-form game First-player and second-player win Game complexity Graphical game Hierarchy of beliefs Information set Normal-form game Preference Sequential game Simultaneous game Simultaneous action selection Solved game Succinct game

Equilibrium
concepts

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

Strategies

Dominant strategies Pure strategy Mixed strategy Strategy-stealing argument Tit for tat Grim trigger Collusion Backward induction Forward induction Markov strategy Bid shading

Classes
of games

Symmetric game Perfect information Repeated game Signaling game Screening game Cheap talk Zero-sum game Mechanism design Bargaining problem Stochastic game Mean field game n-player game Large Poisson game Nontransitive game Global game Strictly determined game Potential game

Games

Go Chess Infinite chess Checkers Tic-tac-toe Prisoner's dilemma Gift-exchange game Optional 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 game 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 Induction puzzles Trust game Princess and Monster game Rendezvous problem

Theorems

Arrow's impossibility theorem Aumann's agreement theorem Folk theorem Minimax theorem Nash's theorem Purification theorem Revelation principle Zermelo's theorem

Key
figures

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