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 a game-theoretic economic model of competition in situations where there are just a few competitors who watch each other, e.g. big companies dividing a market oligopolistically. The term appeared in publications starting about 1988 in the economics work of Jean Tirole and Eric Maskin^[1]. It has been used in the economic analysis of industrial organization.

Definition

A Markov perfect equilibrium is a set of strategies of the players in a formally specified dynamic game which satisfy a number of criteria listed below.

• The strategies have the Markov property of memorylessness, meaning that each player's next move is predicted by the last move of the other player not by earlier histories of moves (except that mixed strategies are allowed so a random element is possible too). The authors call these strategies Markov reaction functions.
• The economists who defined the term^[2] also imposed the requirement that the strategies can depend only on what they called payoff-relevant information which may rule out some strategies that depend on non-substantive moves by the opponent. This restriction simplifies the possible strategies and analysis. 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.^[3]
• The authors took a special interest in symmetric equilibria, meaning the players had strategies, constraints, and opportunities which were mirror images of one another. Symmetry is not part of the definition however.

Industry analogue

This game theoretic concept was designed to characterize competition between firms which had invested heavily into fixed costs and were dominant producers in an industry, forming an oligopoly. The players could be thought of as committed to levels of production capacity or price in the short run, and the strategies would describe their decisions in setting these levels. The firms' objectives were modeled as maximizing the present discounted value of profits. In the early models, the firms faced no exogenous randomness in the outcomes but did not know one another's strategies either.^[4]

Example

This industry example gives the flavor of the concept though it does not include the details necessary to prove it creates a Markov perfect equilibrium.

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 given their varying frequent-flyer programs, the different connections their passengers will make, and so forth. Thus a simple supply and demand model would be unlikely to have equal-price outcome. Both airlines have invested enormously into the equipment, personnel, and permissions. In the near term they are committed to offering this service. Consider that they are therefore engaged, or trapped, in a strategic game against one another.

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 that passengers always choose the cheapest flight and so if the airlines charge different prices, one of them gets zero passengers. (This kind of extreme simplification is necessary to get through the example but could be relaxed in a more thorough study.)

Then if each player assumes that the other player will follow this strategy, there is no higher-payoff alternative strategy for himself. If both airlines followed this strategy, it would form a Nash equilibrium.

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 they do. Then this is a Markov perfect equilibrium. The purpose of specifying it is not (usually) to claim that airlines follow exactly these strategies. rather, this is a model which can succeed in predicting the observed behavior that airlines often charge exactly the same price, when a supply/demand model (for example) would generally not predict that. Thus the model explains (or rationalizes) tacit collusion in an oligopoly with high profits.

The model makes predictions about the behaviors of the airlines if and when the equal-price outcome breaks down, and interpreting and examining these price wars is part of evaluating the accuracy of the Markov perfect model.^[5] In contrasting to another model of tacit collusion, 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 supergame model a price cut may be a punishment to the other player (who perhaps broke the equilibrium with a price cut). The authors say the market share justification is closer to the usual empirical account than the punishment justification, ergo the Markov model predicts accurately.^[6]

Analysis and discussion

Markov perfect equilibria are not stable with respect to small changes in the game itself. A tiny change to payoffs can discontinuously change the set of Markov perfect equilibria, because a state variable with a tiny effect on payoffs can be part of a Markov perfect strategy, but if its effect drops to zero, it cannot be included in a strategy; that is, such a change makes many strategies disappear from the set of Markov perfect strategies.

References

1. ^ Tirole (1988) and Maskin and Tirole (1988)
2. ^ Maskin and Tirole, 1988
3. ^ 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
4. ^ Tirole (1988), p. 254
5. ^ See for example Maskin and Tirole, p.571
6. ^ 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.