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 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]

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]