A game is a situation in which strategic behavior is an important part of decision making.
Non-cooperative game theory is a set of tools for analyzing decision making in situations where strategic behavior is important.
16.1 Some Fundamentals of Game Theory
The players are the decision makers in a game. A strategy is a player’s plan of action in a game. Actions are the particular things that are done according to a player’s strategy for a game. Payoffs are the rewards enjoyed by a player at the end of a game.
Game Trees: Decision Trees for Strategic Decisions
A game tree is an extension of a decision tree that provides a graphical representation of a strategic situation.
A decision rule is a strategy that specifies what action will be taken conditional on what happens earlier in the game. E.g. If Firm A produces “high,” then I will produce “low,” and if Firm A produces “low,” then I will produce “high.”
Dominant Strategy Equilibrium
A dominant strategy is a strategy that works at least as well as any other one, no matter what any one player does.
A dominant strategy equilibrium is an output in a game in which each player follows a dominant strategy. E.g. Firm B should produce “high” no matter what firm A does, and Firm A produces “high” no matter what firm A does.
A perfect equilibrium is a set of strategies that satisfies both the Nash condition and the credibility condition. (The Nash equilibrium states that each firm is choosing the strategy that maximizes its profit, given the strategies of the other firms in the market.)
16.2 Applying Game Theory: Oligopoly with Entry
When the incumbent has the ability to collude with the entrant in post-entry output levels, that ability may make the incumbent worse off. The reason is that the potential entrant will take the possibility (and profitability) of collusion into account when deciding whether to come into the market. Knowing that the post-entry equilibrium will be a collusive one makes entry attractive. Even under the fully collusive duopoly outcome, however, the incumbent earns less than it would if it had retained its monopoly position.
Credible Threats and Commitment
Commitment is the process whereby a player irreversibly alters its payoffs in advance so that it will be in the player’s self-interest to carry out a threatened (or promised) action when the time comes. (In a decision tree for the entry game, this adds another decision to the beginning of the game in which the incumbent, for example, decides whether to make a large capital investment – build a large plant – or a small capital investment – build a small plant.) Though the incumbent would prefer a small plan in the case that entry doesn’t occur, it is logical to build the plant in order to keep the entrant out.
More on Strategic Investment in Oligopoly
The firm may also rely on the “strategic effects” of R&D investment. In both an oligopolistic market and a market with a single firm facing the threat of entry, strategic investment can commit a firm to act more aggressively, which then makes its rivals retreat.
16.3 Games of Imperfect and Incomplete Information
A game of imperfect information is a game in which some player must make a move but is unable to observe the earlier or simultaneous move of some other player. (e.g. Prisoner’s dilemma)
A game of incomplete information is a game in which some player is unsure about some of the underlying characteristics of the game, such as another player’s payoffs. (i.e. You don’t know where you are in the decision tree.)
The Prisoners' Dilemma: A Game of Imperfect Information
Two prisoners are taken in and must simultaneously decide to confess or remain silent. In this game, confess is the dominant strategy for both players. (Even though if both had remained silent, they would each have had less time in jail.)
The prisoners’ dilemma is a strategic situation in which two players each have a dominant strategy, but playing this pair of strategies leads to an outcome in which both sides are worse off than they would be if they cooperated by playing alternative strategies.
A pure strategy is a strategy that specifies a specific action at each decision point. A mixed strategy is a strategy that allows for randomization among actions at some or all decision points.
An example of where a mixed strategy might be used is in the case of a free kick in soccer. The goalie always wants to go the same way as the kicker, but the kicker always wants to go in the opposite direction from the goalie. Hence, there is no equilibrium in pure strategies. There is, however, an equilibrium in mixed strategies: Each player randomly goes left half of the time and right the other half. Neither player can make himself better off by switching his strategy, given the strategy being played by his opponent.
A Bargaining Game of Incomplete Information
An example of incomplete information would be a case in which a seller didn’t know how highly a buyer values a particular good. The seller’s uncertainty about the buyer’s valuation of the good is captured in the decision tree by having Nature make an unobservable move to pick the buyer’s valuation of the good. We can say that Nature chooses a particular outcome with probability B. The seller cannot see the move that Nature has made, and instead he will calculate the expected value based only on the probabilities and his own payoff.
Limiting Pricing: A Game of Incomplete Information
Limit pricing is the practice of setting a high output level, or a low price, to deter entry.
16.4 Repeated Games
There are many situations in which players find themselves repeatedly making the same decisions. One type of agreement that does make use of repeated interaction is known as the grim-trigger strategy. Firms may agree to charge a price, p, each day, as long as no one has cheated in the past. If anyone cheats, the firms “agree” to punish the cheater by setting all future prices at marginal cost. The strategy gets its name from the fact that detection of cheating “triggers” an infinitely long punishment (a “grim” prospect).
Finitely Repeated Games
Finitely repeated games are solved by using backward induction. When there is a unique Nash equilibrium of the stage game, the unique perfect equilibrium of the finitely repeated game is simply the one-shot equilibrium repeated in every period.