Some strategic situations correspond to one-stage games; other correspond to multiple-stage games. In economics, multiple-stage games are more common.
To describe the essential features of a one-stage game, we follow two steps. First, we identify the players and list the actions available to each. Second, for every possible combination of actions (one for every player), we identify each player's payoff, be it a reward or penalty.
Thinking strategically in one-stage games
Strategic situations require us to think about what other people will do. We can try to do so by putting ourselves into others' shoes, but that approach frequently leads to unproductive circular reasoning. Even so, in some situations it is possible to reason out the likely choice of a sensible opponent.
If a player has a dominant strategy he ought to play it, regardless of what he thinks others might do. In some games, like the Prisoners' Dilemma, all players have a dominant strategy.
Sometimes it is possible to reason out the likely choice of a sensible opponent by iteratively deleting dominated strategies.
It's usually a good idea to avoid weakly dominated strategies. In some cases, this dictum leads to a clear choice. For example, it implies that people should bid their true valuations in second-price sealed-bid auctions.
Nash equilibrium in one-stage games
Given a table that summarizes a one-stage game with two players, we can find a Nash equilibrium by looking for a cell that (a) gives the row player the highest payoff in the same column and (b) gives the column player the highest payoff in the same row.
The combination of strategies chosen in a Nash equilibrium is stable. Every participant is content with his choice; no one wants to play anything else. All other outcomes are unstable, in the sense that at least one participant would want to change his strategy.
One justification for focusing on Nash equilibria is that when all players are experienced, they should have reasonably accurate expectations of what the others will do, and should therefore tend to make their best responses to one anothers' choices.
A Nash equilibrium is also a self-enforcing agreement—one in which every party to the agreement has an incentive to abide by it, assuming that others do likewise.
In many games, no strategy dominates any other, even weakly. The concept of Nash equilibrium allows us to analyze strategic behavior in those games.
Depending on the game, Nash equilibrium can involve either good outcomes or bad ones.
In two-player games in which the choices are finely divisible, we can plot curves that show each player's best response as a function of the other's choice. To find the Nash equilibria, we then identify the points where the curves intersect.
In many games involving pure strategies, there are no Nash equilibria, but virtually all games have mixedstrategy equilibria.
Players are most likely to use mixed strategies in situations in which unpredictability is a key to success.
In any Nash equilibrium involving mixed strategies, each player must be indifferent among the choices over which he randomizes. This principle helps us to solve for the equilibria.
Games with multiple stages
We can describe a game with perfect information by drawing a tree diagram that shows the sequence of decisions and indicates the players' payoffs.
The best way to identify credible strategies and sensible Nash equilibria in a game with perfect information is to reason in reverse; that is, to start at the end of the tree diagram and work back to the beginning, identifying the choices that each player will make as we go.
For a finitely repeated game, backward induction implies that punishments and rewards may not be credible, and cooperation may not be possible, in any reasonable Nash equilibrium.
For an infinitely repeated game, punishments and rewards are credible, and cooperation is possible, provided that the players care enough about the future.
Games in which different people have different information
In many strategic settings, people have difficulty predicting each others' choices because they don't share the same information. One player may receive objective information that is not available to some other player, or one player may be uncertain about another's preferences.
In certain types of auctions, unsophisticated bidders tend to overpay whenever they win. A sophisticated bidder anticipates this winner's curse and bids more conservatively.
Other people often learn things about us by observing our actions. As a result, a pattern of behavior can create a reputation. Knowing this, people may intentionally adopt patterns of behavior that help to build or maintain desirable reputations.
