Symmetric and asymmetric
An asymmetric game
|
E |
F |
| E |
1, 2 |
0, 0 |
| F |
0, 0 |
1, 2 |
A symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. If the identities of the players can be changed without changing the payoff to the strategies, then a game is symmetric. Many of the commonly studied 2×2 games are symmetric. The standard representations of Chicken, the Prisoner’s Dilemma, and the Stag hunt are all symmetric games. [1]
Most commonly studied asymmetric games are games where there are not identical strategy sets for both players. For instance, the Ultimatum game and similar the Dictator game have different strategies for each player. It is possible, however, for a game to have identical strategies for both players, yet be asymmetric. For example, the game pictured to the right is asymmetric despite having identical strategy sets for both players.
Zero sum and non-zero sum
A Zero-Sum Game
|
A |
B |
| A |
2, -2 |
-1, 1 |
| B |
-1, 1 |
3, -3 |
In zero-sum games the total benefit to all players in the game, for every combination of strategies, always adds to zero (or more informally put, a player benefits only at the expense of others). Poker exemplifies a zero-sum game, because one wins exactly the amount one’s opponents lose. Other zero sum games include Matching pennies and most classical board games including Go and Chess. Many games studied by game theorists (including the famous Prisoner’s Dilemma) are non-zero-sum games, because some outcomes have net results greater or less than zero. Informally, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another.
It is possible to transform any game into a zero-sum game by adding an additional dummy player (often called “the board”), whose losses compensate the players’ net winnings.
Simultaneous and sequential
Simultaneous games are games where both players move simultaneously, or if they do not move simultaneously, the later players are unaware of the earlier players’ actions (making them effectively simultaneous). Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions. This need not be perfect knowledge about every action of earlier players; it might be very little information. For instance, a player may know that an earlier player did not perform one particular action, while she does not know which of the other available actions the first player actually performed.
The difference between simultaneous and sequential games is captured in the different representations discussed above. Normal form is used to represent simultaneous games, and extensive form is used to represent sequential ones.
Perfect information and imperfect information
A game of imperfect information (the dotted line represents ignorance on the part of player 2)
An important subset of sequential games consists of games of perfect information. A game is one of perfect information if all players know the moves previously made by all other players. Thus, only sequential games can be games of perfect information, since in simultaneous games not every player knows the actions of the others. Most games studied in game theory are imperfect information games, although some interesting games are games of perfect information, including the Ultimatum Game and Centipede Game. Many popular games are games of perfect information including Chess, Go, and Mancala.
Perfect information is often confused with complete information, which is a similar concept. Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions.
Infinitely long games
For obvious reasons, games as studied by economists and real-world game players are generally finished in a finite number of moves. Pure mathematicians are not so constrained, and set theorists in particular study games that last for infinitely many moves, with the winner (or other payoff) not known until after all those moves are completed.
The focus of attention is usually not so much on what is the best way to play such a game, but simply on whether one or the other player has a winning strategy. (It can be proved, using the axiom of choice, that there are games—even with perfect information, and where the only outcomes are “win” or “lose”—for which neither player has a winning strategy.) The existence of such strategies, for cleverly designed games, has important consequences in descriptive set theory.
Notes
- ^ Some scholars would consider certain asymmetric games as examples of these games as well. However, the most common payoffs for each of these games are symmetric.
This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.
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Johnny Moss, Becky Binion, and Puggy Pearson at the 1974 World Series of Poker
