Some examples of "games" include chess, bridge, poker, monopoly, diplomacy or battleship.
[2] The term strategy is typically used to mean a complete algorithm for playing a game, telling a player what to do for every possible situation.
For instance, a game of rock paper scissors comprises a single move by each player—and each player's move is made without knowledge of the other's, not as a response—so each player has the finite strategy set {rock paper scissors}.
For instance, in the ultimatum game, the strategy set for the second player would consist of every possible rule for which offers to accept and which to reject.
The game theorist can use knowledge of the overall problem, that is the friction between two or more players, to limit the strategy spaces, and ease the solution.
A pure strategy provides a complete definition of how a player will play a game.
Pure strategy can be thought about as a singular concrete plan subject to the observations they make during the course of the game of play.
[3] It assumes that if the goalie guesses correctly, the kick is blocked, which is set to the base payoff of 0 for both players.
Their article is well-known as an example of how people in real life use mixed strategies.
In his famous paper, John Forbes Nash proved that there is an equilibrium for every finite game.
For an example of a game that does not have a Nash equilibrium in pure strategies, see Matching pennies.
[4] [5] Game theorist Ariel Rubinstein describes alternative ways of understanding the concept.
The first, due to Harsanyi (1973),[6] is called purification, and supposes that the mixed strategies interpretation merely reflects our lack of knowledge of the players' information and decision-making process.
Apparently random choices are then seen as consequences of non-specified, payoff-irrelevant exogenous factors.
[5] A second interpretation imagines the game players standing for a large population of agents.
For instance, in rock paper scissors an equilibrium in beliefs would have each player believing the other was equally likely to play each strategy.
This interpretation weakens the descriptive power of Nash equilibrium, however, since it is possible in such an equilibrium for each player to actually play a pure strategy of Rock in each play of the game, even though over time the probabilities are those of the mixed strategy.
A famous example of why perfect recall is required for the equivalence is given by Piccione and Rubinstein (1997)[full citation needed] with their Absent-Minded Driver game.
This is fully described in the Absent-Minded Driver game formulated by Piccione and Rubinstein.
On the other hand, looking at the planning-optimal stage only, the maximum payoff is achieved by continuing at both intersections, maximized at p=2/3 (reference).