Strategy (game theory)

In game theory, a player's strategy in a game is a complete plan of action for whatever situation might arise; this fully determines the player's behaviour. A player's strategy will determine the action the player will take at any stage of the game, for every possible history of play up to that stage.

A strategy profile (sometimes called a strategy combination) is a set of strategies for each player which fully specifies all actions in a game. A strategy profile must include one and only one strategy for every player.

The strategy concept is sometimes (wrongly) confused with that of a move. A move is an action taken by a player at some point during the play of a game (e.g., in chess, moving white's Bishop a2 to b3). A strategy on the other hand is a complete algorithm for playing the game, telling a player what to do for every possible situation throughout the game.

Strategy set

A player's strategy set defines what strategies are available for them to play.

A player has a finite strategy set if they have a number of discrete strategies available to them. For instance, in a single game of Rock-paper-scissors, each player has the finite strategy set {rock, paper, scissors}.

A strategy set is infinite otherwise. For instance, an auction with mandated bid increments may have an infinite number of discrete strategies in the strategy set {$10, $20, $30, ...}. Alternatively, the Cake cutting game game has a bounded continuum of strategies in the strategy set {Cut anywhere between zero percent and 100 percent of the cake}.

In a dynamic game, the strategy set consists of the possible rules a player could give to a robot or agent on how to play the game. 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.

In a Bayesian game, the strategy set is similar to that in a dynamic game. It consists of rules for what action to take for any possible private information.

Choosing a strategy set

In applied game theory, the definition of the strategy sets is an important part of the art of making a game simultaneously solvable and meaningful. The game theorist can use knowledge of the overall problem to limit the strategy spaces, and ease the solution.

For instance, strictly speaking in the Ultimatum game a player can have strategies such as: Reject offers of ($1, $3, $5, ..., $19), accept offers of ($0, $2, $4, ..., $20). Including all such strategies makes for a very large strategy space and a somewhat difficult problem. A game theorist might instead believe they can limit the strategy set to: {Reject any offer ≤ x, accept any offer > x; for x in ($0, $1, $2, ..., $20)}.

Pure and mixed strategies

A pure strategy provides a complete definition of how a player will play a game. In particular, it determines the move a player will make for any situation he or she could face. A player's strategy set is the set of pure strategies available to that player.

A mixed strategy is an assignment of a probability to each pure strategy. This allows for a player to randomly select a pure strategy. Since probabilities are continuous, there are infinitely many mixed strategies available to a player, even if their strategy set is finite.

Of course, one can regard a pure strategy as a degenerate case of a mixed strategy, in which that particular pure strategy is selected with probability 1 and every other strategy with probability 0.

A totally mixed strategy is a mixed strategy in which the player assigns a strictly positive probability to every pure strategy. (Totally mixed strategies are important for equilibrium refinement such as trembling hand perfect equilibrium.)

Mixed strategy

Illustration

A B A 1, 1 0, 0 B 0, 0 1, 1 Pure coordination game

Consider the payoff matrix pictured to the right (known as a coordination game). Here one player chooses the row and the other chooses a column. The row player receives the first payoff, the column player the second. If row opts to play A with probability 1 (i.e. play A for sure), then he is said to be playing a pure strategy. If column opts to flip a coin and play A if the coin lands heads and B if the coin lands tails, then she is said to be playing a mixed strategy, and not a pure strategy.

Significance

In his famous paper, John Forbes Nash proved that there is an equilibrium for every finite game. One can divide Nash equilibria into two types. Pure strategy Nash equilibria are Nash equilibria where all players are playing pure strategies. Mixed strategy Nash equilibria are equilibria where at least one player is playing a mixed strategy. While Nash proved that every finite game has a Nash equilibrium, not all have pure strategy Nash equilibria. For an example of a game that does not have a Nash equilibrium in pure strategies, see Matching pennies. However, many games do have pure strategy Nash equilibria (e.g. the Coordination game, the Prisoner's dilemma, the Stag hunt). Further, games can have both pure strategy and mixed strategy equilibria.

A disputed meaning

During the 1980s, the concept of mixed strategies came under heavy fire for being "intuitively problematic".^[1] Randomization, central in mixed strategies, lacks behavioral support. Seldom do people make their choices following a lottery. This behavioral problem is compounded by the cognitive difficulty that people are unable to generate random outcomes without the aid of a random or pseudo-random generator.^[1]

In 1991,^[2] game theorist Ariel Rubinstein described alternative ways of understanding the concept. The first, due to Harsanyi (1973), ^[3] 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 exogeneous factors. However, it is unsatisfying to have results that hang on unspecified factors.^[2]

A second interpretation imagines the game players standing for a large population of agents. Each of the agents chooses a pure strategy, and the payoff depends on the fraction of agents choosing each strategy. The mixed strategy hence represents the distribution of pure strategies chosen by each population. However, this does not provide any justification for the case when players are individual agents.

Later, Aumann and Brandenburger (1995), ^[4] re-interpreted Nash equilibrium as an equilibrium in beliefs, rather than actions. 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 predictive power of Nash equilibrium, however, since it is possible in such an equilibrium for each player to actually play a pure strategy of Rock.

Ever since, game theorists' attitude towards mixed strategies-based results have been ambivalent. Mixed strategies are still widely used for their capacity to provide Nash equilibria in games where no equilibrium in pure strategies exists, but the model does not specify why and how players randomize their decisions.

See also

• Nash equilibrium
• Haven (graph theory)

References

1. ^ ^{a b} Aumann, R. (1985). "What is Game Theory Trying to accomplish?". In Arrow, K.; Honkapohja, S.. Frontiers of Economics. Oxford: Basil Blackwell. pp. 909–924. http://www.ma.huji.ac.il/raumann/pdf/what%20is%20game%20theory.pdf. 
2. ^ ^{a b} Rubinstein, A. (1991). "Comments on the interpretation of Game Theory". Econometrica 59 (4): 909–924. JSTOR 2938166. 
3. ^ Harsanyi, John (1973), "Games with randomly disturbed payoffs: a new rationale for mixed-strategy equilibrium points", Int. J. Game Theory 2: 1–23, doi:10.1007/BF01737554 
4. ^ Aumann, Robert; Brandenburger, Adam (1995), "Epistemic Conditions for Nash Equilibrium", Econometrica (The Econometric Society) 63 (5): 1161–1180, doi:10.2307/2171725, JSTOR 2171725