Strategies, payoffs and rationality in strategic games

The setting: a strategic game. An interaction is strategic when each decision maker, in choosing among the actions open to her, also thinks about how the others will respond, because what she gets depends on their choices as well as her own. The decision makers are the players: each is a participant who chooses a course of action in the game. Three ideas then describe the game: what each player can do (strategies), what each ends up with (payoffs), and how each is assumed to choose (rationality).

Strategies.

• A strategy is a player's plan of action for playing the game. It may be one simple action, or a more involved plan that depends on actions and reactions earlier in the game.
• A pure strategy means the player takes one action with probability one, for certain.
• A mixed strategy means the player does not commit to one action but chooses among her actions according to a probability distribution.
• How strategies are formed depends on the timing. In a simultaneous game all players choose at the same moment, so each must form her strategy on a guess about what the others will choose; in a sequential game the player who moves later can make her strategy depend on the action already taken by the first mover.

Payoffs.

• A payoff is the return a player receives at the end of the game once the chosen strategies are carried out: a gain or a loss.
• It may be monetary (profit, interest) or non-monetary (utility, emotion).
• A player's payoff depends on the combination of strategies chosen by all the players, not on her own choice alone. Payoffs are therefore shown in a table with one cell for each combination of strategies.
• Every player wants her payoff to be as large as possible, since it is what she cares about most.

Rationality.

• Players are assumed to be rational: each aims to maximise her own payoff and picks the strategy that serves her interest best.
• In doing so she takes into account the whole set of possible actions, including the reactions and counter-reactions her choice may provoke; before deciding she weighs her preferences, limits, objectives, constraints and past experience.
• She also believes that her rivals are rational and will themselves choose so as to maximise their payoffs.
• This is completed by common knowledge of the rules: the players know the number of players, the strategies open to each, the payoffs, and that all of them are rational.

Example: two coaching centres in Kota. Our own figures. Two coaching centres each choose a strategy, advertise heavily or don't advertise, at the same time. Monthly profits in ₹ lakh are (Centre A, Centre B):

• Strategies: each centre has two pure strategies, advertise or not.
• Payoffs: each cell gives the profits from one combination of choices; A's profit depends on B's choice as well as its own.
• Rationality: A reasons that if B advertises, advertising gives 8 against 6, and if B does not, advertising gives 14 against 10. Advertising is best for A whatever B does, which makes it a dominant strategy, an optimal choice no matter what the other player does. Expecting B to reason the same way, A predicts (advertise, advertise), with 8 each, although both would earn 10 if neither advertised.

Conclusion. Strategies are the players' plans of action, payoffs are the gains or losses that each combination of strategies brings them, and rationality is the assumption that each player chooses the strategy that maximises her payoff while expecting the others to do the same.