Strategies, payoffs and rationality in strategic games, with a Diwali discount game

The setting: a strategic game. A situation is strategic when a person choosing among actions also considers how others will respond, because what she gets depends on their choices as well as her own. The participants who choose courses of action are the players. A strategic game is described by three things: what each player can do (its strategies), what each player ends up with (its payoffs), and how each player is assumed to choose (rationality).

Strategies.

• A strategy is a player's plan of action for the game. It can be a single action, or a fuller plan that depends on what has already been done and how the others reacted.
• A pure strategy commits the player to one action with certainty, that is, with probability one.
• A mixed strategy leaves the choice to chance: the player picks among her actions according to a probability distribution, for example choosing one action half the time and the other half the time.
• Timing shapes strategy. When players move at the same moment (a simultaneous game), each must build her strategy on a guess about what the others will do. When one moves first and the other follows (a sequential game), the later mover can make her strategy depend on the action she has already seen. In a sequential game a strategy is therefore a complete plan, saying what the player will do in response to each possible earlier move.

Payoffs.

• A payoff is what a player receives at the end of the game when the chosen strategies are carried out: a gain or a loss.
• It can be measured in money, as profit or interest, or be non-monetary, as utility or emotion.
• Each player's payoff depends on the strategies chosen by everyone, not on her own choice alone. That is why a two-player game is set out as a table of payoffs, with one cell for every combination of strategies.
• Payoffs are what the players care about most, so each wants her payoff to be as large as possible.

Rationality.

• Players are assumed to be rational: each tries to maximise her own payoff by choosing the strategy that serves her interest best.
• A rational player looks at the whole set of possible actions, including the reactions and counter-reactions her choice may set off. Before deciding, she takes into account her preferences, limitations, objectives, constraints and past experience.
• She also expects her rivals to be rational, that is, to choose so as to maximise their own payoffs.
• The analysis is completed by common knowledge of the rules: every player knows how many players there are, which strategies each can use, what the payoffs are, and that all are rational.

Example: two sweet shops before Diwali. Our own figures. A large sweet shop (A) and a small one (B) in the same market each decide, at the same time, whether to offer a festival discount or no discount. Profits for the season in ₹ lakh are (A, B):

• Strategies. Each shop has two pure strategies, discount or no discount. Since they choose together, each must form its plan without seeing the other's choice.
• Payoffs. Each cell gives both profits for one pair of choices. B's profit, for example, is ₹4 lakh if it alone discounts but only ₹1 lakh if both discount: its result depends on A's choice as well as its own.
• Rationality, for A. If B discounts, A earns 6 by discounting against 5 by not; if B does not, A earns 8 against 7. Discounting is best for A whatever B does: it is A's dominant strategy, a choice that is best whatever the other player does.
• Rationality and common knowledge, for B. B has no such strategy: discounting is better for it if A does not discount (4 against 3), and not discounting is better if A does (2 against 1). B therefore has to reason about A. Knowing the payoffs and knowing that A is rational, B expects A to discount, and so chooses no discount.
• The outcome. A discounts and B does not, giving profits of (8, 2). Neither shop can do better by changing its choice alone, so this is an equilibrium: once chosen, no player has a reason to depart from its strategy.

Conclusion. A strategy is a player's plan of action, pure or mixed; a payoff is the gain or loss each combination of strategies brings to each player; and rationality is the assumption that every player picks the strategy that maximises its payoff while expecting the others to do the same. The example shows why all three are needed: B's best choice can be found only by working out A's payoffs and trusting A to act rationally.