Strategic games, with a Diwali discount example

What a strategic game is. In many decisions what a person gets depends only on her own choice. In a strategic situation it also depends on what others choose, so each decision maker, when picking an action, thinks about how the others will respond to it. Firms in a market with only a few large sellers are a clear case: each must allow for its rivals' reactions and cannot decide on its own. A strategic game is any such situation of interdependent choice, and the study of these interactions is called game theory. Games of this kind are not confined to chess or cricket; they arise in business, politics, bargaining and daily life.

What makes up a game.

• Players: the participants, each choosing a course of action.
• Strategies: each player's plan of action, which may be a single move or a plan that depends on earlier moves.
• Payoffs: the gain or loss each player ends with once the strategies are played, in money (such as profit) or in other terms (such as satisfaction).
• Rationality: each player is assumed to choose so as to make her payoff as large as possible, and to expect her rivals to do the same. The players are taken to know the rules: who plays, what strategies are open and what the payoffs are.

Kinds of strategic games.

• Simultaneous or sequential: players choose at the same time without seeing each other's choice, or one moves first and the other replies.
• Zero-sum or not: in a game of total conflict one player's gain is the other's loss, while in many others all can gain, as in trade, or all can lose, as in a war.
• One-time or repeated: when the same players meet again and again, reputation and cooperation start to matter.

Example: two sweet shops at Diwali. Our own figures. Two sweet shops on the same street in Jaipur each decide, at the same time, whether to offer a Diwali discount or keep their prices. Profits for the season in ₹ lakh (Shop A, Shop B):

• A's reasoning. If B offers a discount, A earns 6 by discounting and 4 by not, so A discounts. If B keeps its price, A earns 11 by discounting and 9 by not, so again A discounts. Discounting is best for A whatever B does: it is A's dominant strategy, the optimal choice no matter what the other player does. By the same reasoning it is B's dominant strategy too.
• The outcome. Both shops discount and earn ₹6 lakh each. Neither would gain by changing its choice alone, since keeping its price would cut its profit to ₹4 lakh, so (discount, discount) is a Nash equilibrium: each player's choice is its best reply to the other's.
• The lesson. Both would earn ₹9 lakh if neither discounted, yet each, choosing in its own interest and expecting the other to do the same, ends up worse off. The situation is strategic because each shop's profit depends on the other's choice as well as its own (A earns 11 or 6 from discounting according to what B does), even though here each shop's best choice, discounting, is the same whatever the rival does.

Conclusion. A strategic game is a situation in which each player's payoff depends on the choices of others as well as her own, so each must anticipate the others' responses. It is described by its players, strategies and payoffs, played by rational players, and, as the sweet-shop example shows, it can lead rational players to an outcome worse for both than one they could have reached together.