Question

Tourists in Bintan island, a 45-minute ferry ride from Singapore, are demanding the latest espresso drinks from a familiar brand but their small number means demand can only sustain one new coffee shop. You and another popular chain, Starbucks, are considering making the considerable investment necessary to set up a shop in Bintan. Analyse the interaction between the two firms using game theory. Present a payoff matrix to model the situation and analyse it for Nash equilibrium. What can you do to make your best outcome more likely?

Answer 1

A STRATEGIC GAME is a model of interacting decision-makers. In recognition of the interaction, we refer to the decision-makers as players. Each player has a set of possible actions. The model captures interaction between the players by allowing each player to be affected by the actions of all players, not only her own action. Specifically, each player has preferences about the action profile—the list of all the players’ actions. (See Section 17.4, in the mathematical appendix, for a discussion of profiles.) More precisely, a strategic game is defined as follows. (The qualification “with ordinal preferences” distinguishes this notion of a strategic game from a more general notion studied in Chapter 4.) I DEFINITION 11.1 (Strategic game with ordinal preferences) A strategic game (with ordinal preferences) consists of • a set of players • for each player, a set of actions • for each player, preferences over the set of action profiles. 11 12 Chapter 2. Nash Equilibrium: Theory A very wide range of situations may be modeled as strategic games. For example, the players may be firms, the actions prices, and the preferences a reflection of the firms’ profits. Or the players may be candidates for political office, the actions campaign expenditures, and the preferences a reflection of the candidates’ probabilities of winning. Or the players may be animals fighting over some prey, the actions concession times, and the preferences a reflection of whether an animal wins or loses. In this chapter I describe some simple games designed to capture fundamental conflicts present in a variety of situations. The next chapter is devoted to more detailed applications to specific phenomena. As in the model of rational choice by a single decision-maker (Section 1.2), it is frequently convenient to specify the players’ preferences by giving payoff functions that represent them. Bear in mind that these payoffs have only ordinal significance. If a player’s payoffs to the action profiles a, b, and c are 1, 2, and 10, for example, the only conclusion we can draw is that the player prefers c to b and b to a; the numbers do not imply that the player’s preference between c and b is stronger than her preference between a and b. Time is absent from the model. The idea is that each player chooses her action once and for all, and the players choose their actions “simultaneously” in the sense that no player is informed, when she chooses her action, of the action chosen by any other player. (For this reason, a strategic game is sometimes referred to as a “simultaneous move game”.) Nevertheless, an action may involve activities that extend over time, and may take into account an unlimited number of contingencies. An action might specify, for example, “if company X’s stock falls below $10, buy 100 shares; otherwise, do not buy any shares”. (For this reason, an action is sometimes called a “strategy”.) However, the fact that time is absent from the model means that when analyzing a situation as a strategic game, we abstract from the complications that may arise if a player is allowed to change her plan as events unfold: we assume that actions are chosen once and for all. 2.2 Example: the Prisoner’sDilemma One of the most well-known strategic games is the Prisoner’s Dilemma. Its name comes from a story involving suspects in a crime; its importance comes from the huge variety of situations in which the participants face incentives similar to those faced by the suspects in the story. EXAMPLE 12.1 (Prisoner’s Dilemma) Two suspects in a major crime are held in separate cells. There is enough evidence to convict each of them of a minor offense, but not enough evidence to convict either of them of the major crime unless one of them acts as an informer against the other (finks). If they both stay quiet, each will be convicted of the minor offense and spend one year in prison. If one and only one of them finks, she will be freed and used as a witness against the other, who will spend four years in prison. If they both fink, each will spend three years in prison.