(A synergistic relationship) Two individuals are involved in a synergistic relationship. If both…

(A synergistic relationship) Two individuals are involved in a synergistic relationship. If both individuals devote more effort to the relationship, they are both better off. For any given effort of individual j, the return to individual i’s effort first increases, then decreases. Specifically, an effort level is a nonnegative number, and individual i’s preferences (for i = 1, 2) are represented by the payoff function ai(c + aj − ai), where ai is i’s effort level, aj is the other individual’s effort level, and c > 0 is a constant. The following strategic game models this situation. Players The two individuals. Actions Each player’s set of actions is the set of effort levels (nonnegative numbers). Preferences Player i’s preferences are represented by the payoff function ai(c + aj − ai), for i = 1, 2. In particular, each player has infinitely many actions, so that we cannot present the game in a table like those used previously. To find the Nash equilibria of the game, we can construct and analyze the players’ best response functions. Given aj, individual i’s payoff is a quadratic function of ai that is zero when ai = 0 and when ai = c + aj, and reaches a maximum in between. The symmetry of quadratic functions (see Section 17.4) implies that the best response of each individual i to aj is

(If you know calculus, you can reach the same conclusion by setting the derivative of player i’s payoff with respect to ai equal to zero.) The best response functions are shown in Figure 38.1. Player 1’s actions are plotted on the horizontal axis and player 2’s actions are plotted on the vertical axis. Player 1’s best response function associates an action for player 1 with every action for player 2. Thus to interpret the function b1 in the diagram, take a point a2 on the vertical axis, and go across to the line labeled b1 (the steeper of the two lines), then read down to the horizontal axis. The point on the horizontal axis that you reach is b1(a2), the best action for player 1 when player 2 chooses a2. Player 2’s best response function, on the other hand, associates an action for player 2 with every action of player 1. Thus to interpret this function, take a point a1 on the horizontal axis, and go up to b2, then across to the vertical axis. The point on the vertical axis that you reach is b2(a1), the best action for player 2 when player 1 chooses a1

At a point (a1, a2) where the best response functions intersect in the figure, we have a1 = b1(a2), because (a1, a2) is on the graph of b1, player 1’s best response function, and a2 = b2(a1), because (a1, a2) is on the graph of b2, player 1’s best response function. Thus any such point (a1, a2) is a Nash equilibrium. In this game the best response functions intersect at a single point, so there is one Nash equilibrium. In general, they may intersect more than once; every point at which they intersect is a Nash equilibrium. To find the point of intersection of the best response functions precisely, we can solve the two equations:

Substituting the second equation in the first, we get a1 = ½(c + ½ (c + a1)) = 3/4 c + ¼ a1, so that a1 = c. Substituting this value of a1 into the second equation, we get a2 = c. We conclude that the game has a unique Nash equilibrium (a1, a2) = (c, c). (To reach this conclusion, it suffices to solve the two equations; we do not have to draw Figure 38.1. However, the diagram shows us at once that the game has a unique equilibrium, in which both players’ actions exceed ½ c, facts that serve to check the results of our algebra.) In the game in this example, each player has a unique best response to every action of the other player, so that the best response functions are lines. If a player has many best responses to some of the other players’ actions, then her best response function is “thick” at some points; several examples in the next chapter have this property (see, for example, Figure 64.1). Example 37.1 is special also because the game has a unique Nash equilibrium—the best response functions cross once. As we have seen, some games have more than one equilibrium, and others have no equilibrium. A pair of best response functions that illustrates some of the possibilities is shown in Figure 39.1. In this figure the shaded area of player 1’s best re