Chapter 9

Compute the Nash equilibria of the following location game. There are two people who simultaneously select numbers between zero and one. Suppose player 1 chooses \(s_{1}\) and player 2 chooses \(s_{2}\). If \(s_{i}

Find the Nash equilibrium of the following normal-form game: \(S_{1}=[0,1], S_{2}=[0,1], u_{1}\left(s_{1}, s_{2}\right)=3 s_{1}-2 s_{1} s_{2}-2 s_{1}^{2}\), and \(u_{2}\left(s_{1}, s_{2}\right)=\) \(s_{2}+2 s_{1} s_{2}-2 s_{2}^{2}\). (The solution is interior, so you can use calculus.)

Consider a game in which, simultaneously, player 1 selects any real number \(x\) and player 2 selects any real number \(y\). The payoffs are given by: $$ \begin{aligned} &u_{1}(x, y)=2 x-x^{2}+2 x y \\ &u_{2}(x, y)=10 y-2 x y-y^{2} \end{aligned} $$ (a) Calculate and graph each player's best-response function as a function of the opposing player's pure strategy. (b) Find and report the Nash equilibria of the game. (c) Determine the rationalizable strategy profiles for this game.

Is the following statement true or false? If it is true, explain why. If it is false, provide a game that illustrates that it is false. "If a Nash equilibrium is not strict, then it is not efficient."

This exercise asks you to consider what happens when players choose their actions by a simple rule of thumb instead of by reasoning. Suppose that two players play a specific finite simultaneous-move game many times. The first time the game is played, each player selects a pure strategy at random. If player \(i\) has \(m_{i}\) strategies, then she plays each strategy \(s_{i}\) with probability \(1 / m_{i}\). At all subsequent times at which the game is played, however, each player \(i\) plays a best response to the pure strategy actually chosen by the other player the previous time the game was played. If player \(i\) has \(k\) strategies that are best responses, then she randomizes among them, playing each strategy with probability \(1 / k\). (a) Suppose that the game being played is a prisoners' dilemma. Explain what will happen over time. (b) Next suppose that the game being played is the battle of the sexes. In the long run, as the game is played over and over, does play always settle down to a Nash equilibrium? Explain. (c) What if, by chance, the players happen to play a strict Nash equilibrium the first time they play the game? What will be played in the future? Explain how the assumption of a strict Nash equilibrium, rather than a nonstrict Nash equilibrium, makes a difference here. (d) Suppose that, for the game being played, a particular strategy \(s_{i}\) is not rationalizable. Is it possible that this strategy would be played in the long run? Explain carefully.

Consider a two-player game and suppose that \(s^{*}\) and \(t^{*}\) are Nash equilibrium strategy profiles in the game. Must it be the case that \(\left\\{s_{1}^{*}, t_{1}^{*}\right\\} \times\left\\{s_{2}^{*}, t_{2}^{*}\right\\}\) is a weakly congruous strategy set? Explain why or why not.

Consider the following \(n\)-player game. Simultaneously and independently, the players each select either X, Y, or Z. The payoffs are defined as follows. Each player who selects X obtains a payoff equal to \(\gamma\), where \(\gamma\) is the number of players who select Z. Each player who selects Y obtains a payoff of \(2 \alpha\), where \(\alpha\) is the number of players who select X. Each player who selects \(Z\) obtains a payoff of \(3 \beta\), where \(\beta\) is the number of players who select Y. Note that \(\alpha+\beta+\gamma=n\). (a) Suppose \(n=2\). Represent this game in the normal form by drawing the appropriate matrix. (b) In the case of \(n=2\), does this game have a Nash equilibrium? If so, describe it. (c) Suppose \(n=11\). Does this game have a Nash equilibrium? If so, describe an equilibrium and explain how many Nash equilibria there are.

Heather and David (players 1 and 2 ) are partners in a handmade postcard business. They each put costly effort into the business, which then determines their profits. However, unless they each exert at least 1 unit of effort, there are no revenues at all. In particular, each player \(i\) chooses an effort level \(e_{i} \geq 0\). Player \(i\) 's payoff is \(u_{i}\left(e_{i}, e_{j}\right)= \begin{cases}-e_{i} & \text { if } e_{j}<1 \\\ e_{i}\left(e_{j}-1\right)^{2}+e_{i}-\frac{1}{2} e_{i}^{2} & \text { if } e_{j} \geq 1\end{cases}\) where \(j\) denotes the other player. (a) Prove that \((0,0)\) is a Nash equilibrium. (b) Graph the players' best responses as a function of each other's strategies. (c) Find all of the other Nash equilibria.

Suppose you know the following for a particular three-player game: The space of strategy profiles \(S\) is finite. Also, for every \(s \in S\), it is the case that \(u_{2}(s)=3 u_{1}(s), u_{3}(s)=\left[u_{1}(s)\right]^{2}\), and \(u_{1}(s) \in[0,1] .\) (a) Must this game have a Nash equilibrium? Explain your answer. (b) Must this game have an efficient Nash equilibrium? Explain your answer. (c) Suppose that in addition to the information given above, you know that \(s^{*}\) is a Nash equilibrium of the game. Must \(s^{*}\) be an efficient strategy profile? Explain your answer; if you answer "no," then provide a counterexample.