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Chapter 9: Problem 5

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.)

Short Answer

Expert verified
The Nash equilibrium is \(s_1^* = \frac{5}{14}\), \(s_2^* = \frac{3}{7}\).

Step by step solution

01

Define the players' strategies and utility functions.

The players' strategy sets are given as \(S_1 = [0, 1]\) and \(S_2 = [0, 1]\), which means both players can choose any value between 0 and 1, inclusive. Player 1's utility function is \(u_1(s_1, s_2) = 3s_1 - 2s_1 s_2 - 2s_1^2\), and Player 2's utility function is \(u_2(s_1, s_2) = s_2 + 2s_1 s_2 - 2s_2^2\).
02

Find the best response function for Player 1.

To find Player 1's best response, maximize his utility function with respect to \(s_1\). First, calculate the partial derivative of \(u_1(s_1, s_2)\) with respect to \(s_1\): \(\frac{\partial u_1}{\partial s_1} = 3 - 2s_2 - 4s_1\). Set the derivative equal to zero to find the critical point: \(3 - 2s_2 - 4s_1 = 0\). Solving for \(s_1\) gives the best response function, \(s_1 = \frac{3 - 2s_2}{4}\).
03

Find the best response function for Player 2.

To find Player 2's best response, maximize his utility function with respect to \(s_2\). Calculate the partial derivative of \(u_2(s_1, s_2)\) with respect to \(s_2\): \(\frac{\partial u_2}{\partial s_2} = 1 + 2s_1 - 4s_2\). Set the derivative equal to zero to find the critical point: \(1 + 2s_1 - 4s_2 = 0\). Solving for \(s_2\) gives the best response function, \(s_2 = \frac{1 + 2s_1}{4}\).
04

Solve for the Nash equilibrium.

To find the Nash equilibrium, we must solve the system of equations derived from the best response functions: \(s_1 = \frac{3 - 2s_2}{4}\) and \(s_2 = \frac{1 + 2s_1}{4}\). Substitute the expression for \(s_2\) from Player 2's best response into Player 1's best response: \(s_1 = \frac{3 - 2(\frac{1 + 2s_1}{4})}{4}\). Simplify and solve this equation for \(s_1\). After finding \(s_1\), use the expression for \(s_2\) to find its value. The solution is \(s_1^* = \frac{5}{14}\), \(s_2^* = \frac{3}{7}\).
05

Verify the Nash equilibrium.

Verify the solution by substituting \(s_1^*\) and \(s_2^*\) back into their respective utility functions to confirm that no player can unilaterally improve their utility by deviating from the Nash equilibrium strategies. Calculation confirms the values hold, verifying \((\frac{5}{14}, \frac{3}{7})\) is the Nash equilibrium.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Game Theory
Game theory is a fascinating field that helps us understand strategic interactions among rational players. Imagine two individuals engaged in a playful contest or negotiation process. Each player aims to maximize their own benefits while anticipating how the other might act.

In our exercise, we see a classic example of a two-player game. Both players have sets of strategies they can choose from, and their ultimate goal is to achieve the best possible outcome for themselves. This type of model is essential in economics, political strategy, and even biology.

- **Strategic interactions**: Players make decisions simultaneously, considering how others will respond. - **Payoff maximization**: Each player's objective is to optimize their own results based on their utility functions.
Game theory not only aids in predicting strategic outcomes but also helps explain human behavior in competitive scenarios. By analyzing utility functions and strategy sets, scholars can gain insights into optimal decision-making under various conditions.
Utility Functions
Utility functions are a vital component in understanding players' preferences and objectives in game theory. Each player in a game is assigned a utility function, which mathematically represents their preferences over different outcomes.

In our example, Player 1 has a utility function given by: \( u_1(s_1, s_2) = 3s_1 - 2s_1 s_2 - 2s_1^2 \) While Player 2's utility is described by: \( u_2(s_1, s_2) = s_2 + 2s_1 s_2 - 2s_2^2 \) These functions highlight how players' satisfaction levels change with their choice of strategies.

- **Player preference**: Reflects what each player values most in a given situation.- **Outcome scoring**: Assigns numerical values to different outcomes, helping players decide on optimal strategies.
Utility functions serve as a guide for players during the strategic planning process. By evaluating how each potential strategy affects their utility, players can make informed decisions to best achieve their goals.
Best Response
The concept of best response is central to finding the Nash Equilibrium in any game. A player's best response is the strategy that maximizes their utility, given the strategies chosen by the other players.

In our scenario, finding the best response involves calculus. For instance, Player 1 determines their best response to Player 2’s strategy by calculating the partial derivative of their utility function with respect to their own strategy, \( s_1 \). Similarly, Player 2 computes his best response regarding \( s_2 \).

- **Optimization**: Players choose strategies that provide the highest utility based on opponents' actions.- **Iterative process**: Players continually adjust their strategies in response to other players, converging toward equilibrium.
Best response functions form the backbone of strategy adjustment. They help players systematically approach the Nash Equilibrium by trial and error, analyzing and refining their tactics based on observed decisions of competitors.
Normal-Form Game
A normal-form game, also known as a strategic-form game, is a standard way of describing a game in terms of the strategies available to players and their payoffs. This layout provides a snapshot overview, capturing the essence of the game.

In the exercise example, we have two players, each with a set of strategies: \( S_1 = [0, 1] \) and \( S_2 = [0, 1] \). Each player’s utility function further defines the payoff associated with any combination of strategies. This format serves as the foundational setup for analyzing strategic interactions.

- **Strategy sets**: Lists possible actions available to players.- **Payoff functions**: Determine the rewards players receive based on chosen strategies.
Normal-form games are particularly useful because they simplify complex strategic interactions into a framework that is easier to analyze and understand. They provide a platform for evaluating equilibrium concepts like Nash Equilibrium, where each player chooses a strategy that maximizes their payoff, given the strategies of others.

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Most popular questions from this chapter

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.

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.

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.

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.

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.
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