Question

Players \(A\) and \(B\) have found \(\$ 100\) on the sidewalk and are arguing about how it should be split. A passerby suggests the following game: "Each of you state the number of dollars that you wish \(\left(d_{d}, d_{B}\right) .\) If \(d_{A}+d_{B} \wedge 100\) you can keep the figure you name and I'll take the remainder. If \(d_{A}+d_{B}>100,\) I'll keep the \(\$ 100 . "\) Is there a unique Nash equilibrium in this game of continuous strategies?

Short answer

Answer: The unique Nash equilibrium of the game is (50, 50), where both players A and B choose to take $50 each.

Step-by-step solution

Understand the concepts and constraints

In this game, players A and B need to choose the amounts \((d_A, d_B)\) such that \(d_A + d_B \leq 100\), otherwise the passerby takes the entire amount. The Nash equilibrium will be a pair of amounts \((d_A^*, d_B^*)\) such that no player can be better off by changing their strategy, given the other player's strategy.

Setup players' utility functions

The objective of both players is to maximize their utilities. Let's assume that the utility function of player A is \(U_A(d_A, d_B) = d_A\), and the utility function of player B is \(U_B(d_A, d_B) = d_B\). Both players want to maximize their respective utility functions, given the other player's strategy.

Find the Best Response Functions

To find the Nash equilibrium, we need to find the best response functions (BRF) for both players. The BRFs are the amounts that each player will choose to maximize their utility, given the other player's amount. For player A, the BRF is given by the following equation: \(BRF_A(d_B) = \max_{d_A} U_A(d_A, d_B) = \max_{d_A} d_A\) Subject to the constraint: \(d_A + d_B \leq 100\) Similarly, for player B, the BRF is given by the following equation: \(BRF_B(d_A) = \max_{d_B} U_B(d_A, d_B) = \max_{d_B} d_B\) Subject to the constraint: \(d_A + d_B \leq 100\)

Solve for the Nash equilibrium

The Nash equilibrium can be found by solving the BRFs for both players: \((d_A^*, d_B^*) = (BRF_A(d_B^*), BRF_B(d_A^*))\) Using the constraint \(d_A + d_B \leq 100\), we can rewrite the BRFs as follows: \(BRF_A(d_B^*) = 100 - d_B^*\) \(BRF_B(d_A^*) = 100 - d_A^*\) Now, substituting the BRFs for equilibrium amounts \((d_A^*, d_B^*)\), we get: \((d_A^*, d_B^*) = (100 - d_B^*, 100 - d_A^*)\) Since the game is symmetric, i.e., both players face the same constraints and have the same objective, the Nash equilibrium will have both players choosing the same amounts: \(d_A^* = d_B^* = \frac{100}{2} = 50\) Thus, the unique Nash equilibrium of the game is \((d_A^*, d_B^*) = (50, 50)\).