Question

Two people, 1 and 2, are engaged in a joint project. Person 𝑖∈{1,2} puts in effort 𝑥𝑖 (0≤ 𝑥𝑖 ≤ 1), and incurs cost 𝐶_𝑖 (𝑥_𝑖 ) = 𝑥_𝑖 . The monetary outcome of the project is 4𝑥₁𝑥₂ which is split equally between them. Considering the situation as a strategic game, the set of all Nash Equilibria in pure strategies is

• A. {(0, 0), (1, 1)}
• B. \(\{{(0,0),(\frac{1}{4},\frac{1}{4}),(\frac{3}{4},\frac{3}{4}),(1,1)\}}\)
• C. \(\{(0,0),(\frac{1}{2},\frac{1}{2}),(1,1)\}\)
• D. a null set

Answer 1

The Correct Option is C

To solve this problem, we need to analyze the situation as a strategic game involving Nash Equilibrium. Each player (Person 1 and Person 2) must decide on their level of effort (\(x_1\) and \(x_2\)), knowing that the project's output, \(4x_1x_2\), is shared equally. Additionally, each player's cost of effort is represented by \(C_i(x_i) = x_i\).

1. Calculate the Payoff Functions: 
   Each player wants to maximize their payoff, which is given by the equation: \(U_1 = \frac{4x_1x_2}{2} - x_1 = 2x_1x_2 - x_1\) for Player 1, and similarly:
   \(U_2 = \frac{4x_1x_2}{2} - x_2 = 2x_1x_2 - x_2\) for Player 2.
2. Find Best Responses:
   We differentiate the utility functions with respect to each variable and set the derivative to zero to find the best response:
   • For Player 1: \(\frac{\partial U_1}{\partial x_1} = 2x_2 - 1 = 0 \rightarrow x_2 = \frac{1}{2}\), indicating that Player 1's best response depends on Player 2's effort. If \(x_2 = \frac{1}{2}\), then \(x_1 = \frac{1}{2}\) maximizes Player 1's payoff.
   • For Player 2: \(\frac{\partial U_2}{\partial x_2} = 2x_1 - 1 = 0 \rightarrow x_1 = \frac{1}{2}\), indicating Player 2's best response depends on Player 1's effort. If \(x_1 = \frac{1}{2}\), then \(x_2 = \frac{1}{2}\) maximizes Player 2's payoff.
3. Determine Nash Equilibria:
   The Nash Equilibrium occurs where players' actions are mutual best responses. Thus, potential equilibria occur at:
   • The corner solution where \(x_1 = 0\) and \(x_2 = 0\), leading to a utility of \(0\) for both.
   • The solution \((x_1 = \frac{1}{2}, x_2 = \frac{1}{2})\) where their responses are mutual, with utility \(\frac{1}{2}\) for both.
   • The maximum effort solution where \(x_1 = 1\) and \(x_2 = 1\), resulting in payoff \(\frac{3}{2}\) after the cost.

Therefore, the correct answer is: \(\{(0,0),(\frac{1}{2},\frac{1}{2}),(1,1)\}\).