Question

The sum of the payoffs to the players in the Nash equilibrium of the following simultaneous game is ............
 

	Player Y
	C	NC
Player X	X: 50, Y: 50	X: 40, Y: 30
	X: 30, Y: 40	X: 20, Y: 20

Hint:

In a Nash equilibrium, each player’s strategy is the best response to the other player's strategy. The payoff is maximized for both players given their strategies.

Answer 1

We are given the following payoff matrix for the game:

Player X / Player Y	C	NC
C	X: 50, Y: 40	X: 50, Y: 30
NC	X: 30, Y: 20	X: 30, Y: 20

Step 1: To find the Nash equilibrium, look for the strategy pair where neither player can improve their payoff by unilaterally changing their strategy. We analyze each player's best response to the other player's strategy.

- When Player Y chooses \( C \), Player X's best response is to choose \( C \) (because 50 > 30).
- When Player Y chooses \( NC \), Player X's best response is to choose \( C \) (because 50 > 30).
- When Player X chooses \( C \), Player Y's best response is to choose \( C \) (because 40 > 30).
- When Player X chooses \( NC \), Player Y's best response is to choose \( C \) (because 20 >= 20, no change).

Thus, the Nash equilibrium occurs when both players choose \( C \).

Step 2: The payoffs in this equilibrium are:
- Player X: 50,
- Player Y: 40.

The sum of the payoffs is:
\[ 50 + 40 = 90 \]

Final Answer:
\[ \boxed{90} \]