Question

In a 3-player game, player 1 can choose either Up or Down as strategies. Player 2 can chose either Left or Right as strategies. Player 3 can choose either Table 1 or Table 2 as strategies.

Which of the following strategy profile(s) is/are Nash Equilibrium?

• A. (Up, Left, Table 1)
• B. (Down, Right, Table 1)
• C. (Down, Left, Table 2)
• D. (Up, Right, Table 2)

Answer 1

The Correct Option is A, D

To determine the Nash Equilibrium in this 3-player game, we need to identify the strategy profiles where no player has an incentive to deviate, given the strategies of the others.

Step-by-step Analysis: 

1. Consider each table separately for Player 3.
2. For Table 1:
   • Strategy Profile (Up, Left):
     • Payoffs: Player 1 = 3, Player 2 = 2, Player 3 = 5.
     • Check deviations:
       • Player 1: Switching to Down gives payoff 2 (worse).
       • Player 2: Switching to Right gives payoff 1 (worse).
       • Player 3: No deviation as changing table is not evaluated separately.
   • Strategy Profile (Down, Right):
     • Payoffs: Player 1 = 5, Player 2 = 4, Player 3 = 6.
     • Check deviations:
       • Player 1: Switching to Up gives payoff 4 (worse).
       • Player 2: Switching to Left gives payoff 6 (worse).
       • Player 3: No deviation as changing table is not evaluated separately.
3. For Table 2:
   • Strategy Profile (Up, Right):
     • Payoffs: Player 1 = 4, Player 2 = 5, Player 3 = 7.
     • Check deviations:
       • Player 1: Switching to Down gives payoff 3 (worse).
       • Player 2: Switching to Left gives payoff 3 (worse).
       • Player 3: No deviation as changing table is not evaluated separately.

After evaluating these profiles, we conclude that the Nash Equilibria are:

• (Up, Left, Table 1)
• (Up, Right, Table 2)