The game above is a simultaneous three player game between players 1, 2, and 3.

Question

The game above is a simultaneous three player game between players 1, 2, and 3. Player 1 chooses between A and B, Player 2 between C and D and Player 3 between E and F. In the game above, the strategy profile in which Player 1 plays ??????? , Player 2 plays ????????, and Player 3 plays ?????? is a Nash Equilibrium profile.

for Player 2. UESTION 7 Player Ill Player ll Player ll A 2,3,1 3,1,0 B 3,2,1 13,2 Player 133 1,3,2 Player l B 3,10 004 The game above is a simultaneous three player game between players 1.2. and 3. Player 1 chooses betweeh A and B. Player n the game above, the strategy profile in which Piayer 1 plays Nash Equilibrium profie. ,Player 2 plays UESTION B Player ll

Explanation / Answer

Let us assume that left hand side matrix is Y and right hand side matrix is Z

Player 3. Let’s start by evaluating the payoffs for player 3 when Player 2 selects D (first column). For instance, when player 1 chooses A and player 2 selects D (in the top left-hand corner of either matrix), player 3 prefers to respond with matrix A, which gives him a payoff of 3, rather than with B, which only yields a payoff of 1. A similar argument applies for the second arrow, which fixes the other players’ strategy profile at (B,D). Hence, we obtain that player 2’s best responses are BR3(D,A) = Y and BR3(D,B) = Y. If player 2 selects C (in the second column of each matrix), player 3’s pairwise comparisons are BR3(C,A) = Y and BR3(C,B) = Y

Player 2. In particular, we take player 1’s strategy as given (fixing the row) and player 3’s as given. We obtain that player 2’s best responses are BR2 (A, Y) = {D,C}, when player 1 chooses A, BR2(B,Y) = D when player 1 selects B .Visually, notice that we are now fixing our attention on a matrix (strategy of player 3) and on a row (strategy of player 1). Similarly, when player 3 chooses matrix B, we obtain that player 2’s best responses are BR2(A,Z) = D and BR2( B, Z) = C.

Player 1. Let’s finally identify player 1’s best responses, given player’s 2 strategy (fixing the column) and given player 3’s strategy (fixing the matrix).
When player 3 chooses Y (left matrix), player 1’s best responses become
BR1(D,Y) = B and BR1(C,Y) = B

When player 3 chooses Z (right matrix), player 1’s best responses become
BR1(D,Z) = B and BR1(C,Z) = A

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