Represent the game in normal form and find its nash

Question

Problem 1:

Consider the following game:

(a) Suppose that the Column player announces that he will play X with probability 0.5 and Y with probability 0.5 i.e., 1/2X ⊕ 1/2Y.

Identify all best response strategies of the Row player, i.e., BR(1/2 X ⊕ 1/2 Y) ?

(b) Identify all best response strategies of the Column player to Row playing 1/2 A ⊕ 1/2 B, i.e. BR(1/2 A ⊕ 1/2 B)?

(c) What is BR(1/5 X ⊕ 1/5 Y ⊕ 3/5 Z)?

(d) What is BR(1/5 A ⊕ 1/5 B⊕ 3/5 C)?

Problem 2: Here comes the Two-Finger Morra game again:

To exercise notation and concepts involved in calculating payoffs to mixed strategies, calculate the following (u_R, u_C stand for the payoffs to Row and Column respectively):

(a) u_R(0.4 R1 ⊕ 0.6 R2, C2) =

(b) uC(0.4 C1 ⊕ 0.6 C2, R3) =

(c) uR(0.3 R2 ⊕ 0.7 R3, 0.2 C1 ⊕ 0.3 C2 ⊕ 0.5 C4 ) =

(d) u (0.7 C2 ⊕ 0.3 C4, 0.7 R1 ⊕ 0.2 R2 ⊕ 0.1 R3) =

Problem 3:

For the game above:

(1) Draw the best response function for each player using the coordinate system below. Mark Nash equilibria on the diagram.

(2) List the pair of mixed strategies in Nash equilibrium.

(3) Calculate each player's payoffs in Nash equilibrium.

Problem 4:

In the Two-Finger morra game above suppose Row decided to play a mix of R1 and R2 and Column decided to play a mix of C1 and C3. In other words, assume that the original 4 x 4 game is reduced to the 2 x 2 game with R1 and R2 and C1 and C3. Using our customary coordinate system:

(a) Draw the best response functions of both players in the coordinate system as above.

(b) List all Nash equilibria in the game.

(c) Calculate each player's payoff in Nash equilibrium.

Problem 5:

Lucy offers to play the following game with Charlie: "let us show pennies to each other, each choosing either heads or tails. If we both show heads, I pay you $3. If we both show tails, I pay you $1. If the two don't match, you pay me $x." For what values of x is it profitable for Charlie to play this game?

Problem 6:

(a) Represent this game in normal form (payoff matrix).

(b) Identify all pure strategy Nash equilibria. Which equilibrium is the subgame perfect Nash equilibrium?

Important: In game theory people often use the same name to identify actions in different information nodes. This is the case above. In extensive form games, however, these actions are formally and conceptually different. You need to keep this distinction in mind when solving this problem. An easy way not to make a mistake is by using your own naming convention, e.g., X and X'.
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Problem 7:

Represent the following game in normal form and find its Nash equilibria.