a. The TU values are 2 and 4 for the bimatrix game with optimal threats enforcing a mixed strategy. b. The TU values are 2 and 4 with optimal threats enforcing a mixed strategy. c. The TU values are 7 and 8 for the pure Nash Equilibrium. d. The TU values are 0 and 0 for the mixed Nash Equilibrium without threats.
To find the TU (trembling hand perfect) values, associated side payments, and optimal threats for each bimatrix game, we need to analyze the possible strategies and payoffs.
a.
3,2 5,1
0,0 1,4
There is no pure NE (Nash Equilibrium) in this game. The mixed NE involves Player 1 playing (2/3,1/3) and Player 2 playing (1/3,2/3). TU values are 2 and 4. The optimal threats can be made to enforce this mixed strategy.
b.
3,2 1,4
0,0 5,1
No pure NE. The mixed NE is Player 1 (1/3,2/3) and Player 2 (2/3,1/3). TU values are 2 and 4. The optimal threats can be made to enforce this mixed strategy.
c.
7,4 2,8
4,5 5,6
Pure NE is (Row 1, Column 2). TU values are 7 and 8.
d.
-1,0 3,0
-3,0 7,0
No pure NE. The mixed NE is Player 1 (1/3,2/3) and Player 2 (1/3,2/3). TU values are 0 and 0. No side payments are needed for these equilibrium strategies.
In summary:
a. TU values: 2, 4; Optimal threats to enforce mixed strategy.
b. TU values: 2, 4; Optimal threats to enforce mixed strategy.
c. TU values: 7, 8; No threats needed for pure NE.
d. TU values: 0, 0; No threats needed for mixed NE.