You have been asked to solve the following game: Choose the correct analysis bel

Question

You have been asked to solve the following game:

Choose the correct analysis below.

Question 1 options:

A)

Arlene has examined the game and determined that Colin's strategy B dominates his strategy C. She therefore eliminated strategy C and reduced the game. Arlene then noticed that Colin's strategy B also dominates strategy A. She therefore eliminated strategy A as well, meaning that Colin must play strategy B. Rose should therefore choose between her strategy A and strategy C, which each give the same outcome in response to Colin's choice of B. The value of the game is thus 3.

B)

Boyd thinks Arlene is wrong to eliminate strategy A for Colin since it is only weakly dominated by strategy B. Boyd therefore thinks Colin should play strategy B with slightly higher probability than strategy A, but might consider A from time to time. As a result, the best Colin can be sure of getting is 2, and that is the value of the game.

C)

Charlie thinks both Boyd and Arlene are wrong, and that there is no solution involving dominant strategies at all. He carries out a mixed strategy analysis and determines that Colin should play (0.5 A, 0 B, 0.5 C) and that Rose should play any strategy where the probabilities for B and C are the same (for instance, 0.4 A, 0.3 B, 0.3 C will work).

D)

Danielle likes Charlie's solution, but would like to point out that there is indeed a dominant strategy that simplifies the task. Colin's strategy B is dominated by Colin's strategy C (Arlene has it backwards). After Colin's B is eliminated, the mixed strategy calculations give the results described by Charlie.

A)

Arlene has examined the game and determined that Colin's strategy B dominates his strategy C. She therefore eliminated strategy C and reduced the game. Arlene then noticed that Colin's strategy B also dominates strategy A. She therefore eliminated strategy A as well, meaning that Colin must play strategy B. Rose should therefore choose between her strategy A and strategy C, which each give the same outcome in response to Colin's choice of B. The value of the game is thus 3.

B)

Boyd thinks Arlene is wrong to eliminate strategy A for Colin since it is only weakly dominated by strategy B. Boyd therefore thinks Colin should play strategy B with slightly higher probability than strategy A, but might consider A from time to time. As a result, the best Colin can be sure of getting is 2, and that is the value of the game.

C)

Charlie thinks both Boyd and Arlene are wrong, and that there is no solution involving dominant strategies at all. He carries out a mixed strategy analysis and determines that Colin should play (0.5 A, 0 B, 0.5 C) and that Rose should play any strategy where the probabilities for B and C are the same (for instance, 0.4 A, 0.3 B, 0.3 C will work).

D)

Danielle likes Charlie's solution, but would like to point out that there is indeed a dominant strategy that simplifies the task. Colin's strategy B is dominated by Colin's strategy C (Arlene has it backwards). After Colin's B is eliminated, the mixed strategy calculations give the results described by Charlie.

Colin A B C Rose B2 2 0

Explanation / Answer

In zero-sum property if one gains then another one loses. Zero-sum games are a specific example of constant sum games where the sum of each outcome is always zero.

maxmin pure strategy muor(A)

minmax pure strategy muoc(A)

muor(A) = muoc(A) ifand only if A has a saddle point

a)If we eliminate muor(A) or muoc(A) saddle point is not eliminated.

b) If muor(A) not= muoc(A) then saddle point is absent. So saddle point is eliminated.

c) Rose/Colin 1 5

1 (1,-1) (-1,1)

5 (-5,5) (5,-5)

Colin = 1 -1

-5 5

Rose = -1 1

5 -5

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