Question

In a Union-Management negotiation, the following are the annual percentages of wage increases for Union for various combinations of union and management strategies:

Management

M1 M2 M3

U1 1 3 3

U2 4 2 2

Union U3 3 2 3

U4 3 4 1

U5 2 1 2

9a. (5 points) After eliminating all possible dominated strategies, list the Union payoff matrices for the 4 subgames that are developed by taking 3 of the 4 Union strategies to match the 3 Management strategies.

9b. (5 points) Find the best strategy and value of the game for Union with the following payoff matrix for one of the subgames:

Management

M1 M2 M3

U2 4 2 2

Union U3 3 2 3

U4 3 4 1

9c. (10 points) We have solved in class the best strategy and value of the game for Union with the following payoff matrix for one of the subgames:

Management

M1 M2 M3

U1 1 3 3

Union U3 3 2 3

U4 3 4 1

Let q1, q2, and q3 be the respective probabilities for Management to play strategies M1, M2, and M3. Then use the same Principle of Maximin as in deraiving the best mixed strategy for Union to find the best strategy and value of the game for Management. Specifically,

9c1 (5 points) show the three independent linear equations for q1, q2, and q3.

9c2 (5 points) Show the correct solution for these 3 probabilities from these 3 independent linear equations.

Answer 1

Answer:

Given data

Union Management

  

  

  

Let the best strategy for union is then using the rule of min - max

Min from coloumns side is

Max from rows side is

Max (Min) = Max { 2, 2 , 1 }

= 2

Min (Max) = Min { 4, 4, 3}

= 3

and also non dominating so apply mix strategies for union.

let xj's be the variables for union

Mu_{x} V =

s.t

  

  

  

then corresponding dual problem let yj's be the dual variables.

Min u = = Max

s. t

  

j = 1 , 2 , 3

So xj's are the values of for sluck variables then

  

  

and value of the game

then

= 1 / 3

= 5 / 11

= 7 / 33

and value of game = 8 / 3

gc

  

Min from coloumn is

Max from row is

Max Min = 2

Min Max = 3

then min strategy

Max V = Min

s. t

then

and

  

  

  

= 1 / 5

= 2 / 5

= 2 / 5

game value = 13 / 5