Question

An aggressive dictator is deciding where to hide his weapon of mass destruction, while the UN is deciding where to look for it. (The US has gotten the dictator to allow UN inspectors to look for it, in exchange for not attacking the dictator’s country.) The payoff to the dictator of successfully hiding the weapon is 5 and from having it found is 2. For the UN, the payoff to finding the weapon is 9 and from not finding it is 4. The dictator can hide it in facility X, Y or Z. The UN inspectors have limited time and personnel, so they can inspect any two facilities –that is, the UN’s pure strategies are XY, XZ, YZ.
a.Write out the normal form game in matrix form, with players 1 = dictator, 2 = UN inspectors.
b.Find a mixed strategy Nash equilibrium.
c. What is the probability that the weapon is found?

Answer 1

Answer:

(a)

z = UN

Let the Matrix of players 1 = dictator

            X             Y            Z

           9   ²       9   ²       4   ⁵

           9   ²       4   ⁵       9   ²

           4   ⁵       9   ²       9   ²

Above is the normal form of game in matrix form with player 1 = dictator strategy.

Now, the matrix of players inspectors

    XY     9  ²      9  ²      4  ⁵

    XZ     9  ²      4  ⁵      9  ²

    YZ     4  ⁵      9  ²      9  ²

The above strategy is player 2 = UN inspectors strategy.

(b)

Mixed strategy N ash equilibrium

Therefore,

The matrix of player 1 = dictator hiding strategy :

 X   =    9,2    9,2    4,5

  Y   =    9,2    4,5    9,2

  Z   =    4,5    4,5    9,2

The matrix of player 2 = inspector secret strategy

 XY  =   9,2    9,2    4,5

 XZ  =   9,2    4,5    4,5

 YZ  =   4,5    9,2    9,2

(c)

The probability that the weapon is found

=> Seeing the table above, we will calculate 6 of 9 times of inspector calculating weapon.

=> The probability that the weapon is found is

  = 6 / 9

  = 2 / 3

  = 0.67

Explanation: