Question

Consider the following game in extensive form: The King of Siam has many enemies who would like to poison him. His Highness seeks a Royal Taster to taste his food before He eats. The King generously offers a wage w < 10 to anyone who will take the job. The going wage for labor in Siam is w0. The values his life at 10. His Enemies value his death at 10 also, and poisoning attempts cost 5.

Stage 1: The King chooses to Hire or Not Hire a Taster at wage w.
Stage 2: The Enemies choose to Poison or Not Poison.

The game tree with moves and payoffs is as follows:

a. Write down the strategies for Players 1 (King) and 2 (Enemies). Write down the Normal form of the game
b. Find the Nash equilibria of the game.
c. Use Backward Induction to solve the game.
d. Will the King need to offer a wage greater than the going wage wo to attract a willing taster? Why or why not

Answer 1

a). The normal form in the game is given below as :

The King

The Enemies Hire Not hire

To Poison (-5, 10-w) (5, 0)

Not to Poison (0, 10-w) (0, 10)

b). For the enemies there is no such dominant strategy and the king also does not have any dominant strategy. Therefore, there is no pure strategy of Nash equilibrium.

c). The backward induction helps to suggest that there is a spine for the game when the king hires a Royal taster and the enemies do not poisons the king : (Hire, Not poison).

d). Yes the king needs to offer a wage that is greater than going wage,
`$w_0$` in order to attract the willing taster as the value of the life of his more.