Question

Styles aragraph equally talented athletes, expect to compete for the county championship in 1. Jacob and William, two the 400-meter hurdles in the up-coming season. Each plans to train hard, putting in several hours per week. We will use the Tullock model to describe their behavior. For each athlete, the winning is worth 24 hours per week, so we measure the prize as 24 hours. The cost of an hour of effort is, of course, one hour. The probability is as described in the Tullesk model [see page 5 of the slides "A Model of a Contest"]. a. Suppose that Jacob plans to train 10 hours per week, and that William plans to train 15 hours. What is the probability that William will be the county champ? b. What is William's payoff (i.e. (prob x prize) - cost)? Note: the payoff is measured in hours, not money. Suppose that William increases his training time to 20 hours per week. Does hi C. payoff rise or fall? Explain. d. Suppose that William reduces his training time to 10 hours per week. Does his payoff rise or fall (compared to part b)? Explain. e. Is the allocation where Jacob trains 10 and William trains 15 a (Nash) equilibrium? Why or why not?

Answer 1

ANSWER:

Given that,

a)

As, Jacob and William, both are equally talented athletes, probability that William will be the county champ will depend on number of hours William has done in comparison to Jacob.

Probability that William will be the county champ = number of train hours for William / Total number of hours for both

= 15/ (15 +10) = 0.6

b)

William’s payoff = (prob x prize) – cost

= 0.6* 24 - 15 = -0.6

c)

If the  training time increased to 20 hours per week,

Probability that William will be the county champ = 20 /(20 + 10) = 0.667

William’s payoff = (prob x prize) – cost

= 0.667 * 24 - 20 = -3.992

So, the payoff will fall.

d)

If the  training time decreased to 10 hours per week,

Probability that William will be the county champ = 10 /(10 + 10) = 0.5

William’s payoff = (prob x prize) – cost

= 0.5 * 24 - 10 = 2

So, the payoff will rise.

e)

If the training time for William decreased to 14 hours per week,

Probability that William will be the county champ = 14 /(14 + 10) = 0.4167

William’s payoff = (prob x prize) – cost

= 0.4167 * 24 - 14 = -3.9992

which is less than the payoff as in part (b). So, the William strategy would not be for 15 hour training. Hence the allocation where Jacob trains 10 and William trains 15 is not a (Nash) equilibrium.