Question

2. In the classic Dr. Seuss tale The Sneetches, some Sneetches have stars on their bellies and others do not. Or, Seuss puts it: Now, the Star-Belly Sneetches Had bellies with stars. The Plain-Belly Sneetches Had none upon thars. The Star-Belly Sneetches are seen as superior because of their marking that makes them rarer than their plain-bellied neighbors. Suppose the population of Sneetches is N=200, and only 30 sneetches have stars. The payoffs to having a star or plain belly, as a function of the number of Sneetches with stars, n, are: Pstar(n) = 400 - 3n as the more Sneetches with stars, the less it is valued Pplain(n) = 50+ n, as the more star-bellies there are, the more a rare plain belly is more highly valued. a) Calculate current payoffs to show that being a Star-Belly sneetch is more highly valued. Sadly, there's nothing the plain-bellied sneetches can do to change their situation. Enter Sylvester McMonkey McBean with his amazing "star on" machine. Now plain-belly Sneetches can add a star. b) Assuming the "star on" machine is free to use, what is/are the Nash equilibrium(a) number of sneetches with stars? Show clearly how you find this.

Answer 1

(a) The current payoffs can be calculated simply by plugging in n = 30.

Pstar = 400 - 3n = 400 - 3(90) = 130

Pplain = 50 + n = 50 + 30 = 80

As the Pstar > Pplain, therefore it is profitable to have a star on your belly.

(b) Nash equilibrium is achieved when one is indifferent between the strategies i.e. the respective payoffs are the same for both strategies.

Therefore we equate the payoffs to solve for $n^*$ .

400 – 3n* = 50+ n*

4n* = 350

n* = 87,5