Game Theory Problem
Brought to Europe by the explorer James Cook in 1771, a tattoo is made by inserting indelible ink into the dermis layer of the skin. Though that does not sound particularly appealing, tattoos have become quite popular in the United States. According to a survey by Pew Research in 2008, 14% of all Americans have at least one tattoo, and an astounding 36% of Americans between the ages of 18 and 25 do.
In thinking about the decision to have a tattoo, let us suppose there are two types of young people: those who find a tattoo cool, but only if a majority of young people have them—the trendy types; and those who find a tattoo cool, but only if a minority of young people have them—the counterculture types. The payoff to a trendy type from having a tattoo is 10 + 2 x m, where m is the number of all other young people with a tattoo. The payoff to a counterculture type from having a tattoo is 350 - 4 x m. For example, if 30 trendy types have a tattoo and 20 counterculture types have a tattoo, then the payoff to a trendy type with a tattoo is 10 + 2 x 49 = 108 and the payoff to a counter-culture type with a tattoo is 350 - 4 x 49 = 154. For either type of person, the payoff from not having a tattoo is 100. Assume there are a total of 75 young people, with 50 who are trendy and 25 who are counterculture.
a. Is it a Nash equilibrium for only counterculture types to have a tattoo?
b. Is it a Nash equilibrium for only trendy types to have a tattoo?
c. Is it a Nash equilibrium for all young people to have a tattoo?
d. Is it a Nash equilibrium for no one to have a tattoo?
e. Find all Nash equilibria. (Hint: First note that either all T kids have a tattoo or none do. Thus, there are the following types of strategy profiles to consider: 1) no one has a tattoo; 2) some but not all CC types have a tattoo and none of the T types have a tattoo; 3) all of the CC types have a tattoo and none of the T types have a tattoo; 4) all T types have a tattoo but none of the CC types; 5) all T types have a tattoo and some but not all of the CC types; and 6) everyone has a tattoo.)