Simultaneous and Strategic Turnout.Suppose that three voters are faced with the dilemmaof voting for a tax cut. Each voter has a simple choice, since they all prefer that taxesbe cut – the only difficulty is that turning out to vote is somewhat costly.Specifically,each voter receives a benefit ofbfrom the tax cut being passed and incurs a cost ofc < bfrom turning out to vote for the tax cut.All voters make their decision privately andsimultaneously.(a) Express this as a normal form game.(b) Derive all pure strategy Nash equilibria.(c) Derive a mixed strategy Nash equilibrium (i.e., an equilibrium in which at least oneplayer is playing both actions with positive probability).(d) Which equilibrium is more efficient (in terms of maximizing the sum of the players’payoffs)?3.Weak-link Coordination.Consider the followingn-player coordination game. Each playerisimultaneously announces an integer,ai∈A={1,2,3,4,5}. Lettinga= (a1, . . . , an)denote the profile of actions, playeri’s payoff is given byui(a) =aiifai= min[a]min[a]-aiifai>min[a].(a) Derive all pure strategy Nash equilibria.(b) Are there any mixed strategy Nash equilibria (i.e., an equilibrium in which at leastone player is playing both actions with positive probability)?(c) Which pure strategy equilibrium is more efficient (in terms of maximizing the sum ofthe players’ payoffs)?(d) Which pure strategy equilibrium is least efficient (in terms of minimizing the sum ofthe players’ payoffs)? Come up with a brief argument for why this equilibrium mightbe more reasonable than the most efficient one.
1

#### You've reached the end of your free preview.

Want to read the whole page?

- Spring '11
- JohnPaddy
- Game Theory, Nash equilibria, Nash, pure strategy, mixed strategy Nash