Assignment Task: Players: Candidates L and R.
Actions: Each candidate chooses a number between 0 and 1, which represents the tax rate at which the candidate, while on the campaign trail, says that incomes in the highest tax bracket ought to be taxed.
Preferences: A candidate wins if she receives a greater fraction of the vote than her opponent, and ties if she receives half the vote. Voters vote for whichever candidate has adopted the position closest to their own favorite position (the voter's preferred rate, that is, the rate at which the voter believes top incomes ought to be taxed). Assume that the voters' favorite positions are uniformly distributed along the interval [0,1], which means that the median voter's favorite position is 1/2.
If y is the action chosen by the winning candidate, or the position chosen by both candidates, then candidate L's payoff is y, and candidate R's payoff is -y. If the candidates tie by choosing two distinct positions x and y on opposite sides of the median voter, then L's payoff is (x+y)/2 and R's payoff is -(x+y)/2.
Describe the set of Nash equilibria to the game.