Consider the electoral competition game discussed in class, but where the players are the citizens. Any citizen may, at cost c > 0, become a candidate. Assume further that the only position a candidate can take is their own true, favorite position, x, so that a candidate's only decision is whether to stand as a candidate or not. After all citizens have decided (simultaneously) whether to become candidates, each citizen votes for their favorite candidate as in Hotelling's model. Citizens care about the position of the winning candidate; a citizen whose favorite position is x loses |x - x*| if the winning candidate's position is x*. Winning confers a benefit of b. A citizen who becomes a candidate but does not tie for first place earns a payoff of |x - x*| - c, where again x* is the winning candidate's position. Assume that for every position x, there is at least one citizen for whom x is their favorite position.
a. Show that if b ≥ 2c, the game has a Nash equilibrium in which at most one citizen becomes a candidate and describe the position this candidate takes.
b. Now allow b and c to be any values. Could there be an equilibrium in which two candidates with positions different from the median position, m, become candidates?
5. Consider a third-price auction, where the winner is the bidder who submits the highest bid, but he/she only pays the third highest bid. Assume that you compete against two other bidders, whose valuations you are unable to observe, and that your valuation for the object is $10. Show that bidding above your valuation (with a bid of, say $15) can be a best response to the other bidders' bids, while submitting a bid that coincides with your valuation $10 might not be a best response to your opponent's bids.
