by David K. Levine, Andrea Mattozzi and Salvatore Modica on April 23, 2017
In a political
contest such as voting or lobbying groups offer costly effort -
votes, money - and the side that offers the most wins. In voting
effort is provided regardless of the outcome. With bribes only the
winner pays. The tripartite auction theorem says that the groups
do not care whether all pay or only the winner.
We analyze a political contest over a prize between two groups
with costly effort provision. The group offering the greatest
effort wins a prize. The tie breaking rule is endogenous: this
means that a group that is willing to bid a little bit more may be
assumed to win in case of a tie.
Such a contest is called an auction. We examine three different
types of auctions in which the high bidder pays their own bid -
these are called first price auctions. In the first two auctions
only the winner pays: in the ascending bid auction each group sets
a reserve price above which it drops out of the bidding; in the
sealed bid auction both groups simultaneously submit bids. The
third auction is the all-pay auction in which both groups
simultaneously submit bids and each pays their own bid regardless
of whether they win or lose.
Our model of behavior is that of Nash equilibrium. Nash
equilibrium requires that each group makes the best decision for
itself given its beliefs and that its beliefs about the other
group's strategy is correct. We also require that neither group
use a strategy that is weakly dominated. A strategy is weakly
dominated if there is alternative strategy which never does worse
and sometimes does better. Roughly speaking: only a fool would use
a weakly dominated strategy.
A key concept in auction theory is the willingness to bid. This is
the most that a group will bid for a certainty of getting the
prize instead of a certainty of losing. We analyze only the case
in which the willingness to bid of the two groups is not
identical. We refer to the group with the higher willingness to
bid as advantaged, to the one with the lower willingness to bid as
disadvantaged. The surplus is the difference between the value of
the prize to the advantaged group and the cost to the advantaged
group of matching the willingness to bid of the disadvantaged
group.
Tripartite auction theorem: For all three auctions the
disadvantaged group gets zero and the advantaged group gets the
surplus.
It is not difficult to explain why this is the case. Start with
the fairly obvious fact that no group bids more than its
willingness to bid. No matter what the other group does bidding
your willingness to bid is at least as good as bidding more - and
if the other group bids zero you would be better off bidding your
willingness to bid rather than higher. In technical terms
bidding your willingness to bid weakly dominates any strategy of
bidding higher.
Since the disadvantaged group will not bid more than its
willingness to bid, the advantaged group does not get less than
the surplus. It would be foolish to do so since it can do better
by bidding just a bit more than the willingness to bid of the
disadvantaged group.
By contrast the disadvantaged group gets zero. To understand why
this is, think about the lowest bid by either group. Obviously
such a bid is not terribly likely to win. In fact: one of the
groups must lose for sure when it makes this bid - if both
groups had a chance of winning at this lowest bid the advantaged
group should raise its bid a tiny bid to raise its chance of
winning a lot. A group that loses for sure cannot earn more than
zero. We already know the advantaged group earns at least the
surplus so it must be the disadvantaged group that loses for sure
and earns no more than zero. Since either group can guarantee zero
by bidding zero the disadvantaged group must get exactly zero.
Because the disadvantaged group gets zero the highest bid by the
advantaged group must be equal to the willingness to bid of the
disadvantaged group. Otherwise the disadvantaged group could earn
more than zero by slightly beating that highest bid.
When the advantaged group bids the willingness to bid of the
disadvantaged group it cannot get more than the surplus. As we
already know it cannot get less than the surplus we conclude that
it gets at exactly this much.