## Punishment and accuracy level in contests

##### Abstract

In the literature on contests, punishments have received much less attention than
prizes. One possible reason is that punishing the bottom player(s) in a contest
where all contestants are not allowed to quit, while effective in increasing contestants' total effort, often violates individual rationality constraints. But what
will happen in an open contest where all potential contestants can choose whether
or not to participate? In chapter 1, we study a model of this type and allow
the contest designer to punish the bottom participant according to their performances. We conclude that punishment is often not desirable (optimal punishment
is zero) when the contest designer wants to maximize the expected total effort,
while punishment is often desirable (optimal punishment is strictly positive) when
the contest designer wants to maximize the expected highest individual effort.
In the literature on imperfectly discriminating contests, researchers normally
assume that the contest designer has a certain level of accuracy in choosing the winner, which can be represented by the discriminatory power r in the Power Contest
Success Function (the Power CSF, proposed by Tullock in 1980). With symmetric
contestants, it is well known that increasing accuracy (r) always increases total effort when the pure-strategy equilibrium exists. In chapter 2, we look at the cases
where the contestants are heterogeneous in ability. We construct an equilibrium
set on r > 0, where a unique pure-strategy equilibrium exists for any r below a
critical value and a mixed-strategy equilibrium exists for any r above this critical
value. We find that if the contestants are sufficiently different in ability, there always exists an optimal accuracy level for the contest designer. Additionally, as we
increase the difference in their abilities, the optimal accuracy level decreases. The
above conclusions provide an explanation to many phenomena in the real world
and may give guidance in some applications.
In chapter 3, we propose the Power Contest Defeat Function (the Power CDF)which eliminates one player out at a time over successive rounds. We show that
the Power CDF has the same good qualities as the Power Contest Success Function
(the Power CSF) and is more realistic in some cases. We look at both the Power
CSF mechanism (selecting winners in sequence) and the Power CDF mechanism
(selecting losers in sequence) and show that punishments increase expected total
e¤orts signi cantly. More interestingly, we also find that when the contestants'
effort levels are different, the Power CDF mechanism is more accurate in finding
the correct winner (the one who makes the greatest effort) and the Power CSF
mechanism is more accurate in finding the correct loser (the one who makes the
smallest effort).