The skill parameters reflect the abilities of players: the larger the value of a
player’s skill parameter, the more proficient is the player. If the production
cost is according to a linear function,
we can normalize the payoff functions
such that a player’s skill parameter
can be interpreted as the reciprocal of
his or her production cost per unit effort. The game as defined here allows
us to study equilibrium outcomes under different prize allocation mechanisms, such as assigning fixed shares
of a prize budget in decreasing order
of invested efforts, or splitting a prize
budget among players in proportion to
their effort investments. The prize allocation can be interpreted as the winning probabilities for an indivisible
prize item, or as the shares of an infinitely divisible prize. The game allows
us to study equilibrium outcomes for
different types of production costs. For
example, it is common to consider
linear production costs, we refer to as
constant marginal production costs, under
which the production cost per unit effort is constant; in particular, we refer
to unit marginal production cost when
the production cost per unit effort is of
unit value. We may also consider production costs with either decreasing
or increasing marginal costs. It is noteworthy that the game as defined here
formally corresponds to an auction,
where efforts, skills, and production
costs are in correspondence with bids,
valuations, and payments, respectively
We say a game is with complete information if the players have perfect
information about each other’s skill
parameters. A game with complete information can be used as a model of a
contest when the players are informed
about who is going to participate in
the contest and about the skills of
the participants. For example, a situation like this can be found in competition-based software development
platforms such as TopCoder, where
a contest takes place after a registration phase, which reveals identities
of participants. A game is said to be
with incomplete information if the value
of each player’s skill parameter is his
or her private information. In a game
with incomplete information, skill parameters are assumed to be random
variables according to a prior distribution, which is a common knowledge.
The design of skill-rating methods is
based on statistical models of ranking outcomes developed from 1920s
onward. More recent developments
include skill-rating methods that allow
for contests among two or more teams
of players, which are common in online
gaming and online labour platforms.
New results have been recently developed in the area of statistical inference
for statistical models of ranking data,
including new characterizations of the
accuracy of various skill parameter estimators and new iterative methods for
skill parameter estimation.
In this article, we survey some main
results of contest theory. Specifically,
we discuss basic game theory models
of contests that are found in online services. We explain the conditions under
which to optimally allocate prizes to
maximize a given objective, such as the
total effort or the maximum individual
effort, in a strategic equilibrium. We will
focus on games in which players make
simultaneous effort investments; the
games that involve some aspect of sequential play are only briefly discussed.
We consider both games that model a
single contest (see Figure 1) and games
that model a system of two or more simultaneous contests (Figure 2). Simultaneous contests are common in the
context of online crowdsourcing platforms. We explain basic principles of
popular skill rating systems and point
out some new results in this area. We
conclude with an outlook on future research directions.
This article complements existing surveys on the game-theoretic aspects in contest theory, for example,
Corchon, 9 Konrad, 25 and Nitzan. 32 We
provide an overview of some of the topics covered in the book by Vojnović, 42
where the reader may find a more extensive coverage of references.
Strategic Game Models of Contests
The standard game theory framework
for studying contests is based on the
assumption that agents are rational
and strategic players who invest effort
with a selfish goal to maximize their individual payoffs. The payoff of a player
combines the utility of winning a prize
and the cost of production. Specifically,
we consider a normal-form game that
models a contest, defined by:
˲ Set of two or more players:
N={ 1, 2,…,n};
˲ Payoff functions: for any given vec-
tor of efforts b = (b1, b2, … , bn ), the pay-
off of player i is given by
si (b) = vixi (b) – c(bi )
where
˲ v1, v2, … , vn are positive-valued
skill parameters,
˲ x(b) := (x1 (b), x2 (b), … , xn (b)) is
prize allocation, and
˲ c(x) is a production cost function.
Figure 1. Single contest.
Contest
Players
12 n
Figure 2. A system of simultaneous contests: Edges indicate effort investment opportunities.
21
Players
Contests
n
21 m