A tournament solution is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
that maps an
oriented complete graph to a nonempty
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of its
vertices. It can informally be thought of as a way to find the "best" alternatives among all of the alternatives that are "competing" against each other in the tournament. Tournament solutions originate from
social choice theory
Social choice theory or social choice is a theoretical framework for analysis of combining individual opinions, preferences, interests, or welfares to reach a ''collective decision'' or ''social welfare'' in some sense.Amartya Sen (2008). "Soci ...
,
but have also been considered in
sports competition
Competition is a rivalry where two or more parties strive for a common goal which cannot be shared: where one's gain is the other's loss (an example of which is a zero-sum game). Competition can arise between entities such as organisms, indivi ...
,
game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
,
multi-criteria decision analysis
Multiple-criteria decision-making (MCDM) or multiple-criteria decision analysis (MCDA) is a sub-discipline of operations research that explicitly evaluates multiple conflicting criteria in decision making (both in daily life and in settings ...
,
biology
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...
,
webpage ranking, and
dueling bandit problems.
In the context of social choice theory, tournament solutions are closely related to Fishburn's C1 social choice functions,
and thus seek to show who the best candidates are among all candidates.
Definition
A
tournament (graph) is a tuple
where
is a set of vertices (called ''alternatives'') and
is a
connex and asymmetric
binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
over the vertices. In social choice theory, the binary relation typically represents the
pairwise majority comparison between alternatives.
A tournament solution is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
that maps each tournament
to a nonempty subset
of the alternatives
(called the ''choice set''
) and does not distinguish between isomorphic tournaments:
:If
is a
graph isomorphism
In graph theory, an isomorphism of graphs ''G'' and ''H'' is a bijection between the vertex sets of ''G'' and ''H''
: f \colon V(G) \to V(H)
such that any two vertices ''u'' and ''v'' of ''G'' are adjacent in ''G'' if and only if f(u) and f(v) a ...
between two tournaments
and
, then
Examples
Common examples of tournament solutions are:
*
Copeland's method
Copeland's method is a ranked voting method based on a scoring system of pairwise "wins", "losses", and "ties". The method has a long history:
* Ramon Llull described the system in 1299, so it is sometimes referred to as "Llull's method"
* The ...
*
Top cycle
* Slater set
*
Bipartisan set
* Uncovered set
* Banks set
* Minimal covering set
* Tournament equilibrium set
References
{{Reflist
Voting theory