Bishop-Cannings theorem
Encyclopedia
The Bishop–Cannings theorem is a theorem
in evolutionary game theory
. It states that (i) all members of a mixed evolutionarily stable strategy
(ESS) have the same payoff (Theorem 2), and (ii) that none of these can also be a pure ESS (from their Theorem 3). The usefulness of the results comes from the fact that they can be used to directly find ESSes algebra
ically, rather than simulating the game and solving it by iteration
.
The logic of (i) also applies to Nash equilibria
(all strategies in the support of a mixed strategy receive the same payoff).
The theorem was formulated by Tim Bishop and Chris Cannings at Sheffield University, who published it in 1978.
A review is given by John Maynard Smith
in Evolution and the Theory of Games
, with proof in the appendix.
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
in evolutionary game theory
Evolutionary game theory
Evolutionary game theory is the application of Game Theory to evolving populations of lifeforms in biology. EGT is useful in this context by defining a framework of contests, strategies and analytics into which Darwinian competition can be modelled. It originated in 1973 with John Maynard Smith...
. It states that (i) all members of a mixed evolutionarily stable strategy
Evolutionarily stable strategy
In game theory and behavioural ecology, an evolutionarily stable strategy , which is sometimes also called an evolutionary stable strategy, is a strategy which, if adopted by a population of players, cannot be invaded by any alternative strategy that is initially rare. An ESS is an equilibrium...
(ESS) have the same payoff (Theorem 2), and (ii) that none of these can also be a pure ESS (from their Theorem 3). The usefulness of the results comes from the fact that they can be used to directly find ESSes algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
ically, rather than simulating the game and solving it by iteration
Iteration
Iteration means the act of repeating a process usually with the aim of approaching a desired goal or target or result. Each repetition of the process is also called an "iteration," and the results of one iteration are used as the starting point for the next iteration.-Mathematics:Iteration in...
.
The logic of (i) also applies to Nash equilibria
Nash equilibrium
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...
(all strategies in the support of a mixed strategy receive the same payoff).
The theorem was formulated by Tim Bishop and Chris Cannings at Sheffield University, who published it in 1978.
A review is given by John Maynard Smith
John Maynard Smith
John Maynard Smith,His surname was Maynard Smith, not Smith, nor was it hyphenated. F.R.S. was a British theoretical evolutionary biologist and geneticist. Originally an aeronautical engineer during the Second World War, he took a second degree in genetics under the well-known biologist J.B.S....
in Evolution and the Theory of Games
Evolution and the Theory of Games
Evolution and the Theory of Games is a 1982 book by the British evolutionary biologist John Maynard Smith on evolutionary game theory. In it, Maynard Smith summarises work on evolutionary game theory that had developed in the 1970s, to which he made several important contributions...
, with proof in the appendix.