Bishop–Cannings theorem

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[1] (from their Theorem 3). The usefulness of the results comes from the fact that they can be used to directly find ESSes algebraically, 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.[2]

Notes
  1. Bishop, D.T. and Cannings, C. (1978). A generalized war of attrition. Journal of Theoretical Biology 70:85–124.
  2. Maynard Smith, J. 1982 Evolution and the Theory of Games, Cambridge University Press.


gollark: There are *low-cost* ones.
gollark: Rust's pretty fast and has the neat safety thing going on.
gollark: Or you could just use high*er* level languages which make it somewhat harder to randomly corrupt memory or whatever.
gollark: Probably, but at least the logic errors generally lead to "oops that does not work correctly I must now fix it" instead of "oh look, the application is now vulnerable to remote code execution".
gollark: I doubt they can actually pick up on all the exciting variety of memory corruption bugs and such.
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