Wikipedia:Reference desk/Archives/Mathematics/2022 August 24

= August 24 =

Difference between game theory and combinatorial game theory?
Is there any analysis on the difference between game theory and combinatorial game theory? 2A02:908:424:9D60:0:0:0:4A03 (talk) 14:03, 24 August 2022 (UTC)


 * There are many ways to model games. Traditional game theory covers games where choices are made simultaneously and there is a numerical payoff for each player rather than a winner and loser. Meanwhile combinatorial game theory covers games where players take turns and there is a winner and loser. You may want to look at Game theory for the place of combinatorial games within the context of games in general. I'm not sure what you mean by an analysis of the difference; there are different situations which are modeled differently and have different theories. --RDBury (talk) 14:58, 24 August 2022 (UTC)
 * I think the phraseology "combinatorial game theory" is fairly specific to the style of analyzing games developed and popularized by John Horton Conway. It's a little more specific even than RDBury's formulation.  They're games where players take turns, and the first player to have no legal move loses, and this always happens in finite time.  Basically generalizations of nim, or maybe misère nim (not sure which way standard nim goes).  This analysis is closely related to the surreal numbers, which can be considered games of this sort. --Trovatore (talk) 17:21, 24 August 2022 (UTC)
 * One of the problems in terms of terminology is that the theories here really come from different traditions. Strictly speaking, what we think of as game theory comes from economics and human behavior, and many of the aspects of it are colored by that lens.  The kind of games that we see in traditional game theory are things like the Prisoner's dilemma and the Free-rider problem and the Gift-exchange game; what it does is use "game" as a metaphor to analyze human-human interactions in a mathematical way.  Combinatorial game theory comes more from analyzing actual games, like Chess and Go, not as metaphors, but as actual games.  Being rooted in mathematics, both "game theory" and "combinatorial game theory" are about developing generalized mathematical tools and structures that 'could be used' to analyze situations, but then to explore those structures in an abstract way from their application (that's what mathematics does, it generalizes abstractly).  But the application of the two different theories ends up looking different because one is asking "How can I model this economic situation as a game to better understand optimal behavior" and one is asking "What are all the ways one can win at chess?"  The tools of the theories can be applied to broader aspects than that, but the questions they started answering are different, and that of course colors the literature available for looking at them.  -- Jayron 32 18:13, 25 August 2022 (UTC)