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Published online by Cambridge University Press: 27 June 2025
We introduce two equivalent methodologies for defining and computing a position's mean (value of playing Black rather than White) and temperature (value of next move). Both methodologies apply in more generality than the classical one. The first, following the notion of a free market, relies on the transfer of a “tax” between players, determined by continuous competitive auctions. The second relies on a generalized thermograph, which reduces to the classical thermograph when the game is loop-free.
When a sum of games is played optimally according the economic rules described, the mean (which is additive) and the temperature determine the final score precisely.
This framework extends and refines several classical notions. Thus, finite games that are numbers in Conway's sense are now seen to have negative natural temperatures. All games can now be viewed as terminating naturally with integer scores when the temperature reaches —1.
Introduction
At every position of a game such as Go or Domineering, there are two very important questions:
Who is ahead, and by how much?
How big is the next move?
Following Conway [1976], classical abstract combinatorial game theorists answer these questions with a value and an incentive, as discussed in Winning Ways [Berlekamp et al. 1982]. These answers are precisely correct when the objective of the game is to get the last legal move. Values and incentives are themselves games, and can quickly become complicated.
Our ideal economist takes a different view. Following Hanner [1959] and Milnor [1953], he views the game as a contest to accumulate points, which can eventually be converted into cash.
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