March 7, 2022

Game-theoretic probability in physics

Glenn Shafer, business and statistics, Rutgers
Dustin Lazarovici, philosophy, Lausanne
Leah Henderson, philosophy, Groningen
Eddy Keming Chen, philosophy, UC San Diego

In measure-theoretic probability, expected values (prices) may change through time, but the way in which they change is set in advance by a comprehensive probability measure.  Game-theoretic probability generalizes this by introducing players who may take actions and set prices as a betting game proceeds.  In the case of discrete time, this means that a probability tree is replaced by a decision tree.  Statistical testing of the prices is still possible.  But instead of testing a comprehensive probability measure by checking that a simple event of small probability specified in advance does not happen (Cournot’s principle), we test the prices, or the forecaster who sets them, by checking that a simple betting strategy specified in advance does not multiply its capital by a large factor.  See Glenn Shafer’s “Testing by betting” (2021).

Because it provides explicitly for a decision maker, game-theoretic probability accommodates John von Neumann’s axioms for quantum mechanics in a straightforward way, as explained on pp. 189-191 of Shafer and Vovk’s 2001 book.  As further explained on pp. 216-217 of their 2019 book, it also allows a programmer to test whether a quantum computer is performing correctly.

Gurevich and Vovk have argued that the game-theoretic formulation helps take the mystery out of the role of negative probabilities in Wigner’s quasi-probability distribution.  The mystery dissolves because the distribution is nothing more than a rule that tells a forecaster how to set prices.

Advocates of Cournot’s principle in statistical mechanics have pointed out that the Gibbs distribution is needed only to rule out events to which it gives probability near zero (see, e.g., Goldstein et al. 2020).  Any measure equivalent in the sense of absolute continuity would do the same job; the Gibbs distribution is distinguished from the others mainly by its implausible assumption of particles’ initial independence of each other.  Ken Huira (2021) has argued that the game-theoretic formulation can again provide clarification. The distribution does not “generate” outcomes; it merely sets prices.

Questions for the panel:

  1. Is the game-theoretic formulation adequate for understanding probability in physics?  
  2. Is the game-theoretic formulation philosophically acceptable to most theoretical physicists?  If not, is this because they want a generative and creative probability, not merely a probability with predictive value? 
  3. In the modern model for multiple regression, introduced by R. A. Fisher in 1922, explanatory variables that are set by an experimenter are treated as constants (Aldrich 2005). Does a Bohmian account of a programmer’s statistical test of a quantum computer require a similar device?


  1. Glenn Shafer and Vladimir Vovk (2001), Probability and Finance:  It’s Only a Game! Wiley
  2. John Aldrich (2005), Fisher and regressionStatistical Science 20(4):4010417
  3. Glenn Shafer and Vladimir Vovk (2019), Game-Theoretic Foundations for Probability and Finance, Wiley
  4. Yuri Gurevich and Vladimir Vovk (2020), Betting with negative probabilities
  5. Goldstein, S., Lebowitz, J. L., Tumulka, R., & Zanghì, N. (2020). Gibbs and Boltzmann entropy in classical and quantum mechanics. In Statistical Mechanics and Scientific Explanation: Determinism, Indeterminism and Laws of Nature (pp. 519-581).
  6. Ken Huira (2021), Gibbs distribution from sequentially predictive form of the second law
  7. Glenn Shafer (2021), Testing by betting, Testing by betting: a strategy for statistical and scientific communication (PREPUBLICATION VERSION), with discussion and responseJournal of the Royal Statistics Society, Series A 184(2):407-478, 2021.