Glenn Shafer’s paper is a powerful appeal for a wider use of betting ideas and intuitions in statistics. He admits that p-values will never be completely replaced by betting scores, and I discuss it further in Appendix A (one of the two online appendices that I have prepared to meet the word limit). Both p-values and betting scores generalize Cournot’s principle , but they do it in their different ways, and both ways are interesting and valuable.
Other authors have referred to betting scores as Bayes factors and e-values [23, 7]. For simple null hypotheses, betting scores and Bayes factors indeed essentially coincide [7, Section 1, interpretation 3], but for composite null hypotheses they are different notions, and using “Bayes factor” to mean “betting score” is utterly confusing to Bayesians . However, the Bayesian connection still allows us to apply Jeffreys’s [9, Appendix B] rule of thumb to betting scores; namely, a p-value of 5% is roughly equivalent to a betting score of , and a p-value of 1% to a betting score of 10. This agrees beautifully with Shafer’s rule (6), which gives, to two decimal places:
for , instead of Jeffreys’s (slight overshoot);
for , instead of Jeffreys’s (slight undershoot).
The term “e-values” emphasizes the fundamental role of expectation in the definition of betting scores (somewhat similar to the role of probability in the definition of p-values). It appears that the natural habitat for “betting scores” is game-theoretic while for “e-values” it is measure-theoretic; therefore, I will say “e-values” in the online appendices (Appendix A and ), which are based on measure-theoretic probability.
In the second online appendix  I give a new example showing that betting scores are not just about communication; they may allow us to solve real statistical and scientific problems (more examples will be given by my co-author Ruodu Wang). David Cox  discovered that splitting data at random not only allows flexible testing of statistical hypotheses but also achieves high efficiency. A serious objection to the method is that different people analyzing the same data may get very different answers (thus violating “inferential reproducibility” [6, 8]). Using e-values instead of p-values remedies the situation.
Thanks to Ruodu Wang for useful discussions and for sharing with me his much more extensive list of advantages of e-values. This research has been partially supported by Amazon, Astra Zeneca, and Stena Line.
-  James O. Berger and Mohan Delampady. Testing precise hypotheses (with discussion). Statistical Science, 2:317–352, 1987.
-  Jacob Bernoulli. Ars Conjectandi. Thurnisius, Basel, 1713.
-  Antoine-Augustin Cournot. Exposition de la théorie des chances et des probabilités. Hachette, Paris, 1843.
-  David R. Cox. A note on data-splitting for the evaluation of significance levels. Biometrika, 62:441–444, 1975.
-  Annie Duke. Thinking in Bets. Portfolio, New York, 2018.
-  Steven N. Goodman, Daniele Fanelli, and John P. A. Ioannidis. What does research reproducibility mean? Science Translational Medicine, 8:341ps12, 2016.
-  Peter Grünwald, Rianne de Heide, and Wouter M. Koolen. Safe testing. Technical Report arXiv:1906.07801 [math.ST], arXiv.org e-Print archive, June 2020.
-  Leonhard Held and Simon Schwab. Improving the reproducibility of science. Significance, 17(1):10–11, 2020.
-  Harold Jeffreys. Theory of Probability. Oxford University Press, Oxford, third edition, 1961.
-  Erich L. Lehmann and Joseph P. Romano. Testing Statistical Hypotheses. Springer, New York, third edition, 2005.
-  Christian P. Robert. Bayes factors and martingales, 2011. Entry in blog “Xi’an’s Og” for August 11.
-  Thomas Sellke, M. J. Bayarri, and James Berger. Calibration of p-values for testing precise null hypotheses. American Statistician, 55:62–71, 2001.
-  Glenn Shafer. From Cournot’s principle to market efficiency. In Jean-Philippe Touffut, editor, Augustin Cournot: Modelling Economics, chapter 4. Edward Elgar, Cheltenham, 2007.
-  Glenn Shafer. Personal communication. May 8, 2020.
-  Glenn Shafer. Testing by betting: A strategy for statistical and scientific communication. To be read before the Royal Statistical Society on September 9 and to appear as discussion paper in the Journal of the Royal Statistical Society A, 2020.
-  Glenn Shafer, Alexander Shen, Nikolai Vereshchagin, and Vladimir Vovk. Test martingales, Bayes factors, and p-values. Statistical Science, 26:84–101, 2011.
-  Judith ter Schure and Peter Grünwald. Accumulation bias in meta-analysis: the need to consider time in error control. Technical Report arXiv:1905.13494 [stat.ME], arXiv.org e-Print archive, May 2019.
-  Vladimir Vovk. A logic of probability, with application to the foundations of statistics (with discussion). Journal of the Royal Statistical Society B, 55:317–351, 1993.
-  Vladimir Vovk. A note on data splitting with e-values: online appendix to my comment on Glenn Shafer’s “Testing by betting”. Technical Report arXiv:2008.11474 [stat.ME], arXiv.org e-Print archive, August 2020.
-  Vladimir Vovk. Testing randomness online. Technical Report arXiv:1906.09256 [math.PR], arXiv.org e-Print archive, March 2020.
-  Vladimir Vovk, Bin Wang, and Ruodu Wang. Admissible ways of merging p-values under arbitrary dependence. Technical Report arXiv:2007.14208 [math.ST], arXiv.org e-Print archive, July 2020.
-  Vladimir Vovk and Ruodu Wang. True and false discoveries with e-values. Technical Report arXiv:1912.13292 [math.ST], arXiv.org e-Print archive, December 2019.
-  Vladimir Vovk and Ruodu Wang. Combining e-values and p-values. Technical Report arXiv:1912.06116 [math.ST], arXiv.org e-Print archive, May 2020.
-  Vladimir Vovk and Ruodu Wang. Combining p-values via averaging. Biometrika, 2020. To appear, published online.
Appendix A Cournot’s principle, p-values, and e-values
This is an online appendix to the main comment. It is based, to a large degree, on Glenn Shafer’s ideas about the philosophy of statistics. After a brief discussion of p-values and e-values as different extensions of Cournot’s principle, I list some of their advantages and disadvantages.
a.1 Three ways of testing
. If a given event has a small probability, we do not expect it to happen; this is Cournot’s bridge between probability theory and the world. (This bridge was discussed already by James Bernoulli; Cournot’s  contribution was to say that this is the only bridge.) See Figure 1.
Cournot’s principle requires an a priori choice of a rejection region
. Its disadvantage is that it is binary: either the null hypothesis is completely rejected or we find no evidence whatsoever against it. Ap-variable
is a nonnegative random variablesuch that, for any , ; one way to define p-variables is via Shafer’s (3). An e-variable is a nonnegative random variable such that ; one way to define e-variables is via Shafer’s first displayed equation in Section 2. In p-testing, we choose a p-variable in advance and reject the null hypothesis when the observed value of (the p-value) is small, and in e-testing, we choose an e-variable in advance and reject the null hypothesis when the observed value of (the e-value) is large. In both cases, binary testing becomes graduated: now we have a measure of the amount of evidence found against the null hypothesis.
We can embed Cournot’s principle into both p-testing,
and e-testing (as Shafer [15, Section 2.1, (1)] explains),
There are numerous ways to transform p-values to e-values (to calibrate them) and essentially one way () to transform e-values to p-values, as discussed in detail in 
. The idea of calibrating p-values originated in Bayesian statistics ([1, Section 4.2], [18, Section 9], ), and there is a wide range of admissible calibrators. Transforming e-values into p-values is referred to as e-to-p calibration in , where is shown to dominate any e-to-p calibrator [22, Proposition 2.2].
Moving between the p-domain and e-domain is, however, very inefficient. Borrowing the idea of “round-trip efficiency” from energy storage, let us start from the highly statistically significant () p-value , transform it to an e-value using Shafer’s [15, (6)] calibrator
and then transform it back to a p-value using the only admissible e-to-p calibrator: . The resulting p-value of is not even statistically significant ().
a.2 Some comparisons
Both p-values and e-values have important advantages, and I think they should complement (rather than compete with) each other. Let me list a few advantages of each that come first to mind. Advantages of p-values:
P-values can be more robust to our assumptions (perhaps implicit). Suppose, for example, that our null hypothesis is simple. When we have a clear alternative hypothesis (always assumed simple) in mind, the likelihood ratio has a natural property of optimality as e-variable (Shafer [15, Section 2.2]), and the p-variable corresponding to the likelihood ratio as test statistic is also optimal (Neyman–Pearson lemma [10, Section 3.2, Theorem 1]). For some natural classes of alternative hypotheses, the resulting p-value will not depend on the choice of the alternative hypothesis in the class (see, e.g., [10, Chapter 3] for numerous examples; a simple example can be found in [19, Section 4]). This is not true for e-values.
There are many known efficient ways of computing p-values for testing nonparametric hypotheses that are already widely used in science.
In many cases, we know the distribution of p-values under the null hypothesis: it is uniform on the interval . If the null hypothesis is composite, we can test it by testing the simple hypothesis of uniformity for the p-values. A recent application of this idea is the use of conformal martingales for detecting deviations from the IID model .
Advantages of e-values (starting from advantages mentioned by Shafer [15, Section 1]):
E-values appear naturally as a technical tool when applying the duality theorem in deriving admissible functions for combining p-values .