Comments on "Betting interpretations of probability" by Glenn Shafer Philip Dawid University of Cambridge This is an interesting and informative account of the history and current status of the betting approach to probability, which has been sorely undervalued for far too long. I hope that it will help to revive interest in the topic. Glenn mentions de Finetti's use of betting ideas to develop his criterion of coherence: if You are to avoid sure loss to a sceptic who makes a collection of bets at odds You offer, You should ensure that they are based on probabilities obeying the usual axioms. This use of betting arguments to concentrate the mind, before any outcomes are observed, can be described as "ex ante", as opposed to "ex post" uses, in which reality decides on the outcomes and the parties are actually subjected to the appropriate losses or gains. Only in the latter do we need to take into account the actual behaviour of the world, determining our empirical preformance. This important distinction is similar to the dual purpose of penal punishments: ex ante, to keep you and me on the straight and narrow, for fear of what would happen to us if we strayed; ex post, to rain down on malefactors their due punishment. However, I think Glenn is wrong in suggesting that de Finetti's interest was confined to the ex ante argument. He has a body of work (see for example de Finetti, 1962, and Dawid and Galavotti, 2009) which takes ex post empirical assessment seriously; and he famously ran a competition for his students requiring probabilistic forecasting of the results of Italian football games, with rewards and penalties based on the quadratic proper scoring rule. One interesting aspect of betting and game-theoretic approaches to probability is the role of strategies for the various players. For example, a typical theorem of game-theoretic probability is that Sceptic has a strategy that can force some outcome (generally a disjunction of (a) a property of Sceptic's capital, and (b) a property relating Forecaster's forcasts to Reality's outomes), no matter what the other players do. Here it is important that Sceptic be operating a strategy, since this means (i) that we are explicit as to what information he is allowed to access and use, and (ii) we know that he will be able to succeed in all circumstances. It is equally important that the other players are not being required to operate strategies, but can play essentially arbirarily at any point (for example, they are not precluded from taking additional side-information into account, or just making something up on the spur of the moment): by varying these choices, the other players might be able to affect which element of the disjunction will hold, but they can not avoid the disjunction. This universality means that game-theoretic results are actually very much stronger than their counterparts in standard probability theory, which typically follow from considering the special case that both Forecaster and Reality are operating strategies. "Defensive forecasting" involves a somewhat weaker set-up, since now Forecaster is seeking a strategy that will ensure some outcome, but not now for anything that the other players (Sceptic and Reality) may do, since Sceptic is assumed to operate a known strategy. Glenn's talk was about Probability, but there are also some fundamental uses of betting arguments in Statistics, and I wonder how these can be related? For example, Fisher coined the concept of a "relevant subset", which, when it exists, can be used to discredit, (say) a 95% confidence interval, by allowing a system that bets, at the declared odds of 19:1, against the event that the interval contains the target parameter, and has positive expected gain no matter what value that parameter may have. There would appear to be many similarities, but also many differences, between such ideas and the game-theoretic interpretation of probability assignments. If a close connexion can indeed be forged, we might perhaps be able to use ideas from defensive forecasting to construct confidence intervals that are immune from such criticism. Finally, Glenn's analysis of Bayesian conditioning shows the additional clarity that can be gained from applying betting arguments: in particular, we can then more readily interpret what it means to have learned the event A but nothing else. But I am a little puzzled by his assertion that this "has a meaning without a prior protocol". Perhaps he can help clarify the distinction in relation to the following currently discussed problem (see ). "I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?" According to the BBC website the answer is 13/27. However, the question is ill-posed, since we must take into account, not just the content of the information received, but how and why that information came to be supplied. Let us suppose it known that I have two children, and let us assume that, in the absence of further information, each is equally likely to be a girl or a boy, independently, and that the birth day of each is uniformly distributed across the week, independently of each other and of which sexes they are. The figure 13/27 then arises if I am asked "Do you have a boy born on a Tuesday?", and answer "yes". But, if I have two boys, I might have tossed a coin to choose one of them, and announced his birth day; or, always announce the birth day of the first born. In these cases the probability of two boys is 1/3 --- exactly as when no birth day information is supplied. Yet another alternative is that, if I have two boys, I will announce the earliest day in the week (starting from Sunday) that one of them was born. If I announce Tuesday, the probability I have two boys then turns out to be 18/25. To me such problems illustrate the necessity of carefully specifying the protocol governing the supply of information. I should appreciate Glenn's description of the appropriate game-theoretic formulation(s). REFERENCES Dawid, A. P. and Galavotti, M. C. (2009). de Finetti's subjectivism, objective probability, and the empirical validation of probability assessments. In Bruno de Finetti, Radical Probabilist (Galavotti, M. C., ed.), London: College Publications, pp 97-114. de Finetti B (1962) Does it make sense to speak of "good probability appraisers?". In The scientist speculates (Good I. J., ed), Basic Books, New York, pp 357-364.