Update: This post has been substantially revised in response to feedback and incorporating answers to good questions. If something seems ridiculous or nonsensical it is definitely best to assume that is my error rather than the authors'.

__Probability and Finance: It's Only a Game__ by Glenn Shafer and Vladimir Vovk is about math and a new field within math. It is a book-length argument for an extraordinary claim about probability theory. In the course of making this claim they do a lot of what has become my favorite thing to read in math and science - carefully revisit the philosophy and assumptions that underly things we have mostly taken for granted. The claim itself is this: they have used game theory to replace measure theory as the basis for probability.

I'll allow you to digest that for a moment.

The website for the book is here: http://www.probabilityandfinance.com/. The authors have provided a number of reviews, and their own responses; they are diligent in offering corrections and clarifications even in the case of reviews which are completely favorable. This is the first time I followed the back-and-forth in book reviews in any detail, and I found it very helpful to see the criticisms and confusions of better-educated, more-qualified reviewers voiced and addressed. A word on the authors: Vovk was a student of Kolmogorov directly, and Shafer has previously published on mathematical theories of evidence. They also do work in machine learning, on Conformal Prediction. They appear to take seriously the magnitude of their claim, and consistently highlight what is different about their approach and where the advantages lay. The book argues for a game-theoretic basis to probability theory in Part I, and goes on in Part II to apply this same game-theoretic treatment to achieve finance theory.

Of particular interest is the section of the book on historical background, which is mostly Chapter 2. What strikes me is that there is not much overlap between their history and the one we hear about Bayesian probability, particularly from Jaynes' book. They begin in the same place as usual with Pascal and Fermat, but hew strongly to formal mathematics throughout - which is to say that Bayes is not mentioned at all. They do discuss some of the different interpretations of probability, including the interpretation of personal choice; this is the closest I can see to what we understand to be the Bayesian interpretation, but they term it *neo-subjectivism* and trace it to Bruno de Finetti. They relate a lot of interesting context for the development of measure theory, including some details about the relationship between frequentism and computability. Shafer and Vovk's work relies rather on Richard von Mises and Jean Ville, and in particular Ville's work with martingales.

The martingale is one of their core concepts, and it has come to my attention the term has an established meaning in probability and game theory which they do not employ, so it is worth disambiguating. In the book they explicitly rely on a generalization of the martingale betting system, which was originally a strategy of doubling one's bets in order to guarantee that the next bet would make back all the money lost so far. It is from this the name of the stochastic process in probability comes. They privilege the historical origination in many cases, most of which were new to me; this does introduce term confusion sometimes, but they apply it consistently in order to capture (what they claim to be) important subtleties.

They pick up the use of the martingale directly from Jean Ville, who applied it to prove the difference between a frequency which converges via oscillation above and below vs. one which converges only from above - it turns out that if it converges from above, a martingale betting system can make a gambler infinitely rich. This was an important point against von Mises' collectives, with respect to Kolmogorov's measure. Ville generalized the martingale to be *any* system of varying bets, which provided a stronger proof of the impossibility of becoming infinitely rich. The authors argue that the use of the martingale betting system is a *universal *test for randmoness - Ville showed that for any event with probability 1, there is a non-negative martingale which diverges to infinity if that event fails. They extend Ville to say that the probability of an event is the smallest possible initial value for a non-negative martingale that eventually reaches or exceeds 1 if the event happens. They start from that to build their game-theoretic notion of the martingale, which they then rely on to build the rest of their results.

A point which took me by surprise: this book makes little reference to the established field of game theory. In the first this is because they are firmly working in the field of math rather than economics, and in the second this is because they have reached back further than game theory's solidification as a field to work effectively from first principles. This puts the work into an interesting grey zone, where they neither have a strong battery of current results to draw from *or* well-established axioms, but as they state plainly in the introduction they are arguing for the establishment of a new field this does not seem unreasonable to me (the lay reader). The authors make arguments from common sense and appeals to intuition in a way that reminds me heavily of __Probability Theory: The Logic of Science__.

Their basic probability game consists of two players, Skeptic and World: Skeptic is an imaginary scientist, betting imaginary capital, in order to test a theory. Both players have perfect information, and alternate moves. Skeptic bets on what will happen, and World decides what will happen - the theory governs what the payoffs are, and if Skeptic wins too much the theory is cast into doubt. In the simple case, this amounts to Player 1 betting on what Player 2's move will be.

I want to take a moment to dwell on something Shafer and Vovk do right out of the gate which blew my mind, but was so obvious in retrospect I am hurt by not having thought of it before: they *decompose World into multiple players*. My initial interest in game theory came through the wargaming and military strategy aspects of the field, which traditionally suffers from the weakness of modelling whole governments as a single rational agent. I had always reasoned that a more detailed model would make each of a government's moves the outcome of a series of games played among its sub-agents, but dividing it up into different players is a considerably simpler way of getting at the same intuition.

Some of the results they develop for probability include:

- Strong law of large numbers
- Weak law of large numbers
- Law of the iterated logarithm
- Central limit theorem

The second part of the book deals with applying this game theoretic method to problems of finance. My understanding of the background for financial theories is limited, but the impression I have is that they consist of being generated directly against the market data, or sometimes based on probability concepts. That is to say, there *isn't* a general mathematical basis for finance theories. This treatment changes that, but importantly it they develop finance theories directly rather than going through probability first. So if we take a theory in finance which was developed from probability previously, and compare it to this book, we go from this:

Measure -> Probability -> Finance

To this:

Games -> Finance

As a result of not going through probability to get to finance, it looks like the probabilistic elements are *native* to finance theory under this construction - in fact they title this section 'Finance without Probability'. As there was not a unified mathematical grounding before, I am extremely interested to see if the later research in this vein shows the same kind of benefits for putting finance on a coherent mathematical footing as happened with other applied fields. My expectations are not high however; finance is already an area of maximum civilizational effort.

A direct benefit of game-theoretic finance is relaxing the assumption of stochastic price behavior. The authors take aim at stochastic assumptions generally - they argue that the game theoretic approach does not entail a deterministic or randomness assumption, and so they can accomodate both events in the same analysis without issue. This is motivated by meta-theoretical concerns: instead of specifying a full probability measure and making the measure larger and more complicated when contradicted by evidence, they can simply withdraw the wrong predictions. They explicitly tout the minimalism of the approach.

A brief aside: in this same section Shafer and Vovk raise the question of using game theory, which has discrete steps, to model the markets, which take place in continuous time. I assumed this would be child's play - or at least undergraduate's play - as I have a degree in electrical engineering and we do that very thing all the time. Apparently I was wrong in this assumption: they say they solve the problem with Nonstandard Analysis, and go on to say some mathematicians are uncomfortable with this but it is really okay. Perhaps this is because it was only developed in the 1960s?

Some of the results for finance:

- Black-Scholes formula (discrete, continuous, with interest)
- Diffusion processes
- Central limit theorem
- Efficient market hypothesis

This is a math book about math theories but it is heavier on the history, nuance and philosophy than it is the formalism. If you already know measure theory, by the author's own admission you do not gain much utility from this treatment, although you may find the background information interesting and helpful. The finance section, and particularly the discussion of assumptions and information efficiency, seem relevant to prediction markets. The book was published in 2001 and as of yet has no second edition; if you would like to eyeball the papers published afterward to get a better sense of relevance and/or sanity, a list is here. A final caveat - I have not finished the book, so corrections may be forthcoming as needed.