The problem is that in life we are making not just one bet but a series of connected or coherent bets- like an accumulator- such that Pascal's wager involves one in smaller lower fungibility or uncertain arbitrage bets with different time horizons. The intuitive idea I'm trying to get at here is that betting on God might mean your daughter, the dentist in Ireland, dies because the doctors aren't allowed to conduct a procedure that aborts her unviable foetus.
But, even in the simplistic sorts of models Economists use, two related problems inevitably arise. One is that when you offer a fair bet you are assigning a probability to an outcome. If the price of the bet is 'to believe x' then that price does not stay constant as new information becomes available. Bayes' Law shows how that price changes over time. I'm assuming that it is more costly to believe a more improbable thing but even if we drop this assumption there's another reason why this will be the case. That has to do with your ability to 'lay-off' risk or increase the reward for risk without increasing your 'downside' exposure by running a book.
I'm not up on the literature but it seems a reasonable guess that philosophically inclined gamblers in the ancient world- like Yudhishtra, in the Mahabharata, or Ghalib, our Ghazal King- spent a lot of time considering under what circumstances it would be profitable to run a book, that is offer a bunch of bets, consistent with Pascal's wager.
The Italian Probability theorist, de Finetti formalized this notion as follows- A person who has set prices on an array of wagers in such a way that he or she will make a net gain regardless of the outcome, is said to have made a Dutch book
Fudging things a bit, the Dutch book theorem conveys the notion that coherent betting creates Dutch books whose 'fair odds' are probabilities as estimated by the agent.
This raises the question, what is the Dutch Book such that Pascal's wager, by itself, shows us a path to find the 'best' Scripture? This notion is interesting once we admit errancy in Scripture reception as itself a determinant of its content- i.e. the signal is designed to be rationally repairable. Now, Pascal was certainly smart enough to work out whether his Jansenist reading of the Bible was indeed a Dutch Book- let alone the best possible Dutch Book. The fact that he did not make that claim- nobody is taught Probability theory in the Bible- itself tells a stupid bloke like me that it's not a profitable avenue of inquiry..
But what about the Gita? I read it as the 'dual' of the Just King's education in Probability theory. So am I committed to the notion that the Gita is a Dutch book ? One reason why I might indeed be maintaining this position is my belief that the Mahabharata is a series of balanced games with homothetic preferences- i.e. everybody pretty much wants the same sorts of things, has the same information or ability to get that information if they want to, and the guy offering the wager has to give the other fellow first pick. So if I say 5 to 1 we have a White Christmas, you and I have access to the same Weather forecasts and it's your choice as to which state of the world is going to pay-out for you.
Now, assuming that new information about the world- which changes 'the price' of the Pascalian wager- arrives from totally independent sources, let's say all the causal chains involved are totally separate and identically randomly distributed- then it appears common sense to say there is no profitable Dutch book, or such a book is empty. How can you offer a bunch of people as smart as yourself a series of bets and come out ahead regardless of the outcome? Bookmakers and Casinos and Stock brokers and so on make their money on 'the spread'- the margin between the price at which they buy and sell- or else they have 'insider information' or are better at complex maths or something of that sort.
However, this might not be the case because of something inherent in the subjective way we adjust our expectations and calculate probabilities. Of course, if we had some assurance that everything that is knowable is stuff we can know, this does not pose a problem. But what if there are latent variables outside our ken? Well, in practice we know that there are lots of things we can't directly observe or measure but perhaps there's always a good enough workaround so long as things aren't hopelessly entangled.
De Fenetti introduced a distinction between 'independent sequences' which are 'exchangeable' in the sense of being just as random as each other, and exchangeable sequences arising out of dependent sequences. In other words, subjectively there is wiggle room between things being random because all causal sequences are independent and their appearing interchangable though they are in fact not independent at all. It is the 'latent' variable which introduces this wiggle-room and makes me wonder whether I haven't been confusing independence for exchangeability in broader ways.
In other words, just when I was about to say with great confidence that the Gita aint a Dutch book- sure it's great poetry, & good for instilling shradda piety, but no way, no how does it constrain me to embrace Occultation and Occassionalism for purely Rational reasons- I come a cropper because I can't deny latent variables exist nor that Evolution is Probabilistic nor that I'm a dumb fuck knows shit from poetry or piety- i.e. maybe my response to the Gita is Rational and it's only coz I got such low bandwith Rationality that I didn't realize that was the only signal I was getting.
What makes things worse is that 'Evidential decision theory' allows for the possibility of backward causation. In other words, the claim made by Ved Vyasa or Valmiki or Tulsi or whoever, that who ever listens to their work gains salvation without any further mental effort or even volition on their part, turns out to be apodictically true- at least for genuinely stupid people like me. Why?
Well, to quote Prof. Huw Price, the possibility now exists that 'without inconsistency, we might claim to be able to bring about past events. Dummett shows that we can accommodate a belief in backward influence, so long as we are prepared to give up the assumption that before we decide how to act, it is possible for us to find out whether the past event in question has already occurred.'
How this is relevant, is because the random question 'is the Gita a Dutch Book?' has just revealed that I don't know what I believed about the Gita- there's a backward influence of the Dummett kind because it is impossible for me to know whether some element of what made up my 'belief'- or De Finetti 'coherent' speculation- had or indeed has already come to pass.
As I go on to say in this previous blog post- The fact is, it is never possible, on a sufficiently fine-grained phenomenology or theory of the world, to determine that any occurrence is truly 'Past'- which also means Gibbardian 'hyperstates' and judgments made by 'hyperagents' have no road to supervenience with respect to 'prosaic factual properties'; everything is always in a sort of 'mixed inference' or else a Frege-Geachian flux till Beenakker's boundary resolves Hempel's dilemma as the Cosmic cows come home. Thus any Agency and Intentionality-based 'inwardness' we can have knowledge off must be reverse mereological and Time arrow reversed as indeed is what we would expect if our minds evolved on a stochastic fitness landscape.
So, thank You Great Hindu God, yeah, thanks ever so much- why didn't you make me something sensible, like a Mormon or a Scientologist? What? You thought I was Gay? Look I've explained all that. Yes, as a horny 16 year old, I did put an ad in Time Out- 'gay South Indian boy wants to meet gay non-Manglik for gay times'- but I never thought P.Chidambaram would respond. Okay, it was with a cease and desist order, coz I used his wet veshti picture, still, you can't say he doesn't look a bit like Pippa Middleton from behind.