Betting and the Bayesian approach to probability

One sometimes sees the frequentist interpretation of probability justified by a betting argument. It goes something like this: suppose you want to assign a probability p to an event E happening in a given situation S. To do this, we set up the following game: we set up S and we bet on whether the even E will be observed or not, and we do it repeatedly. If you bet £pon E and I bet £(1-p) against you, the expected balance of the game is zero. If the bets are not in the ratio p:1-p, then the game will be skewed in either direction an one of us will make money at the expense of the other. This determines the probability p.

However, there are some of problematic assumptions implicit in this "definition" of probability.

1. We are assuming that the situation S can be replicated at will, as often as necessary. In fact, it is easy to see that this procedure does not apply to the probability of single events (such as whether the Sun will rise tomorrow). This is, in fact, one of the thorny questions in the philosophy of probability which frequentists dismiss as meaningless.
2. We are assuming that the game can go on indefinitely. Now, even though the game is fair and we are expected to break even, the standard deviation of the balance after N bets is £1 times the square root of p(1-p)N. In other wors, it grows without bound and, if the game progresses for long enough, one of the two players is bound to go bankrupt. The problem with this is that bankruptcy is precisely how the right choice of the bet amounts is enforced: if you believe that the probability of E is q>p, you will be happy to bet £q against £(1-q), but you will lose £(p-q) per bet, on average. It should be obvious why the certainty of eventual bankruptcy for players who choose the "right" odds should be problematic.
   

There is an alternative betting scheme which takes care of both problems, at the cost of requiring a subjective interpretation of probability. For those of us who are (at least sympathetic to) Bayesian, this is actually a good thing. The betting scheme is as follows: we assume that there are a large number of players, each with their subjective best estimate of the probabilities involved, and we allow only one round of betting. This allows us to consider the probabilities of single events. We simply ask everyone to bet £1 and divide the proceeds evently among those who bet for the outcome which actually takes place. The probabilities of the various outcomes are simply the fraction of players betting for each of them, and are therefore subjective "market" probabilities.

So far, so good. but suppose now that the event in question is repeatable, and that those who still have £1 left are asked to bet again. They now have information about the probabilities implied by the previous round of betting, and can change their bet accordingly.

Consider the following example. Suppose that there are two possible outcomes, A and B, and that the number of bets in the first round is 11 and 33, respectively. If the outcome is A, for instance, those who bet correctly make £3, and the losers lose £1. The implied odds are 4:1 for A and 4:3 for B. In the second round of betting, there are two kinds of players:

• Those who believe the probability of A is less than 7/22, who will bet for B on the expectation of making £4/3 with probability at least 15/22 and losing £1 with probability at most 7/22, so they expect to make at least £13/22 on average.
  
• Those who believe the probability of A is more than 7/22, who will bet for A on the expectation of making £4 with probability at least 7/22 and losing £1 with probability at most 15/22, therefore also expecting to make at least £13/22 on average.
  

Interestingly, this time around it is likely that more players will bet for A than for B, and everyone thinks that the game is in their favour! Is this leading to a model of the volatility (and stupidity) of the stock market? Assuming a uniform distribution of the estimates of p, we would expect 30 A and 14 B bets in the second round. Those who bet A would make £7/15 (which is les than they expect) if they won, while those who bet B would make £15/7.