Bets and groups
I’ve been thinking about bets recently. As you do. I’m interested in the Dutch book theorem, and central to the proof is having a method to formally represent what a bet is. Borrowing from this paper by Frank Döring, we can think of a bet as a list of ordered pairs of an event and a number: . The are the events and the are the stakes. So if Alice and Bob are betting on whether a coin will land heads or tails, with the winner gaining €1 from the loser the bets they are accepting are as follows: (let’s say Alice bets the coin will land heads) . Bob is taking the other side of the bet, so he is accepting the bet: . Flipping the sign on each stake turns one bet into its “complement” if you like. Taking both bets (i.e. taking Alice’s bet and Bob’s bet) would mean a net profit of 0, however the world turns out. This is a kind of “neutral bet”. We can stipulate that the form a partition of the space of possible outcomes. That is, they are mutually exclusive and exhaustive. A bet has to say what happens in all eventualities. Say you were betting on the roll of a die, and the deal was that you won if a 1,2, or 3 came up and that your opponent won if a 5 or 6 came up. What would happen if a 4 was rolled? The bet as it stands doesn’t say… So for simplicity let’s say bets specify what happens in all eventualities.
Perhaps a better formulation of Alice’s and Bob’s bets is as follows: for Alice and the “signs reversed” version for Bob. The here is supposed to cover all remaining possibilities, and stipulates that in the event that the coin lands on its edge, say, then “all bets are off”.
I like Döring’s framework for bets. It’s wonderfully general. J.Y. Halpern’s book Reasoning about uncertainty has a different framework which is less general, but still sufficient for proving the Dutch book theorem. Döring’s framework allows him to show that no (nontrivial) kinds of conditional probability measure can be justified by a Dutch book argument.A Dutch book in Döring’s framework is a bet where all the stakes are negative.
I was thinking about this framework for talking about bets, and I realised a couple of things. Döring discusses combining bets by taking pairwise intersection of the events, and summing the corresponding stakes, to get you a new bet. The events will still partition the event space, and the net profit will be the same for all outcomes, whether you take each bet individually or take the combined bet. We’ve already seen how there’s a neutral bet, and it should be obvious that combining any bet with the neutral bet doesn’t change anything. And for each bet, flipping the signs on all the stakes gives you a bet which, when combined with the original bet, gives you the neutral bet. Basically, the set of all bets with the rule of combination form a group. (Associativity of the group action follows from associativity of taking intersections and addition of numbers). An abelian group, in fact.
For each event we can define a function that maps a bet to the net gain of the bet if that event occurs is a homomorphism from the group of bets to the additive reals. For each probability function over the events, there is a homomorphism from bets to additive reals that returns the bet’s expected value given that probability distribution.
I don’t know if there is any actual point to this, but I found it neat that you can do this stuff. I wonder if you can put a Dutch book in terms of the existence or otherwise of some group theoretic stucture. A Dutch book in this framework is a bet with all stakes being the same sign (that is, whatever happens, it’s always the same party that benefits). So perhaps one can exploit the fact that this is the same as the maximum stake being negative or the minimum stake being positive, and neither of these functions from bets to additive reals are homomorphisms.