The Dutch book argument of Bruno de Finetti is an argument which is supposed to justify subjective probabilities. What one does in this argument is gives probabilities an operational definition in terms of the amount one is willing to bet on some event. Thus a probability p is mapped to your being willing to make a bet on the event at 1-p to p odds. In the Dutch book argument one shows that if one takes this operational meaning and in addition allows for the person you are betting to take both sides of the bet, then if you do not follow the axiomatic laws of probability, then the person betting against you can construct a Dutch book: a set of bets in which the person you are betting against always wins. For the best explanation and derivation of this result that I know, consult the notes written by Carl Caves: Probabilities as betting odds and the Dutch book.
Now I have many issues with the Dutch book argument, the first and foremost being that it is a ridiculous setup. I mean how often do you place a bet in which you are willing to give both sides of the bet (buy and sell)? “Yes, I would like to either buy or sell a lottery ticket please?” Sure you can do it, but there are many reasons why money has a value outside of the single bet being placed, and therefore buying (giving someone your money and getting paid back if you win the bet) versus selling (recieving money and then having to pay off the bet if you lose) are not symmetric in any world where the unit being exchanged has a temporal value and the bet is placed before the event is resolved. I am, indeed, a one-sided Bayesian. I will leave it up to the reader to construct the axioms of probability by which I work.
Amusingly, at least to me, this objection does not seem to be raised much in the literature on the Dutch book argument. But the other day I found a great quote relevant to this objection which I just have to share. This is from Artificial Intelligence: A Modern Approach by Russell and Norvig. In this book they discuss but don’t prove the de Finetti’s argument. Then they say
One might think that this betting game is rather contrived. For example, what if one refuses to bet? Does that end the argument? The answer is that the betting game is an abstract model for decision-making situation in which every agent is unavoidably involved at every moment. Every action (including inaction) is a kind of bet, and every outcome can be seen as a payoff of the bet. Refusing to bet is like refusing to allow time to pass.
You heard it here first people: if you want to stop time all you have to do is not bet! Crap I have homework due tomorrow what should I do? Well certainly you should not bet, because we all know that refusing to bet is refusing to allow time to pass. ROFL Baysians are so cute when they try to justify themselves.