Suppose you and I agree on a probability estimate…perhaps we both agree there is a 2/3 chance Spain will beat Netherlands in tomorrow’s World Cup. In this case, we could agree on a wager: if Spain beats Netherlands, I pay you $x. If Netherlands beats Spain, you pay me $2x. It is easy to see that my expected loss (or win) is $0, and that the same is true for you. Either of us should be indifferent to taking this bet, and to which side of the bet we are on. We might make this bet just to increase our interest in watching the game, but neither of us would see a money-making opportunity here.
By the way, the relationship between “odds” and the event probability — a 1/3 chance of winning turning into a bet at 2:1 odds — is that if the event probability is p, then a fair bet has odds of (1/p – 1):1.
More interesting, and more relevant to many real-world situations, is the case that we disagree on the probability of an event. If we disagree on the probability, then there should be a bet that we are both happy to make — happy, because each of us thinks we are coming out ahead (in expectation). Consider an event that I think has a 1/3 chance of occurring, but you put the probability at only 1/10. If you offer, say, 5:1 odds — I pay you $1 if the event doesn’t occur, but you pay me $5 if it does — each of us will think this is a good deal. But the same is true at 6:1 odds, or 7:1 odds. I should be willing to accept any odds higher than 2:1, and you should be willing to offer any odds up to 9:1. How should we “split the difference”?
I started pondering this question when I read the details of a wager, or rather a non-wager, that I had previously only heard about in outline: scientists James Annan and Richard Lindzen were unable to agree to terms for a bet about climate change. Lindzen thinks, or claims to think, that the “global temperature anomaly” is likely to be less than 0.2 C twenty years from now, but Annan thinks, or claims to think, it is very likely to be higher. You can imagine a disagreement over the details — since the global temperature anomaly can’t be measured exactly, perhaps you’d want to call off the bet (doing so is called a “push” in betting parlance) if the anomaly is estimated to be, say, between 0.18 and 0.22 C — but surely, given that the probability assessments are so different, there should still be a wager that both sides are eager to make! But in fact, they couldn’t agree on terms.
Chris Hibbert has discussed the issue of agreeing on a bet on his blog, where he mentions that Dan Reeves “argues, convincingly, that the arithmetic mean gives each party the same expectation of gain, and that is what fairness requires.” But Hibbert goes on to say that “the way that bayesians would update their odds is to use the geometric mean of their odds.” I’m not sure of the relevance of this latter statement, when it comes to making a fair bet.
Suppose I think the probability of a given event is a, and you think the probability is b. If the event occurs, you will pay me $x, and if it doesn’t occur, I will pay you $y. We don’t need to know the actual probability in order to figure out how much each of us thinks the bet is worth: I think I will gain ax – (1-a)y, and you think you will gain -bx + (1-b)y. We might say a wager is “reasonable” — the word “fair” is already taken — if I think it’s worth as much to me as you think it is worth to you. Look at it this way: I should be willing to pay up to ax – (1-a)y to participate in this wager, and you should be willing to pay up to -bx + (1-b)y. If those amounts are equal, then we’d each be willing to pay the same amount to participate in this game.
Setting the two terms equal and doing the math, we end up with a reasonable bet if x= y(2-(a+b))/(a+b) or, equivalently, x = y(2/(a+b) – 1). Note that this is the same thing we would get if we agreed that the probability p = (a+b)/2. So, I agree with Dan Reeves and his co-authors: the way to make a reasonable bet is to take the arithmetic mean of the probability estimates.