You think it’ll rain tomorrow in Minneapolis, and I don’t. We thus have a disagreement. Moreover, this is a special kind of disagreement–it’s quite easy to find the answer conclusively: All we have to do is wait a day, and then check the weather report.
Once you reveal your opinion to me (that it’ll rain tomorrow), I have three choices. I can (1) agree with you, (2) argue with you, and (3) bet you.
I hope you see that (2) is right out. There’s no point arguing, since we’ll both know tomorrow. (1) is actually not a bad idea, especially if you might have better information than I. But let’s say I have no reason to think you do. That leaves (3). It’s time to put up or shut up.
But what about the stakes? Should we bet at 1-1 odds? Even though 1-1 is how most people prefer to make informal bets, it’s not usually the fairest way. After all, it fails to account for our different degrees of confidence in our beliefs.
So, how confident am I that it won’t rain? I’d say I’m 75% confident. That means I’d put the odds of its raining tomorrow at 1-3 in favor. You, on the other hand, think it’ll rain with 2-1 confidence (66.67%). Now that we’ve established our beliefs more precisely, we’re in a position to make a fair bet.
But first, what’s a fair bet? A fair bet is one in which each party expects to gain the same amount. To make sure the terms of a bet are fair, we should check our expected winnings to see whether they’re equal.
If we bet at 1-1 odds, what is my expected gain from the bet? Let’s think about it in terms of frequencies: I think it’ll rain once for every three times it doesn’t rain. That means I expect to pay up once, and get paid thrice, for every four trials, on average. Since we’re assuming that the terms are 1-1, the amount I pay and the amount I get are the same (call it $1). Thus I expect to gain $3 and lose $1. My expected gain is $2.
You think it’ll rain twice for every one time it doesn’t rain. You thus expect to gain $2 and lose $1, hence gaining $1. I expect to gain twice as much as you do.
You, being a shrewd bettor, are inclined to refuse this offer of a bet. You want one for which our expected gains will be equal. Well, what terms would you like?