Update See below for main update.
The odds of Father Dougal Maguire (Craggy Island) being the next Pope are 1,000 to 1. You may say this is small, but it’s orders of magnitude a better chance than you have, or has Your Truly, who meets all the qualifications for the office: being a man and Catholic.
Don’t look for me in sitting in St Peter’s chair because, according to the odds maker Paddy Power, even Richard Dawkins has a better chance than I. He’s currently at 666 to 1.
Clearly, Paddy Power is posting these bets (he also has Bono at 1000 to 1) as a novelty, assuming people would pay a Euro or two for the joy of having a slip with Christianity’s self-appointed nemesis’s name printed on it as being the next Pope.
In top spot, as of this writing, Power puts Archbishop Angelo Scola (Italy) in the lead at 3 to 1. This translates into a probability of 0.25 (or 25%). Number two is Cardinal Peter Turkson (Ghana) at 3.5 to 1 (or 7 to 2), which is a probability of 0.222 (22.2%). These two gentleman have been switching places with other for the past couple of weeks. Now one will lead the betting, now the other.
The odds are set by Power with two constraints: the amount of money attracted by each of the various candidates and his (Power’s) goal of making a profit—not to say prophet. (That one is up to the Lord.)
If punters had a fly (or a bug) on the wall of the Conclave, and all their money was flowing to Candidate A, then Power would be forced to change the odds of this candidate. For example, if all the money were going to Turkson, and none to any other candidate, Power would put the odds of Turkson at (say) 1 to 10,000, which means betting 10,000 Euros would win you 1 if Turkson were elected. This would discourage most bettors, which is the point.
But suppose there were only two candidates (Turkson and Scola, say), and that money was flowing to both equally. Then Power would put his odds at close to, but not equaling 1 to 1. Odds of 1 to 1 translate to a probability of 50%. That 50/50 would reflect the “wisdom of the crowd” all right, but it would not allow Power to make any money. This is because all the money that was bet for the losing candidate would have to be paid out to those who picked the winner.
Power, like all bookies, makes his money by creating a “Dutch book” against the bettors. That is, he has to adjust the odds so that, no matter who wins, he takes a profit. That link explains the mathematics of Dutch books, but all we have to know is that the probabilities implied by the odds must add to more than 1. If they do, then Power is guaranteed to make money no matter which man is elected Pope.
This is why if there were only two candidates in the pool, setting them both at 1 to 1, which is 0.50 for each, is not a Dutch book. Obviously, (but not to graduates of New York City high schools) 0.5 + 0.5 = 1, which is not greater than 1. But if both had odds of 1 to 2, which is a probability of 0.66 (66%), then 0.66 + 0.66 = 1.33, which is greater than 1 and thus guarantees a profit for the bookie.
Having the sum greater than 1 implies the amount of money paid out by the bookie must necessarily be less than the amount of money he collected in the form of bets. And this is so no matter who wins. Without going into the guts of it, the further the sum of the implied probabilities are from 1, the more money the bookie keeps.
The sum for Paddy Power’s (current) Pope pool is 1.88, which is very far from 1, and which means that Power will make a profit, and a healthy one, no matter who is picked to serve (this assumes he is also adjusting the odds by the money bet, as discussed above, and which he is surely doing for the top candidates, but probably not for the bottom ones, meaning his profit will be less than the 1.88 implies). Discounting the three novelty bets (Maguire, Dawkins, and Bono) changes the conclusion not at all, since the odds for these non-starters is so long. Indeed, as said above, money for these gents is pure profit for Power (only Bono is eligible).
The mathematics thus hints at a strategy for the bookie: increase the number of candidates. On average, the wider the field the large the sum of the implied probabilities. Of course, the bookie has to have actual money bet on each of the candidates in the field. If he didn’t, then he could just put odds of a million to one on each member of the phone book (a strategy which would push the implied probability sum greater than 1 every time).
For what it’s worth, my money is on Turkson—but there may be a bit of wishcasting there, too.
Update If we assume only the top n candidates are having actual bets placed on them, then the sum of implied probabilities first rises above 1 for n = 8 (it’s 1.03). The lowest rank candidate of the eight has odds of 18 to 1. But this would leave out Cardinal Dolan at 20 to 1, who we can guess is drawing bets. If we include all those candidates 20 to 1 or better, then the sum is 1.22; a good profit.
Update I can’t believe I forgot renormalization! It’s a way to more closely estimate probabilities given the “vig” (one of the insider names for the Dutch book), and assuming calibration and other things that we’ll deal with another day. Idea is to take every implied probability and divide by the sum of probabilities. Assuming the 20 to 1 odds cutoff mentioned in the last update, this means that Scola has probability 0.21 (21%), Turkson 0.18 (18%), and anybody but these two 1 – 0.21 – 0.18 = 0.61 (61%).
If the implied probabilities already sum to 1 then renormalization does nothing, as it shouldn’t.
Update Since Power had (now) Pope Francis at 33 to 1, some are asking how much did Power lose since a long-shot came in. None. Power made a profit on the bets. That’s the point of making Dutch book against his punters. Power wins no matter what. See the link to Dutch Book for the mathematical details.
This is the same reason brokers charge “transaction fees.”