There exist such services already, and they find dozens of arbitrage opportunities on a daily basis. The catch is that the betting houses have methods to detect people systematically making +EV bets, and will stop them from making further wagers.

]]>My point is not that I should have expected Leicester City to win. It’s merely that even without the benefit of hindsight one should have realized that the probability of their doing so was more than 1 in 5000. Whether that makes it worth betting on them is an interesting question: see Domotorp’s comment above (May 3rd, 2:02pm).

]]>And of course, there’s the risk of the person offering the odds already having a proof, and instead of instant fame and recognition, he first wants to pull a tenner out of your pocket…

]]>I don’t know, but I can put some bounds on it. I would definitely bet on it if I was offered odds of 5,000 to 1, since I’ve already seen enough solutions of problems that had a not-in-my-lifetime quality (I’m thinking in particular of Fermat’s Last Theorem and the Poincaré conjecture) to feel that the probability is much higher than that. I think I’d still bet at 100 to 1. At 50 to 1 I’d have to think about it, and at 10 to 1 I wouldn’t be interested.

My argument for 100 to 1 is roughly this: I feel as though it will probably be solved within a century (though I’m far from certain about this). Also, I have the impression that it needs someone to come up with a surprising new idea, after which it may be that it won’t take years of hard technical slog to get the proof to work. If that hunch is correct, then it is not unreasonable to think of the probability distribution on the time of its solution as something like an exponential distribution with a fairly large mean, so that this coming year is not much less likely than some future year merely by virtue of being much sooner.

Having written that, however, I think 100 is perhaps too small and that maybe 200 to 1 is where I’d be fairly happy to bet, while 100 to 1 is roughly on my boundary between where I’d bet and where I wouldn’t.

]]>any logic? ]]>

On the effect of the general betting duty on odds, there is indeed no reason why it should affect the ratio of the odds on team C to the odds on team D. But there will be some shortening of all of the odds together, because the bookie will not simply want to be sure of some gross profit or other. For a given size of business, the bookie will want a certain level of gross profit after the duty has been paid, in order to have enough to pay the staff, pay the rent on the shop, and get a decent return on the capital invested. If the Government is taking a rake-off, someone must lose. The loss may however fall partly on the punters (through shorter odds), partly on the staff (through lower wages), and partly on the bookie (through a lower return on capital and on the proprietor’s own hours working in the business), depending on elasticities of supply and demand. Taxes are like that. They have a habit of imposing burdens in non-obvious places, as well as in obvious places.

]]>Nice!

]]>Yes, I meant it is tax-free for the bettor. Also, I don’t think it affects the odds much, bookies usually try to set them so that they make money regardless of the outcome. (Which is, probably, a good reason why it’s not bad for them to offer Leicester at 5000-1 even if the chances are different as long as not many people bet on them. Is there some statistic about how much money was put on different teams and at what odds to win the league?)

]]>I don’t think it is tax-free, but the tax comes at a later stage. Rather than coming out of the winnings on a given bet (which it used to do), general betting duty is 15% of the bookmaker’s gross profits (bets taken in minus winnings paid out). The effect on odds is not obvious to the punter, but it is still there.

]]>I’m not sure which sportsbook you’re quoting for the 5000-1 odds, but I would suppose that it is parimutuel and the house “sets” the game and takes a percentage off of the top (known as the rake or vigorish) as does the local government in taxes. The odds are then determined by the number of betters and the teams for which they’re betting. If the pool of betters (presumably all having the same information) is “right”, then the odds set are typically “correct” and the proper/fair payouts follow.

In cases where the betting pool is overly biased (perhaps there are more die hard fans for Man United who are betting with their hearts and not their brains) this can create situations known as “overlays” in which careful gamblers who notice the shift can take advantage of the situation and potentially have better-than-usual margins on their bets. Naturally these margins need to be big enough to overcome not only the inherent risk of the initial gamble, but also cover the loss of the houses’s rake and the subsequent taxes. Serious handicappers will use all of the best available information available in making their gambling choices, but will then only proceed to place bets on which they perceive to be overlays. In the long run this will give them a slight edge (not over the house, but) over their fellow gamblers. They also typically place their bets just before the close of betting so that they have the most updated odds available (and can ensure their bet is still an overlay) while keeping in mind that their bet and other last-minute bets can potentially sway the ultimate odds. (example: If someone comes in at the last minute with several million pounds on Leicester City then the odds shift away from them, potentially creating overlays for other teams.)

In most cases, longshots with terrible odds like these, are longshots for a reason, but if a careful gambler has (preferably solid information creating) reason to believe that the odds shouldn’t be as bad as they are, then it creates an overlay which makes the bet more valuable in comparison to all the other gamblers.

For statisticians, think of it as a normal curve whose local x-value along the y-axis is shifting (usually slightly) over time. If there’s a huge bump in the outlying choices just before betting closes, then one’s statistical likelihood of winning is better and it’s a good bet with a better-than-average payoff. In practice, there are so many people gambling (usually intelligently) that these local shifts (sometimes called “noise” in engineering parlance) are fairly small and that in aggregate averaged over time most gambling situations are the mathematical equivalent of a numerical lottery.

As a similar somewhat related example, in US baseball, sabermetricians are gathering large amounts of data to attempt to find otherwise unseen inequities in forecasted player talent and then leverage that to try to win more baseball games. In the early days of applying these methods, one team could statistically do better than others, however over time, as more teams begin using the same methods, the information gap is overcome and everyone is on equal footing again. The trouble in most sports is that coming up with accurate measurements of performance isn’t easy (or even statistically significant) and thereby adds a large enough layer of noise on top of the process that gambling on them is again relatively equivalent to the lottery.

]]>No, gambling is absolutely tax-free in the UK.

]]>“Mathematicians tend to dismiss such bets as being stupid, because the expected payoff is negative.”

Even more, in my experience mathematicians in general (as opposed to probabilists and statisticians) tend not to think beyond expected value calculations when analyzing uncertain events. Many of them seem hardly to be aware that probability theory has even developed more sophisticated ways of looking at things (although some have heard the word “variance”).

]]>And now, to address whether it would have been a good idea to bet £20 on them mid-season if you think that you knew the odds better than others, depends on your utility function, i.e., how much different quantities of money are worth to you. Most bettors would use the Kelly formula, which says that if you think the odds are 100-1 instead of 1000-1, then you should bet about 1% of your bankroll, i.e, the money that you can afford to loose and might occasionally use for betting. So, if you have £2000 to spend on bets (which is realistic), then it would have been a good idea.

*Thanks — I’ve changed “bed” and “championship”.*