I have, as often is the case, retreated to find comfort in mathematics. The football pools were one of the foremost forms of gambling in the UK until the National Lottery launched, nearly twenty years ago. Simplifying, participants attempted to predict which (association) football games from a list would end as a draw - ideally, a score draw (1-1, 2-2, etc.) rather than a 0-0 draw. If a participant picked eight such games from a list of fifty-some - and some weeks there might only be four to find, whereas other weeks might have four times as many - then they would share a prize made up of a reasonably high proportion of the total entry fees. A scoring system shared some of the entry fees as consolation prizes ("second dividend", "third dividend" and so on) among players whose selections were near misses.
The relative difficulty of determining which matches would be drawn in this way made picking such a winning line a very difficult challenge, and a great degree of public interest was placed in trying to make accurate selections. There was a considerable degree of luck in the enterprise and its prominent place within public life was more a historical accident than anything else. Accordingly, it would be usual for participants to select more than eight matches and submit every possible combination of eight matches from the larger number selected. This was, technically imprecisely, referred to as a full permutation. However, the more possible matches were covered in this fashion, the more attempts at the competition were required and the greater the cost. In fact, the number of attempts required increased very rapidly as more matches were named.
The mathematically interesting part was the device of alternate plans that would permit participants to nominate more matches, without the full expense of covering every possible combination within the greater number of selections. These plans would not guarantee that every single combination of eight matches within the wide number of selections would be covered, so it would be possible to pick eight matches that turned out to be score draws and still not end up with a winning line of eight.
On the other hand, the cost of placing such a plan would be much less than full coverage, and a well-designed such plan would offer partial guarantees; picking eight drawn games among your selections might not guarantee you that all eight draws would appear in the same line and win you the biggest price, but a plan might guarantee you would have a line with at least seven draws out of the eight and thus you would win a smaller prize, whichever ones turned out to be drawn. This would mean that you would get to select far more possible draws than you might have done otherwise, but be relying not only on sufficiently many of your selections to be draws but the right combination of draws within your selections to turn out to be on the same line.
Apparently there were between a thousand and fifteen hundred such plans registered by the football pools operating companies of the day. Designing such a plan that struck a balance between permitting people to choose many matches, permitting people to play cheaply and providing an interesting and relevant guarantee of the partial success attained under certain degrees of overall success in the selections strikes me as a very interesting piece of combinatorics and I have no clue how people would go about doing so.
Today I found the Classic Value web site, or perhaps an older site that has been taken down and reinstated as part of some other site. It looks like at one point it permitted you to play football pools games online, though it also looks like it has not done so since at least 2012. (Accordingly, I'm worried that this site, too, may bite the digital dust before long.) This site is more interesting to me than its competitors because it has a much greater degree of detail about many of these partial-cover plans than most of its competitors, and explains the full implications of each one.
For example, to get full coverage of every possible combination of eight matches within 14 selected would take 3,003 entries - but if you had eight draws turn up within the 14, you know there would be a line with all eight of them. On the other hand, the "Lit-Plan B37" plan would let you select 14 matches and only make 398 entries, so roughly one eighth the cost, but you would have to have nine draws to turn up within the 14 in order to know that there would be a line within the plan containing all eight draws and winning you a top dividend.
This Classic Value web site fascinates me because it has details of hundreds of such plans, and discusses their background a little further. I don't know whether the Pools Promoters Association still exists as such and whether it might still be possible to write to The Perms Department at Zetters Pools for further information. I find it a mathematically interesting thought, though, and one that would have been possible within relatively recent living memory, even if relevant no longer.
The lottery games of the world that exist are far less interesting in this way, though there is still a little related interest to be had. For instance, it is possible to place a particular sequence of 163 entries on the National Lottery such that you can be guaranteed that whichever numbers are drawn, at least one of those 163 entries will have at least three of the winning numbers on there - and thus, under current rules, you would be guranteed some return. It's as unlikely as ever that you'd actually end up making a profit, but you'd get some return for your money.
It also strikes me on reflection just how complex and intricate the football pools games really were, particularly in comparison to the much simpler National Lottery that followed it. Perhaps that may be a big part of why the National Lottery has proved so popular over the years. (I don't have the data to hand to compare the popularity of the pools and the National Lottery, but this article covers the popularity and impact of the pools on British society in passing.) On the other hand, some of the lottery games that have developed over time do have some elements of sophistication to them. Considering you have to pick five from fifty and two from eleven, perhaps you could argue that the pools were simpler, plus the selection of matches had at least a veneer of skill to it without becoming inaccessible. It'll be interesting to see where the public taste for different types and complexities of gambling changes over time.
Another discovery on a related search that I considered interesting was this index comparing different UK bookmakers' football gambling offerings in the year 1960. If Ray Winstone had wanted to "have a bang on that" at the age of three, what options might have been available to him? Again, they're quite intricate and interesting from a gaming perspective. Even then it was possible, though highly unlikely, to win many thousands of pounds for a stake of just pennies - and pre-decimalisation pennies, at that. There's also a degree of commonality in appearance between these bookmakers' coupons and the standard format of the football pools coupon that survived over the decades.
The conclusions I draw are that the lack of automation would have forced the coupons to be checked by hand, and surely there must have been an industry of coupon checkers back in the day. The biggest and most interesting simplification is that (e.g., looking at the second half of this William Hill coupon) so many matches are offered with the same odds, both for odds of outright results and for correct scores. By way of comparison, on the most recent William Hill coupon I can find, Billy Hill quote each of Villa, Spurs and the mighty, mighty Black Cats as 21/20 to win their upcoming home matches but this doesn't mean they have identical draw and away odds - home odds of 21/20 might reflect away odds as short as 5/2 or as long as 14/5.
The obvious cunning on William Hill's part is their selection of which matches are available to have which matches placed on them. For instance, the odds for multi-team accumulators in the "nothing barred" long list are much shorter than those of accumulators of similar length in other lists, presumably because the other lists only have relatively tight matches on them. If you want your accumulator to have all the home bankers on it, you're not going to get the good odds on the bet.
It's also really interesting to see the changes in the correct score bets. As I look, Bristol City vs. Coventry on Tuesday is the next upcoming pick-'em game: 6/4 home, 12/5 draw, 6/4 away. This would be the archetypal match to be featured on the correct scores list. These days, with such a tight match, you can get 10/1 against 0-0, 6/1 against 1-1, 14/1 against 2-2 and 50/1 against 3-3, when in 1960 it would have been 16s, 8, 10s and 40s respectively. It would seem to me to suggest this reflects a tightening of scorelines and lowering of scorelines.
Even with the same odds offered on the home and away teams to win overall, there are rather different odds offered against the same scoreline depending on whether the home team win or the away team, which is quite sophisticated. It's also interesting to see that 1-0 (or 0-1) has become the winning result with the shortest odds when back in 1960 the favourite was a 2-1 home win. Again, it reflects changes in formation and tightening of defences. It may also represent more sophistication in the bookmaker's modelling, and a greater degree of sophistication assumed on the part of the potential customer.
The last change I might notice is that increased competition, as well as tightened operations arising at least in part from automation and the lower overheads that come from a simpler legal High Street operation rather than requiring credit betting settled weekly by crossed postal orders (!! - by way of comparison, this was probably about as arcane then as requiring people to obtain, then bet in, Bitcoins would be now) have forced the bookmakers of the world to offer slightly less tight books, as far as I can tell. Very roughly I think you might have been looking at a 120% book for a single match back then (if you could get them to quote such a book, which they only would do as a result of postponements knocking your multiple bets down to a single) and you might be looking at something more like a 110% book now. Still not good, but less bad.
None of which will help you make money betting on football, of course, and will only confirm how strongly the oddsmakers of the world tilt things in their favour. Still interested me, though.
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