For some people, singles - betting on one selection in one event - simply aren't exciting enough, and they need to bet on several selections in several events.
The simplest multiple is the double. The proceeds of winning one bet (a stake of s at odds of d1) are reinvested as your stake in the other (at d2), so you win (s.d1.d2 - s) if both win. A treble is the same with three selections; an accumulator is the same with more.
Another common multiple is the forecast in racing. You predict the first and second place horses/dogs/etc (in order, for a straight forecast, in either order for a reverse forecast which costs twice as much for the same payout*. The winnings are calculated by a Secret Formula known only to the bookies. Exacta and trifecta bets are the same kind of thing, but with three horses.
Now comes the fun bit - exotic bets. The first of these is the patent: three singles, three doubles and a treble on the same three selections, all with the same stake. It thus costs 7 stakes. If all of the selections are at 1.83, you would need to win two to break even (3.66 from the singles, 3.34 from the doubles, minus 7 staked). You need odds of about 7.00 on each to break even with a single win, and three wins will make a profit at any stake at all.
A trixie is similar, but without the singles - so it's four bets: three doubles and a treble. This kind of bet requires at least two of your selections to win to get any kind of payout. At 2.00 each, if two of your selections win, you break even. Three winners, as before, bring you a profit at any odds.
A yankee consists of four selections and every combination of doubles (6), trebles (4) and a 4-way accumulator, making 11 bets altogether. Odds of 3.33 on each selection are enough to break even with two wins, whereas 1.56 would do the trick for three wins.
A lucky 15 adds four single bets to the yankee. At odds of 15.00 each selection, a single winner will break even. Odds of 3.00 will break even with two winners, while 1.52 will guarantee a profit if three of them win.
There are many other types, including the Super Yankee (or Canadian), involving five selections (all combinations except singles - 26 bets). A Heinz is the same thing with six selections (57 bets - hence the name), a Super Heinz the same with seven (120 bets) and a Goliath does the same with eight (247 bets). Lucky 31 and Lucky 63 are the same as Lucky 15 but with five and six selections respectively.
There are two main reasons that exotic bets are interesting. Firstly, for the average punter, there's the chance of a huge payout. If your four singles came in at 2.00, you'd win 8 stakes minus the four you placed. If you'd had a Super 15, you'd have won 80 stakes, minus the 15 you put in.
Secondly, and this is critical for us, they multiply value. This is best explained with a double, or else it gets really complicated. Let's say you have two selections you believe are worth 1.80, but the bookmaker has them available to back at 2.00. On their own, each would give a value of 11% - a nice markup. If you backed them in a double, the true odds of both winning would be 1.80 * 1.80, or 3.24. According to the bookie, the odds of the double are 4.00, giving you a value of 23%, more than double.
Unfortunately, the best odds are usually to be found at betfair, whose structure doesn't lend itself to simultaneous multiples. However, if you're prepared to do a bit of maths, you can figure out equivalent stakes for consecutive events - with the double example above, after the first selection won, you would place the entire winnings on the second selection. I'll see if I can knock up some code to determine optimal staking plans... at some point in the future.
* because it's really two straight forecast bets.
Showing posts with label definitions. Show all posts
Showing posts with label definitions. Show all posts
Sunday, January 28, 2007
Friday, January 26, 2007
ELO, Élö, it's good to be back
So far, I haven't said anything at all about sports or rating systems. The sports thing, well, that's only going to change tangentially. The thrust of this piece is going to describe one of the best-known rating systems, the Élö system. I'm going to get tired of the accents, which may not show up on all systems, so I'll revert to calling it Elo like everyone else.
Elo is primarily used for chess. The basic premise is to compare how well a player did over a given time frame - say the number of games he* won in a month - with how well he ought to have done, given his ranking and his opponents'. At the end of the time period, his rating is adjusted to take those games into account.
That description leaves at least two questions: how do we know how well he ought to have done? and, how do we adjust the rating afterwards?
How well the player ought to have done depends on who he's played - if he's played ten games against Deep Blue, he might be expected to win one, if he's really good. Against my girlfriend's cat Darwin? Probably seven, if Darwin's on form. In fact, the number of games he's expected to win is the sum of the probabilities of each individual game**.
All well and good. But how do we figure out the probabilities of each game, given the ratings? This, unavoidably, is going to require some maths. If you have a rating of, say, 1700, and Darwin (who's only just started playing chess) is ranked at 1500, you have a rating difference (D) of +200. From Darwin's point of view, it's -200. The probability of you beating him is 1/(1 + 10(-0.025 D))***. In this case, that's 1/(1 + 10-0.5), or 1/(1 + 0.32), about 0.76. For Darwin, if you'd like to do the sums, it's 1/(1 + 3.16), or 0.24. Note that the probabilities add up to one, as they ought to****. Over ten games, you'd expect to win 7.60, while Darwin makes do with 2.4. (Incidentally, he plays worse if you throw a ball for him to chase.)
So, let's say you played ten games against Deep Blue - we'll say it has an Elo rating of 2800 - (D = -1100 for you) your probability for each game is very small (0.18% - about one in 550). In ten games, you'd expect to win about 0.02. Let's say you did well against Darwin and picked him off eight times, and Deep Blue creamed you, as is its way. You won eight games (W=8), and your expected number of wins was E=7.62 - you did better than expected by 0.38 wins. Very good.
The last step of the process is to adjust your rating. A certain weighting (K) is given to your recent results - in chess, it's usually about K=12, but in later articles I'll hopefully determine what the best values for tennis are. In this case, your rating goes up by K(W-E), or 4.62 points. Your new rating would be 1705 (rounded off).
* or she. Take that as read pretty much everywhere in this blog.
** For the mathematicians: The expected number of wins is sum(p_i) +/- sqrt(sum(p_i . (1 - p_i))).
*** The 0.025 is arbitrary, but the only difference it makes (so far as I can see) is to the spacing of rankings.
**** ... so long as we exclude the draw as a possible result. Which we do, for simplicity's sake.
Elo is primarily used for chess. The basic premise is to compare how well a player did over a given time frame - say the number of games he* won in a month - with how well he ought to have done, given his ranking and his opponents'. At the end of the time period, his rating is adjusted to take those games into account.
That description leaves at least two questions: how do we know how well he ought to have done? and, how do we adjust the rating afterwards?
How well the player ought to have done depends on who he's played - if he's played ten games against Deep Blue, he might be expected to win one, if he's really good. Against my girlfriend's cat Darwin? Probably seven, if Darwin's on form. In fact, the number of games he's expected to win is the sum of the probabilities of each individual game**.
All well and good. But how do we figure out the probabilities of each game, given the ratings? This, unavoidably, is going to require some maths. If you have a rating of, say, 1700, and Darwin (who's only just started playing chess) is ranked at 1500, you have a rating difference (D) of +200. From Darwin's point of view, it's -200. The probability of you beating him is 1/(1 + 10(-0.025 D))***. In this case, that's 1/(1 + 10-0.5), or 1/(1 + 0.32), about 0.76. For Darwin, if you'd like to do the sums, it's 1/(1 + 3.16), or 0.24. Note that the probabilities add up to one, as they ought to****. Over ten games, you'd expect to win 7.60, while Darwin makes do with 2.4. (Incidentally, he plays worse if you throw a ball for him to chase.)
So, let's say you played ten games against Deep Blue - we'll say it has an Elo rating of 2800 - (D = -1100 for you) your probability for each game is very small (0.18% - about one in 550). In ten games, you'd expect to win about 0.02. Let's say you did well against Darwin and picked him off eight times, and Deep Blue creamed you, as is its way. You won eight games (W=8), and your expected number of wins was E=7.62 - you did better than expected by 0.38 wins. Very good.
The last step of the process is to adjust your rating. A certain weighting (K) is given to your recent results - in chess, it's usually about K=12, but in later articles I'll hopefully determine what the best values for tennis are. In this case, your rating goes up by K(W-E), or 4.62 points. Your new rating would be 1705 (rounded off).
* or she. Take that as read pretty much everywhere in this blog.
** For the mathematicians: The expected number of wins is sum(p_i) +/- sqrt(sum(p_i . (1 - p_i))).
*** The 0.025 is arbitrary, but the only difference it makes (so far as I can see) is to the spacing of rankings.
**** ... so long as we exclude the draw as a possible result. Which we do, for simplicity's sake.
Thursday, January 25, 2007
The Search for Value
Almost any serious article about gambling will mention "value" at some point. Not many of them explain neatly what value is. They will generally give a few examples, assume you've got the point, and move on. So here, once and for all, is the definition of value: the value of the bet is the "real" probability of the result happening divided by the implied probability of the bookmaker's odds. If this number is greater than one, the bet constitutes value.
That seems simple enough. The implied probability might look a bit tricky, but it's not too bad. In decimal odds, the implied probability is simply one divided by the odds - so a team available at 2.00 to win have an implied probability of 0.50, or 50%. If the odds were 1.5, the implied probability would be two-thirds, or about 67%. Odds of 10.00 represent a 10% chance.
All well and good, if you're using decimal odds. What if you prefer the old-fashioned, 100-to-30 style favoured by men in cloth caps making odd gestures? Well, you're going to need to do a little more maths, I'm afraid. Odds in the style x-to-y against simply mean that the bookies imply the selection will lose x races for every y they win. X-to-y on is the other way around - they'll win x for every y they lose. Evens, or 1/1, means wins and losses are (implicitly) equally probable, 2/1 on* represents an implied probability of 2/3, and 9/1 against corresponds to a one in ten chance.
To convert fractional odds like these into implied probabilities, you take the number on the right (left if it's in the style of 2/1 on) and divide it by the sum of the two numbers - so it's the number of races you'd expect to win divided by the total number of races. You can easily see that 9/1 corresponds to 1/10 = 10%. (If you divide one by that, you get the decimal odds, 10.00.)
So much for the maths involved in working out implied probability. There are two burning questions that I can see I haven't answered. One, how do you compute the probability of winning? And two, why is value important? The first is one of the main things The Martingale will be looking into. The short answer is I don't know, but I hope to find out. I can answer the second, though:
A value bet is one where, if you repeated it often enough, you would win more money from winners than you lost from losing. For instance, if a horse was at 10.00 and you figured it had a one-in-eight chance of winning**, over a hundred races you'd expect it to win about 12 or 13 times. Each time, you'd be returned $10 for a $1 stake. So you'd get, let's say $125 back from the bookie after giving him only $100 - a profit of $25. The value (minus one) is the return (here, 0.25 or 25%) you'd expect on your money over the long term if you made bets at that value.
In the next article, I'd better talk about the gambler's fallacy and the law of large numbers.
* This also gets written as 1/2, in which case the first definition is correct - the selection loses one race for every two it wins.
** (1/8) / (1/10) = 10/8 = 1.25, so this is value.
That seems simple enough. The implied probability might look a bit tricky, but it's not too bad. In decimal odds, the implied probability is simply one divided by the odds - so a team available at 2.00 to win have an implied probability of 0.50, or 50%. If the odds were 1.5, the implied probability would be two-thirds, or about 67%. Odds of 10.00 represent a 10% chance.
All well and good, if you're using decimal odds. What if you prefer the old-fashioned, 100-to-30 style favoured by men in cloth caps making odd gestures? Well, you're going to need to do a little more maths, I'm afraid. Odds in the style x-to-y against simply mean that the bookies imply the selection will lose x races for every y they win. X-to-y on is the other way around - they'll win x for every y they lose. Evens, or 1/1, means wins and losses are (implicitly) equally probable, 2/1 on* represents an implied probability of 2/3, and 9/1 against corresponds to a one in ten chance.
To convert fractional odds like these into implied probabilities, you take the number on the right (left if it's in the style of 2/1 on) and divide it by the sum of the two numbers - so it's the number of races you'd expect to win divided by the total number of races. You can easily see that 9/1 corresponds to 1/10 = 10%. (If you divide one by that, you get the decimal odds, 10.00.)
So much for the maths involved in working out implied probability. There are two burning questions that I can see I haven't answered. One, how do you compute the probability of winning? And two, why is value important? The first is one of the main things The Martingale will be looking into. The short answer is I don't know, but I hope to find out. I can answer the second, though:
A value bet is one where, if you repeated it often enough, you would win more money from winners than you lost from losing. For instance, if a horse was at 10.00 and you figured it had a one-in-eight chance of winning**, over a hundred races you'd expect it to win about 12 or 13 times. Each time, you'd be returned $10 for a $1 stake. So you'd get, let's say $125 back from the bookie after giving him only $100 - a profit of $25. The value (minus one) is the return (here, 0.25 or 25%) you'd expect on your money over the long term if you made bets at that value.
In the next article, I'd better talk about the gambler's fallacy and the law of large numbers.
* This also gets written as 1/2, in which case the first definition is correct - the selection loses one race for every two it wins.
** (1/8) / (1/10) = 10/8 = 1.25, so this is value.
Wednesday, January 24, 2007
The Martingale
You're in the casino and place a $1 bet on red. It doesn't come up, so you place a $2 bet on red, so that if you win you pick up $4 - everything you've staked so far, plus a dollar. It misses again, so you bet $4 - the $8 you'll surely win THIS time covers all you've staked - $1 + $2 + $4 = $7, plus a dollar to win. Red's bound to come up eventually, so eventually you'll make your dollar. It's a foolproof system! It's called the Martingale, and - unfortunately - it doesn't work very well.
The word martingale derives from the town on Martigue in Provence, southern France. In Martigue, where trousers were traditionally worn fastened at the back. "A la martingale" soon came to mean 'in a ridiculous fashion' and is applied in exactly that sense to the system above.
The problem with the Martingale is that your bet size gets very big very quickly - as you can see below.
Bet number: Stake size
1: $1
2: $2
3: $4
4: $8
5: $16
6: $32
7: $64
8: $128
9: $256
10: $512
After losing ten bets in a row (which will happen a little fewer than once in a thousand times*), you'll be looking to gamble over $1000 dollars to win just one in return - and the odds you're getting are still a little worse than evens. Yes, you're bound to win your dollar in the end - as long as you have an infinite bankroll. Eventually, you will simply reach a point where you don't have enough to carry on playing, and you have to stomach a devastating loss.
* Actually, in a casino the odds are slightly worse than this because of the house edge. In an American casino with two zeroes, the probability of red is about 0.47 rather than the 50% implied by the odds. That makes the probability of ten losing red bets in a row about one in 610.
The word martingale derives from the town on Martigue in Provence, southern France. In Martigue, where trousers were traditionally worn fastened at the back. "A la martingale" soon came to mean 'in a ridiculous fashion' and is applied in exactly that sense to the system above.
The problem with the Martingale is that your bet size gets very big very quickly - as you can see below.
Bet number: Stake size
1: $1
2: $2
3: $4
4: $8
5: $16
6: $32
7: $64
8: $128
9: $256
10: $512
After losing ten bets in a row (which will happen a little fewer than once in a thousand times*), you'll be looking to gamble over $1000 dollars to win just one in return - and the odds you're getting are still a little worse than evens. Yes, you're bound to win your dollar in the end - as long as you have an infinite bankroll. Eventually, you will simply reach a point where you don't have enough to carry on playing, and you have to stomach a devastating loss.
* Actually, in a casino the odds are slightly worse than this because of the house edge. In an American casino with two zeroes, the probability of red is about 0.47 rather than the 50% implied by the odds. That makes the probability of ten losing red bets in a row about one in 610.
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