for my betting friends

hi there. world cup season's over, so maybe all of you are taking a break from bookie busting.

with knowledge of probability and the strong law of large numbers, i personally don't find it that attractive to gamble anymore. however, this is not to say there is no money to be made in the short run! after all, gamblers are risk lovers, and the utility you get from winning is much higher than that which you have prepared to lose.

and i was preparing to sleep thinking of what i'd learnt over the past few weeks so i've decided to write a few pointers for my brothers out there. it's still pretty sketchy and i will flesh it out over the coming weeks.

lesson 1: risk neutral probabilities.

Let's assume that a bookie doesn't set arbitrary prices. He looks at the total volume of bets coming in and then he prices accordingly, i.e., prices are set by demand for a certain bet on the market.

This means that the market has already priced implicitly it's expected probabilities that each state will occur. let us take the simplest case, asian handicap, where there are two states in the future: eat ball win, or give ball win. we will consider half ball first, and we will take actual probabilities. data is taken from ladbrokes.com

Example: Arsenal (-1.5) 1.94 vs Aston Villa(+1.5) 1.9

let's play around with the numbers.
with probability p, arsenal clears, with probability (1-p), aston villa clears.
sum of probabilities has to = 1. let's see if we can figure out the bookie's spread.

ok. in a fair bet, we should have the following two equations.
1.94p = 1
1.90(1-p) = 1, where 1 is the initial outlay of the bettor.
solving, we get p = 0.515, (1-p) = 0.4736. p+1-p not equal to 1.

what should be the fair price to play the game? clearly one in which the risk-neutral probabilities (implicit in prices quoted) sum to 1, as in real life. call this price x.

1.94p =x, 1.9(1-p) = x => x/1.94 + x/1.9 = 1

solving, x = 0.959. you are charged �1 for a 95.9 pence game. bookie's spread = 4.28%

let's try another game, man utd (-1.5) 1.92 vs fulham (+1.5) 1.92

this is simpler, 2x/1.92 = 1. x= 0.96, charged �1 for a 96 pence game, spread = 4.167..%

lastly, chelsea (-1.5) 1.88 and man city (+1.5) 1.96

x/1.88 + x/1.96 = 1, x= 95.9 pence. bookie's spread = 4.28% Again! for a different set of prices.

let's try two conjectures:

the closer the prices quoted (by the market), the less premium a bookie has to quote.
this is because demand for both sides is likely to be more symmetric, thus the likelihood of an extreme payoff in one direction is less.

conversely, the more asymmetric the prices, the higher the spread, to compensate for risk.

therefore, a three-state game should have a higher spread as it should have higher variation.

we are also assuming that interest rate earned(foregone) by the bookie(bettor), is negligible, but this is hard to estimate unless i work inside the betting company. after all, you leave money in your account which accrues interest which lowers your payoff and increases theirs. but assuming it's a really short period of time it's negligible.

ok. i need more data, and maybe i will work on the ones with more states. until next time, remember, you lose 4.xx% of your bet on average! gamble wisely!