From the postbag: Arbitrage

Splittter writes to say:

Worth saying that, although the arbitrage is seductive as it guarantees profit on each market, to the mathematically inclined gambler it's still only worth doing if all bets involved are value in their own right. Basically because if they aren't, although you'll win every time, over time you'll win less than if you just backed the value ones. I'll leave you to 'do the math' though :)

Alarmingly, my poker-playing friend is quite right. Allow me to thank him for making the first post. Let's say there are two arbitrage situations, one where both bets are value, and one where only one is.

Situation one:
Heads 2.10
Tails 2.10
Total implied probability: 95.23%
The arbitrageur backs both at the same stake, and pulls down 10% of it whatever happens. The value bettor does the same thing, with the same result. Bully for both.

Now let's look at a situation where only one is value:
Heads: 2.25
Tails: 1.95
Total implied probability: 95.72%
The arbitrageur puts GBP1.95 on heads and GBP2.25 on tails, to win GBP4.38 less GBP4.20 = GBP0.18 whoever wins. Over 100 bets, he wins GBP18. The value bettor places GBP4.20 on heads every time. Over 100 bets, he wins GBP9.45 50 times (GBP472.50), minus GBP420 staked - a profit of GBP52.50, almost treble the arbitrageur's return.

The cost of this expected extra profit is volatility. Five per cent of the time, 40 or fewer heads will show up in 100 throws. In the case of 40 heads, he would win only GBP378, less GBP420, a loss of GBP42. How terrible. On the other hand, 5% of the time, there would be 60 or more heads, leaving him with a much increased bankroll- GBP567 - GBP420 = GBP147 profit, more than eight times as much.

Over a much longer time frame, the value bettor would be statistically certain* to make more money.

* that is, over a long enough time frame, you can make it as unlikely as you like that the arbitrageur would win more.