Alex's Adventures in Numberland - Alex Bellos [145]
In fact, this is exactly what Ashley Revell, a 32-year-old from Kent, did in 2004. He sold all his possessions, including his clothes, and in a Las Vegas casino bet the total amount – $135,300 – on red. Had he lost, he would have at least become a C-list TV celebrity, as the bet was being filmed for a TV reality show. But the ball landed on red 7, and he came home with $270,600.
At blackjack, Ed Thorp was presented with a different issue. His card-counting system meant that he could tell at certain points during the game whether he had an advantage over the dealer. Thorp asked himself: what is the best betting strategy when the odds are in your favour?
Imagine there is a bet where the chance of winning is 55 percent and the chance of losing 45 percent. For simplicity, the game pays evens and we play it 500 times. The advantage – the edge – is 10 per cent. In the long run our winnings will work out on average as a $10 profit for every $100 gambled. To maximize our total profit, we obviously need to maximize the combined total of wagers. It is not immediately obvious how this is done, since maximizing wealth requires a minimizing of the risk of losing it all. This is how four betting strategies perform:
Strategy 1: Bet everything. Just like Ashley Revell, put your entire bankroll on the first bet. If you win, you have doubled your money. If you lose, you are bankrupt. If you win, put everything down again for the next bet. The only way you can avoid losing everything is if you win all 500 games. The chance of this happening, if the probability of winning each game is 0.55, is about one in 10130, or 1 followed by 130 zeros. In other words, it is almost certain you will be bankrupt by the 500th game. Obviously, this is not a good long-term strategy.
Stategy 2: Fixed wager. Bet a fixed amount on every bet. If you win, your wealth grows by that fixed amount. If you lose, your wealth shrinks by that amount. Since you win more than you lose, your wealth will increase overall, but it does so only by increasing in jumps of the same fixed amount. As the graph overleaf shows, your money doesn’t grow very fast.
Strategy 3: Martingale. This gives a faster rate than fixed wager, since losses are compensated by doubling up after a loss, but brings with it a much higher risk. With only a few losing bets, you could be bankrupt. Again, this is not a good long-term strategy.
Strategy 4: Proportional betting. In this case, bet a fraction of your bankroll related to the edge you have. There are several variations of proportional betting, but the system where wealth grows fastest is called the Kelly strategy. Kelly tells you to bet the f bankr your bankroll determined by . In this case, the edge is 10 percent and the odds are evens (or 1 to 1), making equal to 10 percent. So, wager 10 percent of the bankroll every bet. If you win, the bankroll will increase by 10 percent, so the next wager will be 10 percent larger than the first. If you lose, the bankroll shrinks by 10 percent, so the second wager will be 10 percent lower than the first.
This is a very safe strategy because if you have a losing streak, the absolute value of the wager shrinks – which means that losses are limited. It also offers large potential rewards since – like compound interest – on a winning streak one’s wealth grows exponentially. It is the best of both worlds: low risk and high return. And just look how it performs: starting slowly but eventually, after around 400 bets, racing way beyond the others.
John Kelly Jr was a Texan mathematician who outlined his famous gambling strategy formula in a 1956 paper, and when Ed Thorp put it into practical use at the blackjack table, the results were striking. ‘As one general said, you get there firstest with the mostest.’ With small edges and judicious money management, huge returns could be achieved. I asked Thorp which method was more important to making money at blackjack – card-counting or using the Kelly criterion. ‘I think the consensus