Kelly Criterion Part 2: Avoid Lotteries and Reverse Lotteries Posted on November 15, 2012 Somewhere along the way someone probably told you not to play the lottery – that it’s a dumb idea. And this is true. The typical state lottery pays out about 50% of the money it takes in as prizes. The other 50% is retained by the state to build parks or educate kids or some such nonsense It’s no exaggeration to say that lotteries are a tax on people that are bad at math – a sort of tax I heartily approve of. There’s an interesting intersection between trading and lotteries you may not have thought about. One aspect of a lottery is the deficient payout – in the typical case $0.50 is paid out for every $1 payed in. In other words playing the lottery has a profit factor of 0.5. Another aspect is the extreme imbalance of payouts – infrequent huge wins paired with frequent small losses. This later aspect is what I want to investigate today – especially the idea of lotteries where the payoff is greater than pay-in. In other words, “good” lotteries. If you haven’t read the profit factor and first Kelly criterionarticles, I suggest you do so before continuing. This is one of those tasks where it’s just easiest to let Excel do the work for you. So I created a series of bets, each with a profit factor of 2.0 that have different probabilities ( P(win) )of winning. Thus they have inversely varying payoff odds to compensate for the varying P(win). These are all in some sense good, profitable bets. The bets with low P(win) represent mild lotteries. The bets with P(win) much greater than 0.5 represent what I term “reverse lotteries” – bets where you frequently win a small amount and infrequently lose a big amount. For each one, I calculated the ideal Kelly bet size, and then what the expected return from one bet in terms of bankroll faction. Here are the results: Payoff Odds P(win) Profit Factor Expectation Kelly Fraction Return in Bankroll % From One Bet If you’re looking for an opportunity to practice your Kelly math, regenerating this table would be a good excercise. Here’s the last column plotted against P(win): What you get is an interesting effect. While all the bets have the same profit factor, they do NOT generate the same rate of bankroll growth. The bets with a P(win) of about 0.5 are much better. The reason for lies in the math of the Kelly Criterion. The Kelly Criterion tries very hard to keep you from going near-broke by scaling back your bets. Both lotteries and reverse lotteries work against that goal – with lotteries, you may go a large number of bets without a win thus risking a large drawdown. Reverse lotteries have the potential for a single big loss to cause a large drawdown. As such in both cases Kelly has to scale back the bet sizes more than you might like. It turns out that both effects are equally bad in terms of bankroll growth rate. Assuming you hold profit factor constant, what you really want is bets that have a P(win) of exactly 0.5 – meaning the typical win would be Profit Factor * typical loss. This math has a lot of implications both for trading system design and for business activities in general. The trading system takeaway is that you should try to design systems that have about equal numbers of wins and losses with bigger wins than losses. Deviating from this a little bit – say withing the range 0.35 < P(win) < 0.65 – doesn’t have much negative effect. But go too much farther one way or the other, and your bankroll growth slows substantially. The general business takeaway is perhaps more interesting if harder to implement. The best business bets, all else equal, are those that fail half the time. This of course assumes constant profit factor – meaning I’m talking about good, positive expectations bets in all cases. But given the choice, you should seek out those medium-risk bets, size them according to Kelly (or less, to reduce volatility), and your business will grow at maximal rate. Obviously bet sizing and bet evaluation is much more difficult in conventional business than it is in gambling or trading, but you should keep the concept in mind and look for this sort of 50% chance of success opportunities. It’s far too easy in business to seek out near-sure things and over-bet on them (for conservative firms), or alternately to bet on colossal long shots with huge payoffs (for startups). Kelly tells you both those strategies are bad ideas – you should avoid both lotteries and reverse lotteries, and seek out opportunities in the middle. http://www.offroadfinance.com/2012/11/15/kelly-criterion-part-2-avoid-lotteries-and-reverse-lotteries/
Nothing wrong with buying ONE ticket from time to time, especially if one ends up in unexpected place, and it has lotto machine. But it must be 1 ticket, since whether it is 1 or 100 odds are roughly the same.
Yes for entertainment it’s okay but don’t expect to get rich or make a living out of buying lottery tickets. It’s said that 100% of the winners have played !
You are talking about a very specific case where the overall payout is rigged against the participant despite the asymmetrical payout. Usually, you see these in one-sided markets like lottery or insurance. Now, imagine a lottery-like payout (e.g. 100x payout for WSB poster who bought a far OTM option) where the pricing is determined by the supply/demand from various market participants. It's much harder to figure out the cost of risk in that situation and making Kelly-like bets becomes near impossible.
also remember if you are addicted to fentanyl and have to work as a prostitute to afford the room you rent winning 500k on the lottery is worth more than to warren buffett. a lecture about the odds, like credit cards, like a lot of nonsense he speaks would demonstrate a lack of basic understanding of how measure spaces work.
You obviously haven't read this: https://highline.huffingtonpost.com/articles/en/lotto-winners/ Like trading options, sometimes you have to think out of the box.
You guys don’t get the point of the article. But I agree. Nothing is un-breakable. Card counting, advantage players. I know this stuff. But the article isn’t about that. If you get a good price for lottery it’s great. Where I am from it’s a sucker play. Even though you can be lucky. The article is about bankroll growth. It’s about time average growth. Large sequential bets.
I actually understood the points you were making. By the way I also enjoyed reading all your posts and appreciate your contributions to ET. You perhaps missed the points I was making: 1. Kelly works only if you have a positive expectancy. 2. To be successful trading option, like the story on lottery winning, you need to find ways to achieve positive expectancy. Randomly trading, whether buying or writing will not get you profits, no matter how you apply Kelly. Best wishes.
Thanks. I don't master my craft but I am trying to get some points. If it can help or even entertain some of you then it's a win/win situation. Certainly missed what you were saying as I've not read you this way. Agree that Kelly or any money management require some kind of an edge to work. No money management ever will optimize your bankroll growth if your expectancy is <= 0 Also you can have bets in your favor but if you don't have enough capital then you're dead. You'll be over betting and hit such a drawdown that you won't be able to bet anymore. But the results the guy finds still bugs me. Because a 50/50 strategy still has the more drawdown. Below are some strategy's max drawdown with different "Risk:Reward @ P(win)" These are not percentage but $ added and subtracted. Multiple Simulations (1M bets run 300 times) This produce the exact reverse of his theory. The more drawdown, the harder to dig yourself out of the hole. But it doesn't refute his conclusions.