tag:blogger.com,1999:blog-5465015914589377788.post7195120449191043892..comments2020-07-11T10:00:38.795-04:00Comments on Michael James on Money: Investing Lessons from Gambling on Coin FlipsMichael Jameshttp://www.blogger.com/profile/10362529610470788243noreply@blogger.comBlogger12125tag:blogger.com,1999:blog-5465015914589377788.post-18198943086335206102016-11-03T10:10:30.435-04:002016-11-03T10:10:30.435-04:00@Blitzer68: If you bet it all on each flip, your ...@Blitzer68: If you bet it all on each flip, your expected return is 20%/flip. But betting 1/3 each time, your expected outcome is 60%(4/3) + 40%(2/3) - 1 = 6.7%. However, expected returns are misleading in this game. Much of the expected return after many flips comes from enormous payoffs with minuscule probabilities. It's better to look at the median outcome (3 heads out of 5). Betting 1/3 each time would leave you with ((4/3)^3)(2/3)^2 = 256/243 times your original bankroll after each 5 flips. After 60 flips, you expect to grow your bankroll by about a factor of 1.9. After 100 flips, the expected growth factor is 2.8. So, it appears you got somewhat lucky. The optimum bet size to grow the bankroll fastest is 20%.Michael Jameshttps://www.blogger.com/profile/10362529610470788243noreply@blogger.comtag:blogger.com,1999:blog-5465015914589377788.post-23508031063286795252016-11-03T08:45:25.007-04:002016-11-03T08:45:25.007-04:00I tried the game again and it took 100 flips to ge...I tried the game again and it took 100 flips to get to $250, dropping as low as $1.23. The game reinforced something else about investing - sometimes we just get lucky and sometimes we don't. It was starting to get painful when my bankroll dropped so low.Jim Turnbullhttps://www.blogger.com/profile/16141660549510415017noreply@blogger.comtag:blogger.com,1999:blog-5465015914589377788.post-46831994785549345022016-11-03T07:56:33.813-04:002016-11-03T07:56:33.813-04:00Or maybe the expected return was 25/3 x 1.1^60 = $...Or maybe the expected return was 25/3 x 1.1^60 = $2.5k. You're the math expert - feel free to correct all this.Jim Turnbullhttps://www.blogger.com/profile/16141660549510415017noreply@blogger.comtag:blogger.com,1999:blog-5465015914589377788.post-69587821206964561822016-11-03T07:43:20.639-04:002016-11-03T07:43:20.639-04:00I played the game before I read your analysis and ...I played the game before I read your analysis and reached $250 in 27 flips betting 1/3 of my bankroll for each flip. I got three tails in a row on flip 16 and dropped to $18.91, but then I got a string of good luck. When I won over $200 I was told the cap was $250, so I only bet enough to reach the cap.<br /><br />I considered betting 1/2 to 1/4 of my bankroll per flip, but settled on 1/3 when I considered that the odds of getting three tails was about 6% (40%^3). I figured if I bet 1/2 with each flip, I might drop too low if I got a bad sequence so I settled on 1/3 as a rough approach. The investment return and time horizon also played a roll in my strategy. I considered that the investment return was good at about 10% return per flip (60% - 40%)/2, and the investment time horizon was long (I figured I could make at least 60 flips in 30 minutes). I assumed I didn't know the cap when I started so I wanted to be aggressive enough to get into the thousands if the cap allowed ($25 x 1.1^60 = ~7k.<br /><br />What did I learn about investing from this? I wish my time horizon and possible investment return was as good as this game. The real world isn't this easy.Jim Turnbullhttps://www.blogger.com/profile/16141660549510415017noreply@blogger.comtag:blogger.com,1999:blog-5465015914589377788.post-68331303591297991982016-11-02T10:58:04.948-04:002016-11-02T10:58:04.948-04:00@Gene: Unfortunately, your outcome is not unusual...@Gene: Unfortunately, your outcome is not unusual using this strategy. One way to understand this is to imagine you start with $32 and bet half your bankroll 5 times. You expect to win 3 times and lose twice. The order of wins and losses doesn't change the final outcome, so let's apply the losses first. Your starting $32 goes to $16, then to $8. Next you win 3 times. Your $8 goes to $12, then $18, and finally $27. So, this typical outcome loses you $5. This means that the median outcome is to lose 5/32 of your bankroll every 5 flips. Starting with $25 and flipping 120 (=5x24) times, your median final bankroll is 25x(27/32)^24 = $0.42. So, you were a little unlucky, but not much. A much better strategy is to be less aggressive.Michael Jameshttps://www.blogger.com/profile/10362529610470788243noreply@blogger.comtag:blogger.com,1999:blog-5465015914589377788.post-34796399412258645012016-11-02T10:49:17.624-04:002016-11-02T10:49:17.624-04:00I fell into ruin, ending up with $0.18. Perhaps 1...I fell into ruin, ending up with $0.18. Perhaps 120 bets? I bet half my bankroll each time, I guess too aggressive. Seemed intuitive that the odds were in my favour and I should bet big. If I had won, I would have felt justified, but now I see that a run of bad luck can ruin good odds.<br /><br />I could tell there was some sort of trick to optimizing outcomes, but I got it wrong. I suppose my error was impatience. If one has fewer chances, maybe more aggressive being is better?genehttps://www.blogger.com/profile/05608927986297939720noreply@blogger.comtag:blogger.com,1999:blog-5465015914589377788.post-12646577623056627332016-11-01T17:05:13.994-04:002016-11-01T17:05:13.994-04:00@Greg: That's one explanation, but I think th...@Greg: That's one explanation, but I think that seeing patterns in randomness is baked into humans. It takes mental effort to convince yourself that the pattern you see is just randomness. Playing craps in Las Vegas, it's very easy to believe in a "hot table". Tables do feel hot at times, even though I know it makes no sense. And it's almost impossible to convince most players that table hotness doesn't exist. In the same way, players of this game who think they see a pattern may well choose to bet on tails, even though we know it's a bad move.Michael Jameshttps://www.blogger.com/profile/10362529610470788243noreply@blogger.comtag:blogger.com,1999:blog-5465015914589377788.post-67958697380528285312016-11-01T16:58:06.654-04:002016-11-01T16:58:06.654-04:00I think anybody who bets on tails simply doesn'...I think anybody who bets on tails simply doesn't understand that past flips have no effect on future flips. Gregnoreply@blogger.comtag:blogger.com,1999:blog-5465015914589377788.post-40743749696973340972016-11-01T13:28:15.084-04:002016-11-01T13:28:15.084-04:00@Martin: While I enjoyed finding optimal strategi...@Martin: While I enjoyed finding optimal strategies and found it very interesting that optimal strategies differ greatly from the Kelly Criterion, what's actually important is the number of subjects who could not help themselves from betting too much and sometimes betting on tails. Almost any reasonable strategy gives great results in this game, but subjects did terribly anyway. I can't see why subjects bet on tails, but I can understand over-sized bets. When I played, I never bet more than 20% of my bankroll, but I did feel some frustration when my progress was thwarted by tails outcomes. It was tempting to bet more and make back the loss quickly. Fortunately, the math of why this is a bad strategy is front of mind for me and I didn't give in. If you bet more than about 39% of your bankroll, odds are that it will shrink over time, even with the 60% bias for heads.Michael Jameshttps://www.blogger.com/profile/10362529610470788243noreply@blogger.comtag:blogger.com,1999:blog-5465015914589377788.post-34926585311788928102016-11-01T13:06:10.245-04:002016-11-01T13:06:10.245-04:00Interesting simply-posed but fairly complex exampl...Interesting simply-posed but fairly complex example, MJ! <br />Think it highlights a couple of interesting points.<br /><br />1) Most people, most of the time, are not in a position (capability, time...) to devise and follow optimal strategies in the face of uncertainty. This is particularly true since little twists may well significantly change the optimal answer (e.g. Monty Hall problem). In investing (your blog) or business decisions (my consulting area), it's therefore often much more important to avoid obvious decision pathologies, e.g. betting on tails, and choose reasonable heuristics, e.g. Kelly criterion even if not fully optimal.<br /><br />2) I think it's a good example of a distinction in risk management that's actually pretty important. There's aleatoric uncertainty, which is uncertainty from randomness (how will the coin tosses go). And there is epistemic uncertainty, lack of knowledge about the problem (what is the cap). Quite often, the best solution to aleatoric uncertainty is optimization, while to epistemic uncertainty it is finding the right heuristics.<br /><br />3) Without having read all your (interesting) analysis in detail, my immediate reaction to your description of the problem was that the best on-the-spot strategy would depend on my subjective judgment on how "remote" the cap as well as the time limit are going to be as likely constraints. The Kelly criterion applies for uncapped, long-run returns. If I have no time limit but strong cap, I should inch up with minuscule bets (to minimize chance of ruin) to discover the cap. If I have limited time but no cap, there's some sort of binomial distribution (more or less?) optimum. If I had just pencil and paper and 5 mins to figure out a reasonable approach, it would be based on my guesstimate whether the experimental design constraints are likely to be so that the (unknown) cap is likely to be more restrictive or the (known) time limit is.Martin Perglerhttp://www.balrisk.comnoreply@blogger.comtag:blogger.com,1999:blog-5465015914589377788.post-47692496679771907172016-11-01T13:00:44.261-04:002016-11-01T13:00:44.261-04:00@Greg: Glad you found it interesting. If the fli...@Greg: Glad you found it interesting. If the flips aren't as advertised, I'm guessing it's a technical problem rather than being deliberate. But most likely it works fine and you just got a slightly unusual sequence of flips. I got to around $150 easily enough by betting 20% of my bankroll each time. Then I just bet $10 at a time until I got to $250. But as I found when I analyzed the game, there are better strategies.Michael Jameshttps://www.blogger.com/profile/10362529610470788243noreply@blogger.comtag:blogger.com,1999:blog-5465015914589377788.post-69375218393817649442016-11-01T12:20:42.590-04:002016-11-01T12:20:42.590-04:00Now that was really interesting. I just played it...Now that was really interesting. I just played it without reading your analysis or thinking about it too much. Still haven't read your analysis. In the first 20 flips I got 6 tails in a row (which took me down less than $5) followed by 8 heads. So I'm suspicious that this is actually modeling a 60% biased coin flip. Seems like there are some pretty long tails in there like in the markets. <br /><br />Eventually I clawed my way up to $120, realizing with 5 minutes left that I could set up the next bet while waiting for the last. I only got in about 140 bets, I think I could have hit the limit if I had got in 300 bets. I bet to much at the beginning, intuitively the lesson I learned from playing is don't bet what you can't afford to lose. <br /><br />I'd be interested to hear your and other's experiences playing this. <br /><br />Now, to read your analysis :). Gregnoreply@blogger.com