Miscellaneous

 Money management and the Kelly Criterion

 From: Stuart Thomson Address: sthomson@armstronglaing.com Date: 4 June 1999 Subject: What stakes are optimum Forum: rec.games.backgammon Google: 7j91hk\$ida\$1@nnrp1.deja.com

```Hi BG'ers!

Has anyone done any analysis on what optimum stakes to use given:

1) Account size (money saved allocated for backgammon).

2) Skill estimation (for example, assuming head to head and a 53 - 47%

Any thoughts?
```

 Chuck Bower  writes: ```This is an interesting (IMO) topic. I don't know of any detailed work that has been done on this subject--money management for head-to- head (and choutte) backgammon play. I can imagine that a computer simulation could be developed which would help answer this question in greater detail than what I am going to present BoE. (I believe John Graas's post was along a similar vein. I'm just more verbose. :) One money management technique which HAS been studied fairly extensively is known as the "Kelly Criterion". First some sketchy history (and, as usual, from memory so I may have messed up some of the details): In the 1950's, a researcher(s) for Ma Bell (probably Bell Labs) wrote journal article about signal routing. I believe it was pub'ed in one of Ma Bell's own journals. The author (or one of the authors) was named 'Kelly'. Someone who read the article (sorry, no name here) realized that the paper's contents actually applied well to money management in some gambling situations (blackjack)? Starting in the early 60's, Edward O. Thorp (of "Beat the Dealer" and "Thorp(e) Count" fame) made the Kelly method popular as a money management technique for casino blackjack (assuming intelligent card counting practice). Although I don't know if it has ever been proven, Thorp (see for example, his book "Mathematics of Gambling", 1984, Lyle Stuart publisher) contends that the Kelly Criterion is an optimal technique for making money as fast as possible under the condition that you are guaranteed never to blow your entire bankroll. There are some conditions on the Kelly method in it's strictest application: 1) Player MUST have an edge! 2) Player has the option of varying his bet amount at each new opportunity (e.g. before each round of cards is dealt). 3) During a particular trial (e.g. during a particular hand of cards) the amount bet does not change. 4) Bets are made "even money"; that is, you are paid the same for a win as you would have to shell out for a loss. (NOTE: there is an expanded Kelly System which takes into account odds being offered, e.g. for horse racing. Although the enhancement is simple, I don't think it applies to my model for backgammon below so I'm not including it. See the above WWW page for details of this enhancement.) Before getting specifically to backgammon, let's create an (artificial) example. (Note: this example is a modification of one in Thorp's "Mathematics of Gambling" book.) Suppose an HONEST (but maybe not particularly bright) person approaches you with the following proposition: You place a bet (size your choice). You roll a fair die. If it comes up 1,2,3, or 4, he gives you the amount of your bet. If it comes up 5 or 6, you lose the amount of your bet. He will play for the next four hours (or less, if YOU decide to quit). You have \$90 in your pocket. How much should you bet on each roll? (I should also say that YOU are honest, meaning you won't bet more than is in your pocket!) Hopefully you realize that betting EVERYTHING each turn is likely to end in ruin for you. But by betting very conservatively you'll blow a chance to make a lot of money. What is the best compromise so that you won't go broke (and have to quit such a lucrative proposition) but still will rake in a large profit? The simple ("oddsless") Kelly Criterion says you should bet a percentage of your current bankroll which is equal to your percentage edge. In the above example, you are a 2::1 favorite on each roll. On average you will win 2/3 of the tosses and lose 1/3. Take the difference (2/3 - 1/3) and that is your 'edge'--1/3. So you should wager 1/3 of your current bankroll at every opportunity. Note that you must CHANGE the amount bet on every new opportunity. Let's take a hypothetical sequnce: Bankroll Bet result 90 30 lose 60 20 win 80 26.67 win 106.67 35.56 win 142.23 ....................... By lowering your bet as your bankroll decreases, you insure that you never go broke. But by raising it as your bankroll increases, you give yourself the maximum opportunity for growth. OK, back to reality--backgammon. First we see that backgammon doesn't fulfill the strict conditions stated above. For one, a game starts off worth a point but usually ends up being worth more than that. The value of any given (money) game is unknown before it begins. Secondly it is not customary in a BG money session for the stake to change, and certainly not every game! Typically two players agree beforehand on a stake and it remains that way thoughout the session (barring certain kinds of 'steaming', of course). This 'unknown' value of each game enters my money management technique in two ways. First, we can assign a typical value to a game. This has been discussed on the newsgroup before, and a good number to use is '3'. That is the standard deviation for money play for Jellyfish, and "not too loose, not too tight" humans as well. The second place the value uncertainty enters is what I will call an 'escrow'. I'm borrowing this term from finance, but most likely I'm abusing its accepted meaning. (Sorry, bankers among you.) Consider the MAXIMUM amount you are likely to lose on a single game over MANY sessions. Experience enters here. I'll take myself as an example. (Note that I tend to be conservative compared to your typical money player in handling the cube.) In my lifetime, I'd guess I've played around 20,000 'money' games of backgammon, head- to-head and choutte. I only recall the cube reaching 32 twice. 16 is a rarity. So in several money sessions, the worst I can imagine is seeing a 16 cube accepted. I set my escrow at twice that = 32. If my bankroll EVER gets less than 32 (my escrow), I must stop playing (or go to the bank machine...). Next I need to estimate my percentage edge. Thanks to the BG ratings formulas and online servers, this is a lot easier than it used to be. I just need to know (or estimate) my opponent's online rating and my own. The differece tells me my edge: (see Kevin Bastian's page: http://www.bkgm.com/articles/McCool/ratings.html) Ratings difference: Edge in a single game 50 3% 100 6% 150 9% 200 11% (I assumed 1-point matches. Note that the relationship between rating difference and edge is close enough to linear that interpolation is reasonable.) I think we have enough info to now speculate on a bankroll size given a known stake: bankroll = (escrow + 300/edge) * stake where 'edge' is in percent, 'escrow' in points, and 'stake' and 'bankroll in some appropriate monetary units. Let's take me as an example (so 'escrow' = 32 points). Say I want to play \$5 per point against someone I estimate to be 50 ratings points weaker than myself: bankroll = (32 + 300/3) * \$5. = \$660. OK, ready to try a chouette? Let's assume n total players so a maximum (when in the box) of n-1 opponents. How does this affect your escrow? You must multiply it by n-1. And how about your "base amount"? 1/n of the time you are playing for n-1 times the stake (per point) and (n-1)/n of the time you are playing for a single stake. So your 'average' stake in a chouette is 1/n * (n-1) + (n-1)/n * 1 = 2(n-1)/n. You must multiply the 300/edge term in the above equation to apply the formula to choutte's: chouette bankroll = [(n-1)*("1-on-1 escrow") + 600*(n-1)/(n*edge)] * stake Again, suppose I get in a chouette where I am 50 ratings points better than the BEST of my opponents (i.e. assume you are ALWAYS playing the best player), for a \$5/point session with a total of four players, I should have: chouette bankroll = [3*32 + 600*3/(4*3)] * \$5 = \$1230. For those who have read this far, I suspect 95% are going to say "you're nuts! I get into money games all the time with nowhere near this much cushion." And I believe you. And maybe my numbers are completely worthless. On the other hand, how often do you have to resort to "IOU's" or writing checks, or getting out of the game prematurely because of an uncomfortable losing streak? And have you ever conciously (let alone unconciously) changed your doubling strategy because the cube was getting too high for your (payability) comfort? Chuck bo...@bigbang.astro.indiana.edu c_ray on FIBS ```

 David Montgomery  writes: ```I asked Danny Kleinman about applying the Kelly criterion to backgammon in a letter to the Chicago Point in the January 1989 issue. Kleinman wrote that Michelin Chabot had written two books applying "Kelly theory" to backgammon, but Kleinman disparaged the books and didn't give any references. Kleinman then admitted that he didn't know about the Kelly criterion and went on to suggest "stakes low enough to absorb a 200 point loss without emotional ruin." This isn't horrible advice, but with Kelly you can do much better. Let r be a random variable, the result of a heads-up backgammon game. Let's say we have the probability distribution for r. Then the Kelly criterion is that we should set our stakes at proportion p of our total bankroll each game, where p maximizes E[log(1+pr)]. If we bet this way, we achieve the greatest expected bankroll growth in the long run. An approximation for p that is often used is E[r]/E[r^2]. For example, let's say you have a .1 ppg advantage over your opponent, and that the variance of the single game result is 10. Then this approximation says that you should set the stakes at .1/10=.01, or 1% of your total bankroll. If your bankroll is \$1000, you do best to play the next game for dimes. A few months back I wrote a program that both calculates the Kelly approximation and explicitly maximizes E[log(1+pr)], given r's probability distribution. For modest edges and/or decent variances, such that the approximation indicates p < .015, the approximation was quite accurate. So if you have a good idea of your edge E[r] and the variance E[r^2] (neither is too hard to estimate), and if E[r]/E[r^2] < .015, then you can fairly easily get a good estimate of what your optimum stake size is. When the approximation indicates a bigger p, it could be substantially too high. Playing around with my program I saw approximation values over .04, but the p that maximized E[log(1+pr)] was in those cases a little over .02. The possibility of extreme results in backgammon, like 16 and 32 cubes, makes it inavisable to set the stake size as high as the approximation suggests. Doing so would make it too likely that you might suffer a big drawdown, after which you would not be able to earn as much. (All of this assumes that we can bet exactly proportion p every game, which in fact we cannot. Because of the inability to fine-tune the stakes after every game, p should almost certainly be somewhat lower than predicted by the Kelly criterion.) Chris Yep, Michael Klein, and Gary Wong gave me a lot of help in figuring out the Kelly criterion and what it means for backgammon. Thanks guys. If anyone has a reference for the Chabot books Kleinman referred to, please let me know. I would very much like to read them. David Montgomery ```

 Chuck Bower  writes: ```I see that a lot more work has been done on this subject than I thought. Appartently, though, very little has been published, or at least made available in form that is readily accessible. I think David's numbers and mine agree pretty closely, which does give me SOME confidence that what I said wasn't totally off-base. Probably the biggest impediment to using my simplified model of applying the Kelly Criterion is that typically you don't (can't?) change the stake after each game. This hurts in two ways: it makes you more likely to go broke (when you are losing) and it doesn't allow you to maximize your earnings (when you are winning). This makes me think there is probably a better money management scheme. And maybe this is where the "total vs. session bankrolls" idea enters. If you have money in reserve (total > session) then you have a chance to adjust your bet size on the NEXT session. As in the true Kelly method, this helps both when you are winning and when you are losing. Besides, if you are getting hammered, maybe you underestimated your opponent.... Finally, using the Kelly Criterion (or some other money management optimization method) is really related to maximizing profit. If you are just sitting down to a friendly game, that might not be your primary goal. In that case, the size of your bankroll (or stake) may be determined by other factors. ```

 David desJardins  writes: ```Chuck writes: > This 'unknown' value of each game enters my money management > technique in two ways. First, we can assign a typical value to a > game. This has been discussed on the newsgroup before, and a good > number to use is '3'. That is the standard deviation for money play > for Jellyfish, and "not too loose, not too tight" humans as well. Patti writes: > I don't think it's that simple, unless you're inclined to settle any > time you get a four cube or higher. Otherwise, you aren't really > accounting for the swings that you'll get when you get gammoned on that > eight cube. Chuck writes: > I agree that setting a bankroll-stake realtionship is not as > simple as my model made it sound. However, I don't understand the > part about "inclined to settle any time you get a four cube or > higher". The central limit theorem says that if you add up a large enough sample from a suitable distribution, then the distribution of the total will be roughly normal. It's not clear that this applies to backgammon, because you can have arbitrarily large payoffs and it's not even clear that the expectation and variance exist. But even aside from that, it's fairly tricky to tackle the question of what is "large enough". It's certainly the case that, for a distribution with "thick tails" such as backgammon payoffs (because paying 16 points for getting gammoned at 8 would be a 5-sigma event with probability less than one in a million, if the payoff distribution were normal with standard deviation 3, the probability of large payoffs at backgammon is much larger than in the normal approximation), the "large enough" that you need for the sum to converge to the normal distribution is increased. > In the example I gave, my bankroll vs. a player rated 50 points below > me was 132 units (points). Thus assuming I was even on all other games, > I could handle FOUR 24-point losses before getting close to my escrow. > As I mentioned, for MY play just one of these would be extremely rare. However rare it is for you, it's a lot less so than the normal approximation would say. ```

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