The Kelly Criterion

The Kelly Criterion ignores some of the realities of normal play, e.g., you do not decrease the value of 1 point in the middle of a session, but it could easily be used by a bot such as Mr. Hyperbot. Mr. Hyperbot could accept challenges from everyone, but could quit after a few games if the opponent does not make as many mistakes as are needed to justify playing for that stake. I doubt Mr. Hyperbot would have difficulty finding backers in case it ran out of money, though, so perhaps its bankroll should be viewed as extremely large.

Mr. Hyperbot is supposed to play perfectly with no table limit, but I doubt it plays perfectly under the real setting of a table limit of, say, 32 points. Does Mr. Hyperbot know not to play for/save the gammon or backgammon when the cube is on 32, and does Mr. Hyperbot ever pass a double to 64 in that situation? I don't know. My guess is that this is a greater concern than following the Kelly Criterion.

Douglas Zare

The Kelly Criterion suggests that you follow proportionate betting. If your bankroll has been cut in half, you need to cut your bets in half. I have never seen anyone lower the stakes in the middle of a money session, though I have seen them raised many times, e.g., by starting with a 2-cube. The natural thing to do if you are taking a beating while following a proportionate betting system is to quit.

Mr. Hyperbot does play money sessions, according to its player information:

"Currently I play moneygames for 25 cents up to $3 per point or 1-point matches for 25 cents up to $15, with members. You can't beat me in the long run, but you can try. "

Mr. Hyperbot is supposed to play perfectly in 1-point matches and in unlimited money play, but there is no reason to suspect that it could play matches of other lengths well. I believe it simply doesn't accept such invitations. I think it does not play perfectly with a table limit, and that this concern is why it will only play for up to $3/point. It's not just a matter of whether it knows to take a cube that is dead, but whether its equity takes into account that you should not run off a worthless backgammon.

Douglas Zare

I forgot to rate the article, as 5 out of 5

From: tobaksa   

Doug:

An excellent article. In blackjack twenty years ago we used a simplified explanation of Kelly, and then worked from there: bet the proportion of your bankroll that was equal to your advantage. That is to say, if you had a 1% edge, you bet 1%. But since the blackjack player makes many bets with negative expectancy (even wonging in and out so that only high counts are played, one still has to do things like split 8s against a ten) typical schemes called for "half Kelly" or less. The average value of the hi-lo true count is about half a percent, so a common betting scheme versus the shoe was to bet one 400th of the bankroll per true. Thus, a player with forty grand might bet $25 in the negatives, $100 at true +2 (player started with a half- percent disadvantage in the old shoe games), $200 at true +3, maxing out at $500. (AKA "true minus one in black.") Lore had it that the risk of going broke before doubling the bankroll was about 2%. The win rate varied depending upon game conditions, but in most shoe games it would be under $100 per hour.

I just tried applying P - (1-P)/N to blackjack, assuming that one were going to win the hand 44/100 (yes, even in positive decks the counter is a probability underdog) but have a 1% edge. Things got really ugly until it dawned on me that the proper value for N would need to be 1.3, and not 1.01. (Contrary to legend, the last math I studied was in 6th grade, so I often reinvent the wheel because I have just broken an axle along the road to understanding.) Once I had the proper value, then 44/100 - (1-44/100)/1.3 yielded about ... 1/100. Hey, this shit really works!

Best,

Jake

Doug R:

I haven't been able to track down the pamphlet you mention, but I have heard of it and would like to read it.

David B:

I think Mr. Hyperbot's creator, Hugh Sconyers, and David Montgomery of Gamesgrid would be the people to ask about how it handles table stakes. I don't think it is an easy problem to fix, requiring the equities be recalculated almost from scratch for each table limit. Of course, that may only take a few days to weeks of computing time now.

I agree that the rating system on GamesGrid ideally would separate backgammon, hypergammon, and nackgammon. I think 1-point nackgammon tends to have larger skill differences than 1-point backgammon, which tends to have larger skill differences than 1-point hypergammon. The ratings formula seems to underestimate the skill in 1-point backgammon matches, too.

Jake J:

Although I'm not very familiar with blackjack, the Kelly Criterion seems to have been studied in blackjack far more than in backgammon, to the extent that some people have computed not just which true counts justify certain actions, but also what bankroll is needed at each count in case the actions change the variance. Of course, in blackjack it may be easier to estimate your advantage.

In blackjack, I think with a 1% advantage you are supposed to bet less than 1% of your bankroll since the average square of a payoff is greater than 1 square bet. The difference between the average square of the payoff and 1 may not be large, but in general I think you are supposed to bet roughly your advantage divided by the average square of your win or loss. This simplifies to the p-(1- p)/n formula when there are only two possible outcomes, but not when there are several possible ways to win or lose.

Mary H:

While many of the qualitative conclusions should agree with common sense, I think most people are inconsistent on the margin, perhaps one day giving up too much equity to avoid a big swing and the next accepting a large bet for little gain. I think some people will refuse a slightly unfavorable settlement on a large cube, then fail to redouble early enough because of the size of the cube.

Two numerical results surprised me:

1) It doesn't take much of a bankroll or an advantage to play for $100 a point. If you average 0.1 ppg, you can do this on a bankroll of just under $10,000. Nevertheless, there are many people who have much larger bankrolls, who believe they have advantages this large, and would not play for anything close to $100/point.

2) The risk of getting your bankroll cut in half does not depend strongly on what your advantage is, if you follow the Kelly Criterion. Having more or less of an advantage just means that the variations happen faster or slower.

I don't think I've ever beavered something my opponent thought was a pass, but a beaver suggests that I'm willing to accept that position for no compensation. It may be tough to find someone willing to pay me a point (or anything close to a point) to take after offering to take for free. Of course, some people don't think about the decisions that way.

Douglas Zare

From: DBrthrtn   

Doh! Don't know how I came up with a Walter Trice / MrHyperBot connection, thanks for straightening me out again (and you do it so gently....). If either Sconyers or Montgomery are reading this thread, then maybe they will chime in about table limits.

The standard deviation of a session should not depend greatly on the advantage you have unless you are playing someone who is wild with the cube. Against most players, the standard deviation is under 3 points per game, so the standard deviation of a session of n games is about 3 squareroot(n) points. You should not be surprised to see results from 2 standard deviations below to 2 standard deviations above the mean, which means being about $4000 above or below average if you play a session of 50 games at $100/point. You can decrease the swings a bit by trying to settle large cubes.

Zbot is coming soon, but I don't think this is the right place to give more details now.

Douglas Zare