Dealing with the perplexing problem of variance in Kelly results

I was concerned that results I was seeing might be skewed by randomness. I decided to change the simulation code to eliminate the random generation of the edge and replace it with a rotating edge from 3% to 7%. This removed one use of the random number function in the program, leaving the only use of it was in determining whether a wager was won or lost (with a 55% chance of winning).  I ran the simulator generating 1,000 cases (a case is equivalent to a season of wagers). I’ve shared the new log file and it can be accessed at bit.ly/2twM3zp.

While looking at the results I noticed something perplexing. It started with noticing that the best Kelly season wasn’t the one with the highest win total. So, I sorted the file by decreasing win total and found the most varied Kelly results occurred in seasons with 61 wins. There were 31 of these and the best Kelly net was $2,513.38, while one with 61 wins had only a $2.09 net Kelly profit. The question is why the difference? Since I had saved the wager details in 1,000 different files. I took a look at both cases, 276 (the good one) and 682 (the bad one).

From my past experiences with using Kelly I suspected that it was caused by how the season started. When you win early the bet size increases and presumably so do your profits. Conversely, when you lose early, your bet size decreases along with your net results. But, even though both cases ended with 61 wins, I assumed case 276 got off to a significantly better start than case 682. It wasn’t until the 50^th game that the good season got ahead of the bad season in wins.

OK, if it wasn’t the season start, then it must have been winning streaks. I had counted these and found that the good season’s longest winning streak was 7 games and the longest losing streak was 4 games. For the bad season the streaks were 9 and 6 respectively.  Not a big difference here.

Next, I looked at the number of winning bets by the edge and generated the following table.

Edge	Won 276	Lost 276	Won 682	Lost 682
3	11	9	14	6
4	9	11	15	5
5	13	7	15	5
6	12	8	9	11
7	16	4	8	12
All	61	39	61	39

Bingo, the winning season had a better won-lost record in the games with the highest edge. Meanwhile, the bad season’s best win-lost record occurred in the games with the lowest edge. The higher the edge, the higher the Kelly bet. Therefore the difference between the good season and the bad season was that the good season won more of the largest bets than the bad season. That explains the overall better net. I found these results disturbing, but more about that in future posts.

Next, I will look at the impact of making partial Kelly wagers. Several users on the Handicappers Think Tank at sportsbookreview.com feel that you are better us 50% of the proposed Kelly bet. They feel this will damp down the fluctuations and reduce the impact of variance. I’ll tweet on Twitter when this post goes up, @ole44bill.