Section 6: The (Very Dangerous) Difference between Simultaneous and Sequential Events
An astute reader will observe that thus far all discussion of the Kelly Criterion has relied on events occurring sequentially. In the real world, bettors usually bet on a whole weekend’s worth of games at once. At best, they can bet on them a few at a time (time slot clusters), but they can never bet sequentially, adjusting their bankroll one game after the other. It would be a huge mistake to bet on simultaneous games as if they were sequential. In fact, betting on enough events simultaneously might even violate the principle that the KC can never reduce a bankroll to zero – if one wagered on 20 simultaneous 55% events at the KC recommended 5.5% per event, it’s hypothetically possible that they could lose all 20 games and go broke.
Fortunately, there are easy to use calculators available for simultaneous event Kelly bet sizing which examine the likelihood of each individual outcome, and then the probability distribution of wins and losses, and then optimize the exact investment percentage for a quarter, half or full Kelly bet.
Consider the case of 10 simultaneous events, five of which are 54% and five which are 56%. Were these events sequential, we would invest 3.39% per game at full Kelly on the first five wagers, and 7.60% per game on the second five. If these ten events occur simultaneously, we would wager only 1.91% per game on the first five and 4.66% per game on the second five, and in total we would have 32.85% of our bankroll riding on these ten simultaneous events. As we calculate wagers for more simultaneous events or for events with higher win probabilities, the amount invested on the entire group approaches, but never hits 100%, and thus as expected we can never go broke.
The growth of advantageous wagers on simultaneous events is obviously much slower but is also much less prone to wild swings of variance. Failing to reduce bet size for simultaneous events, and treating wagers as if they were sequential will undoubtedly result in over betting, wild variance and negative expectation.