It doesn't matter the order of the games. If you play a great game first, then a decent game later, or vice versa, you'd still bet the optimal
percentage of your bankroll. You get a certain percentage, you earn a certain percentage of that, and your overall expected end result is essentially BR*bet_percentage*edge, and that becomes your new bankroll. If you play two different games/bets then BR*bet_percentage1*edge1*bet_percentage2*edge2 is the same thing as BR*bet_percentage2*edge2*bet_percentage1*edge1. It doesn't matter which you play first. Obviously the math looks slightly different if you are comparing the log utility or whatever for Kelly betting, but the end result is the same: it doesn't matter which bet comes first. If you know you are going to bet more than once, you still calculate the optimal bet for each independently.
David Spence said:
Consider a simpler(?) case: you first play one round of Game A. In Game A, you have a .9 probability of losing your wager, and a .1 probability of winning 1000 times your wager. You then play one round of Game B. In Game B, you win 10,000 times your wager with certainty. How much of your $10,000 bankroll would you wager on Game A?
If you bet $991, you have a 10% chance of winning $991,000 on game A and having a final bankroll of $10,010,000,000, and a 90% chance of ending up with $90,090,000. The expected utility of that result is 0.1*log(10,010,000,000)+0.9*log(90,090,000)=8.159. With no bet on game A, we end up with a final BR of $100,000,000 with probability 1. The expected utility of that is 1*log(100,000,000)=8.000, so if your utility function matches the assumptions that go along with the Kelly betting theory, then you should still bet the first positive EV game without regard for possible better games in the future. Think of it this way: yes, you are risking your capital that might be better utilized in a later game, but if you win (and you do have an edge), then you will have even more money to attack that next game.
The caveat here would be if you are sizing your bets based on a total bankroll, but your trip BR is smaller and you can't get any more cash in time to play the better game. Similar issues apply if you are concerned about CTRs or other buy in limits.