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Title: "A Dynamic Kelly Criterion for Games with Maximum Payouts" Abstract: We start with a gambling question: Suppose we had $100 starting capital and had the opportunity to repeatedly play a game with favorable odds. Let's say we have a 60% chance of winning each time. How much of our capital (in percent) should we bet each time? Paradoxically, if we wanted to maximize our expected winnings, the math tells us that we should bet everything each time. But surely, that can’t possibly be the best strategy. Going all in each time would almost certainly lead to ruin in the long run. . . In 1956, J. Kelly proposed a solution to this paradox, using a logarithmic utility function, resulting in a fixed percentage to bet, now well-known as “Kelly’s Criterion." Suppose now that we had the opportunity to play the game with the same favorable odds but that there was a maximum payout M imposed and a finite number N of games to reach it. We will show that in this case, Kelly’s criterion becomes “dynamic,” in other words, it becomes a function of both M and N. Furthermore, we will explore the impact of loosening the logarithmic utility assumption on a crucial metric of success in finance, the risk-reward ratio or “Sharpe ratio."