The Kelly Criterion: A bet allocation strategy that uses information and bias in gambling
Kelly Can't Fail – Win Vector LLC
https://win-vector.com/2024/12/19/kelly-cant-fail/
Mount explains the Kelly Criterion using a card game called 'Next Card Bet' that appears in Peter Winkler's book ' The Best Math Puzzles .'
Here's how to play 'Next Card Bet'.
1: Prepare 52 cards (26 red, 26 black) excluding jokers and shuffle them.
2: Players start the game with a bet of $1.
3: In the game, one card is revealed from the deck, and players bet on whether this card is red or black. Once a card is in play, it cannot be returned to the deck. The payout is 1 to 1.
In this game, you can strategize whether you should bet on red or black by counting the cards that are dealt. For example, if you decide not to bet until the last card, you can safely double your bet.
The Kelly Criterion is a strategy that chooses a betting method that maximizes the expected value of the bet. If the number of red cards remaining in the deck is 'r' and the number of black cards is 'b', the function of the bet fraction (bet_fraction) 'P[draw red] × log(1 + bet_fraction) + P[draw black] × log(1 - bet_fraction)' is maximized when its derivative is zero.
Since the probability of drawing a red card is 'r/(r+b)', we need to solve '(r/(r+b))/(1+bet_fraction)-(b/(r+b))/(1-bet_fraction)=0'. By algebraic calculation, 'bet_fraction = (r-b)/(r+b)', so the bet method when playing 'Next Card Bet' based on the Kelly Criterion is as follows.
・If 'r=b', don't bet
・If the result is “r]b”, you bet |r-b|/(r+b) on “red”.
・In the case of “b]r”, you bet |r-b|/(r+b) on “black”.
If you are sure that the last card is a red or black card, your bet will be doubled. However, if you bet based on the Kelly Criterion, you can increase your bet by an average of 9.08 times over multiple trials. The Kelly Criterion is a strategy that guarantees that you will not go bankrupt (lose all your money) and that you will maximize the expected growth rate of the logarithm of your bet, but it is important to note that it does not usually guarantee anything more than that and that you may actually incur losses.
When playing Next Card Bet based on the Kelly Criterion, players basically bet on the majority color, and every time they lose a bet, the deck becomes unbalanced, making subsequent bets more favorable. If you bet small enough, the profits from winning will offset the losses from losing, so you won't lose. Mount points out that the Kelly Criterion is 'like the exploration and exploitation phase of A/B testing ,' because it sets an appropriate price for information and uncertainty.
Mount's full details on the Kelly Criterion are in his book 'The Great Math Puzzle,' which he highly recommends for anyone wanting to learn more about it or other mathematical problems.
Amazon.co.jp: The Best Mathematical Puzzles: Peter Winkler, Ko Sakai, Hirokazu Iwasawa, Takeshi Ozoegawa: Books
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in Note, Posted by logu_ii