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At some point you should refuse a good bet because the stakes are too high. Personally I think a good measure of the enjoyment one gets from money is the log of the amount. The base of the log does not matter so let's use 10. However we can't take a log less than 10, so let's say the enjoyment is 0 for any amount less than ten. So in your example let's assume you have $0 before winning the $1,000,000 with your first throw. Now you have log(1,000,000) = 6 units of happiness. The expected value of your happiness taking another free throw is 0.75×log(2,000,000) + 0.25×0 = 4.975772. This is less than 6 so in this case you should take the million and walk. However it might be different if you already had some money. Let's say you already have $200,000. Then your happiness by walking is log(1,200,000) = 6.07918. Your happiness by risking the million and taking another shot is 0.75×log(2,200,000) + 0.25×log(200,000) = 6.082075, so you marginally take the second shot. If you were to win that one your choice would be between log(2,200,000) = 6.34242 and 0.75×log(4,200,000)+0.25×log(200,000) = 6.29269. In this case you should not take a third shot and instead walk with the $2,000,000 win. The breakeven point for accepting the first double is an existing wealth of $191,487. To accept two doubles you should have $382,975 in other money. Sept. 11, 2005 What is the best betting system for a player to use when playing the online casino blackjack games that give the PLAYER the slight edge? Are there any systems when there is a positive player edge that can be used to most effectively gain maximum winnings with minimum chance of losses over the long run? Or would the best system be to bet the same amount consistently and follow the optimum basic strategy (and its exceptions for single deck) for all the hands played? I would recommend flat betting. The expected return is the same regardless of how you bet, but flat betting is best for minimizing volatility and ensuring bankroll preservation. Jan. 9, 2005 On a recent Travel Channel show about Las Vegas whales, they surprised me with the statement that casinos sometimes offer loss discounts to whales. In other words, the whale plays on credit, and are charged only a percentage of their total losses at the end of a visit. Does this make it possible to set up a positive-expectation game? Would betting systems start to make sense in this context? Yes, this is true. It is not unusual for "whales" to get a 10% rebate on losses. In my opinion this is a very risky offer to make and a sharp player could easily abuse it and gain an advantage. The kind of player who would make an ideal recipient of this offer is one who grinds out a lot of play in a high house edge game. The kind of player who could best exploit this offer is one who plays a low house edge game, for a short time, and with a wide range of bet size. It sounds paradoxical but under this deal the player must lose to gain any benefit. Thus the player should set a high winning goal and relatively low losing goal. If we can ignore the house edge for the sake of example if the winning marker were $1,000,000 and the losing marker $100,000 then the probability of success is 1/11, as I show in a later question. The expected value after the 10% rebate is (1/11)×$1,000,000 + (10/11)×(0.9×-100,000) = +$9091. A good strategy to achieve a high winning goal quickly would be something like an anti-martingale, or anything where you bet more after you win. Nov. 28, 2004 I know you say that betting strategies don't work because of the negative expectation built in to most games but what about when the player has the advantage? Do betting strategies work under these conditions? Yes! If the player had the advantage a betting system could not help but work in the long run. The reason is the house/player advantage is immutable. Betting systems can not change it. May 22, 2004 I think I read somewhere that if someone could come up with a system that had even only 1% player edge, you could easily turn 1000$ into 1000000$. But some video pokers have an 0.77% player edge, why aren't you turning it into like $770,000 or something? Is it because you can't bet more than 5$ at a time and it would take WAY too much time? Thx. And oh, I said it before, and I'll say it again, LOVE your site!! Thanks! Yes, I said before that if I had a betting system that had just a 1% advantage I could turn $1000 into $1,000,000 by simply grinding out that edge. This would also be possible in video poker but it would take much longer because the 0.77% advantage game (full pay deuces wild) can only be found in the quarter level. Assuming you can play 1000 hands per hour (a speed few can attain) and played perfectly that would result in an average income of $9.63 per hour. To reach $1,000,000 would require working 11.86 years non-stop. $1000 would also be very undercapitalized to play quarter video poker, so the risk of ruin would be quite high. It would be faster to reach the $1,000,000 with the same edge in a table game because the player can bet more. April 11, 2004 I agree with you that there are no system that can beat a negative expectation game. Anyway, I take a look at the cancellation system and keep wondering ... what about using it in a bet like the banker in baccarat, where you have a POSITIVE expectation outcome? In which extension would the commission payed to the house erode your gains in the long run? I apologize for my shady English. - Marcio from Sao Paulo, Brazil Banker is baccarat is not a positive expectation bet. You're confusing the probability of winning the bet with having a positive expectation. Even without a betting system you will probably win any banker bet but you will win less than what you bet, because of the 5% commission. This makes the banker bet a negative expectation bet. No betting system can beat a negative expectation game. May 1, 2001
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Ask the Wizard: Betting Systems for Positive-Expectation Games and Situations