Imspirit writes:

A gambler’s conceit leads to ruin.

Gamber’s conceit is a phrase which describes a gambler’s belief that he is always able to keep his winnings ahead of his losses. When a gambler says, “I’ll stop when I’m ahead,” he is expressing the gambler’s conceit. When a gambler plans to quit playing after making a certain percentage of his bankroll, for example, 10%, he is operating from gambler’s conceit. He thinks he is practicing “good money management,” but underlying this apparently positive discipline is a false sense of invincibility.

Gambler’s ruin is a phrase which describes the fact that a gambler, who has relatively smaller resources, will eventually lose his entire bankroll to an opponent who has relatively greater resources. In context, the opponent is the House, which has in principle an infinite bankroll.

A simple example illustrates how a gambler’s conceit ultimately leads to ruin in a perfectly fair, even-bet game. Aaron has 100 pennies, while Ben has 12,800 pennies. Every time Aaron wins, Ben gives him one of his pennies, and vice versa.

So, Aaron and Ben begin to play. Because the game is entirely fair, each has a perfectly even chance of winning or losing each decision. However, because Ben has more pennies, the chance that Aaron will go bankrupt before Ben does is almost certain. In fact, Aaron has a greater than 99% chance of going broke before Ben does.

Look at it this way. There are only two possible outcomes for Aaron: 1) he will double his 100 pennies, or 2) he will lose all 100 of his pennies. Each possibility occurs with a 50% chance. So, Aaron’s outcomes are like a coin-toss in itself: heads, Aaron doubles his money; or tails, Aaron loses his bankroll. In order for Aaron to win all of Ben’s pennies, this “outcome coin” would need come up “heads” 7 times in a row, netting Aaron 200, 400, 800, 1600, 3200, 6400, and 12,800 pennies, to finally bankrupt Ben. But it only takes 1 “tails” at any time to bankrupt Aaron. The chance of getting 7 “heads” in a row is (0.50)^7 = 0.78%, which means his chance of ruin is more than 99%.

Ben’s chance of winning is higher simply because he has a larger bankroll and can survive more losses (“tails” of the “outcome coin”) than Aaron can.

If Aaron includes a stop-loss or profit-target for each game session, he only prolongs the inevitable and does not change his odds.

Aaron’s chance of success becomes even smaller when the game has a built-in mathematical advantage in Ben’s favor, and of course, when Ben has a practically infinite bankroll. Ben represents the House, and in the long run, its winning against Aaron is virtually guaranteed.

A gambler’s conceit that leads him to believe he can always stay one step ahead of the House is an example of irrational thinking, because in order for the gambler to stay ahead, he must continue playing, but because he continues playing, he increases the chances that he will meet his ruin. This is why the House generously lavishes enticing comps on the winning gambler to encourage him to keep coming back to play. The longer and more often he plays, the greater the chances he will lose it all back to the House. In fact, the chances he will ultimately be ruined rises exponentially the longer he plays.

Clearly, in order for a gambler to consistently stay ahead, he must win more often than he loses. But in a perfectly fair, coin-toss type of game, the best one can hope to achieve is 50% accuracy in the long term. To be consistently more than 50% accurate requires one to be able to see into the future or to alter the odds of the game in one’s favor, both presently impossible. A more sophisticated gambler might employ a betting progression to dig himself out of temporary holes, but eventually, all betting progressions require 1 more unit than is available in the bankroll, resulting in ruin.

[PS: I split the previous discussion into two different topics. You can follow it here:

Regression toward the mean ]