[hrothgar]

gwnn, on Jul 26 2007, 02:42 PM, said:

And you have to take into consideration that in a normal gambling world it is always the case that the ante is lower than the expectancy so the gambler's ruin will have a total of a negative expectancy.

Not sure how you mean to use the word ante, however, this statement doesn't seem quite right.

Lets consider a game of poker with 6 identically skilled players. In this case, the expected value for any player on any hand is going to be zero. By definition, any positive ante is going to be greater than the expected value. If the casino is charging a table fee or a rake it will only increase the skew.

[luke warm]

not speaking for gwen, but it seems that she might be simply comparing the GR to the expectancy of a skilled poker player, and if that's true i certainly hope the ante is lower than what one would expect to win (over the course of a session)... i don't know how richard's example figures in, except maybe in a strict mathematical sense, but i don't think there's any such thing as a table full of identically skilled players

[hrothgar]

luke warm, on Jul 27 2007, 02:04 AM, said:

not speaking for gwen, but it seems that she might be simply comparing the GR to the expectancy of a skilled poker player, and if that's true i certainly hope the ante is lower than what one would expect to win (over the course of a session)... i don't know how richard's example figures in, except maybe in a strict mathematical sense, but i don't think there's any such thing as a table full of identically skilled players

I stand by my original point. The notion of an “ante” is at best tangentially related to “Gambler’s Ruin”. Both “Ante” and “Gambler’s Ruin” have a very specific meaning and the very fact that "Gambler's Ruin applies to games like Blackjack and Craps that don't have any kind of ante should be a dead give away.

An ante is best illustrated using the game “poker”. At the start of each deal, each player contributes a set amount of money into the pot. The amount of the ante could be 25 cents, or a dollar, or whatever. The ante serves a very important purpose. Without an ante, players could adopt an ultra conservative playing style (For example, check every hand, and only and place a bet if you are dealt a straight flush. This would cause the game to degenerate into something ridiculous) Adding an ante to the game accomplishes two ends:

1. Each and every hand eats away at a player’s bankroll. You need to play a more aggressive style or you are going to get nickled and dimed to death.

2. There is suddenly some money in the pot to fight over.

There are variants to the ante system (for example, playing “Dealer antes”, the Dealer contributes the ante for the entire table. Regardless, the basic purpose of the ante structure is to make the game interesting.

Gambler’s ruin is completely different. Gambler’s ruin is a simple function of the fact that players can’t wager infinite amounts of money. These players either have a finite bankroll or (alternatively) there is limit on the maximum amount that one can bet. Let’s look at a simple martingale strategy using LukeWarm’s example. We have a player with a $10,000 bankroll who wants to win $200. Furthermore, lets assume that he is making a “fair” 50-50% bet will an expected value of zero. If he wins, he will take his money and run. If he loses, he’s going to increase his bet sufficiently that he can cover his entire loss plus win the $200 that he originally wanted.

50% of the time, your friend will win $200 on the first roll of the dice. If he wins, he walks away.

If he loses, your friend increases his bet size to $400. 50% of the time, he’ll win and walk away. 50% of the time, he’ll lose and be forced to increase his bet size to $800. (Right now, he’s in the hole $600 for the first two losing bets, plus he still wants to win $200)

Once again, you’re friend will win $200 50% of the time. Unfortunately, if your friend loses again, he needs to increase his bet size to $1600. Guess what happens, 50% of the time, you’re friend gets to walk away with $200. If he loses, he needs to increase the bet size to $3200.

And now, we hit the point where things get ugly. 50% of the time, your friend will finally win his $200. Unfortunately, guess what happens if your friend loses. You’re buddy has now lost $6200. In order to have any chance of recovering his losses and winning the original $200, your friend would now need to bet $6400. Unfortunately, his bankroll is now down to $3800. Game over. Your friend is now forced to walk away from the table after losing $6200.

In order for this to happen, your friend would have to loose 5 bets in a row. The chance that this would happen is very slim. (1 - .5^5) = .03125. 96.9% of the time, your friend gets to walk away with an extra $200. However, the remaining .03125% of the time, he has lost $6200.

96.875% of the time your friend is going to win $200 (96.875% * $200 = $193.75)
3.125% of the time you friend is going to loss $6200 (3.125% * $6200 = $193.75)

Expected value = zero

In actuality, casinos don’t offer “fair” 50% / 50% games. They are in business to make money and a naïve “strategy” like a martingale doesn’t offer any protection.

[Blofeld]

David: Wow! I hadn't thought about the problem enough. I find it very interesting, as well as highly counter-intuitive, that there can be a distribution such that the expectation is always higher when you swap[1]. Thanks for doing the analysis on that.

[1] Though it perhaps shouldn't be a surprise that this kind of distribution leads to counter-intuitive results. After all the "gambler's fallacy" is a guaranteed way of making money even with 99% odds of losing each bet, if there is no limit to the funds available to you (or to the size of stake you can wager).

[gwnn]

This thread is a great re-read. I think some of the newer members will enjoy it too. @hrothgar: sorry, I meant rake, not ante.