Beating Las Vegas : The Gambler's ruin & Winning Strategy

People head out to Vegas with often elaborate schemes to beat the house at a game of black-jack or roulette. Other card games like poker involve a certain degree of understanding of the game's nuances and are not a pure guessing game.

The Gambler's ruin is a true and hard fact buttressed by probabilistic evidence.

Let two players each have a finite number of coins (say, n_1 for player one and n_2 for player two). Now, flip one of the coins (from either player), with each player having 50% probability of winning, and transfer a coin from the loser to the winner. Now repeat the process until one player has all the pennies.

If the process is repeated indefinitely, the probability that one of the two player will eventually lose all his pennies must be 100%. In fact, the chances P_1 and P_2 that players one and two, respectively, will be rendered bankrupt are
P_1 = (n_2)/(n_1+n_2)
(1)
P_2 = (n_1)/(n_1+n_2),
(2)

i.e., your chances of going bankrupt are equal to the ratio of coins your opponent starts out to the total number of pennies.

Therefore, the player starting out with the smallest number of coins has the greatest chance of going bankrupt. Even with equal odds, the longer you gamble, the greater the chance that the player starting out with the most coins wins. Since casinos have more coins than the gamblers, this principle allows casinos to always come out ahead in the long run. And the common practice of playing games with odds skewed in favor of the house makes this outcome just that much quicker.

Its an interesting argument. Your lucky streak is a terminal short-run phase. The longer you stay in a winning streak the less likely you are to cash out. Hence the moment you hit a jackpot, make for the cash counters, because its not going to last.

The analysis above however states that there are games which are "skewed in favor of the house". If they aren't of course, the casino doesn't make much money now does it - but the question is how does any advantage truly convert itself into "house advantage" - as with two players who have no prior knowledge a priori on the true outcome of every event, and with the expectation that the game itself is player in a fair and unbiased manner how could the house hold the upper hand in a game?

The only way then for a Gambler to beat the house without cheating is to
1. Have some sort of advantage in the game, with the game having an inbuilt bias towards the gambler
2. Have more money bankrolled than the casino and expect that in a truly unbiased and fair contest the casino gets bankrupt in the longest run. This of course does not consider the fact that casinos can close the table if they are losing a lot of money, to protect their own interests, since the gambler too is given a chance to walk away with his profit/loss.

In games where the possible set of sample points are large - even if the payout is unimaginably large, the playing field truly screws the gambler for all his money. This is because his chance of hitting it are speciously low, while the payout for the house is a zero sum game. In 38 holes in roulette for example, its possible that the house pays out 34 to 35 times the original amount, while is still a little less than what the house would make whenever it wins - and it would wild 37/38 times.

Very simply put - if you are consistent in your choice each time - your chance of winning is 1 - (37/38)^n - the longer you stay in the better your chances, and depending on how much money you stake, and when the roulette wheel averages out, your payout will cause you that much ruin.

If the house sets a minimum and maximum bet limit, statistically it can never lose money. For the gambler the odds are always 1/38. For house its always 1/38. This is because each spin is a new spin.

One interesting betting system is the Martingale betting system :
The simplest of these strategies was designed for a game in which the gambler wins his stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double his bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake. Since a gambler with infinite wealth will with probability 1 eventually flip heads, the Martingale betting strategy was seen as a sure thing by those who practised it. Of course, none of these practitioners in fact possessed infinite wealth, and the exponential growth of the bets would eventually bankrupt those who choose to use the Martingale.

Hence if you bet x to win >= x, this strategy doesn't seem bad, if the probability of you winning atleast x is a good shot - i.e. you are not unduly biased in losing with respect to the house. Else the Martingale, in my opinion, just doesn't work, when you don't have much of a shot. Since your betting amount doubles each time, it doesn't take too long until you go bankrupt - since your losses mount like crazy. Also the payout amount isn't anything special, you only end up with x at the end of your Martingale run.

Beating the house seems a plausible option iff the game gives the house no more than a 50% chance at winning - and the gambler has a decent enough bankroll, and house doesn't set too high of a limit to wager a bet. The payout is obviously proportional to the amount dropped in, but the odds act solely a multiplier of the initial wager. Winning against the house is never a foolproof strategy, and that's why you have so many people at the slot machines in casinos in Vegas. You might as well lose your $ in quarters than risk it all at the blackjack dealer.