The story about wandering drinkers from the previous article ('Why watch pub crawlers?') is an amusing illustration of the random motion mathematical phenomenon. It is not surprising, however, that the motion of the drunk people has worried others less than the movement of capitals. Historically, the probability theory started with financial problems. For example, in 1650s Blaise Pascal and Christiaan Huygens were trying to predict the moment of casino game ending. The problem can be described in many ways, and we have chosen the most paradoxical.
In casino, a player buys M chips, each of them costs one dollar. The money he pays for the chips is his payment for the participation in the game. Once per minute the dealer flips a coin. If the coin falls tails-up, the dealer takes one of the player’s chips. Otherwise, he gives one chip to the player. The number of casino’s chips is unlimited, so it cannot go bankrupt. The player, in his turn, can. The game continues until the player loses all of his chips, which means he cannot win anything. Until the game ends, the player can eat, drink, chat with other players or entertain himself in any other way for free, at the expense of the casino.
Let’s ask four questions:
- What is the probability that the game will end in N turns?
- What is the median game time?
- What is the average game time?
- Is this game worth playing in the reality, and, if it is, what is the optimal ‘entrance fee’?
This problem is similar to the previous problem about the pub crawlers. One coin flip is comparable to one step made by the drinker. Getting or losing chips is similar to making steps back or forward, while losing all chips can be interpreted as returning to the bar. So, similarly, the probability of losing all chips decreases with the growth of the turns number N. So, Levy flights (anomalously long games) are also applicable to this case, too, which means, an average game will be endless. The only difference is that the player starts with M chips, not with zero, as in the case with the pub crawlers. So, the median game will be around M squared.
What does it mean in practice?
10,000 tramps play Old Harry with casinos
A pauper with one dollar enters the casino (M=1). The problem is now close to the previous case with pub crawlers. The median game time is one turn (the probability of losing everything at the first turn is ½), but the expected average is still infinity. What does it mean to the casino?
Say, there are 100, or 1,000, or even more paupers in the casino. Around a half of them will leave after the first turn, but the rest will stay — and pose a real threat to the casino’s well-being. Just like in the case with the pub crawlers, there will be a small portion of lucky visitors who will play for days, weeks and even years.
The number of such visitors will be comparable to the number of the adventurous drinkers who went far away from the bar. If we take a look at the charts, we will see that the portion of such casino visitors is inversely proportional to the root of N, the number of visitors. Every tenth player plays until the 100th turn, every 100th plays till the 10,000th turn, and so on.
It means that if the casino is visited by 1,000 paupers with one dollar in the pocket, one or two of them will stay in the game for several years! If their number is increased to 10,000, one of them will have the right to enjoy the luxurious life at the casino for hundreds of years. Many other players’ game will, however, still take only a minute.
This problem demonstrates why gamblers should be very conscious when organizing games. Profits and losses are not always assessable at the first sight. While the public crawlers case was a hypothetical anecdote, the situation with the gamblers can be reconstructed in a real casino.
One player with 10,000 dollars devastates a casino
If M>1, the situation can be even worse. The median game time is M squared, so it depends on the initial wealth of the player — just as on the number of paupers and their capital in the case of the luckiest pauper’s game. This is not a coincidence, there is a fundamental link between these cases, but we will touch upon them a bit later. Let’s first assess the game time for different M.
If two friends come to the casino with 10 dollars each, at least one of them will entertain himself at the casino’s expense for more than 1.5 hours (the median game time is 100 turns, or minutes.) If they have 100 dollars each, both of them will stay in the casino for around a month. 10,000 dollars will be enough for hundreds of years!
Start-up capital matters
The discussion comes down to one simple fact: the more money you have, the harder it is to ruin your wealth. In the case of paupers, the long-players needed much luck to make up the fortune, but after that they became almost invulnerable.
In the case of pub crawlers the deviation of the slope from the initial position is proportional to the root of its length. The drinker who has made 100 steps is most likely to be 10 steps away from the bar. The one who’s made 10,000 steps is 100 steps away. Similarly, if a drunk man is 10 steps away from the bar, he will need 100 steps to come back there (it is the median return time).
If the pauper was lucky enough to ‘catch’ 9 heads more than tails, thus having 10 chips, his further game will be the same as the game of a person who bought 10 chips from the very beginning. For both of them the median game time will be 100 minutes. The one who had luck to make a 100-chip fortune, will play for 10,000 minutes.
This conclusion partly explains the growing social inequality and justifies the importance of a company’s underlying strength. A company, or just a wealthy person can save their fortune for centuries, while small start-ups lose their small capitals with a lightning speed.