No one can convince a depraved gambler, because it is a flaw of character. But if you are still a rational person, stop being obsessed with so-called luck. What gamblers can rely on is the blessing of their ancestors, and the bosses behind the casino are such great gods as Gauss, Kelly, and Bernoulli. How could you win the dealer?
Gamblers believe in luck, casinos believe in mathematics.
When the gambling king Stanley Ho took over the Lisboa Casino, the business was booming, but the rational gambling king was still apprehensive. He asked the "God of Gamblers" Ye Han: "If these gamblers always lose, what if they don't come if things go on like this?" Ye Han laughed Said: "A gambler, a gambler for a lifetime, what they worry about is what if the casino is not here."
What Ye Han was talking about was only on the psychological level. The programming design of modern casinos is much more rigorous than that of Ye Han back then. Casinos have concentrated mathematical experience in probability, progression, and limits. An ordinary gambler, as long as he gambles for a long time, he will lose his money in the end. Except for Zhou Xingxing in the movie, Zhou Xingchi in reality doesn't believe in all kinds of so-called winning stunts.
Gamblers will never understand that what they are betting against is not luck, nor the banker. They are competing with masters of mathematics such as Dirichlet, Bernoulli, Gauss, Nash, and Kelly. What is the chance of winning?
What can be seen is probability, what cannot be seen is a trap
Let's talk about the simplest gambling game first: betting on luck and guessing coins.
The rules are like this, toss a coin, heads win and tails lose, you can take away double the money if you win, and lose your principal if you lose, do you want to play? You may think, alas, this game is not bad, fair! It just so happened that luck was good, and the first one won 100 yuan! You are so happy, at this time the dealer tells you, you see you have won so much, and I worked so hard to set up a place, but in the end I didn’t get anything, otherwise, if you win, just leave it to me 2%, even if it's the relief brother, give it a thumbs up! When you hear it, 2% is only so little, take it, it’s not bad money! Well, that's it for now.
However, what you never imagined in your dreams is that it is this small 2%, but in the end, you lose everything and ruin your family.
This small 2-point winning probability seems inconspicuous, but coupled with the "Law of Large Numbers", it becomes a weapon for casinos to make money! The "Law of Large Numbers" was proposed by the mathematician Bernoulli. It is said that n(a) is the number of occurrences of a in n independent repeated experiments, and p is the probability of occurrence of a in each experiment. When n is large enough At this time, for any positive number ε, there is lim{[|(n(a)/n)| p]<ε}=1, the formula is so complicated that 99% of gamblers can’t understand it, it doesn’t matter if you don’t understand it, we Just look at the result, the money that the banker wins in the end = 0.02*a.
The money made by the dealer is ultimately only related to the player's bet size! This is what we often call "flowing water". As long as the players keep playing, the dealer will keep making money! Regardless of whether the player loses or wins, the banker always wins! Why does the casino have a "minimum betting amount", because expanding the "turnover" can maximize profits!
So don't think how smart you are, you have to be thankful that you haven't played long enough, that's why you lose out of ten bets.
As long as you enter the casino, you are a poor man
Let's go one step further, even if the probability of both parties is equal, you are still a loser. This involves "infinite wealth" and "gambler's law of losing light". Hypothesis", in the case of equal probability, whoever has the most capital has the highest winning rate.
You and I bet against each other, you and I each have 5 yuan, until we lose it all. Then you have a 50% chance of winning and a 50% chance of losing.
You bet against me, you have 5 yuan, I have 10 yuan, until you lose all, then the probability of your winning is only 33.3%, and the probability of losing is 66.7% (this involves Gaussian probability theory and Taylor's Series theory), hidden behind is the casino boss Kelly formula, which will be described in detail in the following subsections.
For small retail investors, casinos can generally think that wealth is infinite, you can't win it, but it can eat you. In the eyes of the casino owner, there are only two kinds of people in the world: one is poor now, and the other is poor in the future.
The "Law of Unlimited Wealth" also explains why casinos set maximum bets. It’s not that the boss is kind enough to protect gamblers from going bankrupt, it’s just that the boss has set up a safety barrier to protect himself. Imagine that one day Bill Gates goes to the casino to have fun and spend tens of billions in it at once, the casino boss will really cry Yes, although this kind of thing is unlikely to happen, but it must be guarded against, so the casino designs the highest betting amount according to its own wealth ability, that is, to resist the "infinite wealth theorem"!
Casino Big BOSS Kelly Formula: First tell you how to bet
In fact, the author of the formula, Kelly, is not a senior gambler, but a famous physicist. When he invented this formula, he was a research scientist in the famous Bell Labs. His research direction was still new at that time. Cutting-edge TV signal transmission protocol.
One of the most famous mathematical formulas in casinos
Before we talk about the formula, let’s take a look at a gamble:
Let's say you have $100 to play a coin toss game - if the coin comes up heads, you win $2 for $1; if the coin comes up tails, you lose $1. How much of the principal should you invest each time to maximize the return?
My first feeling is—no way, there will be an answer to this. In fact, it is such a seemingly unsolvable question. The Kelly formula tells you: 25%.
So, what exactly is the Kelly formula?
f*=(bp-q)/b
b = odds (odds = expected profit ÷ possible loss = $2 profit ÷ $1 loss, the odds are 2)
p = probability of success (50% chance of both heads and tails of a coin toss)
q = probability of failure (that is, 1-p, which is also 50% in the game)
Taking the above game as an example, the calculation process is (bp-q)÷b =(2 * 50%-50%)÷ 2= 25%.
From the formula we can get a little inspiration for our investment:
Only when the winning face (bp - q) is positive, can the game bet. This is the most basic principle of all gambling and investment, that is, "never bet if you are not sure".
The winning ratio must be divided by "b" to be the betting capital ratio. That is to say, in the case of the same winning chance, the smaller the odds, the more you can bet. If you don't understand this sentence, let's look at an example:
Using the Kelly formula, we know that the "Small Bot Big" game can only bet 4% of the total funds, but according to the gambling nature of most people, I am afraid they will choose the "Small Bottom" game, and the heavy position or even show hand? However, a rational choice should be It is "big Bo Xiao", because it is much faster, because it can use 40% of the position! So, speaking of this, when we invest in stocks, if we want to increase the short-term position, the best choice may be to consider the low volatility of heavy positions However, for large-cap stocks with a high probability of rising, and for small-cap stocks with violent fluctuations, we must keep low positions.
Is he reliable? I think there are already a large group of mathematicians in the world to support this optimal answer. Let us simply use a picture of GF Securities (000776) to dispel everyone’s doubts (the title is omitted, and there are a total of Five groups of options, 10% of the red curve is naturally the answer calculated by the Kelly formula)
One of the most famous mathematical formulas in financial circles
In fact, investment is like a gamble. We know the winning formula = winning probability * number of operations * participating positions. And if you want to say the most famous person in the financial circle, Buffett must be among them; if you want to say the most famous formula in the financial circle, the Kelly formula must be one of them, and Buffett has also used it to manage funds. Then let's try to apply the Kelly formula to our strategy:
If we can find a profit model, here is an example of our most familiar strategy of chasing the limit board, buying when a stock is about to rise and limit, assuming you are a super expert, you can make a profit every time you hit the board, then your The probability of success is 100%; suppose you are a novice who has just entered the market and loses 9 times out of 10, then your probability of success is 10%. We classify according to different success probabilities from 10% to 100%, and every 10% is divided into one class.
Let's take a look at when the market is good:
The calculation result of the Kelly formula in the above figure shows that when the market is good, if you really chase the daily limit and there is a profit of 4 daily limit, then you can sell as long as you are 30% sure.
Let's take a look at when the market is bad:
The Kelly formula here tells us that when the market is bad, unless you have 80% confidence in winning, you should not make a casual move.
If you think the above formula is a bit complicated, why not consider Buffett’s version of the Kelly formula (excerpt from “Buffett’s Portfolio”):
X=2p-1
p = probability of success
X = % of funds invested
Simple, let’s take the above example as a case. If the market is bad and there is an investment opportunity with an 80% probability of making a profit, then buy 2 * 80% - 1 = 60% of the stock position. If there is a 100% If there is a profitable investment opportunity, then take the full position. Therefore, the thinking of the Buffett version of the formula is simpler, but it seems to be more aggressive than the original version, because the influence of odds is ignored.
If you want to add a stop loss position, you can optimize the formula to:
f*=(b*(1+p)-1)÷(b*stop loss range)
No betting should be placed at any time other than 100% win
Almost all casino games are unfair games for gamblers.
But this kind of unfairness does not mean that the banker is cheating. Modern casinos rely on mathematical rules to earn profits. In a sense, casinos are the most transparent and open places. Stanley Ho was afraid that nine lives would not be enough.
The Kelly formula is not conceived out of thin air. This mathematical model has been verified on Wall Street. In addition to being regarded as a righteous god in casinos, it is also known as the "artifact of money management". It is the favorite of investment tycoons such as Bill Gross, Buffett Relying on this formula also made a lot of money.
In June 1955, an extremely famous TV program called 64000 dollar question appeared in the United States. The answerers accumulate bonuses by answering the questions continuously, and it became popular all over the United States for a while, with a ratings of 85% during prime time, and there are many counterfeit programs from all walks of life. Such a question-and-answer show quickly attracted off-market betting to bet on the winner. The show was recorded in New York, live on the East Coast, and time-lapse on the West Coast. Some scandals broke out in the news at the time. The gamblers on the West Coast learned the results in advance by phone and rushed to place bets before the West Coast live broadcast.
After Kelly watched the news, he thought that the problem of how to maximize the long-term benefits of gamblers who have some inside information but also some noise can be solved by using the formulas of their laboratory on consulting and noise transmission research. Therefore, he launched the prototype of the Kelly formula with a horse racing model.
Kelly's theory is this. For horse racers who have certain inside information, the first natural thought is of course to put in all the funds, but this will cause a miserable situation in case of losing everything. In the problem that Kelly wants to solve, losing all the funds at any one time is obviously not in line with the need to maximize cumulative returns.
What we should really care about is the long-term accumulated income. For the accumulated income, the final result is only related to the number of rounds won or lost, and has nothing to do with the order of winning or losing. So he introduced an optimal investment position ratio to maximize the long-term cumulative return:
bet = edge / odds = expected payoff / return on payoff
edge=bp-q
The edge here can be understood as the probability of winning * odds - the probability of failure in gambling, which is the above-mentioned winning surface. When the number of edge is positive, this is a game worth betting on, and when the edge is 0 or negative, it means that the gambler does not have edge and should not bet.
The odds are odds, and we can understand it as a public estimate of probability, which is public information.
We can use Kelly to simulate such a situation: Xiao Ming now has a starting capital of 100 yuan, and he will now flip a coin 4 times, and each time he flips a coin for heads, he will get 6 times the capital return (1 with 5) , when he flips a coin tails, accompany the light. May I ask how Xiaoming should allocate the funds for each bet in order to maximize his profit after 4 coin tosses?
According to the Kelly formula calculation, we can establish such a probabilities of 50%, edge = 0.5*5-0.5 = 2, odds is 5, the best position is 40%, we can see that there are 16 possible positions in the end Of the results (4 throws), 12.96 and 8100 appeared 1 time, 64.8 and 1620 appeared 4 times, 324 appeared 6 times, and the payoff of 16 results was 324. The purpose of the Kelly formula is to maximize the payoff from these outcomes.
Since the Kelly formula focuses on long-term rate of return and risk control, it naturally attracts investors to apply it in investment. For example, after the famous legendary mathematician Edward Thorp read Kelly’s thesis, he first taught himself Fortran and developed a set of algorithms for blackjack on an IBM mainframe (interested students can go to the movie 21, the card counting in the movie The way to get edge is the source of edge), and brought Kelly's mentor to Las Vegas to attract a lot of money.
Conclusion: The only rule to win: Do not gamble
Nobody can convince a depraved gambler because it's a character flaw.
But if you are still a rational person, stop being obsessed with so-called luck.
What gamblers can rely on is the blessing of their ancestors, and the bosses behind the casino are such great gods as Gauss, Kelly, and Bernoulli.
How could you win the dealer?
In terms of rationality, no one is more rational than a casino owner.
When it comes to mathematics, no one is more proficient in mathematics than the experts hired by casino owners.
In terms of gambling capital, no one has more capital than the casino owner.
If you really want to win this game, there is only one rule: don't gamble.
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