1% chance to get 1 million, 5% chance to get 1 million, what do you choose? Different people have completely different choices. There is no right or wrong choice, but different choices determine different life fates. 24 hours a day, you will face hundreds of choices, and your life is the superposition of countless choices.

this article is very long, but it is worth your time to read it deeply. It tells a lot about the way of thinking that affects your life.

The author's old saying

Source: Lonely Brain

1

Eight answers to an interesting question

This question is more interesting than expected, with the following eight answers:

1) According to the expectation theory, the green button is worth 5 million;

2) Many people are still willing to choose the confirmed 1 million, because they can't stand the 5% chance of getting nothing;

3) In other words, if a person can't bear "nothing", then the choice on the right is equivalent to "you have a 5% chance of getting a billion dollars and a 5% chance of dying". Of course you can't bear to die, not to mention a 5% chance;

4) think openly that if you have the right to choose, you can sell the right option worth 5 million to someone who can bear it, for example, 2 million (or even higher);

5) Continue to optimize the last item. Considering the possibility of "finding someone who is willing to buy your right to choose", you can sell this right with only 1 million yuan (low down payment), but ask the buyer to share 1 million yuan with you.

6) Further, you can make this option into a lottery ticket for public offering, and chop it up for retail, and print 2 million copies for two yuan each. The first prize is 1 million. Contrast 5, the risk is lower and the income is greater;

7) In view of 6' s successful business model, we began to raise the next 1 million yuan as the first prize to make it a business.

8) According to the P/E valuation, it raised 2 billion yuan and went public with a market value of 1 billion yuan.

2

Three risk decision concepts

From 1 million to 1 billion, let's jump out of the brain teaser game and study the serious mathematical principles behind it.

There are three concepts of risk decision-making in economics: expected value, expected utility and prospect theory.

expected value: in probability theory and statistics, the expected value of a discrete random variable (or mathematical expectation, or mean, also called expectation for short, called expected value in physics) is the sum of the probability of each possible result multiplied by its result.

in other words, the expected value is the average value of the equivalent "expectation" calculated by the results of repeated random experiments under the same opportunity. (from Wikipedia)

For example, when you roll a six-sided dice, the expected value of its points is 3.5, which is calculated as follows:

Expected utility: In microeconomics, game theory and decision theory, expected utility is a utility theory, which means that under the risk situation, the choice made by an individual is to pursue the maximization of a certain number of expected values. This hypothesis is used to explain the expected value in gambling and insurance. (This concept was born to solve the "St. Petersburg Paradox")

Prospect theory: In 197s, Kahneman and Tvoski systematically studied the prospect theory. For a long time, mainstream economics has assumed that everyone is "rational" when making decisions, but this is not the case in reality; The prospect theory adds people's asymmetric psychological utility to the conditions such as earning loss and the probability of occurrence, and successfully explains many seemingly irrational phenomena.

Based on the above theoretical basis, I want to throw out some interesting conclusions:

1) The anti-human principle that "every step is made according to the overall optimal probability" is the first secret of successful people in the traditional sense;

2) The poor sell their "probability right" to the rich at a low price, and the probability right is a more hidden and larger surplus value exploitation (which does not mean that I agree with the concept of surplus value);

3) At present, the popular artificial intelligence relies on each step to calculate the optimal probability independently and in cold blood, thus defeating human beings. For example, alpha dogs;

4) However, irrationality and impulsiveness may become the last bastion of mankind. (I will write this separately in the future)

Let's go through the basic concepts first.

3

Expectation Theory (the basic decision-making tool for the wise)

According to the expectation theory, it is the same thing to get 5 million yuan with a 1% chance and 1 million yuan with a 5% chance.

Bayesian theorem is one of the most frequently used simple formulas for smart decision makers.

Description: "Multiply the loss probability by the amount of possible loss, then multiply the profit probability by the amount of possible profit, and finally subtract the former from the latter. This is what we have been trying to do. This algorithm is not perfect, but it is as simple as that. (By Buffett)

Example A: (Biography from Rubin, former CEO of Goldman Sachs)

After the merger of the two companies was announced, the stock price of Unevis was $3.5 (compared with $24.5 before the merger was announced).

This means that if the merger is settled, the share price from arbitrage trading may rise by $3, because the share price of Unevis Company will be worth $33.5 (.675 × the share price of Brady Company).

If the merger is not successful, the stock of Unevis may fall back to about $24.5 per share. The stock we bought may fall by about $6.

we set the probability of success of the merger at about 85% and the probability of failure at 15%. On the basis of the expected value, the possible increase of the stock price is $3 times 85%, while the risk of falling is $6 times 15%.

USD 3× 85% = USD 2.55 (possibly rising)

-USD 6× 15% =-USD .9 (possibly falling)

So, the expected value = USD 1.65

This USD 1.65 is what we hope to get by putting aside the company's capital of USD 3.5 for three months. This gives a possible rate of return of 5.5%, or 22% on an annual basis. A lower rate of return than this is our bottom line. We don't think it's worth paying our company's capital for an annual return of less than 2%.

Rubin specifically explained that this is what he does every day. It seems like gambling, and he often loses. But what he wants to make sure is to make money most of the time.

For example B: (from the author of Black Swan)

Taleb said at the investment seminar, "I believe there is a high probability that the market will rise slightly next week, with a rising probability of about 7%. But he shorted the S&P 5 futures in large quantities, betting that the market would fall.

in his opinion, it is more likely that the market will go up (I am optimistic about the market outlook), but it is better to sell short (I think the result is bad), because if the market falls, it may drop greatly.

The analysis is as follows:

Suppose there is a 7% probability that the market will go up and a 3% probability that it will go down next week.

but if it goes up, it will only go up by 1%, and if it goes down, it may go down by 1%.

The expected future result is:

7%×1%+3%×(-1%)= -2.3%.

Therefore, you should bet down, and there is a greater chance of making a profit by short selling stocks.

As Munger said, what Buffett does every day is to work out this simple math problem. It is not so much a mathematical ability as a mode of thinking. It's easy to know, but extremely difficult to do.

For example, C:

Probability sometimes seems "counterintuitive".

A taxi caused an accident on a rainy night. A witness at the scene said that he saw the car was blue. It is known that:

1) the accuracy of this witness in identifying blue and green taxis is 8%;

2) The taxis in this area are 85% green and 15% blue.

Excuse me, what is the probability that the taxi was blue?

a: the probability that the car is green but regarded as blue is (.85×.2), and the probability that the car is blue and regarded as blue is (.15×.8), so the probability that the car is really blue is (.15× .8)/(.85× .2)+(.15× .8). ）。 That is, the car is more likely to be green.

is it a little different from your brain intuition? Although our brain work is amazing, it is very immature in some mathematical intuition.

however, the expected value theory can't answer why the red button is worth as little as 1 million, and many people still choose it.

4

Expected utility theory (ambition or fear)

In his paper in 1738, daniel bernoulli challenged the expectation of the amount as the decision-making standard with the concept of utility. The paper mainly includes two principles:

a, the principle of diminishing marginal utility: one's possession of wealth is better, that is, the first derivative of utility function is greater than zero; With the increase of wealth, the increase rate of satisfaction is decreasing, and the second derivative of utility function is less than zero.

b, maximum utility principle: under the conditions of risk and uncertainty, the personal decision-making behavior criterion is to obtain the maximum expected utility value rather than the maximum expected amount value.

back to the case of wentou. Choose the red button, immediately realize 1 million, and give up the option worth 5 million. On the one hand, it is because it is "satisfied" with 1 million. As far as its wealth is concerned, 1 million has brought about an order of magnitude change, which can solve the biggest problem at present and is enough to be satisfied.

and what's the use of 5 yuan for one more order of magnitude? It may not be imagined;

on the other hand, I want to avoid the 5% zeroing risk of the green button. The fear of zeroing is far greater than the expectation of getting 49 million more.

to be exact, choosing the red button interweaves the comprehensive functions of "expected utility theory" and "prospect theory".

5

Prospect Theory

Don't be a Normal Fool quotes Kahneman's summary that he won the Nobel Prize for the prospect theory:

1) People are risk-averse when they get it;

2) When it is lost, the rational person is risk-averse, while the "normal fool" is risk-preferred;

3) Rational decision makers' judgment of gains and losses is not influenced by reference points, while "normal fools" often decide on gains and losses according to reference points; (For example, a rational decision-maker won't have to wait until he returns to his original position to throw away a stock that should be thrown away)

4) Normal fools usually avoid losses.

As behavioral economics studies, social, cognitive and emotional factors will make people make less "rational" choices.

For example, the base of wealth, as a reference point, largely determines whether people press red and green.

6

The probability weight given up by a fool

A fool doesn't know the basic common sense of probability and can't count as an expectation (based on one of the three theories).

Myth 1: I don't understand the law of large numbers

In mathematics and statistics, the law of large numbers, also known as the law of large numbers, is a law that describes the results of repeated experiments for quite a few times.

according to this law, the larger the number of samples, the closer the average is to the expected value. Stupid people always want to make money in casinos, and casinos are just the firm winners of the law of large numbers.

Myth 2: Gambler's Fallacy

Tworsky and Kahneman concluded:

In real life, people mistakenly associate the independent probability between each random trial. Take the coin toss as an example. We know that the probability of getting the pros and cons every time is 1/2, but some people will think that if you get the pros several times in a row, the probability of getting the tails next time will be even greater.

people often think that it conforms to the expected probability distribution as a whole, and it will also conform to the same probability locally. This phenomenon of applying the law errors obtained from large samples to small samples is called "law of decimals".

Looking back on the stock market crash in 215, it was bargain hunting that dealt a fatal blow to investors. After falling so hard, there should be a decent rebound. This is also a kind of gambler fallacy.

Myth 3: Survivor bias

Its meaning is that the statistical analysis based on the survivors of the event is biased, because the losers (or "victims") are not included in the sample (silent evidence in Black Swan), so the whole represented by the survivors is biased (even wrong).

myth 4: vividness effect

people attach too much importance to evidence that is more vivid and easier to extract from memory.

who should say "bon voyage" to whom? Friend B drives A 2km to the airport, from where A will fly to a city 75km away.

When leaving, friend B will say to A, "Have a safe trip". Ironically, B's 2-kilometer drive home is more than three times more likely to die in a traffic accident than A's flight.

However, influenced by the "vividness effect", it is still B who wishes for A..

7

The poor give up the probability right

The poor are eager to realize cash, unable to meet the delay and have low expectations for utility.

Cederhill, a Harvard professor, stated in his book Scarcity:

We are caught in the dilemma of scarcity. Once everyone is faced with scarcity, whether time or money is scarce, we will go into a state of "watching", which will lead to our scarcity mentality, which will easily lead to short-sightedness and borrowing from the future. In the end, we are getting poorer and busier.

I once chatted with a man, and he said, what we lack most is actually that a dad tells himself that you are awesome.

Apart from genes and resources, there may be the following reasons why there are a large number of talented people in a scholarly family or a wealthy family:

1) Having a high enough reference point, not being taken away by small interests, being more able to bear risks (in fact, with low probability), thus capturing high returns;

2) the demonstration effect of a group of people around;

3) Inner motivation that is ignited.

They are less likely to sell their probability rights "cheaply" than the poor.

So:

1) At the key decision-making point of the gap between the rich and the poor, the poor give up their probabilistic rights and interests;

2) The secret of the so-called winner is to stick to the dominant probability and not change the principle of life betting even if you are frustrated repeatedly;

3) Buying lottery tickets is the most expensive self-abandonment about the probability option, so it is called IQ tax.

if you have more money, you will invest in value; if you have less money, you will gamble. this