- #1

kairama15

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- TL;DR Summary
- Interesting things happen when a bunch of people randomly give and take a dollar from one another over and over.

I wasn't sure which forum to post this topic to, so feel free to move it, mods. The topic seems to cover a lot of fields of study: economics, physics, statistics, calculus, etc. :)

I was listening to a YouTube video the other day, and the speaker presented situation where there are a thousand people, each given 10 dollars. Two random people are selected to flip a coin. The loser of the coin toss gives the other person one dollar (and if they don't have any money at all, nothing happens since they can't give a dollar to the other person). Next, two new random people are selected, flip a coin, and a dollar is given in a similar fashion. This process is repeated over and over. The video link is:

.

(Skip ahead to 5:20 if you'd to watch the part that describes this situation.)

I made a computer program to imitate this situation because I was quite interested in it. The link to the program (the program runs in the web browser) is : https://www.openprocessing.org/sketch/950688 .

My attempt at trying to figure this out:

Note 1: After the equilibrium state occurs, the rate at which people lose money is proportional to the number of people that have 0 dollars. This is because if there are a lot of people that have 0 dollars, a person that wins a coin toss has a good chance of not even getting a dollar as a result. This 'force' makes it harder to obtain more money as more people have less money.

Note 2: I noticed that, in the video link, the speaker described a 'pareto destribution'. I feel that this is the improper distribution to describe this phenomenon. You may notice that the columns that describe how many people have a certain amount of money don't quite fit the pareto distribution overlaid on the speaker's graph. I believe this distribution is more appropriately described as an exponential distribution of the form:

y = K*e^(-L*x)

where K is the number of people that have 0 dollars at the equilibrium state (since when x=0, y=K), L is a constant describing the rate that the distribution approaches 0, x is the amount of money held, and y is the number of people that hold that amount of money. On the program, I overlaid an exponential distribution instead of a pareto distribution and it fits the graph nicely.

Note 3: If the total amount of people "n" must all be in the graph, then the integral of the distribution K*e^(-L*x) dx from 0 to infinity must be equal to n. Solving this simple equation, I get: n=K/L. So, the rate that the distribution decreases "L" is related to the number of people that have 0 dollars "K". Interesting!

Note 4: There is something relating to statistical thermodynamics going on here. If I simplify the problem to only 3 people, each given 1 dollar, there is something that stands out. I notice that:

-there is only 1 arrangement of the money where all 3 people have 1 dollar

-3 ways to arrange the money where one person has all 3 dollars and the other two people have nothing

-6 ways to arrange the money where someone has 2 dollars, someone has 1 dollar, and someone has 0 dollar.

So, if there is one arrangement of the money that just by chance appears more often, then it is the likeliest to occur. This similarity to microstates and entropy may be related to this problem. A very similar situation using energy states of molecules is described here: https://ch301.cm.utexas.edu/section2.php?target=thermo/second-law/microstates-boltzmann.html

If this situation is exaggerated to 1000 people instead of 3, then the system may be acting to increase its entropy by going to its most chaotic state. After realizing this relation to entropy, I noticed that the speaker's program actually even has a graph charting entropy as the program runs! So the solution to finding how many people have 0 dollars may be seriously related to Boltzman's study of microstates.

I was listening to a YouTube video the other day, and the speaker presented situation where there are a thousand people, each given 10 dollars. Two random people are selected to flip a coin. The loser of the coin toss gives the other person one dollar (and if they don't have any money at all, nothing happens since they can't give a dollar to the other person). Next, two new random people are selected, flip a coin, and a dollar is given in a similar fashion. This process is repeated over and over. The video link is:

.

(Skip ahead to 5:20 if you'd to watch the part that describes this situation.)

I made a computer program to imitate this situation because I was quite interested in it. The link to the program (the program runs in the web browser) is : https://www.openprocessing.org/sketch/950688 .

**(after some time goes by). I imagine the number of people who will have 0 dollars will likely be a function of how many people there are in the system and the initial money given to each person.***I am interested in figuring out how many people (on average) will have 0 dollars after the system reaches its equilibrium state*My attempt at trying to figure this out:

Note 1: After the equilibrium state occurs, the rate at which people lose money is proportional to the number of people that have 0 dollars. This is because if there are a lot of people that have 0 dollars, a person that wins a coin toss has a good chance of not even getting a dollar as a result. This 'force' makes it harder to obtain more money as more people have less money.

Note 2: I noticed that, in the video link, the speaker described a 'pareto destribution'. I feel that this is the improper distribution to describe this phenomenon. You may notice that the columns that describe how many people have a certain amount of money don't quite fit the pareto distribution overlaid on the speaker's graph. I believe this distribution is more appropriately described as an exponential distribution of the form:

y = K*e^(-L*x)

where K is the number of people that have 0 dollars at the equilibrium state (since when x=0, y=K), L is a constant describing the rate that the distribution approaches 0, x is the amount of money held, and y is the number of people that hold that amount of money. On the program, I overlaid an exponential distribution instead of a pareto distribution and it fits the graph nicely.

Note 3: If the total amount of people "n" must all be in the graph, then the integral of the distribution K*e^(-L*x) dx from 0 to infinity must be equal to n. Solving this simple equation, I get: n=K/L. So, the rate that the distribution decreases "L" is related to the number of people that have 0 dollars "K". Interesting!

Note 4: There is something relating to statistical thermodynamics going on here. If I simplify the problem to only 3 people, each given 1 dollar, there is something that stands out. I notice that:

-there is only 1 arrangement of the money where all 3 people have 1 dollar

-3 ways to arrange the money where one person has all 3 dollars and the other two people have nothing

-6 ways to arrange the money where someone has 2 dollars, someone has 1 dollar, and someone has 0 dollar.

So, if there is one arrangement of the money that just by chance appears more often, then it is the likeliest to occur. This similarity to microstates and entropy may be related to this problem. A very similar situation using energy states of molecules is described here: https://ch301.cm.utexas.edu/section2.php?target=thermo/second-law/microstates-boltzmann.html

If this situation is exaggerated to 1000 people instead of 3, then the system may be acting to increase its entropy by going to its most chaotic state. After realizing this relation to entropy, I noticed that the speaker's program actually even has a graph charting entropy as the program runs! So the solution to finding how many people have 0 dollars may be seriously related to Boltzman's study of microstates.

**Any encouraging insights about this interesting phenomenon are greatly appreciated.***So, how may we elucidate the number of people who have 0 dollars at the equilibrium state?*
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