This is an irreducible Markov chain (i.e. it's possible to go from any state to any other with positive probability in a certain number of steps) on a finite state space (because money never enters or leaves the system). This means that every state is recurrent, i.e. will happen if you wait long enough.
This means you should eventually see each person hoard all of the money in turn. Try it with 5 people and 5 dollars each. Then the system only has 29 choose 25 = 23751 states[1], which should give your computer enough time to hit them all.
Edit: Here, Python "proof"
from random import randint
numpeople = 5
initmoney = 5
ledger = [initmoney]*numpeople
while True:
recipients = []
for i in range(numpeople):
if ledger[i] == 0:
recipient = None
else:
recipient = randint(0, numpeople-2)
if recipient >= i:
recipient += 1
recipients.append(recipient)
for i, recipient in enumerate(recipients):
if recipient is not None:
ledger[i] -= 1
ledger[recipient] += 1
if max(ledger) == initmoney*numpeople - 1:
print ledger
I do not have the time to do this right now, but - as this is a recurrent, finite, state machine, there is a unique stable distribution on states by application of the Perron Frobenius theorem[0] to the state transition probability matrix; that stable distribution on states is a strictly positive eigenvector with sum 1 and eigenvalue 1 (and can thus easily[1] be computed from the sparse 23751x23751 transition matrix)
This, is, by the way, somewhat similar to how Google's original PageRank[2] worked: Each page has some "worthiness capital", and at each point in time, it randomly gives one "worthiness coin" to one of the outgoing links. Give each page some starting capital (e.g, 1,000,000 coins), run the game for a very long time, and rank pages by the capital they managed to collect. There is a big difference, though - each page is not equivalent to a person in the 5-person game, but to a state in the 23751-state space; also, PageRank had damping factors, sink escapes, and other changes, but the basic idea is the same, and was actually (at least) 20 years old at the time that Page and Brin applied it to links of the "href" kind.
I agree with most of your comment, but this part is misleading:
> This means that every state is recurrent, i.e. will happen if you wait long enough.
It doesn't mean that the expected frequency of every state in the chain is equal. Moreover, the frequencies are very different.
The interesting part of the experiment is not that each state is reachable. The state where someone has all the money - 1 is reachable. The state where all have the same amount of money again is reachable. But if you leave these people alone in a room for some time and open the door unexpectedly, you will almost sure find out that they have very different amount of money.
I ran an numerical experiment and I get that the average distribution of money is for 5 persons with 5 coins each is:
11.04 6.38 4.03 2.44 1.11
i.e. if you enter the room at random, the one with more money will have on average 11.04 coins, not almost 24 coins
If you count the number of times they have
11±1 6±1 4±1 3±1 1±1
coins, you get a 21% frequency.
If you count the times someone has 24 coins, you get only a 0.00125% frequency. If you count how many times someone has at least 90% of the coins you get 0.0131% frequency (10x more).
This is difficult to compare, because each classification has a different amount of states, but nevertheless, it's clear that an almost linear distribution is more common than a very concentrated distribution.
For 10 persons with 10 coins I get that the average is
> The point is not that some people become rich and never lose their top position. This runs infinitely and will contain every possible sequence of good and bad luck for every person.
> The richest will become the poorest, everyone will experience every rank, and so on.
> The interesting thing is that this simple simulation arrives at a stationary distribution with a skewed, exponential shape. This is due to the boundary at zero wealth ...
It should be fairly anti-symmetric, since the dynamics of losing are similar to those of winning. So instead of exponential looking, it's shaped like exp(rank) - exp(-rank). (just confirmed this with a quick simulation)
With participants able to go into debt indefinitely, the process becomes a random walk.
The result should be each participant being up or down from 0 at a rate of proportionated sqrt of t, the number of iterations of the system. Implying a wealth distribution which would become ever-more skewed over time.
Is that what "skewed" means? To my mind a widening bell curve which remained symmetric (which is what we're talking about, right?) would not be "skewed".
Over time, the average participant would have either far more coins in credit than they began with or be in debt for far more coins than they started with.
Whether you have what statistician call "skewed" or not, you have what most people would call skewed, unequal, extreme.
"Varied" or "widely distributed" seem like better terms. To my mind, "skew" is what we have in the original problem (with a zero-bound) - it only describes asymmetric distributions.
Removing the boundary doesn't really make sense -- if you allow people to go into negative dollars but still be able to give a dollar each tick, then there's no longer a finite amount of money in the system.
You're kidding, right? No way a generic observation about a markov chain (which doesn't even model interest) implies that "uncapped debt reduces wealth mobility".
So, with typical debt, interest rates are higher for debt than for savings. Does seem that a) removing the cap, and b) introducing interest on savings < debt would trap some people in poverty?
> I agree with most of your comment, but this part is misleading:
>> This means that every state is recurrent, i.e. will happen if you wait long enough.
> It doesn't mean that the expected frequency of every state in the chain is equal. Moreover, the frequencies are very different.
I do not find it misleading; I do not see how you equate "if you wait long enough it will happen" with "every state is equally likely" or "the average waiting time to get to any state is the same" (which may or may not be equal depending on your where you consider your starting point).
But there's no previous state that can result in one person (I'll name them X) having all the money and everyone else having no money. There are 2 cases:
1) X had some money on the previous turn, giving a dollar to Y. Then Y has at least 1 dollar.
2) X had no money on the previous turn. Then by pigeonhole principle there exists a person Y who has at least 2 dollars. Then Y has at least 1 dollar on the current turn.
Oops, right. I'm counting the number of states wrong. But the general point stands, you'll see everyone hit the maximum amount of money possible for one individual.
I wonder what their point is though? Like, are we supposed to be surprised with the shape of the distribution?
Also, while I agree this is theoretically possible to "see everyone hit the maximum amount of money possible for one individual"... given enough individuals the possibility of someone ending up with nearly all the money is close to zero. I've run a simulation with 500 individuals (here: https://github.com/subroutines/DollaBillz ) and it seems like the distribution of money reaches a steady-state. What should we make of that?
I'm disinclined to attribute them with having much of any point at all. Their discussion is too brief, and their presentation style is more clickbait thance science. They tout a counterintuitive result (It's really not, at least not for anyone used to thinking about data or state changes, certainly not once you think about what the 4th or 5th state change looks like) And then it ends the headline with the overused, even in for content farm clickbait, "you'll never guess what happens next". This tells me they're less interested in discussing an interesting result than getting eyeballs on the page via a shallow observation passed off as insight. If there's a deeper point to be made, they haven't made it.
Thank you. Finally someone said it. This was not the least bit counterintuitive to anyone familiar with randomness.
We're they trying to imply that people should assume the distribution would remain uniform?. That strikes me as more "insane" than intuitive.
It's simply a function of how the tokens are being exchanged that would almost require some actors get the short straw, and since this is happening 100 times per round, that randomness ensure some actors get progressively shorter straws over time, because randomness is neither fair, not even. It's randomness.
No, they're making a simple digital model that conclusively disproves the notion that 'because disparity in wealth exists, meritocracy exists'. Knowingly or not, that's the point being made.
If you want an even more insanely staggered distribution, let people with resources increase their chances of being given a dollar. This establishes that even in the complete absence of merit, a striking distribution will arise and stably persist, and it's created out of the mechanical behavior constraint of 'possibly being broke and having no dollar to give'. That alone creates what looks like a 'meritocracy' with 'high performers'.
Add literally any form of actual merit, and this gets more extreme still. The funny thing is, the merit-less distribution is not wildly out of line with what people think the distribution should be WITH merit.
Think of it as "entropy mostly increases". It's theoretically possible that a bowl of soup left to itself could become hot in one corner and cold everywhere else, but it's vanishingly improbable compared to a mostly uniform temperature distribution.
Ok right. It just seemed like the article and jordigh were making contrasting points. Although neither make an explicit broad claim (in which case, what are we doing here other than running an arbitrary MCMC and then stating a Markov property), the article is saying hey look, rich people get richer, while jordigh is saying, nah "you should eventually see each person hoard all of the money in turn.".
So which is it, probably? I think this might be more complex than "everyone will hord all the money eventually", given that the most probable number of sign changes in a 1D random walk is zero.
Yes, the second law of thermodynamics is a statistical law. It is physically possible that all the particles in an ideal gas container will be moving east all at the same time, for example (it doesn't break any law of mechanics), or that a temperature fluctuation will spontaneously appear in a material in thermal equilibrium. It's just that the probability for such spontaneous entropy decreases happening is astronomically small.
I'm not a physicist and I probably could have phrased that a little better. The example of a bowl of soup randomly and unevenly changing temperature implies to me that the physical system behaves randomly which I do not believe it does.
You don't have a detailed knowledge about the state of every molecule in the bowl soup and its environment, so you have to rely on statistical mechanics. And the current macroscopic state that you observe could correspond to a particular microscopical state that evolves in surprising ways. But the probability is so low that it just doesn't happen.
To make things worse, quantum mechanics is intrinsically random (at least as far as we know) and having a detailed knowledge of the state of the system allowing for certain prediction of its evolution is impossible in principle (and not just in practice).
A dose of realism would have anyone with >$x be able to reduce the amount or frequency of giving away money. Or be able to spend some of their wealth on 'marketing' and increase the likelihood of being the recipient of any particular dollar being given away.
Not to demonize accumulation of wealth but just to model reality where rules can sometimes be changed by those with enough wealth accumulated.
Assuredly, but this is the same as with a deck of cards. If you apply a shuffle that you know creates a closed markov chain (say, moving the top card to a random position within the deck), then you will reach a steady state. But, given enough shuffling, you'll also return to a new deck order
True, but I think Grue3's remark is more about the fact that this Markov chain is not irreducible than the number of states being wrong. Because you can go from the state where one person has all the money to the other states but not the other way around.
In this case, the unreachable states don't partition the space in a meaningful manner, nor does the system start in an unreachable state, so the only effect is that those states occur with probability zero instead of with very low probability in the stationary distribution. There are also very few (N) of those states compared to the total number of states (about N^(2N) if I'm estimating right), so removing them has very little effect on the overall statistics.
For a more interesting example, suppose that on each round, everyone with money picked a random person and gave them 2 dollars (and take N even). Now you'd have a Markov chain that is reducible in a nontrivial manner -- think about the parity constraints.
While that moves the pieces back to their original position, it does not reset to the original game state. After moving the knights back, it would only require 48 additional moves without a pawn moving before the game ends in stalemate, where it required 50 moves at the start of the game.
Because both players have already taken two turns, a player can claim a draw after only 48 moves. If the movement of the knights was repeated 25 times, for instance, then either player could claim a draw. However, this is not possible at the beginning of the game, so simply moving the knights back does not actually restore the game state.
It's just built into the rules: 50 moves without a capture results in a stalemate (https://en.wikipedia.org/wiki/Fifty-move_rule). I guess you could say that the two moves 'go' into the implicit state counter that keeps track of the number of moves without capture.
I guess what sounds authoritative is that I'm using jargon while trying to explain the jargon at the same time, as if I knew what I was talking about. I've since edited my comment. I know it's stars and bars, but I keep messing up the count. Anyway, it's a finite amount, it's not huge, and your count is wrong too by a sister comment. :-)
the fun is in actually starting from every initial condition and then evolving the simulation out for as many rounds as your computational resources allow....
then do some statistical analysis to categorize the various initial conditions and their distributions.
then inject perturbations in the simulation like "death taxes" and other redistribution policies.
I built up a lot of mathematica code for anyone to play with.
one of my favs was to start 1 person with all the money and see how long it takes for them to no longer be the richest person in the game.... you know like millions of ticks of the clock.
> This is due to the boundary at zero wealth which, we imagine, people don’t consider when they think about the problem quickly.
So I wanted to see if this is true. (I ran 5000 ticks, 45 people, 45 initial dollars each). I used my own program.
I removed the condition that you must have at least $1 to redistribute it (in other words I allowed negative wealth) and ended up with this distribution:
While it affects the shape, it is far from the case that it looks like some brownian motion near the starting wealth. When sorted, it still has a basically unequally distributed shape.
There's two asymmetries going on. The first is the one you played with, which is zero hard boundary. But there's also the asymmetry of losing zero to one dollars every round, while gaining zero to n-1 dollars every round. I bet you can minimize the skew by applying a progressive rate to how much is given away, such that you can also lose zero to n-1 dollars every round.
I don't think this is relevant, though. It really doesn't matter that the process reaches every configuration given enough time; the post is about the limiting behaviour. Take roulette for example. If played long enough it will pass through every configuration with 100% certainty; yet its limit is -∞ dollars.
It's not possible for one person to ever have all of the money unless they have the possibility of not giving money away. Imagine the most optimistic state scenario just prior to hopefully having a state where one person has all the money. In this scenario everyone but our lucky rich guy has one dollar and the rich guy has all the rest. On state transition everyone gives the rich guy a dollar, but the rich guy still has to give a dollar to someone.
This is not surprising. Random doesn't mean homogeneous. Notice that in this model there are rich people and poor people, but advance many iterations and the rich people aren't the same people as they were.
If you want to see the real problem, don't let the people with no money get away with not paying the dollar. Make them borrow it with interest from someone with money.
Actually, without interest people would have no incentive to lend to others. Any loan carries a risk of not getting your money back. If interest was illegal then the rational move would be to never loan out any money.
> Stores would extend credit to customers because they want to make sales.
Not GP but: much less credit would be extended because many stores wouldn't be able to do much with bookings. One reason credit cards are so ubiquitous is because they allow even tiny stores to take advantage of extending credit without having to worry about all the stuff that comes with it (not getting paid soon, fraud etc.). So yeah, sure, some people would still lend to close friends and family, but in general the credit market would get orders of magnitude smaller, and I don't think that's good for anyone.
> But regardless, how was that supposed to dispute any part of the post you responded to?
Again, not GP, but he is replying to your comment implying that access to money at interest would make things worse for the lower class in this model and that making that change would make it more realistic. It may make it worse for the lower class in this model because you can't inflate the number of dollars, but in real life there's a lot of good reasons to think credit markets are extremely helpful for everyone, including the lower class.
Murabaha is largely a semantic hand wave that really doesn't fundamentally differ from the credit/mortgage model we have. The lender makes profit, frequently there are late fees, the government gets involved if the loan is not repayed (with varying consequences based on the country), the underlying asset gets seized etc. The major difference is that you don't call the profit 'interest.'
It's really basically just a way for Islamic countries to claim that they are adhering to Sharia law while not being completely left in the dust because credit is so completely vital to a modern economy.
In addition to what wdewind said, the restriction on charging interest and late fees is problematic. What to do about defaults and late payments? In the Gulf, the solution is to put debtors in prison.
> It may make it worse for the lower class in this model because you can't inflate the number of dollars, but in real life there's a lot of good reasons to think credit markets are extremely helpful for everyone, including the lower class.
There are a lot of good reasons to think that they aren't, too.
If you model credit as something you give to one person then it looks like a win -- now that person can afford a home when they couldn't before. But if you make credit generally available then it causes price inflation. That not only means higher costs, you have to pay interest on the inflated price until you've paid off the principal. Which middle income people can generally never do, because if they could then the people in the percentile below them could have gotten an interest-only loan and outbid them for the house.
The result is that you get to claim a PR win because more people "own a home", except that the homes are really owned by the bank and the only difference from renting is that the payments are called interest instead of rent.
The number of people who own a home outright goes down because many middle income people could otherwise have bought a home without borrowing, but against the wildly inflated prices they have to take out a mortgage. Imagine the cost of a home was little more than a down payment currently is.
And this carries down through the generations because children who inherit a home now also inherit a loan in nearly the entire amount, the money used to pay inflated medical bills that are only so expensive because credit availability allows them to be.
It's perhaps not a coincidence that the last few decades have seen ever increasing credit availability while the middle class shrinks.
I'm not really sure what you're arguing here. As I've mentioned in other posts, there's obviously a balance, but suggesting that the credit markets could go away entirely and be supplanted by "friends and family" loans is a gross misunderstanding of the credit markets, and certainly would not at all benefit the lower class. The entire premise of TARP was that this is true and economists on both sides of the aisle largely agree with it.
> It's perhaps not a coincidence that the last few decades have seen ever increasing credit availability while the middle class shrinks.
I don't think so. I think it has much more to do with tax law.
> The entire premise of TARP was that this is true and economists on both sides of the aisle largely agree with it.
The premise of TARP was that allowing large institutions to fail and trigger a cascade of bankruptcies would have resulted in too much collateral damage to innocent parties who reasonably assumed that large financial institutions and insurers would meet their obligations.
Putting restrictions on bank lending is very different than allowing them to default at scale.
> I don't think so. I think it has much more to do with tax law.
That is a widespread belief but it has no basis. The rich have never paid high taxes in the US, even at times with nominally high tax rates. During the period in the 20th century that the highest individual rates were >80%, there were also so many loopholes that rich families regularly paid no taxes. And if the taxes actually paid haven't gone down then it has to be something else.
But "the power to tax is the power to destroy" as I'm sure you know. What difference in result do you imagine between a law discouraging lending and a law taxing it heavily enough to prevent wealth concentration?
Also, it's good to remember what 'marginal rates' are. I always say, if you want to argue as if marginal rates are the base rate, let's ask your accountant if they'd like to adopt your line of argument. I daresay they'd have a pretty clear idea of what's at stake.
No, because it means much less economic growth. There is obviously a balance: too much lending and the economy collapses the other way. But too little lending is really bad for everyone too, and there are pretty fair reasons to think it might impact the lower class the most. Getting rid of (relatively) easy access to credit would be completely catastrophic for the global economy.
You're assuming the alternative to lending at interest is nothing rather than something.
A loan really does three things.
First, it creates new money. When you borrow money from a creditor, notice that they generally don't mail you a bundle of dollar bills. What really happens is that they credit your account with the loan amount, and at the same time create a loan account with a negative balance which is what you owe them. These cancel out, which causes their books to balance even though it is simultaneously the case that the amount of cash in their vault hasn't changed and the amount of money in your bank account has. This is why increased lending causes price inflation.
Second, it causes there to be more money in the hands of borrowers. This is where you get economic growth; people have more money to spend.
And third, it causes the borrower to owe the lender interest.
Notice that the first two are the only ones that are actually good for anything. Even the inflation is generally a cost, but in moderate amounts it's beneficial because it counteracts the natural deflationary tendencies of a growing economy. All you really want is for people to have money to spend.
But you don't actually need any lenders for that. If all you want is to create new money and give it to people, the government can create money by fiat and just hand it out to all citizens, or pay for government with it in lieu of taxing people. It's the same thing, there is just no interest.
> You're assuming the alternative to lending at interest is nothing rather than something.
No, I'm arguing with what you proposed, which is this:
> > People would still lend money to their friends and family. Stores would extend credit to customers because they want to make sales.
which is not in anyway a meaningful alternative to the actual credit market.
> Notice that the first two are the only ones that are actually good for anything.
Except that the first two only exist because of the third.
> But you don't actually need any lenders for that. If all you want is to create new money and give it to people, the government can create money by fiat and just hand it out to all citizens, or pay for government with it in lieu of taxing people. It's the same thing, there is just no interest.
It's the same thing, there is just no incentive whatsoever for the behavior to happen, so it reality it's not at all the same thing. This would cause massive devaluation of currency.
> It's the same thing, there is just no incentive whatsoever for the behavior to happen, so it reality it's not at all the same thing.
What do you mean? The government creates money all the time. They love it -- they get to spend money without raising taxes. The only downside at all is that in the extreme it causes too much inflation.
If we made it harder for banks to make loans but then created correspondingly more money and used it to lower taxes on people so they would have more money and not need to borrow as much, what economic difference do you see other than interest not being paid to banks?
It may be easier to imagine if you think of the government as a bank that makes interest-only loans at 0% interest.
> This would cause massive devaluation of currency.
Bank lending already does that. Look at housing and education prices (where the inflationary effect of lending is strongest), or even the general value of the dollar over time. The dollar has lost more than 90% of its value since 1950 and far more of the increase in the money supply has been due to banks than the government.
$1 in new loans and $1 in government-created money cause the same amount of inflation. If you replaced one with the other the only difference would be that no interest would be paid to the bank.
The vast majority of new money is not created by the government, and that fact seems to be missing from your argument. If we no longer had credit we would be shrinking the available money supply by about 90%. This would be completely catastrophic for not only the US but the entire global economy.
Maybe you could be more specific about what you're proposing because I feel like I am not really understanding it.
> Maybe you could be more specific about what you're proposing because I feel like I am not really understanding it.
Cause less money to be created by banks and more by the government, and have correspondingly lower taxes (or even negative taxes) on middle income people.
>But you don't actually need any lenders for that. If all you want is to create new money and give it to people, the government can create money by fiat and just hand it out to all citizens, or pay for government with it in lieu of taxing people. It's the same thing, there is just no interest.
Oh boy. You might want to take an economics class there bud, what you're talking about would just create the same situation Zimbabwe is in. More fiat without anything backing it leads to inflation.
And why would "more economic growth" be a good thing?
I don't want endlessly slaving to make the pie bigger (while ignoring second order effects, from quality of life, to environmental issues, to debt etc).
How about we learn to enjoy a moderate rate of economic growth, or even a steady economy, and focus on improving other aspects of life besides bank accounts?
>But too little lending is really bad for everyone too, and there are pretty fair reasons to think it might impact the lower class the most.
Under the current model though. Whereas e.g. better distribution of wealth might help them more than more lending.
> And why would "more economic growth" be a good thing?
Because billions of people still live in poverty worldwide.
> I don't want endlessly slaving to make the pie bigger (while ignoring second order effects, from quality of life, to environmental issues, to debt etc).
No one does. Economic growth, although not by itself, helps alleviate a lot of these issues. There are other factors that also impact this, and I'd argue a lot of the economic issues we face aren't from too much growth but from these other factors (taxation etc.) not being handled properly. So in general more growth is advantageous, but we have to other things going right as well.
> How about we learn to enjoy a moderate rate of economic growth, or even a steady economy, and focus on improving other aspects of life besides bank accounts?
Well a lot of people disagree about what things like quality of life and adequate environmental care are. One of the beauties of capitalism is that it allows people with wide disagreements about what's important in life to coexist and even benefit each other.
Edit: sorry I didn't see this before:
> Under the current model though. Whereas e.g. better distribution of wealth might help them more than more lending.
Maybe, but if that better distribution of wealth comes at the cost of shrinking the overall system (which there is a real reason to think would happen: this is why people and corporations bank outside of the US frequently already) then I think it's hard to conclusively argue either way.
Again this is not to say I don't see problems with our current economy, I just think "it's growing too much" is not one of them.
But the causes of poverty are not an insufficiency of economic growth, it's the distribution of the fruits of economic growth. In many places the wealth and/or political power (which are to some extent proxies for each other) are hoarded by a small number of people who have no interest in distributing them.
In their view it's better to be a big fish in a small pond than a small fish in a larger pond. Of course this isn't a foolproof strategy (dictators and kings sometimes come to ignominious ends) but holding onto as much wealth and power as possible while keeping the majority of people disenfranchised is a much safer strategy than spreading the benefits as widely as possible and hoping that everyone will love you enough to assure your future security.
Well a lot of people disagree about what things like quality of life and adequate environmental care are. One of the beauties of capitalism is that it allows people with wide disagreements about what's important in life to coexist and even benefit each other.
Yes, but it doesn't assure that, and that's where it gets ugly. If too much of capital is controlled by someone who doesn't care about the environment and is fine with cutting corners or actively polluting, then the less well-off people suffer and die, sometimes horribly. If that problem gets bad enough, then everyone could suffer as the whole environment is wrecked - like the population of a country whose leader foolishly enters a war and gets bombed, or a world where reckless fossil fuel production/use continues past the point of long-term sustainability.
You'll recall that Keynes said 'in the long run, we are all dead.' Capitalism would be great if everyone lived long enough to have their day at the top of the pile, just as it is enjoyable to play Monopoly as a board game because while you might lose one game today you might have the pleasure of winning tomorrow. But people are not economic abstractions; if you're poor and you can't compete effectively in a capitalistic economy then there's a good chance that your life will be terrible, and then you'll die.
I mostly agree with all of this and felt like it was covered by:
> Economic growth, although not by itself, helps alleviate a lot of these issues. There are other factors that also impact this, and I'd argue a lot of the economic issues we face aren't from too much growth but from these other factors (taxation etc.) not being handled properly. So in general more growth is advantageous, but we have to other things going right as well.
Agreed, I just don't feel that an insufficiency of economic activity is the issue right now, as compared to a few years back when it appeared the world might fall into a deflationary spiral or seize up in a liquidity crisis.
Technically speaking, banning the charging of interest does not necessarily need to mean that the yield on credit extended drops to zero (as Islamic banking or zero-coupon bonds show).
So first off, I want to question the premises that this is a widespread thing, and that it is an effective business practice. The whole point of credit cards is it allows the bar to be out of this business, and most of them gladly accept credit cards so they can do that.
Second of all, especially in smaller communities where this practice is more likely to occur, there is a social capital gained by doing it which is not necessarily directly monetary, but I think it's pretty easy to see the rational self interest argument for "being part of a community" outside of just purely monetary gain.
> It is very rational to lend money with zero risk if that gets you more business.
Well it's definitely not zero risk, and when credit cards exist, which deal with all the risk, it's not rational to lend money without credit cards.
With tabs, the bar management has to do risk management: to decide who is allowed to keep a tab and how large each tab can get. This is hard to do without access to customers financial data. With credit cards, that risk management is done by the credit card network (mainly the issuer bank). The merchant pays a fee, and in return it gets cash at a predictable timeline with relatively little risk.
If you have ever spent a non-trivial amount of time in a bar, you can see that people with credit cards run up tabs due to a myriad of events that include intentional bad decisions or irresponsibility (like being too drunk to remember to pay before jetting off to another destination). These tabs are generally held for a number of hours, days or weeks (depending on familiarity with the parties involved) before being elevated to a more serious consideration.
That's like arguing that restaurants are extending credit because they charge you at the end of the meal. The "keep it open and then you forget to pay" type tabs are very different than having a long running tab at a bar, which is not super common and is what I thought we were talking about. One is much closer to credit than another. Regardless, neither are significantly meaningful in the larger credit market.
Is it normal in <your country> for the bar to take a credit card when opening a tab?
That's the normal method in Britain -- your credit card is generally put in a little booklet behind the bar, so they can charge it if you forget at the end of the night¹. Alternatively, they may take a payment upfront -- for example, if a large group of teenagers have booked a private area of a nightclub and put £500 up for a tab. The drinks continue until the upfront money is spent.
I assume the situation is a little more relaxed for the regulars in a rural pub.
¹ They won't have your PIN, but they can still run the transaction -- though it's easier for the cardholder for dispute.
that's not really a 'tab' in the original sense as implied by the OP.
a tab is where you dont pay at all for an extended amount bof time, and settle afterwards. for example, you might settle monthly, or yearly. i do think this kind of tab doesn't happen very often.
What a weird request. For someone to be confident that a practice is not widespread, he does not need to know where it is confined to. We don't expect him to have visited all localities.
I am confident that there is a goat behind one of the doors because the host has told me as much. That doesn't mean I can tell you with confidence which door it is behind.
There are loans without interest. An extra cost/profit is calculated up front (that's e.g. how Murabaha works in islamic banking which prohibits interest). The law still enforces the person to pay back the original amount (or face the repercussions).
(Plus people lend money and tools and whatever all the time with neither interest nor legal guarantees).
I agree with that. Seeing more of this thread, I would clarify that I agree that interest rewards people for accepting risk in addition to temporarily losing the use of their money, and I support that.
I just want to point out that the phenomenon of interest-free lending as charity is a real one.
Assuming the number of $ initialized remains constant, wouldn't all of the money end up in with a single player leaving all other players in perpetual debt?
Since the money required to pay the interest is never created, this would create a negative sum game, as players would be charging for money (interest) that doesn't exist.
This would certainly be interesting. One could even keep loading this up with different tax system. Also money could be injected into the system both at a small random rate externally and rarely as an innovation boom following a substantial loan.
IMO, this reveals more about human intuition regarding randomness than whatever financial point it purports to make. It's still a useful point though.
Reminds me of the load balancing literature. There, the explicit goal is to evenly divide the burden across your fleet of servers, and having wide distributions is a problem on both ends: you're paying for servers to sit idle, and some are over burdened and giving customers a bad experience (high pageload times).
Here's the video result: https://www.youtube.com/watch?v=94Vc7gf3ONY Much tighter distribution, though you need to be able to identify the size of people's bank accounts. In this model, it's very rare for anyone to give the richest anything, unless you magically choose two people randomly tied for richest.
It converges there pretty quickly. So I put together another video starting out with an imbalanced curve (maybe due to a rolling app deploy) smoothing back out. As you might expect, it gets there rather quickly:
https://www.youtube.com/watch?v=wvnwx4r2ERA
What is happening here is fairly simple: Since each state in the system occurs with the same probability, the distribution that arises is the one that maximizes the entropy. Since the total amount of money is conserved, this means that the mean of the distribution is fixed. The maximum entropy distribution in this case has a geometric shape, which is skewed:
Yep, it's just a simple counting argument, you see the distribution that has exponentially more states than any other, subject to constraints. Specifically, you are allowing the range [0,∞). If you had constrained it to a fixed range of money (e.g. there is corrective feedback when you get too much money), then you would have seen a more uniform distribution.
As others have noted, every state in the state space will eventually be explored and so the appearance of convergence is a little misleading. Eventually the richest person will lose all the money and someone else will take that place.
An interesting question is why we happen to see the system in such an unequal state when we look at it after e.g. 10000 iterations. The reason for this, I believe, is entropy. There are just vastly more ways to distribute the money unequally than equally (thinking of these as macro states). For example, there is only one (macro) state where everyone has $100, but there are 100! states where the money is distributed in a perfectly unequal way (e.g. 0, 1, 2, ..., 99, 5050 or any other way where all the people can be distinguished). So, if you randomly peek at the system at any given time it'd be very unlikely that you'd see it in any way other than a highly unequal state. To be fair, this argument neglects the dynamics (perhaps the transition probabilities to those states are sufficiently small that the n! multiplicity is watered down, although I suspect this is not the case.)
In this simulation, the amount of money you have does not affect how much you have to give away per round (until you get to $0, in which case you give away nothing) or how much you will get per round (until some people get to $0, in which case the money you get is less on average). So in most cases, with whatever you have, you are expected to see a gain distribution with zero mean. Since it's not a self correcting process other than at the $0 end, it's not surprising to see (1) a snapshot with uneven distribution, (2) no one dropping out permanently.
Actually I don't know what is the point they are trying to make with this video. There is neither anything surprising in the general sense nor a solid statement about what is specifically seen.
The point is that it's counterintuitive even to intelligent people, and that matters because politics and economics are largely driven by the accumulation of intuitive decisions.
I went a little further and ran a simulation [^1] with random payments in the range from $1 to a fixed percentage of the payer's wealth.
When this percentage is in the range 2-10%, the wealth inequality is significantly smaller than in the base case (i.e. all payments are $1). Then it starts growing again.
I doubt it would change very much. I wrote a program a few years ago similar to this [1] and I tried several different amounts of "payment" (fixed and random) and each time the result was the same.
What I did not track was the number of times a "person" became rich (for example, how many times a "person" had over 90% of the money) as that did not occur to me.
Worth reading the comments under the post – e.g. people testing it and seeing that values for individual people go from 0 to very high and back over time, which you can't see in the histogram.
Thanks for pointing this out. It's still fascinating to see how after running the simulation for some time, the ranked histogram appears to be geometric (? [1]) at any given time point, while the wealth of each individual players will go up and down over time.
Happens in real life, too. The income level of the top 10% and the top 1% and the top 0.1% are relatively stable in aggregate, but who is in each of those categories changes drastically over time.
> Studies by the Urban Institute and the US Treasury have both found that about half of the families who start in either the top or the bottom quintile of the income distribution are still there after a decade, and that only 3 to 6 percent rise from bottom to top or fall from top to bottom. - https://en.wikipedia.org/wiki/Socio-economic_mobility_in_the...
Now, you could look at how people/families move around the distribution over longer timescales (multiple generations) but we don't really have the data yet.
IMO, 3-6% moving all the way from bottom to top or top to bottom is actually quite good. I'd be extremely concerned if it was 20%, as that would imply the system was nearly random. I'd also be concerned if it was 0%, as that would imply the system was static. 3-6% implies there is mobility, but it takes both hard work and luck to make it all the way from bottom to top.
Likewise, half of families staying in either end quintile also seems healthy to me -- it implies half of families leave those quintiles, too.
In an ideal world, what range would you like to see for those numbers?
This kind of simulation results in an exponential distribution, which is fairly equal compared all things considered. In an exponential distribution, most people have roughly the same order of magnitude of wealth (10s to 100s of dollars). In real life, the bottom 99% follow an exponential distribution pretty closely, while the 1% follow a pareto distribution, which is WAY More unequal. The transition is very sharp too, and has been studied in econophysics models.
If you have $1, you have to give it away, but you receive less than $1 back on average (because some people have no money to give you). So the poor get poorer. But wait, by the same argument the rich get poorer and this simulates a system of stochastic handouts to the very poorest - those with $0.
Alternatively, it simulates a lottery. From the point of view of any individual, every turn you spend $1 on a lottery ticket and get a prize of between 0 and $99. Viewed that way, it's not surprising that some people end up rich in the simulation.
This is a bit too "spherical cow in a vacuum" for my tastes. As a quick experiment, I included population growth (using average population growth rate of the US over the last 50 years) and GDP growth (again using US average for last 50 years). Assumptions being that new population start with $0 and GDP growth applies equally to all as a percentage of their then-current wealth. It substantially enhanced the inequality.
In the simple run, median/mean was 67%, but with compounded growth it was only 7.9%. Moreover, the standard deviation went from 1.31 times the median (87% of the mean) to 17.83 times the median (141% of the mean).
without any cap it doesn't look fun for the unlucky person. Take a look at my other comments on this thread, there's a link to a gist that generates an animation with the same starting conditions as OP, but forcing the person with $0 to borrow and introducing an interest rate of 10%. At some point the system destabilizes and the debt of the unlucky person goes out of control.
This reminds me of "power of two" balls and bins, which is useful for load balancing. Throwing work onto a random machine doesn't work well. Picking two random machines, and throwing the work on to the less loaded machine is asymptotically better.
I think you'd see something very similar with this type of simulation.
It makes sense intuitively: one bit of information (i.e. which machine to assign work to) splits the space into two options and you pick the better one at every comparison
The only reason why anyone might find this surprising or counterintuitive is that they're using a terrible visualization. Dollars should be on the x-axis, and number of people should be on the y-axis. If you had that, you'd see this slowly converge to a standard distribution (bell curve).
Of course it's not uniform. The number of dollars handed to any given person each tick is a random variable drawn from some distribution (too tired to think of which right now). An individual's final worth will probably be selected from some sort of Gaussian, due to the trials being repeated and summed.
Question is, why did they have to hide the solution in a video? Are we allergic to static images and written text now?
I'm usually opposed to video evidence, but in this case, I think the video was an especially good means of illustrating the aggregation of resources to the right in a way that a series of static images would not have captured.
A d3 animation might have done the job just as effectively, but depending on the author's skills, might have taken a very long time to create, comparatively.
I fundamentally disagree that the resulting distribution isn't more or less equal. It's basically randomly distributed around 45, with a pretty small standard deviation to boot...
Of course it's randomly distributed around 45 -- conservation of money at work there.
As for the std deviation, the normal rule of thumb is "95 percent of samples fall within 2 std deviations". So basically, all but the highest and lowest. Well maybe two highest, because of the 0 dollar lower bound. A standard deviation of half your starting pool seems pretty high, even if we all have equal chances of being at the far end of the spectrum of results.
The difference I see, is that in the OP game every player is guaranteed to give out $1 at each step, while in the video, a player is chosen at random to give out $1. I don't know how this might change the long-term distribution.
Insightful point, especially considering what the difference might mean on the physics side. Aka a time evolved system of uniformly every particle that has energy donating some energy to another random particle vs random particle donating energy to another random particle.
Modified the problem from this post to also include debt. If someone doesn't have the funds to pay during the turn, they have to borrow money from the person they should be giving it to, and pay back with interest.
NUM_PEOPLE = 45
INITIAL_BANK = 45 # everyone gets 45 dollars
INTEREST_RATE = 0.1 # 10% per turn that you didn't pay
ROUNDS = 5000
RENT = 1 # how much to pay each round
everybody's cash goes to zero on step 1008. From that point on all trades are debt manipulations only
Surely not in the highly improbable case of everyone starting with 1 dollar, giving money to someone else in a chain that leads back to themselves? Theoretically such a system could continue indefinitely? (a gives to b gives to c gives to.... a)?
then we'd indeed have a balanced situation where everybody keeps their initial balance. However, we do sample with replacement, and at some point a lucky winner is going to start hoarding cash. Until such time when no one has cash to pay the winner, and he's ultrawealthy... on paper
We see this result because as a percentage of their total wealth, the poor are giving away a larger portion. Try this experiment again, but instead of having everyone give $1 to a random person, have everyone give 1% to a random person.
So I've repeated the experimental model. Yes, if at each step everyone gives $1 to a random person, the same thing happens as in the article: https://www.youtube.com/watch?v=Hu0vcn_-vX4 .
Intuitively, I would expect that we can collapse the result of doing this type of dollar scrambling over many iterations of the simulation into a single operation which takes a sequence of dollars and partitions it into N randomly-sized partitions for N people.
For instance, we can arrange the dollars into a sequence and then insert N-1 randomly-placed divisions to chop up the sequence. Then each of N people in the corresponding people sequence gets their corresponding (possibly empty) piece of the dollar sequence.
It is not intuitive at all to expect the pieces to be more or less identically long (which corresponds to the wrong intuition that everyone will have more or less their equal share of the dollars). That would mean that the chops are evenly spaced and not random.
Because expectation is linear and the total amount of money in play never changes, we know that the sum of the expectations for each player over a given turn is equal to 0. If no players have run out of money, this means that the expected change for each player over a single turn is zero.
If instead we have m players with money and b broke players, each player still has an equal expected number of dollars received, and the m players with money each expect to give 1 dollar. Summing this, we have a total expected change of (m + b)E(received) - m, which must equal zero, meaning E(Received) = m/(m + b), so players with money expect a change of -b/(m+b) and players without money expect a change of m/(m+b).
This tells us that the expectation for a turn is basically always zero and never gets above zero in a way that allows accumulation of wealth for a single player. So over long periods of time we should expect this to look like a drunk walk with a weird distribution.
I don't have R installed and don't want to spend the rest of teh day setting up packages and trying to get animations underway, but I'd like to explore two other scenarios:
a. Same rules (100 people, 100 dollars), but everyone has to give away 1% of their wealth instead of $1 each round.
b. Same rules as the original, but anyone who has $0 at the end of a round 'dies' and is no longer able to participate.
We have simulation games like Simcity, Civilization, the Sims and so on. I wonder why there aren't better economic simulation games. I know of various academic tools but I mean games that would be accessible and enjoyable for consumers while also allowing them to easily explore simulation spaces. You can mess around with tax policy in simCity type games, for example, but it's really primitive. Academic economic simulation tools tend not to be very engaging (since they're not built to entertain) and also have ugly visuals and user interfaces.
Don't need any simulations to tell you what (b) would look like. Over the long run, you would be left with only 2 players since everyone else would eventually die out.
I'm not saying I'm smarter than your average PhD but it seemed intuitive that it would concentrate wealth, basically because you can only give out 1 dollar per round even if you have >1 dollars.
Having said that I would not draw too much inference about economics and human society from this. There is no such "1 dollar per round" rule in life.
It does not stably concentrate wealth. It does create short-term inequality of wealth. Every player will be the richest at some point if you play long enough.
This model essentially represents an economy wherein every person spends exactly what they make, and in the real world, there are plenty of people who behave that way (at every position on the wealth spectrum).
However, it also ignores the fact that for the most part, people are not forced to live that way. Sure, there's a certain minimum cost associated with simply staying alive, but there's also tremendous potential for people to save and/or invest (a portion of) their money rather than spend it, which in turn may increase their likelihood of being on the receiving end of other people's "random" expenses. It also ignores the fact that for every dollar spent in this economy, we can assume that a dollar of value is received in return. If a person spends a dollar truly randomly (or frivolously), then that expense has an opportunity cost, and in many cases (above a baseline for basic living expenses), that dollar would be better spent/invested in activities that will increase the person's future money-making power.
I suppose if all those points were included in the model, we'd probably see even more inequality in the wealth distribution at the top end, but on the middle-to-low end, there's no reason it couldn't be much flatter; when people fall on hard times (as all will eventually), they can make the difficult-but-necessary decision to spend less than they make, thereby slowing or reversing any downward trend. Philanthropic activity would also help the unlucky few on the extreme lower edge of the distribution to bounce back quickly.
Of course, the real problem in this scenario is that we should never underestimate the number of people who just don't care - or those who spend their money more "randomly" than not, with little concern for the future. Those are the same people who later on will be lining the streets protesting about how "unfair" the system is when they're finally broke.
As a brief summary, K is what's called a "vector-oriented" language; functional, very concise, naturally parallel. The canonical implementation(s) of K are commercial products sold by Kx Systems, and offer extremely high performance. I personally maintain an open-source clone implemented in JavaScript which is comparatively quite slow but is suitable for tinkering and prototyping.
The above program was implemented in iKe, a browser-based livecoding environment I built on top of my interpreter:
The concision and flexibility of K is a huge boon for this kind of interactive development. I wrote my simulation in a few minutes, and most of that time was spent deciding how I wanted to draw my histograms.
If you have any more specific questions I'd be happy to try to answer them.
I don't understand the math here. But, it seems people are saying there is equal probability that any state can arise. Is that correct? If so, I'm unclear why that's true given that the state where one person has all the dollars seems like it can only arise from a much smaller set of states (where only one person, and there would be 99 options, had one dollar and one person had all the rest). In comparison, wouldn't there be lots of times when the next state (say where all the people had similar amounts of money) would be easier to get to from multiple states. I'm saying it simplistically seems like the extreme states are harder to get to than the "middle" states. Is than an erroneous assumption? Is there some technique I can use to reason about this problem in a different way?
Interesting phenomenon! To me it makes intuitive sense when viewed as a uniform distribution of money between 0 and ~$100. Bottom graph approaching linear.
In that light these kinds of posts can be a bit disappointing: "here, have a simulation to prove the phenomenon". Yes but how does this work in general? As others have noted, is it stable over time? Which factors play a role in the speed of convergence? What if one introduces tax per transaction, giving the returns to the poorest? What would be a fair percentage?
Very interesting. I'd like to see this simulation run with different (pseudo-)RNGs. I wonder if and how the quality of the randomness impacts the outcome.
I doubt the PRNG will have an effect, unless it's a bad one. For any individual, the flow out is fixed at 1 dollar per round, but the flow in is unbounded. This favors wealth accumulation. If five people happen to select the same donor target, that person will take five rounds to get rid of that money, during which time they are likely to receive more.
Meanwhile if someone only has one dollar, they lose their entire wealth if no one targets them immediately. Only large amounts have staying power.
There is a second-order effect that as money becomes more concentrated there are less donations per round. If one person has five dollars, four dollars are not being donated. This makes being targeted for multiple donations less likely as time goes on.
If you run this for very long time frames everyone should end up both broke and wealthy. It's basically just several bounded random walks which will diverge in both directions and then flip around.
And really this is just another case where random does not seem random to us.
I want to believe that this is true because there are no absorbing states, but it could be false. Let's not forget Pólya's theorem: random walks on integer lattices are recurrent at dimension 1 and 2 (always come back to the origin) and transient in any higher dimension (eventually never come back).
But the number of states for the wealth distribution problem isn't infinite, hm, because the total amount of money never changes. Yeah, I think I just convinced myself that you may be right, no state is absorbent so they must all be positive recurrent.
Maybe that's how we can restore our intuition on this problem. Everyone should be wealthy and poor, but it may take very long time to see the wealth flip around. While we wait for that to happen, we'll see gross wealth inequality.
The finite states argument feels kind of wrong. You're usually interested in the general behavior for n people, and the number of states is exponential in n, which means that you need time exponential in n to repeat yourself. What you're seeing here is the early behavior of the system. Your arguments is a bit like saying every computer is actually a finite state machine, because of the finite memory it's going to repeat itself eventually. The argument is fundamentally flawed because you inflated one part of the question (time of running the question) while keeping the more significant part (number of particiants) fixed.
It favors wealth accumulation, but the winner isn't locked in because eventually any person will lose all they have if they don't get any income for a long stretch.
I wouldn't expect it to make a detectable difference, unless the rng was as low quality as the one featured in Dilbert [0]
The simulation has to make 100 decisions every turn, sometimes slightly fewer. A typical way to do this is to get 100 32-bit numbers from the RNG, even though you really need fewer bits of entropy than that. If the RNG had cycles of exactly 100, or a factor of 100, that could cause a feedback loop here. But a typical PRNG, even a simple one, has a period length of billions.
I hoped to hear a positive response. At the same time, I was trying to impart a lesson that it took young-engineer-me a lot of time to really grasp, which is that you can trust PRNGs and randomness as engineering building blocks (and reciprocal-2-to-the-power-of-a-lot events are unintuitively rare).
On the first:
PRNGs work because they are not random in ways that are very weakly correlated with things you might want to simulate, but there are cases where you do get correlation, and I'm not really sure what those cases are, and read some technical explanations but nothing intuitive enough to be able to look at a real world scenario like this and intuitively grasp that here is one of those rare situations where an LCG is not enough but a Mersenne Twister is, or harder - a Mersenne Twister is not enough but real random data sampled from some physical system and normalized by XORing to a Mersenne Twister is. If you're one of those people that have this skill, maybe you can explain it to me intuitively.
On the second:
The best example is trusting cryptographic hashes. When I was young I saw systems that use a file's hash as that file's unique identifier, that were built on the completely unflexible assumption that there will never, ever be a collision, because that would completely bork the system. I had a bad feeling whenever I saw this assumption being made, because, I mean, yeah, hashes have many bits, but can't it happen even once, somewhere around the world, that two files would be found that have the same hash? There are still so many more files than hashes...
And the answer is no. It can't. Oncce a decade or so your hash algorithm will be successfully broken and you'll need to replace it with a stronger one, but it probably never happened in the entire history of humanity and never will that two files were created that have the same cryptographic hash by coincidence. It's completely reasonable to make this assumption. Just as it is completely reasonable to use a long number as an authentication token, etc' etc'.
We humans simply don't have a mechanism in our brain for representing "a nonzero chance that might happen but is so small that it is inconceivable that it would ever happen even once in the history of the world". So we see "possible" and feel deep in our bones "might happen". And we're worse engineers for it.
Well, yes, and as your sibling commenters contributed useful and insightful replies it indicates the question is interesting enough, at least to them, to be worth contributing thought to.
If you're really just looking for an unsubtle way to point out there is some better term of art that isn't salient enough to me to presently recall, perhaps you might wish instead to helpfully point it out for my benefit and that of other readers.
Let's consider a different problem.
There are 100 non-negative numbers summing up to 10000.
Let's choose a random combination of numbers satisfying this condition. Intuitively, it's quite clear that typical case will be very different from equilibrium (where every number=100). I fail to see how the transfer of 1 dollar at a time is qualitatively different. So the result is quite intuitive - contrary to what the article suggests.
A random distribution of points in a circle is different from a set of points calculated by using a random angle and distance from the center, even though they're both "random". How you construct your randomness is significant.
Except that, in the real-world, wealthy people will spend more money than the poor. I wonder what we outcume would be if each agent spends a random amount of money that depends on their current wealth (say, anything between $1 and 5% of the total).
Also wealthy people have much bigger gains from investment (what money they already have), which will ultimately cause the distribution to be much more exponential than the one from the simulation.
I suspect giving away a proportion would have no significant effect on the result, because poor and wealthy have still the same likelihood of receiving money.
True, but typically he will spend more than _one_ single person with a dollar.
I created a gist (https://gist.github.com/lou1306/1041ed6cd4eed433cfabf45f666b...) to try and prove my point a little better: when people are only allowed to pay $1 to one another, the resulting distribution is quite unfair (as shown in the original post).
However, when payments can be as big as, say, 2-10% of the player's wealth, there is a noticeable decrease in the Gini coefficient.
I believe happens because the states where everyone has close to the same amount of money have lower entropy e.g. there is only 1 state where everyone has equal money, but there are by far more states where the funds are exponentially distributed.
If it makes you feel better, in this model eventually everyone will spend time as both rich and poor... probably not so much like the real world. :)
Given the uneven distribution of wealth being more likely then an even distribution of wealth per tick. If you now add the rule that someone having less than $100 is unhappy and someone having at least $100 is happy you end up with lower overall happiness on average compared to the initial state were erveyone has $100 and would be happy with that.
Lots of complicated explanations why this doesn't strongly tend toward being very even. Here's my attempt to state it simply.
Every round, each person (except those who are broke) definitely gives away $1. They may or may not get a dollar. They might even get more than a dollar.
So it's not surprising that some get a bit ahead and others fall a bit behind.
> gives a dollar to one randomly chosen other person
I wonder is the person randomly chosen by the machine or is it randomly chosen by the person who gives away the dollar? If it's the second case - I wonder how much does personal attraction matters here. Unconsciously you would want to give something to a person who you find more attractive...
> If on quick reflection you thought “more or less equally”, you are not alone. I asked 5 super-smart PhDs this question and they all had the same initial intuition.
Mmm ... What kind of PhD did they ask? Mathematicians have a strong bias for exact solutions, and not too much intuition for this kind of problems. Have they tried asking Physicists?
It's not surprising that the distribution is not even. I expect that the spread increase in time. I guess something like
average + k1 * sqrt(t) * erf((n- middle) * k2)
where t is the number of simulation steps, and k1 and k2 are some magic constants that I'm too lazy to estimate. ("Proof": Everything is a Gaussian.)
I was surprised that at some point the distribution was almost linear. Perhaps k2 is not a constant, and the correct guesss of the estimation is
This estimation fails when t is big. (And it's probably incorrect anyway.)
After some time the person with more money apparently starts to increase the amount of money. I'd like to see a longer simulation to see if after some time the first one accumulates most of the money.
We mathematicians have as much intuition as anyone else. I think of this as a random walk on a very large but finite state space with positive n-step transition probabilities between all states for all large enough n. It's an aperiodic irreducible Markov chain. So we should have ergodicity. My intuition now tells me that this means every state is positive recurrent, i.e. for each person x should see x hoard all of the money at some step. Given enough time, a very long time, we should see everyone eventually have all of the money and eventually have none of it. The intermediate steps may look very skewed, but we should see all steps happen.
I'm a bit wary about concluding that just because there's positive probability between all states that we should see all states. At least for infinite state spaces, we know this fails (Pólya's theorem on random walks in dimension 3 or higher). But my intuition tells me that the finite state space should mean that all states are positive recurrent.
Edit: Aha, here's the relevant theorem which confirms my intuition: in a finite-state Markov chain, every closed class is recurrent:
Untangling the jargon, if you have a random process in which you can go from any state to any other state with positive probability after a certain number of steps, and there are only finitely many states, then the probability of getting to any state as the number of steps goes to infinity is 1.
In spite all the states have a probability 1 to be reached infinitely many times, the frequency of each kind of state in the chain is not equal. Some states will have a much higher frequency than the others.
In this case, the states where everyone has a number of coins that is between average-2 and average+2 will be extremely infrequent. They will be reached, but you probably have to wait a lot of time to see them.
But there will be some family of cases that appear very frequently after enough time. I'm not sure if the most common case is
1) Some of the persons has 99% of the money and the rest have a tiny amount of money
2) A few persons have almost all the money.
3) There is some kind of heavy tail distribution were everyone is expected to have some money. If you order them by money you will see something like a wiggly line.
I vote for 1), but I'm not sure at all.
This is similar to the typical thermodynamics problems. Imagine that you have 100 boxes in a line, were you can distribute 100000 "atoms" (or coins). In each step the coins can go to neighbor box at random. This is not the same problem, so it has a different solution.
But it's also ergodic and you can see all kind of weird distribution of the coins/atoms if toy wait enough time.
If you wait enough time you can see for example that all the coins/atom went to the leftmost box and all the other are empty. But this is not common at all.
Most of the time all the boxes will have a number of coins/atoms that is close to the average. (Not exactly the average obviously.)
But this is a different problem with the same state space (100 boxes/persons, 10000 atoms/coins). The rules in the Markov chain are different, so the expected frequency of the states is different.
In the thermodynamic problem, the most frequent states are when the atoms are almost evenly distributed. In the problem of the article, I suspect that the most frequent state is when someone has almost all the money.
After some time the person with more money apparently starts to increase the amount of money. I'd like to see a longer simulation to see if after some time the first one accumulates most of the money.
They don't - if you track individual players for a long time, every player ends up having periods of both very high and near-zero wealth.
I love this! If I had to ELIA5 it, I would say: if you get lucky you get rich quickly. If you get unlucky you get poor slowly. So some people will get lucky and some will get unlucky, but the first to get unlucky will get below 100 and the first to get lucky will get above 100, explaining the initial behavior.
Put another way, the key is that for n people and n dollars, you can receive up to n-1 dollars in a given turn, but you can only ever lose 1. Therefore, whoever is lucky early will accumulate faster than they can distribute.
The lack of money "accumulates" because you always have a chance to receive money, but when you run out of money you cannot give more. So there is a lower cap but no more upper cap.
In the real world this is even worse. Once you're out of money your chances of getting money go down drastically.
> How does the distribution look? Play the movie above to see.
How about I don't play the movie, and you use these magic things called "words" that let you describe experimental results. Maybe we can call these descriptions "abstracts" and put them at the beginning of articles.
I don't see why it's counterintuitive. If we know that "random" distribution will place values on a curve, then why would it not follow that the money would distribute this way and lead to inequality.
But I like the article! Excited to share to students.
I don't understand the video -- with 45 players, wouldn't one have $89 after round 1? But that's not what we see -- it takes hundreds of rounds before the first person breaks $80.
I guessed an exponential distribution of wealth right after the problem was posed. Apparently years of crying hot tears over agent-based economic simulations served my intuition well.
"Dollar studies" is perhaps the second bullshitest part of psychology after "infant-looking-times studies".
Artificial cartoon-like setups and biased interpretations cannot be considered scientific or even accurate.
Babies are merely confused and overwhelmed, dollar studies are over-simplified and sterile and does not take complex emotional and hormonal patterns (which are much more powerful than rationality biological driving forces in real life) into account.
Snap-judgements, jumping to conclusions, escaping from emotional pressure and reduction of cognitive load is what derives people's behavior. Any sales or ad professional would confirm this.
You're missing a subtle point: not only is the distribution exponential, but also as time progresses various agents exchange their positions in top and bottom categories. So if one introduced a taxation system that redistributed equality to create a uniform distribution, you would also enforce utter social immobility. I doubt that is desirable (basically you'd be recreating the drabbest kind of communist regime known to mankind).
"Imagine a room full of 100 people with 100 dollars each. With every tick of the clock, the set of people with money each give a dollar to one randomly chosen other person. The set of people with zero dollars simply lose the opportunity to give away a dollar and thus decrease the overall chances that any participant will receive a dollar. After some time progresses, how will the money be distributed?"
To me this now seems intuitively obvious. Since the game creator has declared by fiat that the number "0" gets a special branch in the program, it stands to reason that the resulting distribution will change once any participant hits zero and triggers that code-path.
For example-- suppose you start with game with one person who has $10,000 and the remaining 99 people have zero. (I don't think you can arrive at that state game organically, but it doesn't matter for the point I'm making.) The first tick through the game all that happens is that the ten-thousandaire gives one dollar to one marginally lucky participant while the other 99 do-- nothing at all. Essentially the 99 all begin by losing a turn.
E.g., imagine an implementation where each tick loops through the participants and the ten-thousandaire happens to be the first participant. If you are the last in line, the only opportunity you have to collect a dollar is when the ten-thousandaire gives away a dollar on the very first iteration of the loop. For the rest of the 99 iterations of the loop you have exactly 0% chance of receiving a dollar. That means on the first tick you and all the other zero-aires would have 1 in 99 chance of receiving a dollar. If on the other hand you started out with everyone having a dollar you would have [some immensely larger chance of receiving a dollar that someone who isn't lazy like me can probably calculate here].
But it's actually worse on the first tick of this example for the ten-thousandaire. That player has a 100% chance of losing a dollar and a 0% chance of gaining one.
I believe this "problem" essentially describes the conditions of an economic depression. It's also a bit of a visual/conceptual illusion because we're likely to look at that chart and see accumulation of dollars as "winning" and zeros as "losing". But if you instead measure liquidity it should be clear that those who accumulated dollars suffer decreased liquidity even worse than the zero-aires.
Also if you remove the special case for zero then the intuition about "more or less equal" distribution should be true.
Edit: change "lose a turn" to "lose the opportunity to give away a dollar" to guard against pedantry. Also, changed "tick/turn" to "tick".
It is not randomly chosen by the person who gives away the dollar. Personal attraction matters here. Unconsciously you would want to give something to a person who you find more attractive.
Since people are incapable of making truly random decisions, I reckon the person with the most pleasing physical appearance will receive most of the money.
1. everybody need to give 1, but may receive 0 (99/100 probability) or 1 (1/100 probability)
2. when somebody run out of money, the total give out is no longer 100 (before this, 100 changed hand in every tick), and the probability of receiving money also changed.
So this is history dependent. The simulation need to run many rounds then compare the results of all rounds.
This is the relevant theorem:
https://math.stackexchange.com/questions/543051/markov-chain...
This means you should eventually see each person hoard all of the money in turn. Try it with 5 people and 5 dollars each. Then the system only has 29 choose 25 = 23751 states[1], which should give your computer enough time to hit them all.
Edit: Here, Python "proof"
----[1] https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)...