Counterintuitive problem: People in a room keep giving dollars to random others
decisionsciencenews.com
decisionsciencenews.com
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"
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
[1] https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)...
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.
[0] https://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theor...
[1] for a suitable definition of easily.
> 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
28.77 19.11 14.25 10.98 8.53 6.57 4.93 3.53 2.27 1.06
> 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 ...
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.
Whether you have what statistician call "skewed" or not, you have what most people would call skewed, unequal, extreme.
>> 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).
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.
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?
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.
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.
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.
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).
Not to demonize accumulation of wealth but just to model reality where rules can sometimes be changed by those with enough wealth accumulated.
The way I interpreted the solution was that every individual will at some point have the max possible amount of money (which is roughly $100).
I don't think jordigh is trying to imply that a single individual will have a majority of the total wealth
(minus 1 because you give away $1 on every turn).
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.
here's how that looks in a 100 person with 100 bucks game (10,000 money supply)
https://github.com/un1crom/givemoneyaway/blob/master/qqplots...
but no you cannot get back to that initial condition.
but in the real world you most definitely can start that way.
chess is like that too... the initial condition can never be regained.
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.
Not true. The problem is in one-to-one correspondence with the non-negative solutions to
a + b + c + d + e = 25
which is in one-to-one correspondence with the positive solutions to
a + b + c + d + e = 30
which is a simple counting problem---(29 choose 4) = 23751.
In my opinion, you should really try to prove your math to yourself before answering with such an authoritative tone.
Why is my solution incorrect, though?
* Oh, I see why my solution is wrong. It's never the case that one person has all the money. I should take some of my own advice.
23751 - 5 = 23746
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.
https://github.com/un1crom/givemoneyaway
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.
https://github.com/un1crom/givemoneyaway/blob/master/qqplots...
enjoy!
> 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:
http://i.imgur.com/SctNJWi.png
vs with the >$0 redistribution condition:
http://i.imgur.com/wRVZxlO.png
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.
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.
People would still lend money to their friends and family. Stores would extend credit to customers because they want to make sales.
But regardless, how was that supposed to dispute any part of the post you responded to?
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.
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.
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.
> 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 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?
Isn't that a good thing? In lots of places, e.g. Germany for one, people almost never buy with credit.
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.
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.
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.
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.
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.
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.
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.
> 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.
That depends on available resources and distribution -- not economic growth.
Besides, what's really bad about poverty is not access to lack of access to West-level consumption but lack of basics -- from health care to housing.
How about clients paying suppliers many months after stuff was delivered?
It is very rational to lend money with zero risk if that gets you more business.
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.
Really? Where do you think it's confined to?
Credit cards are not an equivalent service, they're a means to pay the tab...
https://en.wikipedia.org/wiki/Diners_Club_International#Orig...
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 a credit card you don't need a bar 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.
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.
(Plus people lend money and tools and whatever all the time with neither interest nor legal guarantees).
Like with credit cards, most of the money on loans is not paid on the default interest and with the perfect payment schedule, but on late payments.
or only upon collateral.
I just want to point out that the phenomenon of interest-free lending as charity is a real one.
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.
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).
By way of illustration, I took the code and made a simple modification to it, implementing power of 2 random choice (http://www.eecs.harvard.edu/~michaelm/postscripts/tpds2001.p...).
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.
https://en.wikipedia.org/wiki/Maximum_entropy_probability_di...
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.)
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.
On only 5% of days was the spread greater than 80 -> 120.
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.
[^1]: https://gist.github.com/lou1306/1041ed6cd4eed433cfabf45f666b...
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.
can be a long time
Dollars received is additive process of 99 independent random variables from [0,1].
Dollars given is -1 or 0 if there is none.
The process has memory because givers can run out of money.
https://en.wikipedia.org/wiki/Birth%E2%80%93death_process#/m...
[1] Didn't check the math.
Or for example -
> 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.
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?
Brief introduction to econophysics for the mathematically inclined: https://arxiv.org/abs/0709.3662
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.
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).
https://gist.github.com/anonymous/4f6c546a776a8ab142235c8617...
Wonder what it would be like taken further - forcing population that have $0 to take a loan from the wealthiest people and pay them interest.
TIL what an economic crisis is and how it happens
I think you'd see something very similar with this type of simulation.
Question is, why did they have to hide the solution in a video? Are we allergic to static images and written text now?
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.
https://en.wikipedia.org/wiki/Maximum_entropy_probability_di...
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.
https://www.youtube.com/watch?v=SCUnoxJ5pho&feature=youtu.be...
Check out the source and the produced animation here: https://gist.github.com/kirillseva/961fa1f5b5d64254e0117caf1...
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
edit(formatting)
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
There are increasingly more ways to distribute wealth unevenly, than evenly.
Reduce the problem to the simplest case of 3 persons: `a`, `b`, and `c`. Person `a` has decision to give to either `b` or `c`. Then use combinatorics:
from itertools import product
decisions_a = ['ab', 'ac']
decisions_b = ['ba', 'bc']
decisions_c = ['ca', 'cb']
for combination in product(decisions_a, decisions_b, decisions_c):
print combination
>>> ('ab', 'ba', 'ca') # uneven
>>> ('ab', 'ba', 'cb') # uneven
>>> ('ab', 'bc', 'ca') # even
>>> ('ab', 'bc', 'cb') # uneven
>>> ('ac', 'ba', 'ca') # uneven
>>> ('ac', 'ba', 'cb') # even
>>> ('ac', 'bc', 'ca') # uneven
>>> ('ac', 'bc', 'cb') # unevenhttps://en.wikipedia.org/wiki/Zipf%27s_law
Vsauce made a video about it: https://www.youtube.com/watch?v=fCn8zs912OE
If, instead, at each step, everyone gives 1% of their currently held amount, this happens: https://www.youtube.com/watch?v=N8Ce3eQTA9c .
The results are radically different. Draw your own conclusions.
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.
If we distribute randomly placed chops into section of the real number line, we end up with string lengths that follow an exponential distribution: https://en.wikipedia.org/wiki/Exponential_distribution
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.
Randomly placing chops is a https://en.wikipedia.org/wiki/Poisson_process
However, part of the intuition here (possibly wrong) is that Poisson is applicable to a crudely discrete process like this, in some approximate way.
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.
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.
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.
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.
http://johnearnest.github.io/ok/ike/ike.html?gist=f974a638c4...
Source, for the curious:
p: 72 / players
s: p#100 / score
c: {(+/(!#x)=/:(+/t)?#x)-t:0<x} / change (per round)
b: {[c;p;y;x](p+0,2*y;;x#c)} / draw bar (color;pos;y;x)
g: {x'[!#y;_60*y%|/y]} / draw graph (bar;data)
tick: {{s+::c s}'!20} / iterate several steps/frame
draw: {g[b[3;10 10];s@<s],g[b[2;90 10];s]} / graph sorted, raw valueThe above program was implemented in iKe, a browser-based livecoding environment I built on top of my interpreter:
https://github.com/JohnEarnest/ok/tree/gh-pages/ike
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.
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?
Can anyone with insight offer comment?
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.
And really this is just another case where random does not seem random to us.
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.
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.
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.
Of course it could be a very insightful question in disguise but since you don't offer any reason for it...
Probably looks like Boltzman distribution describing velocity of ideal gas.
https://github.com/subroutines/DollaBillz/
The writeup there includes a clip simulating 500 people (each starting with 100 dollars) and some distribution fits.
Is capitalism mostly noise?
Further, if you do running average over very long time frames the averages will converge. Aka weathly will become poor then wealthy again.
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.
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.
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. :)
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.
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...
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
average + k1 * sqrt(t) * erf((n- middle) * sqrt(t) * k2')
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.
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:
https://math.stackexchange.com/questions/543051/markov-chain...
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.
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.
In the real world this is even worse. Once you're out of money your chances of getting money go down drastically.
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.
But I like the article! Excited to share to students.
It's not one chosen person getting all the money from that round.
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.
And then simulate how this tax rule works in a capitalist market model.
I'm curious.
It's really cool though to have a skewed distribution at each time step and a uniform distribution over the entire time series.
It's not exactly the same as just being ergodic. The probability that the distribution is uniform at a particular time step is very small.
"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".
One of the most annoying and overused phrases in the universe.
You mean we don't naturally end up with 1% of the people having 99% of the dollars??
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.