## Friday, October 27, 2017

### Fair coin, foul outcome

From the Mathematics of Inequality article I quoted yesterday:

This is essentially a coin toss, and you might think that in the end both sides would end up with the same amount of wealth. But it turns out those who have more keep getting more. Even if both agents have the same wealth to begin with, one will eventually begin to dominate the other, even though the coin is fair.

Yeah okay, but I wanted a better explanation. I Googled yard sale model. The first hit was The Growth of Oligarchy in a Yard-Sale Model of Asset Exchange: A Logistic Equation for Wealth Condensation (PDF, 15 pages) by Bruce Boghosian, same guy. The idea quoted above is restated in his PDF:
In the absence of any kind of wealth redistribution, Boghosian et al. proved that all of the wealth in the system is eventually held by a single agent. This is due to a subtle but inexorable bias in favor of the wealthy in the rules of the YSM: Because a fraction of the poorer agent’s wealth is traded, the wealthy do not stake as large a fraction of their wealth in any given transaction, and therefore can lose more frequently without risking their status. This is ultimately due to the multiplicative nature of the transactions on the agents’ wealth, as pointed out by Moukarzel.

So: In any exchange of two equal values, the wealthier party puts a smaller percentage of his wealth at risk than the other guy. The worst outcome for the rich guy is not as bad as the worst outcome for the poor guy. Now, play this game over and over again. Even if the odds are even, the rich guy stands to do better.

Into a spreadsheet, so I can see if I believe this story.

Something else in the article has me confused. What is this "transfer of wealth" that occurs independent of the exchange of equal values? How much is it? Again, from the article:

A simplified version [of the Yard Sale Model] goes something like this: Two people enter into a series of transactions, and both have the same probability of winning some amount of wealth from the other, just as in a free-market transaction.

What is this "small amount of wealth" -- slop in the connection? A dollar's worth of money exchanged for a dollar's worth give or take of stuff? And the better or luckier or wealthier dealmaker (or all of these in one) ends up with the difference?

Maybe that's it. The article quotes economist Michael Ash, who refers to "mutually acceptable, apparently equal exchanges" (emphasis added). Apparently equal, but there's a little slop. Okay, that's good enough for now.

For my spreadsheet I figure there is a poor guy who starts out with wealth equal to 100 and a rich guy who starts with ten times as much. I figure a transaction equal to 10% of the poor guy's wealth, and a wealth transfer equal to 10% of the transaction amount. I don't pretend these numbers are realistic. I just want to see how they work out. Does the wealthy guy always get the slop? I want to see this for myself.

I set up different scenarios. First, the rich guy always wins the wealth transfer:

 Graph #1: The Rich Guy Always Wins
The blue line shows the rich guy, who begins with 1000 units of wealth. His wealth climbs to 1100 as the poor guy's wealth drops from 100 to zero. By eye, that transfer of wealth is complete after about 400 transactions. (The numbers on the horizontal axis indicate the number of transactions.)

On this graph, with each transaction, one percent of the poor guy's remaining wealth is subtracted from his total and added to the rich guy's total. The rich guy wins every time.

On the next graph, the guy we call the "poor guy" always wins the wealth transfer:

 Graph #2: The "Poor Guy" (the one who starts out poor) Always Wins
The guy we call the poor guy starts out poor, but ends up with more wealth than the guy we call the rich guy. This time the so-called poor guy's wealth grows from 100 to 1100, while the so-called rich guy's wealth falls from 1000 to zero.

By eye, on this graph the transfer of wealth is complete after less than 250 transactions. The transfer happens faster on this graph than on Graph #1. Why? Because the transfer is one percent of the wealth of the guy we call the poor guy, but that guy doesn't stay poor. As the transfer of wealth continues, the amount lost by the so-called rich guy becomes bigger and bigger. So the "rich" guy's wealth evaporates quickly.

I finessed the calculation, and made another graph. On the next graph, one percent of the wealth of whichever guy is wealthier gets transferred to whichever guy is less wealthy:

 Graph #3: The Actually Poorer Guy (whichever one is poorer) Wins
The transfers of wealth bring the rich guy's number down and the poor guy's number up until they meet. After that, the transfers take from the guy with a little more, and give to the guy with a little less. This causes the red and blue lines to stabilize around the 550 level.

Finally, the next graph shows the "fair coin" example that the article is about. It shows what happens when the rich guy and poor guy take turns receiving the transfer. The blue line shows a relentless increase in the rich guy's wealth. The red line shows an equally relentless decline in the poor guy's wealth:

 Graph #4: Alternating Winners (rich guy, poor guy, rich guy, poor guy ...)
The blue wealth begins at 1000 (low side of the left end of the blue line) and rises to just below 1005 (low side of the right end of the blue line; find the value on the left-hand scale).

The red wealth begins at 100 (high side of the left end of the red line; find the value on the right-hand scale). It falls to just over 95 (high side of the right end of the red line, again on the right-hand scale).

After a thousand transactions the wealthy guy's wealth increases by approximately five wealth-units, while the poor guy's wealth decreases by the same amount. Remember, the wealthy guy receives the wealth transfer for the first, third, fifth, and all the odd-numbered transactions. And the poor guy receives the transfer for the second, fourth, sixth, and all the even-numbered transactions.

Okay, I accept the claim made by the article, that even using a fair coin, "without redistribution, wealth becomes increasingly more concentrated, and inequality grows until almost all assets are held by an extremely small percent of people." In this model, at least.

But why (you may ask) why are those lines so thick? I didn't know, either. So I cropped off the left and right sides of the graph, just keeping the middle section where the lines cross. I zoomed-in on it to get a better look:

 Graph #5: A Closer Look at the Thick Lines for the Alternating Winners
You can see here that the "thick" red and blue lines are actually made up of thin lines that zig-zag rapidly up and down. We can say each "zig" is a transaction where the wealthy guy receives the transfer. And each "zag" is a transaction where the poor guy receives the transfer.

The "top edge" of the "thick" blue line is really the sequence of values that trace out the wealth position of the wealthy guy after he receives the transfers. The "bottom edge" represents the wealth position of the wealthy guy after the poor guy receives the transfers. (The thick red line represents comparable values for the poor guy.)

If you look at the position of a thin "zig" line relative to the one just before, the later line is just a very little higher (or a very little lower) than the earlier line. That tiny difference is the "slop" that causes wealth to concentrate.

The thin lines show that after each consecutive pair of transactions, there is almost no change in either guy's wealth. But transactions occur all the time. And the overall result of very many very tiny transfers of wealth, given enough time, turns out to be a very large transfer of wealth. Reminds me of Richard Prior in the Superman movie, turning rounding-errors into huge amounts of money.

But it's not just one guy doing it. It's everybody involved in transactions, and all the transactions. And it turns out that those who have more keep getting more. And without redistribution, wealth becomes increasingly more concentrated, and inequality grows until almost all assets are held by an extremely small percent of people.

The Mathematics of Inequality It's an interesting concept.

Woke up early again. I get it now.

Consider a two-transaction sequence. On average in a two-transaction sequence, because the coin is fair, the rich guy and the poor guy should each win one of the wealth transfers. There are two ways this can happen:

1. The rich guy wins the first transfer, and the poor guy wins the second. Or
2. The poor guy wins the first transfer, and the rich guy wins the second.

If the rich guy wins first, the first transfer reduces the poor guy's wealth. So the second wealth transfer is smaller. The poor guy loses the larger transfer and wins the smaller one. Therefore, his wealth is reduced.

If the poor guy wins first, the first wealth transfer increases the poor guy's wealth. So the second wealth transfer is larger than the first. The poor guy loses this larger transfer. Therefore, his wealth is reduced.

Either way, the poor guy loses.

Poor guy!

// The Excel file

jim said...

Hi Art
I see several problems with this yard sale model

Real yard sales are not zero sum.
The adage that prevails is 'one man's junk is another man's treasure" (implying both sides win).
It also should be noted that this YSM is an economic model for non-GDP transactions. Wealth generated by actual production of goods and services is ignored.

Your analysis at the end exposes the fallacy of the conclusions derived from the model.
The reason the poor guy loses is that the amount won/lost on each transaction is a function of the poor guy's net worth.
If you change that assumption so that the amount won/lost is a function of the rich guy's wealth (let's say 1%) then I think you will find the rich guy getting poorer in the long run.

Jerry said...

Jim, I don't think that's true at all. The poor guy will only have to lose a couple of coin tosses before he's out of the running entirely.

(I think the "you win, I win, you win, I win" in Art's model is an important flaw, actually. It's an important part of how it works, that sometimes you win a few in a row, or you lose a few in a row. Because e.g. losing a few in a row will mean game over for the poor guy. So it's sort of only a matter of time until the poor guy loses, unless he gets extremely lucky.)

All that (the model, etc) aside,
It seems obvious that, absent any rules, the big guy will beat the little guy.
That's why civilization needs some kind of rules.

jim said...

" losing a few in a row will mean game over for the poor guy.?"

If the amount won/lost is 10% of the poor guy's net worth then he will always have 90% left so in theory he can never go broke. in practice, if the 90% left is less than a penny, he is broke.

If however you make the amount won/lost 1% of the rich guy wealth then he might go broke if he loses the first 10 in a row, but if he wins half the first 10 and loses half, regardless of the order, the poor guy will win more than he loses. If he wins the first five then the 5 losses that follow will be smaller and if he loses the first five then the five wins that follow will be larger. Eventually yhe poor guy will become the rich guy.

Oilfield Trash said...

The math is not the only thing that gets you.

As in most things not all the players are equal, coin toss analogies for win loss is a very rare event when it comes to wealth building.

Asymmetric information exist and almost all economic transactions involve information asymmetries.

Trash and treasure discussing to me means the seller does not understand the value the item has to the buyer. He is making a mistake due to his lack of knowledge of his buyer and what he is selling.

I know people who make a great living exploiting asymmetric information for finding valuables buried in boxes of trash.

When I play Hold'em at a table with out much information on the players, I have to assume your I am the worse player and stay away from the big stacks. I play the math more than the player.

You only take them on when start to get information and behavior tells on the other players. Now you can play math and the player (Asymmetric information). It can make a coin flip decision into something more favorable to you.

If you can not figure out who the fish is quit that table because it is you.

People who spend 15% if their disposable income on lottery tickets comes to mind. You can not explain to them that they would be better off financially (accumulate wealth) if they just put that money in a bank savings account.

The Arthurian said...

OT: "... almost all economic transactions involve information asymmetries. Trash and treasure discussing to me means the seller does not understand the value the item has to the buyer."

When I read that I thought: like selling manhattan for \$24...