The infamous coin toss
ergodicityeconomics.com
ergodicityeconomics.com
Investors: 100,000 Iterations: 100
Average worth after 100 iterations: $83.923
Average net worth increases. However the distribution of wealth is skewed dramatically.
Winners: 13,704 (net worth of more than $1 at the end) Investors worth < $0.01: 53,935
What That Guy was worth (the investor who made the most money):
$1,171,830.00
He flipped 71 heads and 29 tails.
Median wealth of all investors: $0.00515378
What could have been made more clear in the presentation. Average wealth increases; but average log(wealth) decreases. Investors on the long improbable upper tail make huge amounts of money; investors on the lower tail just don't make much difference to the average, when they lose 40% of $0.01. What causes the wild income distribution: the fact that log(wealth) decreases on average. This doesn't decrease the total money supply, but it does increase the skewed distribution of wealth.
It has nothing whatsoever to do with anything related to flipping consecutive heads and tails. And a great deal to do with the fact that the log(gain) is 0.405465 and the log loss is -0.510826. So the average investors log(wealth) decreases over time even though the average total wealth increases.
Most figures are wildly variable from trial to trial. Most especially, what That Guy makes is highly variable. If he rolls 72 heads and 28 tails instead of 71 heads and 29 tails, then he's worth $2,565,000 more, and single-handedly adds $25.65 to the average net worth! The number of people who lose money is surprisingly, depressingly consistent.
The obvious conclusion: increasing wealth concentration is an unavoidable consequence of capitalism that probably has very little to do with merit.
The only solution: eat the rich.
My bad I thought you made a coin flip simulation, not a capitalism simulation.
At what point do we stop wagging the finger at the ghost of capitalism and just say it's mathematics? You don't hear people crying for the upheaval of mathematics
Then the response of advocates against this situation is to either commit to more structures to prevent that outcome towards diluted (non-perfect) capitalism or other social forms (socialism, etc...)
The usual response against that is: what is wrong with being a winner? I did something right, let me have my well-deserved money.
The problem with someone getting well-deserved money is that the money starts working against everybody else, so essentially this money gives everybody else negative money.
Regardless, the problem is very rarely that you “earned” money. Almost everyone is fine with you getting wages or perhaps entrepreneurial income.
The problem is that wealth allows people to become more wealthy without really doing anything. Wealth compounds so that the rich get richer. And often the risk of financial loss is so low in a diversified portfolio that it’s basically a guaranteed win. See economic rent.
IMO what people want is neither an income tax, nor a wealth tax, but an income tax rated on wealth. Wealth does get created over time, but despite this, the wealthy also take a continually larger share of the total pie. Society should be the one to get the bulk of passive wealth accumulation gains. Not individuals who happened to be wealthy.
If you have a billion dollars, your income tax rate should be like 99%.
A) If your wealth is in cash in the bank, you are providing the bank with deposits. The bank can now make cheaper loans which helps others. But if these loans default, your bank fails and you get bailed-in. Assuming no government intervention comes to save you.
B) If your wealth is in stocks or corporate bonds, you're helping to finance companies that are making stuff / providing services that people might need. But markets can go down, companies fail, and as long as there's no bail-outs, you should be able to lose it all.
C) If your wealth is in government bonds, you're obviously financing the government. But if there's no buyer-of-last-resort (central bank) for your bonds, then one dodgy policy decision by the government, a large bout of deficit spending, or a failed bond auction will wreck the value of your bonds.
D) If your wealth is only in stuff, land etc. then your wealth shouldn't grow - in fact it should fall as these deteriorate in quality, and you're taking a big hit in the form of opportunity cost by not doing anything productive with the land/stuff. If you're having to maintain this 'stuff' then you're losing money, think cost of ownership, replacing capital etc.
E) If your wealth comes from rent, then you're taking a risk, and expect costly demands from tenants and to have to keep up with a competitive market. This I accept is something of a grey area because uncompetitive markets / monopolies can exist.
F) If you're wealth is in patents, copyrights fees, royalties etc... then there may be an argument here. This is up to the government to change, similar to E but without as much risk.
The trouble with the above A-E is the inherently inflationary system we have now where markets are never allowed to go down and deflation, bank failures, and default is prevented with stimulus. It props up stocks, bonds, means assets and stuff gets more expensive nominally without doing anything, it keeps banks afloat and means bail-ins of depositors are unlikely. Rich people who acquire assets are on to a guaranteed win like you say.
The solution is not to take away the incentive to produce more wealth by a punitive income tax, but to take away the asset inflation that makes asset holders rich by doing nothing, which also makes cash holders poorer and punished for being prudent with their meagre savings.
The economy improves and so assets do become more valuable, inevitably. We should be taxing that inevitable passive growth of society’s progress to benefit society.
"The economy improves and so assets do become more valuable, inevitably." More pricey, no guarantees about more valuable though. Some assets may be entirely Ponzi-like and have little value, but a high price.
And anyway, if the economy is improving, homeownership / rent would be more affordable, families would only need one parent to work, the number of hours worked to buy S&P500 would be more reasonable, things like healthcare and education would be getting relatively cheaper. But artificially cheap margin debt, speculation, and financial chicanery backstopped by the central bank have lead to a lack of productive investment. Unusually high immigration and population growth without a corresponding increase in productivity causes higher prices too. Everything is getting harder from the point of view of Mr & Ms NoAssets 20-somethings who earn a wage.
The fixation of the tax-happy is on making the rich poorer in a dollar-net-worth sense. But there is no focus on making the poor richer in a purchasing power sense (what you can buy).
This requires a gradual deflation, higher reserve requirements, saving and careful investment. All currently very unpopular with the asset-rich, banker class, and thoroughly captured mainstream economics profession, who want you drowning in debt buying ever more expensive stuff, and not deferring consumption to the future.
This is exactly why we need progressive asset taxes. Income tax should be almost abolished, it can be something like 10% on all types of income and then nobody needs to bother with tax evasions on that.
5-15% VAT or sales tax to generate revenue for the places where people congregate and do their business, the rest of society should be funded by asset taxes.
Income tax fails in its progressive goals not because it is inherently bad (like VAT) but because the tax laws are written by the richest players. (Ask a poor person how much influence they’ve had on tax law, and then ask a top 5 accountancy firm. This isn’t a controversial statement.)
I would support a low flat income tax for two reasons. It is simple, if all income is taxed at 10% you just pay that and there are no opportunities (or frankly the need) for some fancy loopholes to avoid it. The second reason is that I do like the idea that everyone pays tax, it is part of being a citizen with all the rights and responsibilites that comes with it.
If “all things were equal” and there were no assumptions and we were all starting at an equal position there “may” be some merit in this position. But that’s just… nothing, absolutely nothing, like where we are. And if it seems like it’s fairly similar to where we are: your education has been very blinkered. I say that with compassion, btw. The scale of current inequality is so astronomical it’s beyond comprehension.
If I’m so smart what is my simple and obvious solution to this global catastrophe? It’s pretty damn hard to put it all on me to solve in one hacker news comment. Sheesh.
Yes, this is known. I just asked you to clarify what it was I said that is so naieve?
I think when the person above suggested eating the rich you literally believed them and you’ve been on a misguided tear ever since. There’s little I can do to help you, but by god I’m glad I tried. Best wishes.
I’m sorry if I got on your nerve with the “worldly and educated” jab, but you were (and have continued to be) quite condescending in your responses.
Have a nice day :)
Backing up a little and answering your next question.
The naivety is that you say “I know vat is regressive but if it’s a low amount…”
Ok.
First thought I had here was:
What a middle class person considers a low amount of tax on a necessity is not the same as what a poor person considers a low amount of tax on a necessity.
At that point - just pondering that, which is only a preliminary consideration, I think - this person, I like them and I care for them but they are a bit naive. I’m not “adversarial” I’m just a bit sorry, thinking, this dude can’t see past their privilege. Fair enough. I make the same mistake multiple times per day. Genuinely.
But then I consider just how extraordinary the gap is, not between low and middle class people but between low+middle class people and “wealthy” people. Low and middle have very different opinions on the impact of 10c on a loaf of bread. But those opinions are utterly insignificant compared to the value a wealthy person places on that same 10c. Logarithmic difference.
There’s so much more to it -/ but yes just saying “vat is regressive but…” is to me a betrayal of a kind of (on the one hand) admirable level of optimism (I genuinely appreciate that) but also a naive/blinkered/privileged eye rolling “here we go again” level of ignorance that is Just. So. Exhausting.
Again - if all things were equal, if people were all starting from the same starting point - then we could “genuinely” engage in simplistic first principles discussions that treat all market participant as frictionless spheres in a vacuum etc — but we’re very far past that point… 10,000 plus years of civilisation beyond that.
Thank you for reading this far. Best wishes.
To answer your point, the thing about prices and taxes is that the price will be basically whatever the market will pay. It is not determined by the cost of service except at the lower end. So a low VAT (compared to no VAT) will come mostly out of the profits and not so much from a price increase.
This is especially true when the current situation is in most places that there is a rather high VAT, so there the lower VAT will actually reduce prices a little bit, but again not by the whole difference because the price of necessities isn’t determined by the cost but by demand.
In any situation however I remain of the opinion that until we get a progressive wealth tax we’ll continue on the same path of increasing inequality that ultimately destroys democracy.
I too am optimistic that markets can work and there are situations where the price will be what the market will pay.
But what I observe is - from the consumer side: a rapid rise of monopolies
From the producer side: an even more rapid rise of monopsonies (this is the sneakier consequence of monopolies that most of us don’t see, but which affects us most)
Those two items damage the premise of free markets
But add this:
From the government pov - these companies are beyond taxation: they can shift profits off shore and or manipulate tax policy such that the playing field is no where near level.
A simplified version of this:
- two tax brackets, 60% tax over all income below 40 000, 40% tax over all income above 40 000. - A deduction from taxable-income for everyone of about 25 000 - Negative income after deductions means you 'pay negative tax', i.e. you get money from the state - All other subsidies to people come in the form of more deductions from taxable income.
This is effectively a universal basic income of 15 000. It remains a progressive system, and ensures that taxable income deductions are actually valuable to people short on cash.
The numbers obivously need tweaking.
- owning media - lobbying the government - owning the means of production
It could take that dire situation and make it drastically worse. I mean, in theory.
Who should own media? Private people, or the state?
> lobbying the government
If lobbying the government is effective, it's either because the lobbying was correcting a wrong, or the government is allowing itself to be corrupted.
> owning the means of production
The "means of production" was barely relevant when Marx found out what a factory was from his factory-owning mate. A plumber owns "the means of production" when he owns his own tools. It's far too low resolution a phrase to be useful, except for anyone who finds it useful to teach people to hate another group of people.
I mean of course — who should own media.. we have precisely two choices, and no possibility to imagine anything else! What you’ve put forward is known as a false dichotomy. That is a situation where you present two options as if they are the only options.
Oh wow — in your response Re government you have literally stated a dichotomy. Alright. I feel bad now. I have to let you know that there are more than two options (your preferred and a horrible alternative) in just about every situation in life. Please my friend - go out into the world and seek out the alternatives - they are infinite. This false dichotomy business is harmful to everyone around you.
I agree that we should move beyond Marxist thought .. but I think we should be post Marxist not pre.
https://hellgatenyc.com/about-us
We could also restrict ownership by audience size. So any one person or non-media organization cannot own or control multiple media organizations if their total audience size is above X people. Audience size would be prorated for partial ownership or control (i.e., owning 1% of a newspaper that serves 100K people would count 1K people towards one's total). This is an off-the-cuff alternative, so assume the regulations would be written by actual lawyers who consider edge cases.
Agree. That would be private people owning it, though.
This is wrong by definition. There is an actual "true" dichotomy between public ownership and non-public ownership. Two criteria:
- "jointly exhaustive": the two subsets contain all possible scenarios
- "mutually exclusive": the two subsets do not share any possible scenarios
These criteria are met with the subsets of "the state" (meaning "public") and "private people" (meaning "non-public").
To be clear - the two options you’ve presented are not actual options that anyone is arguing for. So presenting a single alternative would be to fall into an obvious trap. I ought not to have replied to disingenuous debate in the first place I get that.
But perhaps someone else can - in future - spot a false dichotomy and say “oh - avoid this nonsense.”
I am giving a true dichotomy here, either:
1. You believe there are precisely two options
2. You do not believe there are precisely two options.
Which is it?
E.g. this could have gone:
LeonB: what about businesses owned by many people?
robertlagrant: ah, yes - what I was saying would put them in the "private ownership" bucket.
Or whatever it is that you're actually thinking of, but are reluctant to say.
The irony of you determinedly pinning me down, when I'm happy to converse, while you will do anything but answer what I was asking, is hopefully not lost on you.
Your question was:
>Who should own media? Private people, or the state?
The reason why no one ought to answer a question like that is because it is presented in a format identical to a “bad faith” question.
You may have meant it genuinely but it looks like the old “Have you stopped beating your wife?” lawyer trap.
I chose not to answer the question because it was recognisable as a “false dichotomy”. Then I explained my reasoning and so on.
I wasn’t “doing anything but answer”ing what you were ostensibly asking. Avoiding it was easy. Trying to inform you and anyone else reading of why such a question is never worth answering, is what took a lot of effort. That’s just par for the course if one chooses to respond to a question that has the format of a bad faith argument. I don’t mind.
So (to carry on your schtick) apparently you feel that if “the undeserving rich” are, as I suggested, in any way restricted from owning media, the only other bucket is state controlled media. Well I’m sorry you feel that way. Perhaps a democracy could have some laws or something, without turning into a fascist state. Seriously - the general category of “functional democracy” provides millions of shades “between” oligarchy and totalitarianism - but even saying “between” is giving too much credence to a bad faith preposition.
I’ll go through it for you again if you wish.
Over here at a specific extreme we have a totalitarian regime controlling all the media. Over here at a different extreme we have a small number of rich people owning everything including control of media.
Now we can construct a 1 dimensional line and say that these are two extremes on a single axis. And we can go even further and say, as you have, that all possibilities fall into these two buckets.
Consider though that this world we live in and the world of political possibilities is not a one dimensional world. Nor is it two dimensional or three dimensional. It is “n” dimensional for some inconceivably large “n”. There are many different ways that power can be distributed, channeled and many more ways in which benefits and taxation can be organised - and or other mechanisms and feedback systems other than these. A functional democracy can avoid being either of those two specific buckets. That one dimensional line and those two specific buckets are not in any way inevitable.
> Your question was:
> >Who should own media? Private people, or the state?
Well, no. My question was: can you give a third option?
> Then I explained my reasoning and so on.
You didn't explain your reasoning. You reiterated that it was too obvious to need to explain your reasoning, and left it as a claim.
Ad hominem.
It’s also snarky.
Ad hominem includes many types attacks on a person that aren’t about the person but how they presented the argument. Ie mocking how slow someone spoke, or in this case how many words they used: https://en.wikipedia.org/wiki/Ad_hominem
> Nowadays, except within specialized philosophical usages, the usage of the term ad hominem signifies a straight attack at the character and ethos of a person, in an attempt to refute their argument.
I personally would have apologized a while ago, but you do you.
Well, it does if I'm being a bit snarky, and not using that snark as a way of dismissing a claim. I'm saying there is no justification for the claim, and that it's taken far more words to claim the thing than it would be to supply a simple disproving example.
> or more importantly prevent what you said from both being a logical fallacy
No, something being not ad hominem doesn't prevent it from being a logical fallacy, but that's not how reason works. You disprove. You don't say, "Well this criticism failed but that doesn't prove it's valid!"
> I personally would have apologized a while ago, but you do you.
But... this is also rude. More so than a little bit of snark.
Start again. What are you trying to achieve here?
No, being snarky in no way changes what’s going on.
> a simple disproving example
Which I provided an hour ago and you haven’t responded, thus disproving your argument here with an example.
> is also rude.
If you acknowledge you’ve been rude then apology is appropriate. Simply ignoring rudeness is poor manors as otherwise people can’t improve. Instead proper manors is to bring up the mistake and offer a minor correction. In person subtle body language is useful, but in text the only option is to be more explicit though still indirect. Thus what I said was quite literally the opposite of being rude.
You were very patient.
That’s an “ad hominem” attack because it is against me.
Please refrain from doing that to people, whether me or others.
Please quote where I claimed that.
Its a fallacy but its not ad hominem. (Its non sequitur, but than most fallacies are specialized forms of non sequitur, so that doesn’t really narrow it down much beyond saying it is fallacious.) There's not, AFAIK, a common catchy name specifically for “argument to the peripheral stylistic features of the opponent’s argument”, though there maybe should be, because in practice its a common fallacy.
That’s why it was. Ad hominem is based on what is being attacked not how.
Attacking the person not the argument is deeper than simply dismissing something because of their political party, race, or educational history. When you attack stylistic elements of an argument you’re attacking the person as someone who made an argument in that form.
No, you aren’t.
“This argument is false because ofn the way it is presented” is fallacious, but it does not rely on the reader rearranging it into “This argument is wrong because anyone who would present an argument this way is a bad person whose arguments should be dismissed”.
Its definitely a non sequitur, but calling it ad hominem is interpolating things that aren't there.
A website isn’t literally a person, but what’s being attacked is the thing making the argument. It kind of break down because this isn’t directly equivalent to two people debating before the Roman senate, but western philosophy really didn’t care much about such superficial differences.
I can’t think of the phrase but there is the platonic ideal of an argument and individual who voiced it. Something about after an argument has been spoken it, as in the argument, needs to be addressed directly not the person who said it.
PS: Similar idea in a different context don’t judge a book by its cover. Not directly relevant, but I find it fascinating how often the idea of superficial as unimportant is used while people focus on it so much. Sort of a humanity has changed less than you might think in the last several thousand years kind of thing.
(1) Government ownership, (2) ownership by a single rich person, (3) ownership by a nonprofit, (4) ownership by the employees, (5) ownership by the readers…
However, suggesting just one additional options focuses excessive attention to that specific option. Mentioning how much Twitter has changed going from a public company to a private one owned by a single wealthy person is only barely related to the underlying argument here. Namely, that you need to consider many options and their individual tradeoffs not just 2 bad options.
You can have private ownership without the owners automatically being rich. Further the owners could be rich without qualifying as undeserving rich by whatever metric the post’s author had in mind.
That said, thanks for providing my point ;-)
This whole conversation has been about my "false dichotomy" of state and private ownership.
E.g. from LeonB:
> I mean of course — who should own media.. we have precisely two choices, and no possibility to imagine anything else! What you’ve put forward is known as a false dichotomy. That is a situation where you present two options as if they are the only options.
That’s not equivalent to all forms of non governmental ownership in that context. Charities are not owned by private people quite literally as they can’t liquidate assets etc. You can be convinced of Theft if you take assets from a charity you set up.
If you want to now redefine your point as all forms of non government ownership then that’s simply a strawman arguemnt in terms of the person you where replying to.
The danger in media control is not the public/private axis, but the monopoly/ plurality axis and the truth/falsehood axis.
> If lobbying the government is effective, it's either because the lobbying was correcting a wrong, or the government is allowing itself to be corrupted.
This seems like "correcting a wrong" or "not correcting a wrong and therefore the government is allowing itself to be corrupted". The second part has an argument that effective lobbying which does not correct a wrong is necessarily government corruption. There seems to be a regular old dichotomy here (https://en.wikipedia.org/wiki/Dichotomy). Not all dichotomies are false.
But I also want to highlight a part of the comment:
> Please my friend - go out into the world and seek out the alternatives - they are infinite.
If these alternatives are so infinite, it should be easy to name one. That is something that is yet to be provided in this thread, seemingly despite your confidence that this dichotomy is false.
Both plumbers and factory owners do indeed have opportunities to exploit people. But my plumber friend can only exploit one customer at a time, where as a factory owner multiplies that ability. Multiplication is a powerful operator.
A plumber can exploit a customer by charging an exorbitantly high rate for work. They can choose to exploit their position of power as a supplier of services where a customer has an urgent need.
Exploitation is not charging a high rate for work. It is charging a higher rate when a customer is not in a position to shop around.
It's hard to know where to start with things like this. How does the plumber exploit employees? Why are you assuming anyone exploits anyone by default?
Customers don’t get exploited in this way. They generally are also employees of some company that does exploit them, but that is a separate relationship.
A plumber owns a fraction of the means of production so small it’s not worth mentioning. How much of the means of production is owned by megacorps? A sizeable amount.
How much of the means of production is owned by singular entities in the context of local economies? Sometimes over 90%.
At the end of the day if some guy owns a fabulous machine and will pay you a non living wage to press a button on it while he reaps the rewards you will grow resentful. Especially if the shittier prior machine required more skilled interactions and higher wages.
Don't look at history when people did this, though. Just do it based on this simulation.
deep breath
Hahahahaha
Collapse that chain and you replace value creation as a form of mild power with political ability as a way to access direct and high levels of power and things start going wrong fast[0][1][2].
[0] https://www.britannica.com/topic/Stalinism [1] https://en.wikipedia.org/wiki/Great_Leap_Forward#Consequence... [2] https://en.wikipedia.org/wiki/Venezuela#Suspension_of_consti...
This assumption is doing a LOT of heavy lifting here.
> This assumption is doing a LOT of heavy lifting here.
Value is determined by the people. I personally find Pokemon & MTG cards to be worthless, but other people do, and they're where that value is derived.
The difference between the two is the number of sources of said power.
The power that flows from value creation is “mild”, because there are many avenues/paths for creating value. Value creation is inherently derived from what at least one person sees as valuable in of itself, as thus less concentrated. The changes brought by said value creation is limited to the people that see its value & congregate around it, and when a better/more valuable creation turns up, their decision to leave or stay with the existing creation is ultimately theirs.
The power that flows from political acumen is “direct and high”, is because that power has the ability to control everyone's lives, and that there is only one place where said power is allowed to accumulate: The State. Power inherently concentrates when only one apparatus is allowed to accumulate power. This makes manipulations & corruption of that power much easier to perform, since it inherently concentrates into few hands/figures.
Not really. P implies Q is not the same as Q implies P.
The fact that you can get a large gap between rich an poor with pure coin toss does not mean that when you have such a gap the gap resulted from pure randomness. It could have resulted from any combination of merit, hard work, inheritance, cheating, and luck. Your conclusion simply does not follow.
In this version people do not end up at some small fraction of the initial balance but instead they go negative, i.e. exit the simulation to homelessness.
The ability to keep flipping the coin without starving is absolutely the primary determinant.
In this analogy the coin flip results in 150% your net worth or 60%. In real life, we make thousands of decision per day, but very few people make a single decision like that in their entire life. And this analogy goes for 1000 such decisions.
In the real world, most of our decisions are not like a coin toss actually: if I do this homework, I might become a bit smarter and boost my lifelong learnings by 0.001%, and if I don't I will enjoy an extra half an hou of TikTok. The outcome is anything but a coin toss, still most kids go for the second choice. The kids who go consistently for the first will end up in the top 1%, but unfortunately, they'll hear from their TikTok-loving friends that their success is just due to pure coin flips.
The idea that the choice of homework vs TikTok is the choice of wealth vs destitution is a valid concept for relatively low values of income, generally within the realm of working for other people. This is because performance on homework is not necessary and not sufficient for actual wealth, only for a relatively safe employment outcome.
What we are talking about is making major bets on all the wealth you have accrued to date. To the point where a single major bet moves the global average wealth.
If your homework told you how to build the next Amazon/Tesla/Facebook after having enough money to start it, then it is relevant to this wealth discussion. Otherwise it is just a myth that got you to a relatively healthy, stable income.
What’s the mathematical rationale (or intuition) for looking at the log values?
I don’t quite understand what they represent.
e.g.
x = 60, 10, 10, 10, 10
avg. x = 20
avg log(x) = 1.1556 (which is 10^1.1556 = 14.3)
Exponential trends become linear.
The geometric random walk where you multiply by X or divide by Y with equal probability becomes an additive random walk where you add log(X) or substract log(Y) and you can represent that as a standard random walk (where the up and down steps have the same magnitude and probability) plus a trend.
There is a more mathematical way akin to the original derivation of the Kelly criterion [1]. Let's compare two strategies, max(EV) and max(log(EV)). Our game is coin-toss with heads=+50%/tails=-40% payouts and infinite repetitions; max(EV) would always go all-in, while max(log(EV)) would only bet a fraction of its current wealth at each turn.
- After the first turn, there are two cases; if heads, max(EV) has a higher payout, if tails, max(log(EV)) has a higher payout. On average, max(EV) has a higher payout because it went all-in, and there's a positive EV per toss.
- After two turns, there are four cases; in one of them (heads-heads), max(EV) has a higher payout, and in the other three, max(log(EV)) does. The latter is hence the "safer" choice. However, in the 25% chance that max(EV) wins, it has significantly more money than max(log(EV)), so the average (not median) payout is higher for max(EV).
- This repeats, and after more and more repetitions, the chance that max(log(EV)) has a higher payout converges towards 1. Note that the payout for max(EV) in the (very!) rare case that it gets lucky skyrockets at an even faster speed, so on average, max(EV) still has a higher payout.
- Turns out this is not just true for max(EV) and max(log(EV)), but in fact max(log(EV)) always maximizes the chance of higher payouts, given enough repetitions.
So, in other words, max(EV) maximizes the average payout, while max(log(EV)) maximizes the chance of having a higher payout than any other strategy. The original Kelly paper has more precise maths, but I hope this helps to guide you.
The log change intuitively shows this.... log (0.5) = -0.3 & log (2) = 0.3
If you lose three quarters of your wealth, you have to quadruple your wealth to get back to where you where.
Again, log intuitively shows this.... log (0.25) = -0.6 & log (4) =0.6
An intuition is that log(wealth) represents utility. An intuition for why log(wealth) represents utility is that wealth has decreasing marginal returns. The logarithm is a function that fits this. If you play around with it a bit you might see that it conforms to your intuitions: increasing your net worth from $100k to $200k increases your utility significantly, maybe about as much as increasing your net worth from $1m to $2m, and much more than increasing your net worth from $1m to $1.1m.
In the real world there are usually side effects to these 'bets'. For example, a person, who is already rich, uses their wealth to purchase a factory and produce goods at a dollar price in the hope to earn more wealth. Those goods might be rejected by the market, but if accepted, they receive dollars and the rest of the world receives goods. The factory owner can only receive the dollars if others can afford to buy them. This means there is another road to ruin - fail to produce anything people want at a price they can afford and you'll soon go bust.
The buyers who receive goods in exchange for money, particularly durable labor saving goods like dishwashers/fridges/cars/computers, are surely more wealthy in some sense? In addition, the factory owner might make tools/machinery that enables other entrepreneurs to make stuff too.
Also consider ongoing costs, that capital has to be replaced (things break!), unexpected bumps in the road, finite lifespan, waste, and so on.
Which leads me to our current system: the rich have got richer through financialization, government lobbying, zero-interest financing et.c, that has allowed them to acquire assets that appreciate without a substantial increase in production.
The solution is to make rich people take much bigger, potentially ruinous, risks with their wealth on production instead of financial chicanery. More abundant goods and technology are the payoff while failure leads to liquidation of their assets, making assets more accessible to everyone else who might need them.
Deflation, and crashes are a feature, not a bug. Inflationism, stimulus, cheap money et.c helps protect workers in the short term during a downturn, but prolonged stimulus makes assets unaffordable. Worse it causes huge misallocations of capital into unproductive endeavors at no risk to the investor.
My belief is you don't 'prop-up' the big fish using the government/central bank. Then yes, you may still have inequality, but it won't be nearly as static, but more dynamic. i.e there may always be an extreme power law wealth distribution, but what's more important is there's regular turnover at the top and nobody spends too long in the top or the tail.
The only? Maybe not bet 100% of your wealth every time? That's what Kelly criterion is about.
Making 4 separate bets in each round with 25% of your wealth produces the following results:
Investors: 100,000 Iterations: 100 Portfolio size: 4
Avg wealth: $130.041 Median wealth: $11.6259
Winners: 85,902 Investestors worth < $0.01: 119
Max wealth: $86,177.5 heads: 241 tails: 159
Especially since, as someone indicates in another top level comment, the percentages are not even chosen properly, gambling on absolute money is a dumb choice, one should only gamble on log-money. So, a bunch of people making a dumb gamble with no merit involved produce a distribution.
I don't understand why anybody at all cares about this - it's just garbage in, garbage out. The only real-world connection one could make is... don't play lottery, I guess? Oh wait I do understand why people care - that's because this "supports" a social conclusion they already pre-supposed.
I ran the same simulation as you. After 100 iterations, the richest one had $71. After another 1,000 iterations, the richest one had $78 (of a total $133). After another 10,000 iterations, the richest one had 10^-165 dollars. They were still an extreme outlier with 95% of the total wealth, but he had almost nothing.
With high probability.
In concrete terms I guess that means such things as re-distributive taxes ( either on death or through life ) - pooled pension funds, insurance, family and friends etc etc.
Since it is actually a very not-normal distribution, the typical concepts of "mean", "median", and "std deviation" (which AFAIK isn't even really mentioned and would be meaningful) are warped to the point of meaninglessness.
Of course economics being essentially a social science with some math and therefore defaults to thinking in typical normal distribution assumptions, won't naturally deal with that sort of pattern.
And while I agree game theory (and this is a game) is relevant to economic study, the fact is that real world "games" like wealth distribution has a vast chasm of chaos/randomness/nonlinearity that blocks things like this or the "pirate distribution" game and others.
Distrubingly, the games are parroted in some either overt or subvert manner to justify the distribution of wealth in the current 30 year era.
"It's the natural way of economics and mathematics, how dare you chop off my head in the guillotine!"
And yet "Some men want to watch the world burn". The elite also love the "great man" theory of history, because they are, of course, great men. But bring up the revolutionaries, and suddenly they don't like those pet ideas.
And to put the phenomena in much simpler words: flipping a coin once is good, having $100 bucks you either lose $40 or get $60, on average you get $10, and you win half of the time. The problem is if you have to flip a coin at least twice, because then the accumulated multiplier of your value, rather than .6 / 1.5 becomes:
.6 / .9 / .9 / 2.25
So while on average you gain 16.25% of value, you are a winner only ¼ of the time, and it gets better/worse depending on average gain / chance to gain.
…And apparently, what I learned just now, since your chance to gain approaches 0, for a finite population and infinite time, the chance approaches 0, therefore rendering the average gain to also be 0.
However, if you were to make it “double your money” (+100%), it would become clear that the only fair downside would be “halve your money” (-50%). For these values, the “trick” becomes much more obvious: that increases in repeated games need to be far greater in percentage terms than decreases (i.e. not just a 10% difference) in order to balance out.
Arithmetic mean: sum all values, then divide by the number of values.
Geometric mean: multiply all values, then take the nth root, where n is the number of values.
Even a simple calculation will show the truth:
-40% = 0.6
+50% = 1.5
0.6 * 1.5 = 0.9
Conclusion: With every coin-toss you lose 10% of wealth on average.
1.5 * 1.5 = 2.25
0.6 * 1.5 = 0.9
1.5 * 0.6 = 0.9
0.6 * 0.6 = 0.36
=> 1.1025x total
It's counter-intuitive because even though you almost always lose, you still win (linear) wealth on average (but not median). The difference is that if you have unlimited tosses available, you don't care about maximizing EV after X tosses. Instead you care about minimizing your risk of losing it all.
They are also saying parent is not correct in saying that it also loses on a population basis, showing that the average across a population is still winning.
(2.25 * 0.9 * 0.9 * 0.36)^(1/4)
=0.9
Which is below 1!
To test the geometric mean with the earlier example ("+100% is equivalent to -50%"):
(2.0 * 0.5)^(1/2)
=1, as expected.
If you're curious, I wrote a fairly lengthy explanation in another comment [2] about why your calculation is preferable in many cases, but to maximize average wealth, you really do want to go all-in.
That depends on how you define it. The arithmetic mean is usually presupposed when talking about expected values, but the arithmetic mean is not the only mean, and in fact it is inappropriate to use in some cases, so arithmetic expected value is likewise not the only "expected value", and in fact inappropriate sometimes. So yes, talking about non-arithmetic EV should be just as common as talking about non-arithmetic mean, even though it isn't.
> Instead, you maximized the geometric growth rate, which is what the Kelly criterion does; but it doesn't maximize your average outcome.
Well, it doesn't maximize the arithmetic average, but it does maximize the geometric average. :P There is no point in assuming we have only one kind of average and one kind of expectation. A suboptimal common usage of terms can in fact be a hindrance to thinking straight. You use logarithms to keep talking about the arithmetic mean/average, but in my opinion logarithms just obfuscate the fact that we are in geometric mean territory.
I agree, this is a semantic discussion. I'm using Wikipedia's definition. https://en.wikipedia.org/wiki/Expected_value
It would be kind-of obvious that if you play a sequence of games you need two heads for each tails just to break even. (If you're lucky and get a streak of heads you can win big though.)
The arithmetic expectation is wrong in this case.
In SPP, EV approaches infinity as the bank's resources approach infinity. Put bounds on the bank's resources, and you find that even with trillions of dollars your EV is less than $50.
Here, not only are we assuming that the bank's resources are infinite, we're also assuming that the population is large enough that there are always enough lucky players to compensate for the unlucky ones. Put bounds on the size of the population, and you see that everyone goes bust in all but a tiny fraction of cases. Put bounds on the bank, and even that tiny fraction can't compensate for all the losers, and EV is negative.
the exponential example is just much more useful, since there are plenty of systems easily described by compound growth
> We have thus arrived at the intriguing result that wealth averaged over many systems grows at 5% per round, but wealth averaged in one system over a long time shrinks at about 5% per round.
Wealth averaged over many systems doesn’t grow by 5%. It shrinks just like the average. The EV calculation is just wrong. For any finite starting wealth between the players and the bank, there is a number of iterations where the EV turns negative.
If you say, well let the starting wealth be infinite, I say, okay? If you have infinite dollars there are a lot of tricks for making infinite more dollars. It doesn’t work in the real world.
If we put bounds on the bank in SPP, the first coin toss would still have positive EV. In the new ergodicity problem, even with bounds on the bank, it is unclear whether the "first" coin toss is worth taking.
Not if the bank starts with $0. 0 is just as valid a bound as a trillion. You can’t calculate the EV without knowing how much the bank has, and once you know that, you realize the naive calculation for EV is wrong.
If I win I get 1.5 times my money, if I lose I'm left with less than 1/1.5 times. When I lose, I lose more than I win.
I agree it's counterintuitive how any individual (on average) will lose over time, yet the entire system grows.
A related concept in finance/trading is “drawdown”. A single trade can have a positive expected value. But over time, if you take a loss you have to get a bigger win to end up back where you started, because you have less capital to work with.
Yes, the article shows that almost certainly, any individual's wealth will approach 0 from repeatedly taking this gamble. However, the comments I replied to say:
> But this is purely a result of the distribution of returns from a single toss.
which I don't understand.
If the loser got 0.6666c instead of 0.6c, and the winner got $1.50, then over time you'd break even, on average.
And yet the expected return would apparently be 1.08333. If think the conclusion is that 'expected return' is a fallacy, you just can't add proabability-outcomes in this way to get an 'expected outcome'.
For what definition of "average"? Yes, the most likely outcome would be to break even. But if we mean "expected value" when we say average, then on average, that repeated wager would be massively profitable for us (in terms of wealth). Maybe a better way to view it is that it would be massively unprofitable for the casino offering it (this is more straightforward since the casino's outcomes are more ergodic than the individual's).
---
Maybe a more clarifying discussion would be: what if you could accept these wagers (let's say 0.6x and 1.5x) if you can use a bet amount of your choice each time? That is, you don't have to wager your entire life savings, but you could choose to make the bet with $1, with the outcomes being $0.60 and $1.50.
Then, this wager is clearly a free money machine, right? We can come to the conclusion that it is a free money machine by looking at its expected value. That's why I disagree that "you lose more than you gain" and I disagree that "this is purely a result of the distribution of returns from a single toss".
---
I guess I should concede that I agree with your reasoning, if we make the reasonable assumption that utility is the logarithm of wealth. In that case, I agree that I would be indifferent to taking a 0.666 and 1.5 bet for my entire networth, and that I would not take a 0.6 and 1.5 bet for my entire networth. However, I still contend that we can analyze the value of these wagers using expected value -- we simply look at the expected value of what we actually care about (utility), not wealth.
In the $60/$150 bet the 'expected value' is $105, but no-one is getting $105. What is that number? It indicates a 'probable limit' to the total money in the system, but says nothing about what any individual should expect, or about the actual total size of the system, which will obviously tend towards the aggregate individual outcomes. And that is clearly exposed when you start repeating the bet.
In fact 'expected value' is quite clearly 0.6 * 1.5 = 0.9, ie $90, or I think more generally: (outcome-1 * probability-1 * ... outcome-n * probability-n) / (probability-1 * ... probability-n)
It's maybe counter-intuitive that the 'limit' of the system should keep growing, while the 'actual' size of the system tends toward the 'real' aggregate expected value, but I think it's much less counter-intuitive than what I think is claimed here, that repeating a bet turns it from being a 'good' bet to a 'bad' bet.
It was always a bad bet, just measured wrong.
Howver, that's what 'expected value' means: https://en.wikipedia.org/wiki/Expected_value
It does! You take insane amounts of risks (that you want to avoid), but the expected value stays positive, even if you repeat it a million times. You might have a 99.9999999999...% chance of going broke, but you'll also have a 1/2^1mil chance of making an absurd amount of money; on average (and not median), you will come out ahead.
I don't see how this is true. Winning adds 0.5x of your current bankroll. Losing subtracts 0.4x of your current bankroll. When you win, you win more than you lose. The gamble has positive expected value.
You need to gain 67% ($400 on $600 or $667 on $1000) to break even before or after a 40% loss. The math is 1/(1-0.4)-1=0.67 or (1-0.4)*(1+0.67)=1.
> But this is purely a result of the distribution of returns from a single toss.
Your examples contain two tosses.
HH: 2.25! HT: 0.9 TH: 0.9 TT: 0.36.
Expectation: 4.41/4 = 1.1025. The losses due to HT/HT get more than offset by the massive gain from HH over TT.
I can imagine there are arguments to be made about median expected value, and the effect on concentration of wealth. But whatever they are, they aren't being made.
Put another way, for each HH pair you should expect a TT pair, and a HHxTT or TTxHH sequence is worth the same as HTxHT, losing the usual 10% every two tosses. 2.250.36=0.81=0.90.9.
The thing that still confuses me is, why the heck is the EV 1.05? It seems to be expressing something true - if you were to split your money into a thousand piles and "play" each individually, you make money overall.
The article, just like the poster above you, characterizes a series of bets into win/loss pairs that add up to a 0.9 return per pair. There are lots of sequences that can be characterized this way. However there will be a few sequences that contain many, many heads and win a lot of money. There are of course also a few sequences with many tails, but their loss cannot decrease below zero so it is contained.
So it's a little bit like a lottery ticket, where the positive gains are extremely concentrated into a very small lucky group. The more rounds of the gamble you play, the smaller the lucky winning group gets, and the larger their wealth.
This is exactly the catch of the experiment. It's proposing you cannot split your money in independent experiments. You need to pick a history and stick with it. That's what ergodicity is about.
Hence, no one owns the average (EV) money of all possible outcomes and it's a pointless metric, even though it doesn't say something that's mathematically false.
2.25 * 0.36 = 0.81 = 0.9 * 0.9.
HH: 2.25 HT: 0.9 TH: 0.9 TT: 0.36.
Expectation: (2.25 * 0.9 * 0.9 * 0.36)^(1/4) = 0.9. So you expect to lose 10%. The losses don't get offset by the wins. Intuition: Possibility of losing 100% is not offset by a possibility of winning 100%, so the arithmetic mean is wrong.
Your stake should grow or shrink by the same factor if you win or lose.
If you win, your stake is multiplied by 1.5.
If you lose, it should then be divided by 1.5 (multiplied by 0.66..).
But is in fact multiplied by only 0.6.
> If you lose, it should then be divided by 1.5 (multiplied by 0.66..).
In what sense "should" this be true?
Of course the actual amounts, percentages, proportions look bigger for the win compared to the loss, but that's just maths.
In the given game, the collective loses money just as the individual does. The collective wealth is the summation of the individuals' wealths, and both the collective and individual wealths drop over time. Write a simulation and try it if you don't believe me. I did, because I couldn't work out how the collective wealth could grow while the individual wealths drop, and the answer is that it does not.
The only way it looks like the collective makes a gain is if you try to say that x_i(t) = x_i(t-1)*1.05, but that's a simplification that doesn't hold in either the individual or the collective case.
But if you forget the part about the supposed misalignment between individual and collective, then the idea that the expected value of 1 iteration can be positive while the expected value of repeated iteration can be negative is very fascinating!
I agree the distinction is not at all between individual and collective, but rather between one-off and repeated games.
No, they don't. If 100 people starts with $100 (i.e. a total wealth of $10,000), and half of them win as expected, the total wealth at the end of first round will be $10,500.
Granted it won't be a 5% increase for the next round, and eventually collective wealth will also approach 0, but that's the point. 100 people playing 1 round is not the same as 1 person playing 100 rounds.
The interesting part is that they expect to lose money over time even though they expect to gain money if they only play once. But that holds for both the individuals and the collective.
A less confusing game with the same mechanism is "flip a coin, if it's heads I give you 1000x your initial investment, if it's tails I wipe you out completely" - you expect to make a massive gain on any given turn, but if you play enough rounds you'll eventually lose everything.
The collective wealth calculation just makes it easier to visualise.
Round 1: 100x$100 (total $10000) -> 50x$60 + 50x$150 (total $10500)
Gain of $500 total
Round 2: 50x$60 ($3000) -> 25x$90 + 25x$36 ($3150)
50x$150 ($7500) -> 25x$225 + 25x$90 ($7875)
Gain of $150+$375=$575.
Round 3: 25x$36 ($900) -> half $54 and half $21.60 ($945 total) And so on.
Each round, the collective wealth goes up 5%, no matter how many rounds you run.
Think of it this way: If everyone's individual wealth approaches zero, why would the total go up?
Just run the following simulation in a python REPL:
import random
POPULATION = 100
INITIAL_MONEY = 1000
ROUNDS = 10000
wallets = [INITIAL_MONEY for _ in range(POPULATION)]
for iteration in range(ROUNDS):
for person in range(POPULATION):
if random.random() > 0.5:
wallets[person] = wallets[person] * 1.5
else:
wallets[person] = wallets[person] * 0.6
print(f"{sum(wallets)=}")And this isn't true.
I could agree with that.
(However, if you add "in expectation" it will be true for one round and for each round no matter how many rounds you run.)
However you agree that it's true that "in one round, the collective wealth goes up 5% in expectation", right?
(I agree that "for a finite population and starting wealth, the collective wealth will tend to 0 given enough rounds ALMOST SURELY". The last bit is important though. Otherwise it's like saying that if you flip a coin enough times you will get heads - which is not exactly true.)
Just like you cannot be absolutely certain to get heads if you flip a coin enough times even though it will happen almost surely.
Also, "5% per round" assumes an infinite number of people. If you run this with 100 people, by the 3rd round, the expected group sizes are non-integers, so you will diverge from the "infinite people" case.
If you change the rules to, say, +100% and -90%, they lose even when population > 20 * rounds. I'm sure there is a statistical formula that can be used to estimate the optimal number of rounds.
The intuition for why this happens is that the losses of the majority are made up by the big gains of a minority. You can even see that after two iterations:
Case 1: two heads -- 225% wealth Case 2: heads, tails -- 90% wealth Case 3: tails, heads -- 90% wealth Case 4: two tails -- 36% wealth
So, most people are losing money but a single lucky guy makes enough money that the expected value is still positive.
If you run a simulation with a low sample size, you might not see this effect because the "lucky guy" case is rare (and gets more rare with more iterations).
(The question can be reformulated such that this could be the case.)
> that the expected value of 1 iteration can be positive while the expected value of repeated iteration can be negative is very fascinating!
is indeed untrue. Thanks!
The reason for this is that the very rare outliers will accumulate truly enormous wealth, while the rest of the players will never lose more than they start off with. You can see this in the first graph after say 250 rounds - the richest player has around 10,000x his initial wealth, even as the other 1000(or so) have very little - so on average they have 10x what they started with. And the reason no one is rich after 1000 rounds is simply that there were not enough samples. If there were say a billion samples, you might expect one to come in with say a trillion times its starting wealth, overcompensating for the other 999,999,999 losers - on average.
The median person does get poorer over time though, a small percent get extrmeely wealthy: https://imgur.com/a/v2AVyrv https://imgur.com/a/7pPfKk6
An ever smaller slice of huge winners is pulling up the average, but in any finite population this slice disappears in finite time. Only in an infinite population can it persist indefinitely.
All of this is fairly straightforward for a mature student of probability. If enough folks want, maybe I could write up the details to prove these things.
ROSENCRANTZ: (interrupting) Do it.
lim players -> inf
mean wealth -> starting_wealth * (win_chance * (1 + win_gain) + lose_chance * (1 - lose_loss)^rounds)
win_change = lose_change = 0.5
win_gain = 1.5
lose_loss = 0.4
(0.5 * (1 + 0.5) + 0.5 * (1 - 0.4)) = 1.05
lim rounds -> inf
wealth -> 0
ln(1.5) = 0.41 ln(0.6) = -0.51
For the game to be profitable, it must satisfy
ln(1 + win_gain) > -ln(1 - lose_loss) <=> 1 + win_gain > 1/(1 - lose_loss)
So for a gain of 50% the loss must be less than 33.3%.
The first part shows the expected value of the return function, which is positive (from the gambler's perspective).
The second part uses f(E(x)): the gambling function applied to expected number of heads and tails (50/50), which is a loss on average. But that's disingenuous because it silently thwacks off the outliers where a gambler wins almost every time or loses almost every time, which is hugely positive on average.
Essentially any nonlinear return function would have the same inequality. It's just stated trickily here and the analysis is deliberately misleading.
The author seems to be confusing mode and mean; the modal path does approach zero.
The point is that you as an individual playing a repeated game don't get to meaningfully sample the expected value of the distribution. You only get to sample once, and you will almost surely (i.e. with probability approaching 1 as n goes to infinity) sample a point in the distribution where you lose nearly all of your money.
And the answer is yes - with probability approaching 1 as n increases (ie excluding a portion of the distribution whose measure decreases to 0), the random process matches a deterministic process which is described by "you lose 5% each round".
I asked a friend who is an econ professor "Why does this Peters guy explain this so poorly" and his response was more or less, yes, all of economics has been wondering that too since he first published his Nature Physics paper on this a decade ago.
This quote tells you all you need to know about the author's ability to understand things:
"my second criticism is more severe and I’m unable to resolve it: in maximizing the expectation value — an ensemble average over all possible outcomes of the gamble — expected utility theory implicitly assumes that individuals can interact with copies of themselves, effectively in parallel universes (the other members of the ensemble)."
Try 225% * 90% * 90% * 36% to get the expected value.
[Edit: delete bad math]
heads: double your money tails: lose all your money
in which case the expected value is always $1, as you have a 1/2^n chance of having $2^n dollars after n rounds, and 0 otherwise.
The point of discussing ergodicity here, however, is whether you can describe the behavior of the iterated distribution deterministically if you exclude a portion of that distribution which has measure zero.
The funny thing though is that despite the wheel being painted half green half orange, the actual odds are 75% green and 25% orange. But calculating how much of your money to bet at each iteration isn't super straightforward; in the end it turns out that to maximize your profit you should always bet exactly half of your money on green.
Thus, +50% and -40% are not the right arithmetic pairs. It should've been +50% and -33%. It's even more intuitive to say 3/2 and 2/3. Waiting for a matician who can explain better, but this whole story is more of a parlor trick than anything else.
I don't think it is really. You're right in saying that 2/3 and 3/2 are the right numbers to ensure that the population wealth stays the same. But the point of the exercise is to show that the population wealth can increase while the wealth of almost any individual in the population decreases.
This is the author's jumping-off point to argue that the standard arguments in economics about average welfare can be far removed from the experience of any single worker in the economy.
Quick correction:
The article uses 0.6 and 1.5. In this setting, the population wealth increases and the wealth of almost any individual decreases.
In the comment you replied to, they propose 0.66 and 1.5. In this setting, the population wealth doesn't stay the same, it increases even more than before.
(In order for population wealth to stay the same, the payoffs should be 0.5 and 1.5)
> the expected value ... grows by 5% in every round ... but, crucially, a head and a tail experienced sequentially is different from two different agents experiencing them ... we have lost ... approximately 5% per round
W_t = exp(k E[X_t])
Whereas the wealth of any individual grows like
W_t = E[k exp(X_t)]
There's a result called Jensen's inequality that says that
f(E[X]) > E[f(X)]
for any random variable X and any convex function f (exp is one such function). In a sense, I think this all just falls out of Jensen's inequality.
> For historical reasons, this average is also called the expected value
Why is this “for historical” reasons?
https://web.universiteitleiden.nl/fsw/verduin/stathist/1stwo...
EXPECTATION. According to A. W. F. Edwards, expectatio occurs in 1657 in Huygens's De Ratiociniis in Ludo Alae (David 1995).
According to Burton (p. 461), the word expectatio first appears in van Schooten's translation of a tract by Huygens.
The two references above point to the same text as Huygens's De Ratiociniis in Ludo Alae was a translation by van Schooten. NB The word expectatio is used quite frequently throughout the text.
This is the Latin translation by Van Schooten of the first proposition:
Si a vel b expectem, quorum utriusque aeque facile mihi obtingere possit. expectatio mea dicenda est (a+b)/2
This is the Dutch text of Huygens' Van Rekeningh in Spelen van Geluck. This text was published in 1660 but already written in 1656.
Als ick gelijcke kans hebbe om a of b te hebben, dit is my so veel weerdt als (a+b)/2
The literal translation of the Dutch text is: If I have an equal chance to get either a or b, this to me is worth as much as (a+b)/2. There is no explicit mention of expectation only of value, but as the rest of the explanation of the first proposition is concentrated on the possible outcomes of a game of chance, expectation is implicitly around.
Expectation appears in English in Browne's 1714 translation of Huygens's De Ratiociniis in Ludo Alae (David 1995).
This is Browne's 1714 translation of the first proposition:
If I expect a or b, and have an equal chance of gaining either of them, my Expectation is worth (a+b)/2
Random walks with downward drift don't inevitably go up arbitrarily high like ones without drift.
heads +100%
tails -75%
If you play five times in parallel you're as likely to make money as to lose money but when you gain you gain 45% on average and when you lose you lose 20% on average. Overall, the expected gain is 12.5%.
If you play five times in sequence you're unlikely to make money (less than 20% probability) but then you make a 300% gain (four heads) or even a 3100% gain (five heads). Overall, the expected gain is 12.5% compounded five times which is 80%.
As the number of coin flips grows it is less and less likely that you lose if you do it in parallel (the expected gain is 12.5% and the distribution of returns converges to that value) and more and more likely that you lose if you do it in sequence (the expected gain is again 12.5% compounded but the distribution of returns is an increasingly skewed lognormal).
Exercise for the reader: take this interesting but simple and well-known fact and make a career out of it from repeating it again, and again, and again, and again, and again, and again, and again...