Expected value doesn't mean jack shit if the game can only be played once.
> Expected value (also known as EV, expectation, average, or mean value) is a long-run average value of random variables.
If you can only press a button once - you should take the guaranteed money in almost all circumstances (assuming you have finances that look like most Americans - if you're already a millionaire... do what you want, this game doesn't matter much to you).
Basically - This is a dire misunderstanding of how statistics works in general. The population at large might be better off pressing the 50% at 50 million button (because then you are running this game many times and you will likely achieve the expected value) - but as an individual, who can only roll the dice once, you are much better off just taking the immediate and guaranteed win.
And that's not even accounting for the drop off in marginal value of each dollar as you accumulate them - that first million is far more impactful than the next 49.
This is a key observation in more practical concerns like retirement planning. Often, maximizing expected value isn't actually what you want. For somebody with a comfortable retirement portfolio you care a lot more about not running out of money than ending up with a huge amount when you die. So you'll choose strategies that might have worse expected values but limit the frequency of worst case scenarios.
Rory Sutherland (behavioural science chap) made a similar point on travel. He says that when he must get to the airport on time, he takes the back roads that get him there in a guaranteed 30 minutes rather than take the freeway that will take 15 minutes 95% of the time but could be heavily congested (and inescapable) otherwise. Sometimes urgency and efficiency are at odds!
I remember looking at real-time traffic reports, but they were not future time, which is what I needed. I had an appointment to make but 500 feet in front of me...horrible crash. No way out. Stuck for 45 minutes. Had I had future time instead of real-time, that would be the best solution.
Had I taken the back roads, if there was an accident, there would be tons of options to get to where I needed to go.
If you need to be somewhere at a certain time, do the tried and true 100% guaranteed way.
You can turn around in the road if you encounter an accident on the back roads. Encounter one on the highway and you may be stuck until it's cleared. I always go back roads if I need to make a flight.
This can be captures pretty well by taking the logarithm of each outcome's dollar figure before computing the expected value, if you ever find yourself wanting to calculate how to balance a portfolio.
Isn’t it expected utility that matters, not expected value? From that standpoint taking $1 million guaranteed is rational unless you already have high net worth.
That's an additional concern, but even if you have a flat "utility" scale you can use for comparisons, _expected_ value still isn't necessarily the thing you want to optimize.
Maximizing the min/max outcomes are common alternative preferences. Like, suppose I have a 1% chance of being tortured for a year and a 99% chance at being the next God-Emporer. I can't actually average those futures; I'll be in one or the other, and I might want to have a 0% chance of torture or if the odds are flipped maybe I'm okay with being tortured for the tiny chance at being God-Emporer.
Such preferences are hard to cover under the umbrella of maximizing expected utility because it introduces (neg)infinite utility to certain outcomes. Instead recognizing that you might be optimizing something else is a cleaner way to handle the problem.
> that first million is far more impactful than the next 49.
This is in fact the reason you should take the million.
How many times you get to play the game is irrelevant. Your whole life is filled with potential but uncertain payoffs, and you should maximise expected utility every time (where utility is not the same as dollars).
No it’s not, if you play the game 20 times you’re almost certain to win 50 million and probably a lot more. Unless your utility function is flat after 20 million it does matter.
If you play the game 20 games you'll still be better off pressing the 1 million button 5-10 times, at the start if you don't know in advance how many presses you get, or at the end if you do and haven't won big yet.
You've only got a one in a million chance of losing 20 times in a row.
If you lose 19 times in a row, sure you can take the sure million on the last try. But it doesn't make any sense to take the sure million at the beginning.
After you've pushed "give me $1 million" on your first go, your utility function looks very different to how it did before (assuming you're not already a millionaire).
I replied to someone claiming that you should always go for certainty no matter how many games you play. If you get to play the game 49 times it doesn’t make much sense to go for certainty 49 times because 2^-49 (or 2^-20) is really small.
It's an interesting question (assuming you know the number of times you get to play up front).
In reality, with these numbers, the best strategy for most people who aren't already very wealthy would probably be to get a sure-fire nest egg and then play the odds.
If you have to play the same every time, I'm not sure. Again, with these numbers, the utility function is looking pretty flat after $20 million for the vast majority of people. And "almost certain" != certain.
Then this is worthless as a decision making system since your decisions will always already be the one that maximizes expected utility via this unknowable transformation.
No, that doesn't follow. You might be acting in a short-sighted or impulsive way that doesn't maximize your utility. You can't just post-hoc declare that your true utility function is whatever would have led you to the decision you made.
And well, if everyone played the game, then the population at large would still be better off taking the million. I can well imagine there being fewer social problems if we all get a million fun bucks versus half of us getting fifty million. But then that's a different effect kicking in.
Personally, a million would affect my life positively (I'd buy a house), 50 million negatively (I'd stop working).
My job is what I want to work on. But I'm lazy, and I know myself well enough to know that if I didn't have to go through the things I dislike about my job (hello doing performance reviews), I'd stop doing it all. And it would ultimately be to my detriment. I wish it were otherwise, but there we go.
I'm probably much older then you. My advice is stop thinking you are lazy. If $50 million is enough for you to be comfortable for the rest of your life then take that - nothing is more valuable than time. You'll find something rewarding to do with your time. Even if you just enjoy your life, it's fine!
Not the person you replied to, but I would 100% stop working. Even when I find things I want to do (not work), those rarely last more than a month, usually a week.
Anything that expects me to wake up at the same time every day, or working for a set amount of hours, or prevents me from stopping or taking a break (weeks, not hours) when I get bored of working on it is out of the picture. That leaves zero work options as far as I'm aware.
The problem with the analysis in the article and with your analysis is that the expected utility of the player is not the same as the expected amount of money. Different people have different "utilities of money" reflecting their different risk tolerances, incomes, satiation rates (diminishing marginal utility), etc. The expected value analysis is the correct one if you use the right "value".
If you are only playing the game once, then any rational agent should attempt to maximize expected utility. Here "rational" just means that preferences are consistent in a particular way. For the purposes of this game played just once, almost all humans are rational. When humans play multiple times, they quickly lose the ability to calculate and make rational decisions.
Econ 101 covers expected utility, and it's one of the few pieces of useful econ theory. It's like people write these articles without an elementary understanding of the theory which might be able to sensibly explain the situation.
But the article does go over the utility and explicitly states that for many people the utility of a guaranteed 1 million dollars is greater than a 50/50 chance at 50 million, so I'm not sure what "people" you're talking about or if you even bothered to read the article.
The people who write articles like this, call it "why people make dumb decisions", and don't appear to understand this is a well understood area.
It's written as though they stumbled across this esoteric idea written by some dude 100 years ago. There is a whole set of papers, Bernoulli is one of the guys who did research on it and there are a bunch more. There is a whole field, the ideas have been pulled together. It's introduced in any half decent microeconomics class.
What's next? An article suggesting maybe we can predict how long it will take an apple to hit the ground? That some guy named Newtown penned a few useful ideas on it back in the day? And pretending "physics" isn't a field?
“Go big or go home” comes to mind. Most people I know would take the 50/50 chance. In the worst case, nothing in their life changes. If they take the million dollars, something is going to change :)
I’m also reminded that “people are happier when a choice is made for them” or some other thing I’ve heard thrown around.
And expected utility (and decreasing marginal utility of money) does a good job of explaining why most people would change behaviors as you scale the numbers involved even if you keep the ratio of expected values the same.
> Expected value doesn't mean jack shit if the game can only be played once.
Thinking like this was the mistake I've made.
While you can play a given game only once, your life will have plenty of such games. So there definitely is a relevance to "expected value". And this is easily to simulate with a program. The expected value of the wealth for those who take the chance when the "local expected value" is better than the certain outcome does tend to be higher.
Well, life doesn’t always give many chances to play a game. You can only work at so many failed startups, or have so many failed long-term romantic relationships before you’ve used your best years! Someone else already made the point about the risk of walking away empty handed, but I’m just pointing out that some domains allow for many retries and some don’t.
This represents the trap of over-rationalisation which is so prevalent in the Western world. You cannot devise universal rational guidelines suitable for every situation and every subjective experience. There is a multitude of various different factors involved in every particular situation. The lean and precise rational model breaks badly simply because it doesn’t (and can’t) account for all the factors.
> Well, life doesn’t always give many chances to play a game.
Disagree. Sure - you don't get many games involving millions of dollars, but you do get many for smaller amounts.
I could put all my extra money into paying off a low interest mortgage (guaranteed return), or I could put it in an index fund (higher average return, with no guarantees, and a potential for a loss).
And working at startups: Not sure the expected value is high there. May be higher than working at a FAANG. I doubt it.
I assume you agree that some kinds of opportunities are limited. Not trying new foods because you're afraid of wasting your money would be silly, or not saying hi to your neighbor because they might ignore you would be silly, but some things are very complicated. I'm thinking of: surgeries, mate selection, college degrees, white-collar crime, etc. I'm just saying that utility and loss aversion come into play, and "life is long" can't always save the day.
On startups, I think there are people who have been in situations where they have an expected value greater than something like a FAANG $300k/year over 3 years scenario (e.g. they own a large stake in a close-to-IPO company). And they should maybe still walk away, if the 50% chance of a tiny IPO payout would destroy their self esteem and make them feel even further behind their high-salary peers. (Also keep in mind that not everyone lands jobs at FAANG companies, so it shouldn't be super hard to find people who lucked into a startup where their EV is higher than their market salary over a few years). In other words: even if a startup somehow has higher EV, you may want to ignore the EV.
This is a much more profound statement than it seems at first and I wholeheartedly agree with it. Not only that but the gains compound over time.
It's not about the expected value of any one opportunity, it's about the expected value among every opportunity you will encounter in your life. This also implies that one should do what they can to expose themselves to said opportunities especially while they're young.
A very common one: You have a debt to pay off (typically mortgage). Should you put all your extra money to pay it off early or should you pay the minimum and invest the rest?
As another commenter pointed out: Most investments involve this. In the RE circles you often have the same dilemma: Buy a house for rental in a LCOL area where you get (mostly) guaranteed net income, or buy in a place like California where the rent income won't cover all the expenses, but you feel you can pay the difference and rely on profiting off the hoped appreciation.
Insurance is also a good example someone else pointed out.
Even: Get a guaranteed low paying job as a relatively unskilled worker, or get into deep debt to go into medical school, do a residency, and earn a lot. The latter can have significant risk: Some people don't do well enough to get a residency. Others get the residency but don't have what it takes to complete it. In both cases you're left with a huge amount of debt.
Everytime you book additional insurances that cover small amounts of money. Like a airplane ticket insurance (that only covers the fee of the ticket if you cancel). Or a additional rental car insurance.
Assuming that Insurance companies are not stupid and only offer an Insurance that is +ev for them, that means its -ev for you. If you are in the financial situation that 1-5k$ wont ruin you its rational to NOT take these kind of insurances.
Every spot in life that you encounter that can be seen purely from an EV perspektive should be played as that. Only exceptions are longtail ruinous outcomes, like House Fire insurance, Health Insurance. Thats why in many western nations these types of insurances are mandatory.
Investing has a degree of this as well. And, in practice, most rational investors will diversify based on a number of factors into fairly safe but low return assets and into potentially higher return but riskier ones.
The investor's situation, I believe, is very much different from the common person's. The investor put him or herself in the position of doing tons and tons of financial transactions and investments, etc, like that. He or she put him or herself in a situation such that EV-reasoning makes sense. It seems to be that this isn't the situation for the common person.
But I agree... If you are an investor, or maybe a professional poker player, then you'd have put yourself in a position that favors reasoning guided by EV.
There are other ones as well, non-money related. For example, in sports. I believe basketball players probably try to do this. There are so many shots. They're probably using EV to guide their strategy and practice.
Five Thirty Eight writes about this from time to time. Three points shots in basketball. Going for it on fourth down. Going for a two point conversion. You can work out the stats for all this sort of thing--and there are apparently biases for various reasons why coaches/players don't always follow the EV strategy.
I think I get it, but I'm not so sure I'm convinced. Those examples, however, don't resonate with me (don't have a car, nor a license to drive one; nor I own a house; I've been inside an airplane only once).
However, I believe I've done similar things with used electronics. I tend to favor buying a really cheap used ones for [sometimes] 1/5 of the price instead of a new one. It could break or be of low quality, but chances of that are small and thus (over time -- making an EV-ish calculation), I spend less money on electronics.
I also believe I do this in buying new products. In many situations, I can pay extra for an extra year or two of 'guarantee' (not sure if the right term is 'guarantee' or 'insurance'). However, very often, the first 6 months or 1 year of guarantee is given and has its cost embedded in the price of the product. The question becomes: how likely it is for the product to fail given it hasn't failed for the first year. I believe the chances are small so I don't buy it. I guess it's also an EV kind of calculation (just like you gave as an example).
However, those don't seem that common, really. Maybe it's just the kind of life that I live.
Is the situation 100%1M vs. 50%50M supposed to exemplify these ones? These not-so-frequent ones for small amount of money?
Another thing is that expected value has to do with a limit in this situation:
(1/n) x SUM [j = 1 to n] outcome(j) -> E for n -> oo
(there is an ergodicity assumption going on here -- which doesn't always hold in practice). That limit can be E while the first idk how many hundreds of values of outcome(j) be very distinct from E.
How many times will things like that happen in your lifetime? Some dozen? What if you separate away the large-scale ones (like the 100%1M vs 50%50M)? The small-scale ones will be more frequent and you just blindly follow the EV approach to them. The large scale ones will be extremely rare, and maybe another approach is better. No?
>In many situations, I can pay extra for an extra year or two of 'guarantee' (not sure if the right term is 'guarantee' or 'insurance'). However, very often, the first 6 months or 1 year of guarantee is given and has its cost embedded in the price of the product. T
Extended warranty which is basically insurance. Leaving aside the fact that some credit cards provide it for you anyway and things like that. Yes, for most purchases, this is a bad deal because the expected value is almost certainly negative and--probably--if something does break you can replace it.
Here we're talking about losses rather than gains. The certainty of small losses (extended warranty purchases) vs. the chance of a relatively large loss. But it's the same idea with a negative sign.
One of the thing that isn't obvious to me that seems to be for many people is the decision to maximize expected value instead of best worst case scenario. In this situation, given how exceptional the 100%1M vs. 50%50M situation is and how the 1M will definitely kill your financial problems, it really does seem like you'd like to pick the strategy that maximizes your worst case scenario (if choice=red, worst-case=1M; if choice=green, worst-case=0). I understand the reasoning behind expected values, I guess, it's just that it's not clear to me it is of any use here.
To me, the choice looks like "solve your financial issues with the red button; 100% chance" vs. "solve your financial issues and get extra money you won't really need, but with 50% chance through the green button".
I'd have a hard time choosing the green button.
It's curious because I'm a mathematician. I feel like I should know this better, but I've never really studied probability, much less statistics or economics.
(edit)
Another issue is what would it mean, in practice, that "50%" statement? I guess it means that if you'd play the game long enough, 50M would come out roughly half the times (by counting). This could mean a system in which the first 10 always fails, the second 10 always succeed, and the ones after that have their results based on a fair dice (1,2,3->50M; 4,5,6->0). This would certainly fit the frequency "definition". In practice, these probabilities don't mean a clean neat thing very often. Another issue is that the definition of that 50% means if you played that game long enough, you'd observe the half-half split, but you'll play it only once. Again, there is a statement about a limit (a statement about a_n, for n large), but you're only looking at a_1 (it often seems to me that people believe that information about EV transfers to information about a_1 -- it really does not). Even though I can mostly think of artificial examples (stuff like the one above), I'm not sure it'd be clear [in an actual situation] what is the meaning of that '50%'.
If the $1m "solves your financial problems" or is otherwise life-changing, you should almost certainly take the sure thing. As other discussions suggest, once you get into maybe the $3m-$5m net worth range, you presumably already don't have financial problems and another $1m is nice but not really transformative whereas $50m would be even though not a sure thing.
Even for a one time event, at some point it makes more sense to place the bet depending on a number of factors.
If it's hard to conceive of in this scenario, pick numbers about which it's easier to have intuition. What if you could take $10 for certain vs. a 50% chance of getting $500? Or pick some other values with the same ratio. 50% in this case just means a coin flip. You're right that no one gets the expected value. They get zero or they get $50m. But that may be a good bet depending on circumstances.
EV is such a nonsense measure anyway once you step outside the realm of pure theory.
For example, the EV in this example (50% chance of $50m, or $0) is $25m. The EV of a 2.5% chance of $1 billion is also $25m, but your probability of getting nothing is 20 times higher. Is it more rational to choose this over the certainty of $1m? I don't think so. Is it rational to chose a 0.0025% chance of $1 trillion over $1m? At that point I think even the most avowedly rational economist would choose the cash.
Increase the payout with a correspondingly lower probability and you're basically lowering the expected utility--down to some point where it crosses the sure-fire payout.
I'm not sure on your argument. I belive it's rational to make a decision that you would repeat every time you are presented with the decision, no matter if you knew beforehand how much times that decision would be presented to you.
Let's account for the marginal value of each dollar to set the number 50 to be some number that equated to triple the utility of the first million.
Why would it make less sense to choose the 50%? Assuming you would definitely take a 99.9999% chance of 50 million over 100% of 1 million, at what percentage do you switch over to the higher percentage?
You can derive what sort of variance a person will tolerate if they have a "utility function" quantifying the marginal value of each dollar. If you are risk averse (which people usually when they have a small net worth) then your utility curve will be concave. Think like sqrt(x) function.
OTOH, insurance companies can afford to have a roughly linearly utility function (because, as pointed out, they play the game much more often than others), which is why they are in business.
The most profound comment was not in this HN thread, nor by a user in the Twitter thread: the most profound comment was the one quoted in the article itself, by Bernoulli!:
> Bernoulli once wrote, “The utility [of probabilistic decisions] is dependent on the particular circumstances of the person making the estimate. There is no reason to assume that the risks anticipated by each [individual] must be deemed equal in value.”
Risk over non-fungibles (body parts, sentimentally valued heirloom pieces, ...) are obviously subjectively valued. But even for platonic (ideal) fungibles like fiat money the risks depend on the person because modeling reality as if everyone is treated equal in commerce or has equal access to and treatment in the courts etc. is a very strong assumption to make.
Let us first assume contracts are never reneged etc, and let us thus first assume agreements are rigorously respected.
Let us further assume the subject has the usual goal of maximizing its capital, here denoted in dollars.
Since currencies are a social construct and only hold value in the context of a society, we assume the subject is in prolonged contact with a society that values this currency. (If not the subject doesn't care which answer to give.)
Contrary to all the comments here in HN (nonlinear utility etc.), if the goal of the subject is to maximize capital, then the correct answer (assuming absence of things like conscientious objection) is unconditionally the green button with the highest Expectation Value, non-linear utility functions, or one-time-ness of the offer, or subject poverty be damned!
To understand why: even if the offer is one-time, and even if the subject can not afford the regret of missing out, the subject is still in contact with society. This society has companies regularly dealing with large sums, and optimizing expectation value.
THE SUBJECT CAN SIMPLY GO TO A BANK AND TRADE THE HIGH ROI FOR STABILITY WITH THE BANK:
For example the subject and bank can agree to the following:
* Bank pays subject $20 million
* subject presses green button
* If subject receives $50 million, it forwards this to the bank, otherwise nothing
In this scenario the subject wins $20 million unconditionally, and the bank spent $20 M with an expected return of $25 M, so the bank sees an expected ROI of a handsome +25%.
The real catch is not personal utility function, one-time-ness, etc ... but reliability of contracts and trustworthiness of the system enforcing them, which one can read between the lines of Bernoulli's comments.
All the comments and observations about how poor people "should" take the certainty with the lower amount is just echoing the indoctrinated "learn and embrace your lowly position in society" wheither thats low in rewards, or low in reliability of fair enforcement of the law.
I find it hard to read intellectually capable people concoct artificial examples to make people distrust mathematical rigor when it can be entirely relied upon. The real element of unreliability is in the systems under which we are subjugated.
There's diminishing returns on the utility of money. If you're living paycheck to paycheck that guaranteed million is gonna give you a higher expected return of utility than the next 49 million combined. I disagree with the title calling it a "dumb" financial decision. It can be perfectly rational to take the million.
There's also "purely statistical" problems with the argument the 50 million at 0.50 probability is the better decision: it makes an erroneous ergodicity assumption. In other words, it assumes that the expectation of a one-shot decision over people is the same as it is for the decision within-person. Sure, if you were making the choice over and over and over again, it would be better to repeatedly hit the green button. But that's not the situation. The situation is a mixture of outcomes, so for any given individual the expected outcome might be zero.
There's also I think an implicit stationarity assumption built in, that if you say you'll pay me X amount over time, that you'll actually do that, that inflation wont eat it into oblivion, etc. It's a classic case of theoretical models not working in reality.
This is kind of the point of the essay, but I think it could have been made more rigorously (as people here are pointing out).
It's a classic spherical cow model. It has no bearing on reality, although this pay structure is becoming more and more common; albeit with less then 50% odds.
How many companies have you seen with "contests" to get ideas or content. The vast majority of people end up with nothing.
>How many companies have you seen with "contests" to get ideas or content. The vast majority of people end up with nothing.
Yeah, but you see the same thing with giveaways at trade shows, for example. Even for a fixed giveaway budget, there's an argument to be made for having a drawing for a nice prize, rather than giving everyone some cheap swag.
Exactly. The fact that this article doesn't even mention the concept of marginal utility, and acknowledge that it's mathematical rather than "psychological", is borderline irresponsible.
"If you don’t have a dime to your name you should take the guaranteed million dollars all day, every day.
But what if you have some money? What if you’re already a millionaire? At that level of wealth taking the 50/50 shot at $50 million might be far more tempting."
The takeaway is that a wealthier person can take greater risks without endangering their livelihood.
A wealthy person, could, for example risk buying an older used car that would potentially need costly repairs. In case it needs these repairs, they will suffer some financial losses but would still be able to derive utility from the car. In case it doesn’t need them, they get rewarded for the risk with a functional car that costs considerably less than a new one.
For a broke person the same decision is much harder. Not being able to repair the car would unlock undesirable 2-nd and 3-rd order effects, like, not being able to go to work.
No, not even then. You shouldn't play games other people arrange in any way, except ie educationally, or for the pleasure of winning and nothing else. Not with money.
Prizes for challenges, maybe. Chess tournaments with a buy-in, that I respect because no luck involved, meaning no tipping the scales no cheating.
So the problem in gambling is when you just lost a big bet of money you didn't need and the only way to recover is putting up a tiny bit of money you do need. By nature gambling--and by the way it's sold--is designed to fuck with that fine line.
No, it doesn't mention it, and in fact the quote you copypasted does not contain any mention of it. The author is oblivious to the concept and is erroneously concluding it to be merely a psychological effect.
Marginal utility does let you view the pressing of the red button as an action a rational actor could take. That said, when you get into concepts like prospect theory in behavioral economics, there is definitely a psychological aspect as well.
Hear, hear! Tired of seeing these tirelessly dumb takes about expected value theory's supposed flaws or its psychological dimension.
Poker players make bets based on bank roll size and very good understanding of expected value. They either use Kelly criterion[1] or develop competing heuristics for value at risk[2].
Plenty of areas where psychology adds an interesting human dimension to decision making, but this and other risk-neutrality scenarios are not part of this category!
A useful formula is the Kelly Criterion [0]. I'm abusing the logic and probably going to apply this wrong, but...
I think this counts as a 24:1 bet (we notionally have $1 million, we can gamble to get another $24). The Kelly bet is 0.5 - 0.5/24 ~= 0.5. So we would want to put about half our wealth into this gamble and that implies it starts becoming attractive around the time we have $2 million to invest. Up till then we might take the gamble but we don't have enough money to really feel comfortable.
This is an interesting way of looking at it that I hadn't thought of--turning the Kelly criterion around to ask "what would my bankroll have to be to make this bet worth it" rather than "what size bet should I make given the bankroll I have."
It's worth noting that it can often make sense to be more conservative than the Kelly criterion would suggest, depending on your risk tolerance. So I would consider your calculation a lower bound.
That feels about right. Depends on age, future earning potential, objectives, circumstances, etc. But it seems like somewhere in the $3-5 million net worth range is where most people would seriously start considering rolling the dice but you could probably imagine it as low as $2m if someone were confident that $1m today wouldn't really be life-changing.
Yeah like why are the numbers so enormous? Game shows typically require signing up, a lot of waiting, backgrounders, castings even, they don't treat you great, and they tease money but never pay it. Like one time per game show, not per season, per show.
And it fucks people's reasoning about money up, you can't go into a raise negotiation with those game show numbers getting your personal finances and your leverage and expectations out of whack. Same as fashion magazines, just seeing a pretty shirt and seeing it costs $45000 for instance instantly fucks your notion of the value of a dollar.
The comment I'm replying to is showing a shade of grey, but this is a valid perspective, and reminds me of how Feynman attacked poorly worded, vague, or academic questions. I disagree that this reasoning is tangential to the question; it's an ill-defined question, and this comment's take pokes at that in the right place.
I don't know how that article is written without mentioning utility theory in economics and the concept of diminishing marginal utility of money and the risk aversion it implies. No need to even bring in behavioral economics.
Agreed. The article claims the scenario is “mathematics vs circumstance”, but really it’s really naive math vs math which takes into account the nonlinear utility.
On a tangent, and I’m just spitballing here, how is this for a business idea. I: have a PhD (which is not to say I’m smart, it is just to say I have been exposed to lots of facts that other people may not have been exposed to), and I have a bunch of ideas that may or may not be good ones, but I’m too risk-averse to act on any of them and start a business. You: have a bunch of money and are open to ideas. So you pay me, say, $100 bucks to just vomit my ideas out during a 15 minute phone call. Like a cheap loot box of ideas, most of which have a low probability of success, but there may be a nugget of gold in there.
The thing you learn from start-ups is that ideas are worth nothing. It's execution that matters.
But ... if you were an oracle (religious, not database) who sometimes foretold the future, would there be a marketplace for your ideas? In 2003 if you described a social network would that be valuable information? I tend to think no since there were social networks before Facebook but FB were lucky and executed very well.
Now I'm wondering what the expected value is if you were to auction this button pressing opportunity. Intuitively it seems to be over $1m (and less than $25m) so if I'm right then that's better than pressing the red button.
Assuming there's no risk to payouts being made/no fraud/etc. then presumably yes. It's probably related to hedges against commodity price increases/ foreign exchange fluctuation, etc. Not quite the same thing but somewhat similar in principle.
Also made me think of the parallel universe to the show “Silicon Valley” where Richard Hendricks sells Pied Piper to Gavin Belson for like $10 million and happily retires in La Veta, Colorado.
Nassim Taleb famously destroys those lines of reasoning in his books: game theory is unpractical for most people, because it almost always ignore variables that don't exist in a lab but are crucial IRL.
Comments have been explaining which ones already apply to this article, I'm not going to repeat them.
But there is an another example from Taleb that always makes me smile:
- If the other player tosses a coin and gets 9 tail in a row, what are the chances of getting tail on the next toss?
Honestly I'd hit the red button. I'd rather take a guaranteed payoff of my mortgage and all other debt, with plenty left over for a few neat toys, than chance walking away with nothing.
I’m also quite puzzled that nobody mentioned yet that if you were offered a chance like this in real life, it would likely be the only time in your life that you get a chance like that. Unless you get a repeat, or you are rich, it would be foolish to not press the red button.
I wouldn't, because I can likely pay off my mortgage without it and still have money left over for some toys. My mortgage isn't a big burden. But I can understand it for folks who can't save much due to a mortgage.
The problem with the scenario is that the disparity is so high: $1M vs expected value of $25M. 50% is high enough that for people like me, it's clearly a green button option.
But how about this:
Guaranteed $1M vs a 4% chance of winning $50M. Now the expected value is $2M - still a lot higher than $1M. But ... 4% chance? Suddenly the guaranteed $1M is a lot more attractive.
The utility function, and to some degree, attitude towards risk is going to differ a lot among individuals. And I imagine that among those reading this here, some are probably going "A million is a nice sum but it's not really life changing whereas $50 million would let me retire right now."
And you can scale the numbers up or down and at some point almost everyone will choose red or choose green respectively.
This is a nice concrete refutation of the fallacious reasoning in Pascal’s Mugging [0]. You can take this argument to its absurd conclusion by setting probability p arbitrarily small and the payout X arbitrarily large, such that p*X is arbitrarily greater than $1M, e.g. a 1/100000 chance of winning 100 trillion dollars.
There are 20 people in line ahead of you. Each one of them hits the green button, and you physically see that half of them made $25 million.
Would you not be tempted to hit the green button?
$1 million will make a big difference to me, but in many cities it's not enough to retire on - especially with children. While $25M isn't worth 25x more to me, it's certainly worth a heck of a lot more than $1M.
Yeah this is really weird. The author knows one "mathematical" measure, 'expected value', and calls that "mathematics". Ignoring the fact that mathematics have given us way more measures such as variance and the other moments that are just as important and "mathematical". Then microeconomics have given us 'utility' etc.
Finding one mathematical concept and saying "mathematics say this" is so strange.
Surprise surprise, people are not perfect emotionless economic units!
I don’t think it’s a good look for the “Director of Institutional Asset Management at Ritholtz Wealth Management” to call perfectly sensible decisions by people whose net worth is many orders of magnitude smaller than his “dumb” just because he can afford to pass up a guaranteed million.
The dumb is in the author thinking these are dumb decisions. $50M is nowhere near 50x as valuable as $1M, thus the green button is nowhere near 25x the value of the red button. For most people pushing red is the smart decision, not the dumb one.
I constantly run into situations where I spend money in ways that are financially non optimal, but socially good (in my mind).
An easy to understand example is, I believe I should pay more in taxes and everyone as wealthy as I am should too.
I rent an apartment, but I rent it out at the cost it takes to maintain it in good condition, because I think profiting off rent is unethical. This means I'm generally renting much much cheaper than local rents, and my tenants can therefore build savings.
No, I don't think you understand. I'm not charging below market rate, I'm charging at cost. Whatever it takes to maintain the building and provide utilities, etc. The space is worth, perhaps, $2400 a month. My last tenant paid $600 a month. She needed a place to stay for a year while she built up a down payment. Being able to stay with us meant she could save tens of thousands and she was able to embark on her own.
Could I have charged $1000 and pocketed a little profit? Of course. But it would have come directly at her ability to succeed. I think that's deeply unethical. I think it's morally repugnant to profit from housing.
But the point is that I believe we should all chip in more to help each other out. If you make, eg, $750k a year like I do an increase in taxes isn't really going to hurt your ability to live comfortably. I'm confident I could travel anywhere in the world, buy a second home, etc. I could still do those things if I payed more in taxes. Just... Not as often.
seriously flawed perspective. it's a 50% chance of nothing versus a 100% chance of a life-changing amount of money. if it was $1K:$25K or $100K:$2.5M then you'd take the risk.
It's a catchy headline, but the "decisions" used as examples, aren't really "dumb" under the complete set of facts.
Really, what this is about is that the typical mathematics used to discuss a certain type of financial decision (mostly things like investments) uses an incomplete model that doesn't consider appropriately the actual values involved -- for example, failing to consider the wildly nonlinear curve of the marginal value of one dollar.
It's just a bad title. Choosing a guaranteed 1M instead of a 50% chance at 25M isn't particularly dumb.
"Dumb" decisions might be playing the lottery, or spending a windfall instead of saving it. But even those dumb decisions have reasonable psychological underpinnings for the person doing them.
Another example of a 'dumb' decision: torpedoing a career to preserve relationships.
My prospects are abysmal, my savings insufficient and I'm still dysfunctional, but I'm better off than I would've been in many ways if I had not decided to give my loved ones (and my mental health) higher priority. I like to think I can make a comeback one day, but it's okay if I don't.
Admittedly, I'm not an expert at this stuff, but it seems like strictly using expected values to calculate optimum decisions can get you into some strange situations, like infinite expected value [1]. For some reason the author brings up lump sums vs annuities, which I don't think is at all comparable to betting (annuities from the US govt are guaranteed payments). That aside, a number of people have already mentioned Kelly criterion [2]. This strategy would tell you that you should take the guaranteed $1 million, but this is a long-run strategy. I personally would take the $1 million because it is guaranteed. I'm also not sure if relying on math for a one-off event like this makes sense.
Other commenters have mentioned marginal utility, but this article is basically explaining minimax [0] decision making - people (and chess AIs) tend to pick the option that minimizes worst case losses.
"Emotion may lead you to make bad financial decisions. For example, people who feel sad will pay more, sometimes four times more, for a consumer product than those who do not feel sad."
All the classic economic models for choice that I've seen fail to consider that the perception of not only risk but reward are BOTH nonlinear. IMO, this has been an Achilles heel of classic price and game theory. The rise of behavioral economics in recent decades would seem to agree with this iconoclysm.
If I need $1 million right now or else a loved one dies, then it doesn't matter how big the reward of a riskier alternative choice may be. I take the million NOW. If the additional reward is a victim of decreasing value as that offer rises, it's only rational for the decider to show diminished interest in choosing the greater reward (even if the marginal odds are only a tiny amount less likely).
Disregarding the reward curve of the individual is going to consistently misjudge economic choice and will surely be a poor basis for any economic model.
Not sure what you mean by this. Even in the most basic rational choice analysis, where the players are agents seeking to maximize a utility function, the "reward" can be nonlinear. I'm not familiar with any economic model that unintentionally restricts the agent's utility to be linear. Sometimes you assume that agents have a linear "utility of money", but everyone in economics knows that this is a taylor expansion around a small region where the linearity assumption is reasonable, and not an actual fact about human preferences.
For those who say they would press the red button ...
* Imagine the payout on the red button were not $1M but $100K or $50K or $10K. Is there any point as it diminishes toward zero that would make you switch buttons?
* Imagine the payout on the green button were not $50M but $100M or $500M or $1B. Is there any point as it increases toward infinity that would make you switch buttons?
For those who say they would press the green button ...
* Imagine the payout on the red button were not $1M but $2M or $5M or $10M. Is there any point as it increases toward $50M that would make you switch buttons?
* Imagine the odds on the green button were not 1:2 but 1:3 or 1:5 or 1:10. At what point, as the odds diminish, would you switch buttons?
You can certainly fiddle with numbers to the point where you can basically force a given person to go with green or go with red. In general, as you get into certain payouts that aren't a big deal for an individual they'll tend to go with higher expected value at least up to a point. But as the odds get longer, most people will tend to go with certainty as long as it's a reasonable amount.
I see this is being downvoted for some reason, but I have the same question. I get that if you keep replaying the game the expected value materialises over averages, but if you have one chance it doesn’t sound right that you should expect 25M if the two outcomes are zero or 50M? And that this is so true and obvious that it is dumb to take the 1M?
Only if you have reason to believe that the events are not independent. And anyway the way the original scenario is phrased suggests that it is a one off.
Given my current financial situation, 1 million would let me retire immediately. What I see when I look at those buttons are: 100% chance of being able to retire early vs 50% chance of being able to retire early.
Unless you're expecting to less than maybe 10 years to live, I wonder how anyone can assume $1 million would be enough to retire, given the uncertainty about the rate of inflation in the next few years.
This is similar to the choice between a salary (quite predictable) and a startup (maybe a lot, might just as well be zero). Or, in a company, between doing consulting or in-house product development.
The discussion here seems to be whether, if we can capture all of the relevant details, a certain person is making a rationally optimal decision. Taking this to its logical conclusion we're fitting math to a process of decision making and adjudicating which criteria are considered rational and which are not. Sure there is mathematics involved here, but it reads a lot more like a question of who is our isn't allowed agency, in this case in their economic and financial decisions.
I still don’t get it how hitting the green button (50% at 50 mullion) is the “rational” choice, it isn’t. 1 million in your pocket, no matter what, is exponentially and life-changing (for the majority of us) better than a 50% of getting nothing. Maybe if the value behind the red button would have been smaller (let’s say $1000 or even $100) then things would have been different, but, again an $1 million in one’s pocket no matter what is life-changing for most of us.
You can witness people buying lotery scratch cards every day in the UK and wonder why people are so dumb given the odds of actually winning a big prize.
But then bear in mind that this person maybe has a big bill to pay and only £5 to their name, do they keep the £5 knowing that it isn't going make any difference or take a wild chance that will?
I once was down to my last dollar, and I bought my one and only lottery ticket ever. What could it hurt? I couldn't do anything to help myself with one dollar. I lost.
I buy scratchy lottos sometimes (the cheap 1-2 dollar ones) because I find the act of scratching them to be tactilely pleasing. $1-2 for a couple minutes of fun and the possibility of winning enough to get myself some candy for 'free' is nice.
If you think about it, isn't buying insurance pretty similar (especially if you pay extra for really rare insurance policies, like lightning strike insurance).
You're essentially making a bet with your insurance company that xxxx will happen. Just like with the lottery, the expected monetary value of insurance is always negative (it has to be, otherwise the insurance company won't make money), but the utility value of that insurance is different for each person.
E.g. for me, insurance on a phone doesn't make sense, since I easily buy a another cheap phone if mine breaks. But for somebody with less money, those few hundred dollars might have a much higher utility.
Those 5 pounds also give you the opportunity to just dream a little once in a while, as in “what would I do with all the money if I really win?”, it’s like a drug, takes you out of your not-ok (from a financial perspective) life.
Indeed, I have an ongoing subscription with zero expectation of winning but I do sometimes enjoy thinking 'what if'.
Considering how much money I throw away on streaming subscriptions I barely use, books I never get beyond the first chapter of, food I buy that ends up in the trash, etc, it represents quite good value for money.
What people say they will do on a hypothetical is not the same thing as what they will do if actually faced with the decision.
The way I've been able to deal with this personally is by thinking "what would I do if this was Monopoly money?" and then reconcile that with my emotional decision.
If I was flat broke, living on the street, or in debt even, I would find investors to pay, say 5 @ 200k each, for me to press the green button and reward them 1mm each in case of payout.
Isn't the best solution here to find a wealthy investor and sell him the option to press the green button priced at $20M? You get $20M, and they get an instrument with an EV of $25M at the cost of $20M.
You would need an auction to get a good payout. If you only find one smart wealthy investor, they can offer you a take-it-or-leave-it $2 million.
Your choice then becomes take $1 million from the red button, or sell the green button for $2 million.
I imagine most people would take the 2 million, especially because I suspect most people are poor at negotiating when life-changing amounts of money are involved. I have seen many naive people do silly house trades (or missing out on good trades).
If you could get 10 friends to agree on such an arrangement would you still have one press the red button to ensure everyone at least gets something? Or take the 1/1024 chance of getting nothing at all in return for the likelihood of everyone getting 25 million?
tl;dr: doubling winnings on each throw of heads and paying out on the first tail is a game with expected winnings diverging to positive infinity, yet probably no one would pay more than a few bucks to enter
The rational behaviour is to make everyone press the green button and then give away a million dollars to anyone who didn't get a prize but the obvious problem is that no such thing happens. Instead of cooperating some people insist on getting the full 50 million dollars as if they deserve it and were destined to get the money while the plebs who didn't get anything also deserve to stay poor.
In other words, the problem is that humans are cruel to each other and peace of mind vs other cruel people is worth more than a higher reward.
Co-op is definitelly best outcome. Find 20 people to each press green and divide equally total amount. U r then in worst case scenario better then taking 1m
> Expected value (also known as EV, expectation, average, or mean value) is a long-run average value of random variables.
If you can only press a button once - you should take the guaranteed money in almost all circumstances (assuming you have finances that look like most Americans - if you're already a millionaire... do what you want, this game doesn't matter much to you).
Basically - This is a dire misunderstanding of how statistics works in general. The population at large might be better off pressing the 50% at 50 million button (because then you are running this game many times and you will likely achieve the expected value) - but as an individual, who can only roll the dice once, you are much better off just taking the immediate and guaranteed win.
And that's not even accounting for the drop off in marginal value of each dollar as you accumulate them - that first million is far more impactful than the next 49.