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.
It's how it is.
So there's that.
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.
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).
You can get any result you want if you just rewrite the problem conditions ◔_◔
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.
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).
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.
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.
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?
I’m also reminded that “people are happier when a choice is made for them” or some other thing I’ve heard thrown around.
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.
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.
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.
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.
What are you talking about? Which life will have plenty of such games? In what way is that true?
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.
But what situation? How is it that a person's life has many of these chances in large enough volumes to make expected values worth it?
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.
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.
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?
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.
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%'.
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.
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.
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?
"Variance" and "risk tolerance"
There is only one of me, not a Large Number
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.
> 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.
> 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.