6 ms·

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-r

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.

jstanley 4y ago
> 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).
- Bootvis 4y ago
  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.
  - orlp 4y ago
    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.
    - gweinberg 4y ago
      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.
  - ghaff 4y ago
    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.
  - anigbrowl 4y ago
    if you play the game 20 times you’re almost certain to win 50 million You can get any result you want if you just rewrite the problem conditions ◔_◔
    - Bootvis 4y ago
      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.
  - jstanley 4y ago
    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).
- UncleMeat 4y ago
  Not so. It can be throughly reasonable to make a choice that has lower expected value but has a distribution that fits your needs more closely.
  - CrazyStat 4y ago
    If one option doesn't fit your needs then it has lower utility.
    - UncleMeat 4y ago
      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.
      - CrazyStat 4y ago
        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.
BeetleB 4y ago
> 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.
- phao 4y ago
  > your life will have plenty of such games What are you talking about? Which life will have plenty of such games? In what way is that true?
  - thfuran 4y ago
    In the way that these are analogies for actual situations, not just pure whiteroom thought experiments.
    - phao 4y ago
      I suspected that much (that it was an analogy for some kind of actual situation). 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?
      - UKR_anon 4y ago
        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.
drewg123 4y ago
Why not form a "company" with 10 friends, and pool the winnings?
yjk 4y ago
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?
- lupire 4y ago
  > Why "Variance" and "risk tolerance" There is only one of me, not a Large Number
mrwh 4y ago
Agreed. 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).
- rblatz 4y ago
  Would you stop working? Or would you take a break until you found something you truly wanted to work on?
  - mrwh 4y ago
    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.
    - djmips 4y ago
      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!
  - xboxnolifes 4y ago
    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.
UncleMeat 4y ago
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.
- whatshisface 4y ago
  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.
  - fxtentacle 4y ago
    Great idea. So you weight x10 the EV as +1 point.
- petercooper 4y ago
  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!
  - chmod600 4y ago
    Depends a lot on what the back roads are like, though. The freeway can be more reliable a lot of the time.
    - incone123 4y ago
      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.
anigbrowl 4y ago
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.
- ghaff 4y ago
  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.
- infinite8s 4y ago
  That just does to show that expected value is not the rational metric, but expected utility is.
seanmcdirmid 4y ago
Unless you already have a million, then 50% chance at 50 million can make more personal sense than another million.
mgraczyk 4y ago
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.
- danielmarkbruce 4y ago
  Exactly. 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.
  - ghaff 4y ago
    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.
  - Kranar 4y ago
    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.
    - withinboredom 4y ago
      “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.
abnry 4y ago
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.
DoctorOetker 4y ago
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.

jstanley 4y ago

> 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).

Bootvis 4y ago
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.
- orlp 4y ago
  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.
  - gweinberg 4y ago
    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.
- ghaff 4y ago
  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.
- anigbrowl 4y ago
  if you play the game 20 times you’re almost certain to win 50 million You can get any result you want if you just rewrite the problem conditions ◔_◔
  - Bootvis 4y ago
    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.
- jstanley 4y ago
  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).
UncleMeat 4y ago
Not so. It can be throughly reasonable to make a choice that has lower expected value but has a distribution that fits your needs more closely.
- CrazyStat 4y ago
  If one option doesn't fit your needs then it has lower utility.
  - UncleMeat 4y ago
    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.
    - CrazyStat 4y ago
      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.

BeetleB 4y ago

> 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.

phao 4y ago
> your life will have plenty of such games What are you talking about? Which life will have plenty of such games? In what way is that true?
- thfuran 4y ago
  In the way that these are analogies for actual situations, not just pure whiteroom thought experiments.
  - phao 4y ago
    I suspected that much (that it was an analogy for some kind of actual situation). 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?
    - UKR_anon 4y ago
      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.

drewg123 4y ago

Why not form a "company" with 10 friends, and pool the winnings?

yjk 4y ago

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?

lupire 4y ago
> Why "Variance" and "risk tolerance" There is only one of me, not a Large Number

mrwh 4y ago

Agreed. 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).

rblatz 4y ago
Would you stop working? Or would you take a break until you found something you truly wanted to work on?
- mrwh 4y ago
  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.
  - djmips 4y ago
    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!
- xboxnolifes 4y ago
  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.

UncleMeat 4y ago

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.

whatshisface 4y ago
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.
- fxtentacle 4y ago
  Great idea. So you weight x10 the EV as +1 point.
petercooper 4y ago
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!
- chmod600 4y ago
  Depends a lot on what the back roads are like, though. The freeway can be more reliable a lot of the time.
  - incone123 4y ago
    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.

anigbrowl 4y ago

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.

ghaff 4y ago
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.
infinite8s 4y ago
That just does to show that expected value is not the rational metric, but expected utility is.

seanmcdirmid 4y ago

Unless you already have a million, then 50% chance at 50 million can make more personal sense than another million.

mgraczyk 4y ago

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.

danielmarkbruce 4y ago
Exactly. 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.
- ghaff 4y ago
  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.
- Kranar 4y ago
  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.
  - withinboredom 4y ago
    “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.

abnry 4y ago

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.

DoctorOetker 4y ago

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.