Reference Frames for Expected Value
Puzzle 1: George mortgages his house to invest in lottery tickets. He wins and becomes a millionaire. Did he make a good choice?
Puzzle 2: The U.S. president questions if he should bluff a nuclear war or concede to the USSR. He bluffs and it just barely works. Although there were several close calls for nuclear catastrophe, everything works out ok. Was this ethical?
One interpretation of consequentialism is that decisions that produce good outcomes are good decisions, rather than decisions that produce good expected outcomes.12 One would be ethical if their actions end up with positive outcomes, disregarding the intentions of those actions. For instance, a terrorist who accidentally foils an otherwise catastrophic terrorist plan would have done a very ‘morally good’ action.3 This general view seems to be surprisingly common.4
This seems intuitively strange to many, it definitely is to me. Instead, ‘expected value’ seems to be a better way of both making decisions and judging the decisions made by others. However, while ‘expected value’ can be useful for individual decision making, I make the case that it is very difficult to use to judge other people’s decisions in a meaningful way.5 This is because ‘expected value’ is typically defined in reference to a specific set of information and intelligence rather than an objective truth about the world.
Two questions to help guide this:
- Should we judge previous actions based on ‘expected’ or ‘actual’ value?
- Should we make future decisions to optimize ‘expected’ or ‘actual’ value?
I believe these are in a sense quite simple, but require some consideration to definitions.6
Optimizing Future Decisions: Actual vs. Expected Value
The second question is the easiest of the two, so I’ll begin with that one. The simple answer is that this is a question of defining ‘expected value’. Once we do so the question kind of goes away.
There is nothing fundamentally different between expected value and actual value. A more fair comparison may be ‘expected value from the perspective of the decision maker’ with ‘expected value from a later, more accurate prospective’.
Expected value converges on actual value with lots of information. Said differently, actual value is expected value with complete information.
In the case of an individual purchasing lottery tickets successfully (Puzzle 1), the ‘actual value’ is still not exact from our point of view. While we may know how much money was won, or what profit was made. We also don’t know what the counterfactual would have been. It is still theoretically possible that in the worlds where George wouldn’t have purchased the lottery tickets, he would have been substantially better off. While the fact that we have imperfect information doesn’t matter too much, I think it demonstrates that presenting a description of the outcome as ‘actual value’ is incomplete. ‘Actual value’ exists only theoretically, even after the fact.7
So this question becomes, then ‘should one make a decision to optimize value using the information and knowledge available to them, or using perfect knowledge and information?’ Obviously, in this case, ‘perfect knowledge’ is inaccessible to them (or the ‘expected value’ and ‘actual value’ would be the same exact thing). I believe it should be quite apparent that in this case, the best one can do (and should do) is make the best decision using their available information.
This question is similar to asking ‘should you drive your car as quickly as your car can drive, or much faster than your car can drive?’ Obviously you may like to drive faster, but that’s by definition not an option. Another question: ‘should you do well in life or should you become an all-powerful dragon king?’
Judging Previous Decisions: Actual vs. Expected Value
Judging previous decisions can get tricky.
Let’s study the lottery example again. A person purchases a lottery ticket and wins. For simplicity, let’s say the decision to purchase the ticket was done only to optimize money.
The question is, what is the expected value of purchasing the lottery ticket? How does this change depending on information and knowledge?
In general purchasing a lottery ticket can be expected to be a net loss in earnings, and thus a bad decision. However, if one was sure they would win, it would be a pretty good idea. Given the knowledge that the player won, the player made a good decision. Winning the lottery clearly is better than not playing once.
More interesting is considering the limitation not in information about the outcome but about knowledge of probability. Say the player thought that they were likely win the lottery, that it was a good purchase. This may seem insane to someone familiar with probability and the lottery system, but not everyone is familiar with these things.
From the point of view of the player, the lottery ticket purchase had net-positive utility. From the point of view of a person with knowledge of the lottery and/or statistics, the purchase had net-negative utility. From the point of view of any of these two groups, after they know that the lottery will be a success, it was a net positive decision.
| No Knowledge of Outcome | Knowledge of Outcome | |
|---|---|---|
| ‘Intelligent’ Person with Knowledge of Probability | Negative | Positive |
| Lottery Player | Positive | Positive |
Expected Value of purchasing a Lottery Ticket from different Reference Points
To make things a bit more interesting, imagine that there’s a genius out there with a computer simulation of our exact universe. This person can tell which lottery ticket will win in advance because they can run the simulations. To this ‘genius’ it’s obvious that the purchase is a net-positive outcome.
| No Knowledge of Outcome | Knowledge of Outcome | |
|---|---|---|
| Genius | Positive | Positive |
| ‘Intelligent’ Person with Knowledge of Probability | Negative | Positive |
| Lottery Player | Positive | Positive |
Expected Value of purchasing a Lottery Ticket from different Reference Points
So what is the expected value of purchasing the lottery ticket? The answer is that the ‘expected value’ is completely dependent on the ‘reference frame’, or a specific set of information and intelligence. From the reference frame of the ‘intelligent person’ this was low in expected value, so was a bad decision. From that of the genius, it was a good decision. And from the player, a good decision.
Judging
So how do we judge this poor (well, soon rich) lottery player? They made a good decision respective to the results, respective to the genius, and compared to their own knowledge. Should we say ‘oh, this person should have had slightly more knowledge, but not too much knowledge, and thus they made a bad choice’? What does that even mean?
Perhaps we could judge the player for not reading into lottery facts before playing. Wasn’t it irresponsible for falling for such a simple fallacy? Or perhaps the person was ‘lazy’ to not learn probability in the first place.
Well, things like these seem like intuitions to me. We may have the intuitions to us that the lottery is a poor choice. We may find facts to prove these intuitions accurate. But the gambler my not have these intuitions. It seems unfair to consider any intuitions ‘obvious’ to those who do not share them.
One might also say that the gambler probably knew it was a bad idea, but let his or her ‘inner irrationalities’ control the decision process. Perhaps they were trying to take an ‘easy way out’ of some sort. However, these seem quite judgmental as well. If a person experiences strong emotional responses; fear, anger, laziness; those inner struggles would change their expected value calculation. It might be a really bad, heuristically-driven ‘calculation’, but it would be the best they would have at that time.
Free Will Bounded Expected Value
We are getting to the question of free will and determinism. After all, if there is any sort of free will, perhaps we have the ability to make decisions that are sub-optimal by our expected value functions. Perhaps we commonly do so (else it wouldn’t be much in the sense of ‘free’ will.)
This would be interesting because it would imply an ‘expected result’ that the person should have calculated, even if they didn’t actually do so. We need to understand the person’s actions and understanding, not in terms of what we know, or what they knew, but what they should have figured out given their knowledge.
This would require a very well specified Free Will Boundary of some sort. A line around a few thought processes, parts of the brain, and resource constraints, which could produce a thereby optimal expected result calculation. Anything less than this ‘optimal given Free Will Boundary’ expected value calculation would be fair game for judging.
Conclusion: Should we Even Judge People or Decisions Anyway?
So, deciding to make future decisions based on expected value seems reasonable. The main question in this essay, the harder question, is if we can judge previous decisions based on their respective expected values, and how to possibly come up with the relevant expected values to do so.
I think that we naturally judge people. We have old and modern heroes and villains. Judging people is simply something that humans do. However, I believe that on close inspection this is very challenging if not impossible to do reasonably and precisely.
Perhaps we should attempt to stop placing so much emphasis on individualism and just try to do the best we can while not judging others nor other decisions much. Considerations of judging may be interesting, but the main take away may be the complexity itself, indicated that judgements are very subjective and incredibly messy.
That said, it can still be useful to analyze previous decisions or individuals. That seems like one of the best ways to update our priors of the world. We just need to remember not to treat it personally.
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Dorsey, Dale. “Consequentialism, Metaphysical Realism, and the Argument from Cluelessness.” University of Kansas Department of Philosophy http://people.ku.edu/~ddorsey/cluelessness.pdf ↩
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Sinhababu, Neiladri. “Moral Luck.” Tedx Presentation http://www.youtube.com/watch?v=RQ7j7TD8PWc ↩
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This is assuming the terrorists are trying to produce ‘disutility’ or a value separate from ‘utility’. I feel like from their perspective, maximizing an intrinsic value dissimilar from our notion of utility would be maximizing ‘expected value’. But analyzing the morality of people with alternative value systems is a very different matter. ↩
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These people tend not to like consequentialism much. ↩
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I don’t want to impose what I deem to be a false individualistic appeal, so consider this to mean that one would have a difficult time judging anyone at any time except for their spontaneous consciousness. ↩
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I bring them up because they are what I considered and have talked to others about before understanding what makes them frustrating to answer. Basically, they are nice starting points for getting towards answering the questions that were meant to be asked instead. ↩
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This is true for essentially all physical activities. Thought experiments or very simple simulations may be exempt. ↩
Why I haven't signed up for cryonics
(OR)
How I'm now on the fence about whether to sign up for cryonics
I'm not currently signed up for cryonics. In my social circle, that makes me a bit of an oddity. I disagree with Eliezer Yudkowsky; heaven forbid.
My true rejection is that I don't feel a visceral urge to sign up. When I query my brain on why, what I get is that I don't feel that upset about me personally dying. It would suck, sure. It would suck a lot. But it wouldn't suck infinitely. I've seen a lot of people die. It's sad and wasteful and upsetting, but not like a civilization collapsing. It's neutral from a point of pleasure vs suffering for the dead person, and negative for the family, but they cope with it and find a bit of meaning and move on.
(I'm desensitized. I have to be, to stay sane in a job where I watch people die on a day to day basis. This is a bias; I'm just not convinced that it's a bias in a negative direction.)
I think the deeper cause behind my rejection may be that I don't have enough to protect. Individuals may be unique, but as an individual, I'm fairly replaceable. All the things I'm currently doing can and are being done by other people. I'm not the sole support person in anyone's life, and if I were, I would be trying really, really hard to fix the situation. Part of me is convinced that wanting to personally survive and thinking that I deserve to is selfish and un-virtuous or something. (EDIT: or that it's non-altruistic to value my life above the amount Givewell thinks is reasonable to save a life–about $5,000. My revealed preference is that I obviously value my life more than this.)
However, I don't think cryonics is wrong, or bad. It has obvious upsides, like being the only chance an average citizen has right now to do something that might lead to them not permanently dying. I say "average citizen" because people working on biological life extension and immortality research are arguably doing something about not dying.
When queried, my brain tells me that it's doing an expected-value calculation and the expected value of cryonics to me is is too low to justify the costs; it's unlikely to succeed and the only reason some people have positive expected value for it is that they're multiplying that tiny number by the huge, huge number that they place on the value of my life. And my number doesn't feel big enough to outweigh those odds at that price.
Putting some numbers in that
If my brain thinks this is a matter of expected-value calculations, I ought to do one. With actual numbers, even if they're made-up, and actual multiplication.
So: my death feels bad, but not infinitely bad. Obvious thing to do: assign a monetary value. Through a variety of helpful thought experiments (how much would I pay to cure a fatal illness if I were the only person in the world with it and research wouldn't help anyone but me and I could otherwise donate the money to EA charities; does the awesomeness of 3 million dewormings outway the suckiness of my death; is my death more or less sucky than the destruction of a high-end MRI machine), I've converged on a subjective value for my life of about $1 million. Like, give or take a lot.
Cryonics feels unlikely to work for me. I think the basic principle is sound, but if someone were to tell me that cryonics had been shown to work for a human, I would be surprised. That's not a number, though, so I took the final result of Steve Harris' calculations here (inspired by the Sagan-Drake equation). His optimistic number is a 0.15 chance of success, or 1 in 7; his pessimistic number is 0.0023, or less than 1/400. My brain thinks 15% is too high and 0.23% sounds reasonable, but I'll use his numbers for upper and lower bounds.
I started out trying to calculate the expected cost by some convoluted method where I was going to estimate my expected chance of dying each year and repeatedly subtract it from one and multiply by the amount I'd pay each year to calculate how much I could expect pay in total. Benquo pointed out to me that calculation like this are usually done using perpetuities, or PV calculations, so I made one in Excel and plugged in some numbers, approximating the Alcor annual membership fee as $600. Assuming my own discount rate is somewhere between 2% and 5%, I ran two calculations with those numbers. For 2%, the total expected, time-discounted cost would be $30,000; for a 5% discount rate, $12,000.
Excel also lets you do calculations on perpetuities that aren't perpetual, so I plugged in 62 years, the time by which I'll have a 50% chance of dying according to this actuarial table. It didn't change the final results much; $11,417 for a 5% discount rate and $21,000 for the 2% discount rate.
That's not including the life insurance payout you need to pay for the actual freezing. So, life insurance premiums. Benquo's plan is five years of $2200 a year and then nothing from then on, which apparently isn't uncommon among plans for young healthy people. I could probably get something as good or better; I'm younger. So, $11,00 for total life insurance premiums. If I went with permanent annual payment, I could do a perpetuity calculation instead.
In short: around $40,000 total, rounding up.
What's my final number?
There are two numbers I can output. When I started this article, one of them seemed like the obvious end product, so I calculated that. When I went back to finish this article days later, I walked through all the calculations again while writing the actual paragraphs, did what seemed obvious, ended up with a different number, and realized I'd calculated a different thing. So I'm not sure which one is right, although I suspect they're symmetrical.
If I multiply the value of my life by the success chance of cryonics, I get a number that represents (I think) the monetary value of cryonics to me, given my factual beliefs and values. It would go up if the value of my life to me went up, or if the chances of cryonics succeeding went up. I can compare it directly to the actual cost of cryonics.
I take $1 million and plug in either 0.15 or 0.00023, and I get $150,000 as an upper bound and $2300 as a lower bound, to compare to a total cost somewhere in the ballpark of $40,000.
If I take the price of cryonics and divide it by the chance of success (because if I sign up, I'm optimistically paying for 100 worlds of which I survive in 15, or pessimistically paying for 10,000 worlds in which I survive in 23), I get the total expected cost per my life being saved, which I can compare to the figure I place on the value of my life. It goes down if the cost of cryonics goes down or the chances of success go up.
I plug in my numbers and get a lower bound of $267,000 and an upper bound of 17 million.
In both those cases, the optimistic success estimates make it seem worthwhile and the pessimistic success estimates don't, and my personal estimate of cryonics succeeding falls closer to pessimism. But it's close. It's a lot closer than I thought it would be.
Updating somewhat in favour that I'll end up signed up for cryonics.
Fine-tuning and next steps
I could get better numbers for the value of my life to me. It's kind of squicky to think about, but that's a bad reason. I could ask other people about their numbers and compare what they're accomplishing in their lives to my own life. I could do more thought experiments to better acquaint my brain with how much value $1 million actually is, because scope insensitivity. I could do upper and lower bounds.
I could include the cost of organizations cheaper than Alcor as a lower bound; the info is all here and the calculation wouldn't be too nasty but I have work in 7 hours and need to get to bed.
I could do my own version of the cryonics success equation, plugging in my own estimates. (Although I suspect this data is less informed and less valuable than what's already there).
I could ask what other people think. Thus, write this post.
VNM expected utility theory: uses, abuses, and interpretation
When interpreted convservatively, the von Neumann-Morgenstern rationality axioms and utility theorem are an indispensible tool for the normative study of rationality, deserving of many thought experiments and attentive decision theory. It's one more reason I'm glad to be born after the 1940s. Yet there is apprehension about its validity, aside from merely confusing it with Bentham utilitarianism (as highlighted by Matt Simpson). I want to describe not only what VNM utility is really meant for, but a contextual reinterpretation of its meaning, so that it may hopefully be used more frequently, confidently, and appropriately.
- Preliminary discussion and precautions
- Sharing decision utility is sharing power, not welfare
- Contextual Strength (CS) of preferences, and VNM-preference as "strong" preference
- Hausner (lexicographic) decision utility
- The independence axiom isn't bad either
- Application to earlier LessWrong discussions of utility
1. Preliminary discussion and precautions
The idea of John von Neumann and Oskar Mogernstern is that, if you behave a certain way, then it turns out you're maximizing the expected value of a particular function. Very cool! And their description of "a certain way" is very compelling: a list of four, reasonable-seeming axioms. If you haven't already, check out the Von Neumann-Morgenstern utility theorem, a mathematical result which makes their claim rigorous, and true.
VNM utility is a decision utility, in that it aims to characterize the decision-making of a rational agent. One great feature is that it implicitly accounts for risk aversion: not risking $100 for a 10% chance to win $1000 and 90% chance to win $0 just means that for you, utility($100) > 10%utility($1000) + 90%utility($0).
But as the Wikipedia article explains nicely, VNM utility is:
- not designed to predict the behavior of "irrational" individuals (like real people in a real economy);
- not designed to characterize well-being, but to characterize decisions;
- not designed to measure the value of items, but the value of outcomes;
- only defined up to a scalar multiple and additive constant (acting with utility function U(X) is the same as acting with a·U(X)+b, if a>0);
- not designed to be added up or compared between a number of individuals;
- not something that can be "sacrificed" in favor of others in a meaningful way.
[ETA] Additionally, in the VNM theorem the probabilities are understood to be known to the agent as they are presented, and to come from a source of randomness whose outcomes are not significant to the agent. Without these assumptions, its proof doesn't work.
Because of (4), one often considers marginal utilities of the form U(X)-U(Y), to cancel the ambiguity in the additive constant b. This is totally legitimate, and faithful to the mathematical conception of VNM utility.
Because of (5), people often "normalize" VNM utility to eliminate ambiguity in both constants, so that utilities are unique numbers that can be added accross multiple agents. One way is to declare that every person in some situation values $1 at 1 utilon (a fictional unit of measure of utility), and $0 at 0. I think a more meaningful and applicable normalization is to fix mean and variance with respect to certain outcomes (next section).
Because of (6), characterizing the altruism of a VNM-rational agent by how he sacrifices his own VNM utility is the wrong approach. Indeed, such a sacrifice is a contradiction. Kahneman suggests1, and I agree, that something else should be added or substracted to determine the total, comparative, or average well-being of individuals. I'd call it "welfare", to avoid confusing it with VNM utility. Kahneman calls it E-utility, for "experienced utility", a connotation I'll avoid. Intuitively, this is certainly something you could sacrifice for others, or have more of compared to others. True, a given person's VNM utility is likely highly correlated with her personal "welfare", but I wouldn't consider it an accurate approximation.
So if not collective welfare, then what could cross-agent comparisons or sums of VNM utilities indicate? Well, they're meant to characterize decisions, so one meaningful application is to collective decision-making:
Post retracted: If you follow expected utility, expect to be money-pumped
This post has been retracted because it is in error. Trying to shore it up just involved a variant of the St Petersburg Paradox and a small point on pricing contracts that is not enough to make a proper blog post.
I apologise.
Edit: Some people have asked that I keep the original up to illustrate the confusion I was under. I unfortunately don't have a copy, but I'll try and recreate the idea, and illustrate where I went wrong.
The original idea was that if I were to offer you a contract L that gained £1 with 50% probability or £2 with 50% probability, then if your utility function wasn't linear in money, you would generally value L at having a value other that £1.50. Then I could sell or buy large amounts of these contracts from you at your stated price, and use the law of large number to ensure that I valued each contract at £1.50, thus making a certain profit.
The first flaw consisted in the case where your utility is concave in cash ("risk averse"). In that case, I can't buy L from you unless you already have L. And each time I buy it from you, the mean quantity of cash you have goes down, but your utility goes up, since you do not like the uncertainty inherent in L. So I get richer, but you get more utility, and once you've sold all L's you have, I cannot make anything more out of you.
If your utility is convex in cash ("risk loving"), then I can sell you L forever, at more than £1.50. And your money will generally go down, as I drain it from you. However, though the median amount of cash you have goes down, your utility goes up, since you get a chance - however tiny - of huge amounts of cash, and the utility generated by this sum swamps the fact you are most likely ending up with nothing. If I could go on forever, then I can drain you entirely, as this is a biased random walk on a one-dimensional axis. But I would need infinite ressources to do this.
The major error was to reason like an investor, rather than a utility maximiser. Investors are very interested in putting prices on objects. And if you assign the wrong price to L while investing, someone will take advantage of you and arbitrage you. I might return to this in a subsequent post; but the issue is that even if your utility is concave or convex in money, you would put a price of £1.50 on L if L were an easily traded commodity with a lot of investors also pricing it at £1.50.
Expected utility without the independence axiom
John von Neumann and Oskar Morgenstern developed a system of four axioms that they claimed any rational decision maker must follow. The major consequence of these axioms is that when faced with a decision, you should always act solely to increase your expected utility. All four axioms have been attacked at various times and from various directions; but three of them are very solid. The fourth - independence - is the most controversial.
To understand the axioms, let A, B and C be lotteries - processes that result in different outcomes, positive or negative, with a certain probability of each. For 0<p<1, the mixed lottery pA + (1-p)B implies that you have p chances of being in lottery A, and (1-p) chances of being in lottery B. Then writing A>B means that you prefer lottery A to lottery B, A<B is the reverse and A=B means that you are indifferent between the two. Then the von Neumann-Morgenstern axioms are:
- (Completeness) For every A and B either A<B, A>B or A=B.
- (Transitivity) For every A, B and C with A>B and B>C, then A>C.
- (Continuity) For every A>B>C then there exist a probability p with B=pA + (1-p)C.
- (Independence) For every A, B and C with A>B, and for every 0<t≤1, then tA + (1-t)C > tB + (1-t)C.
In this post, I'll try and prove that even without the Independence axiom, you should continue to use expected utility in most situations. This requires some mild extra conditions, of course. The problem is that although these conditions are considerably weaker than Independence, they are harder to phrase. So please bear with me here.
The whole insight in this post rests on the fact that a lottery that has 99.999% chance of giving you £1 is very close to being a lottery that gives you £1 with certainty. I want to express this fact by looking at the narrowness of the probability distribution, using the standard deviation. However, this narrowness is not an intrinsic property of the distribution, but of our utility function. Even in the example above, if I decide that receiving £1 gives me a utility of one, while receiving zero gives me a utility of minus ten billion, then I no longer have a narrow distribution, but a wide one. So, unlike the traditional set-up, we have to assume a utility function as being given. Once this is chosen, this allows us to talk about the mean and standard deviation of a lottery.
Then if you define c(μ) as the lottery giving you a certain return of μ, you can use the following axiom instead of independence:
- (Standard deviation bound) For all ε>0, there exists a δ>0 such that for all μ>0, then any lottery B with mean μ and standard deviation less that μδ has B>c((1-ε)μ).
This seems complicated, but all that it says, in mathematical terms, is that if we have a probability distribution that is "narrow enough" around its mean μ, then we should value it are being very close to a certain return of μ. The narrowness is expressed in terms of its standard deviation - a lottery with zero SD is a guaranteed return of μ, and as the SD gets larger, the distribution gets wider, and the chances of getting values far away from μ increases. So risk, in other words, scales (approximately) with the SD.
Extreme risks: when not to use expected utility
Would you prefer a 50% chance of gaining €10, one chance in a million off gaining €5 million, or a guaranteed €5? The standard position on Less Wrong is that the answer depends solely on the difference between cash and utility. If your utility scales less-than-linearly with money, you are risk averse and should choose the last option; if it scales more-than-linearly, you are risk-loving and should choose the second one. If we replaced €’s with utils in the example above, then it would simply be irrational to prefer one option over the others.
There are mathematical proofs of that result, but there are also strong intuitive arguments for it. What’s the best way of seeing this? Imagine that X1 and X2 were two probability distributions, with mean u1 and u2 and variances v1 and v2. If the two distributions are independent, then the sum X1 + X2 has mean u1 + u2, and variance v1 + v2.
Now if we multiply the returns of any distribution by a constant r, the mean scales by r and variance scales by r2. Consequently if we have n probability distributions X1, X2, ... , Xn representing n equally expensive investments, the expected average return is (Σni=1 ui)/n, while the variance of this average is (Σni=1 vi)/n2. If the vn are bounded, then once we make n large enough, that variance must tend to zero. So if you have many investments, your averaged actual returns will be, with high probability, very close to your expected returns.
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