He's misunderstanding. For it to "make sense" for you to pay $9,000 for a 1% shot of winning a million, you don't need to have to be able to make that same wager many times. You have to look at the long term.
Life will present you with many different wagers with positive expectation, but it often won't present you with the same one multiple times. Maybe it will give you that shot, then the next time give you a +EV chance to invest in the startup, then later give you the opportunity to play the St Petersburg game for $2, etc. Keep taking the +EV wagers over your lifetime and you'll end up way ahead, even if you never take the same exact one twice.
You don't need a good sample size for an individual wager to be a good idea, just for all wagers combined. All of that assumes, of course, that taking any one wager doesn't prevent you from many others. Taking the $9k wager if you have $9k to your name will. Taking it if you have $10m to your name won't and therefore is a no-brainer.
*Edited to account for Kelly Criterion. Thanks Aston
Not overspending out of your bankroll to take a wager, +EV or not, is pretty important. You probably wanna stick pretty near to (or beneath) the Kelly Criterion (http://en.wikipedia.org/wiki/Kelly_criterion). In the case of Aaron's game, you should only be willing to put down $9000 for that game if you have on the order of $8.9 million to gamble with.
How far beneath the Kelly criterion depends on the game. In gambling scenarios you can estimate the risk relatively easily. In investment scenarios, it's much harder, so there's an additional 'risk model uncertainty,' representing whether your risk model is off. In fact most investors famous for using the Kelly Criterion (Warren Buffett, Bill Gross - on Bonds) stick to the domains that they know.
It's not a no-brainer. Does "winning one million" means that you have to do absolutely no work, spend no time at all and the money is tax-free? The answer most likely will be no. Taxes alone make the expected value negative.
The more important problem is that many times it's hard to determine whether a "wager" has a positive or negative expectation, compounded by the fact that the positive value has to be utility to you rather than just money.
I think you're missing the point of the exercise. It's like you're assigned a physics problem about point masses on frictionless surfaces and you answer the question by saying it's irrelevant in the real world because masses are distributed over volumes and surfaces are never frictionless.
Yes, of course you have to factor in taxes and time and so on and so forth. But that's not what the question was about in the first place. You never see idealized physics problems in the real world either but what you learn from naive, simplified applications of classical physics still gets you somewhere in real world problems. Same with EV calculations.
The point is that EV is an incomplete measure of value when it comes to evaluating gambles, not that you have to pay taxes. EV doesn't work on one-off situations, like if you're on a game show with Howie Mandel.
No, I understand that point perfectly. My point was that in order for you to be able to evaluate whether a gamble makes sense repeatedly, you have to be sure that each instance has a positive EV.
Say that you planned to do the 9k with %1 chance of a million payout enough times to take advantage of probabilities. You could have found out much later that the gamble had a negative EV. This is my point: the parent made it sound like you can evaluate all decisions as if the EV was easy to establish. In reality that's a rare case.
In reality, there are no point masses or frictionless surfaces, either. Maybe you perfectly understand the point and you're moving on to various niggling contingencies. That's fine. The article is trying to explain something more basic than what you're talking about. You're doing the equivalent of butting into a freshman physics lecture and pointing out everything they've oversimplified, which is just plain obnoxious.
I'm not commenting on the article. I agree with the article, which in fact talks only about wagers with quantifiable EV. My comment was a reply to mattmaroon's observation that over the course of a lifetime someone can find enough wagers with a positive value to eventually come out ahead. Please read the entire thread.
I have read the entire thread and I just don't understand the point of your comment. Of course people have to pay taxes and spend time on investments. That doesn't make diversified investing a fool's errand. It's like we're trying to build a go-kart and you're saying "it'll never work because there's friction." Maybe there's friction, and we'll work out what that friction is, just as we'll calculate the post-tax EV of our investments.
To summarize: life usually doesn't offer a succession of quantified wagers like one would ideally like. Nothing to do with friction, you just don't know the expected value of anything other than lotteries or other chance games where a trustworthy entity backs them up. You simply can't assume that you will always be able to pick "real life" wagers with positive expected values. The closest you can get are "traditional investments" for which you can look at past returns and hope for the future to repeat the past. For new situations, there simply isn't enough data.
If you decide to start a startup today, what's your expected value? Do you have access to data about all startups ever started, costs involved and payouts? How does that change if you segment it by startup type, decade, etc? Can you say for sure that starting it has a positive expected value in terms of utility to you as opposed to the probabilistic certainty of the expected value of the 9k wager?
Taxes alone don't necessarily make it EV negative. For me it wouldn't be since I already have substantial gambling income to deduct from. (Unless you're a professional gambler, you can't deduct more than you earn.) But you're right, for most people that one particular wager would, assuming they were going to get a 1099 if they won. (Most money made from gambling goes unclaimed, as it doesn't take place in a casino.)
Startup investing is somewhat the opposite I suppose, where taxes actually makes it more profitable. If you lose, you can deduct the loss, if you win, you usually pay the much lower capital gains taxes. (Unless I'm misunderstanding the tax ramifications of a losing angel investment, which I may certainly be.)
Taxes were just an example. My issue is when you say:
Life will present you with many different wagers with positive expectations
and then you give the example of a startup. The problem is that you make it sound as if it's always easy to identify the wagers with positive expectation. Of course if you pick enough wagers with positive expectation you will come out ahead, but other than for chance games you simply don't know the expected value of your gamble. The closest you can get is by looking at past returns for certain well-known investments and hope that the future returns will be similar. For most "wagers" there isn't enough data.
There's a concept for this called utility. It has nothing to do with how many times you get to play, but rather with how much use you have for the money.
If the $10000 means little to you, you might even take a negative expected value for a chance at a big score (people do this every day with $1 lottery tickets), regardless of how many times you get to play.
Edit: The reason that repeating the game has some allure is that you are changing the game from a low probability of winning a large payout to a higher probability of winning a distribution of payouts computable via the binomial expansion. It also changes the utility equation.
The principle is that I'm willing to buy a lottery ticket, even though the odds are slanted (the agency takes a large cut) because the $1 the ticket costs me is meaningless to my life, but the $1 million I might win would fundamentally change my life.
The is why people buy insurance: the $1000 I spend every year on homeowner's insurance hurts only slightly, but the amount that I'd lose to a fire would destroy me financially.
It's also the rationale for burglary: the thief is very likely to make a large sum of money, while he only has a small chance of a devastating loss (i.e., going to jail).
The question is all wrapped up in the relative size and likelihood of the events, where the size is judged relative to the difference that it makes to you.
If "repeating the game ... changes the utility equation" from negative to positive, how can you possibly say "it has nothing to do with how many times you get to play"?
It's be a mistake to say it has nothing to do with that, but the number of times you get to play is subsumed into the utility equation. It would be irrelevant if utility was linear.
The point is that $10,000 to me is worth more than 1% of the value to me of $1,000,000. Whereas I would probably pay $1.10 for a 1% chance of winning $100, though not on a regular basis.
If utility dictates that I should not risk $10,000 on a 99% chance of losing it all, it is unlikely that I have the bankroll to play the game as many times as would be required to overcome my risk of losing everything. And by playing multiple times, I have introduced a scenario with significant probability of losing much more than $10,000, which is obviously catastrophic if losing $10,000 means so much to me.
The utility is certainly different, but I wouldn't necessarily claim it is better. It could in fact be worse, but I haven't studied it enough to assert either.
Exactly. It's important to remember that, while in theory you might win a particular game on average, that game does not exist in a vacuum. Suppose you're playing a game wherein the gambling is only a component, and you also have some other use for the currency (say, it's the metric that causes you to win). The player in the lead should typically not bet all his money on any gamble, no matter how good the expected payoff.
edit: Of course, there are exceptions to that last rule of thumb. It's just a common case.
While one can fold this type of difficulty into expected utility theory, this is not (apparently) true of all situations. People still overweight loss, overweight certainty, and overweight extremely low probabilities.
The classic paper by Daniel Kahneman and Amos Tversky presents a pretty thorough critique of expected utility theory as a model for decision making, as an empirical question. c.f.
www.hss.caltech.edu/~camerer/Ec101/ProspectTheory.pdf
I was a little confused when I read the section on the St. Petersburg Lottery, so I looked around to see what it was all about.
In order to find the expected value, you simply multiply the probability of the event by its expected payout. Then to find the total expected value, you sum all of the possible outcomes. So the probability of the first flip being tails (1/2) times the payout ($1), gives the expected value of that flip ($0.50). The rest of the flips follow in a similar way: 2nd (1/4)x($2)=0.50, 3rd (1/8)x($2)=0.50, 4th (1/16)x($4)=0.50, etc.
If you take the sum of these values to an infinite number of flips (.50+.50+.50+...), you end up with an expected value of infinity. On the other hand, even if the bank is ridiculously wealthy [1] (i.e. US GDP), the expected payout is ridiculously small. That, to me, is the crux of the fallacy. It reminds me of the legend of the creator of chess who asked for one grain of rice for the first square of the chessboard, two for the second, four for the third, etc. which becomes greater than the world production of rice after a few dozen squares.
This is a paradox only because we are allowing infinite winnings. That's not possible in the real world. The solution is to end the game when the 'casino' can no longer pay the amount on the table.
In this case, the expected value is simply .50 * log 2 C, where C is the total cash available.
From my limited experince (you HN regulars correct me if I am wrong) VC does not behave like the article say.
It is true that the VC diversifies on different inversions, not to put all the eggs in the same basket and to invest on different markets/ideas
But VCs do not invest on a lot of startups. The companies I know have a very limited portfolio (sometimes less than ten). They do not have big inversions, I suppose because they can not handle the risk or the workload
If someone -- or more interestingly, some speculative project -- offers you a 1% chance at $1 million, for only $9,000, as many times as you want to take it, then the issue becomes one of finance.
If your society has strong contracts, the corporate abstraction, norms of honesty and disclosure, and a pool of wealthy investors (or operational investment groups) which can weather a few bad runs, you'll likely raise the money, and everyone wins.
If your society fails in one or more of those dimensions, it may be impossible to raise the money. Too little respect for contracts/corporations (such as via ex post facto wealth redistribution), too much dishonesty/secrecy, or too few wealthy people/groups could make the costs of assembling the investment exceed the returns. Thus the valuable project is never undertaken, and society is poorer.
Life will present you with many different wagers with positive expectation, but it often won't present you with the same one multiple times. Maybe it will give you that shot, then the next time give you a +EV chance to invest in the startup, then later give you the opportunity to play the St Petersburg game for $2, etc. Keep taking the +EV wagers over your lifetime and you'll end up way ahead, even if you never take the same exact one twice.
You don't need a good sample size for an individual wager to be a good idea, just for all wagers combined. All of that assumes, of course, that taking any one wager doesn't prevent you from many others. Taking the $9k wager if you have $9k to your name will. Taking it if you have $10m to your name won't and therefore is a no-brainer.
*Edited to account for Kelly Criterion. Thanks Aston