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I think it's hard to enjoy gambling if you are sure you'll lose money, which is how I feel like. I may be over pessimistic.
Don't get over-excited. You are still losing money in a less than fair-odds situation.
And since most people don't stop gambling until they have some deficit from gambling, casinos usually make more than the odds give them.
In fact, people take such gambles (with negative expectation but with high probability of winning) everyday.
They fly on airplanes and drive to work.
In our world people do not place infinite value on their own lives.
There is nothing in what I wrote that implies people value their lives infinitely. People just need to value their lives highly enough such that flying on an airplane (with its probability of crashing) has a negative expected value.
Again, from Nick Bostrom's article:
"Pascal: I must confess: I’ve been having doubts about the mathematics of infinity. Infinite values lead to many strange conclusions and paradoxes. You know the reasoning that has come to be known as ‘Pascal’s Wager’? Between you and me, some of the critiques I’ve seen have made me wonder whether I might not be somehow confused about infinities or about the existence of infinite values . . .
Mugger: I assure you, my powers are strictly finite. The offer before you does not involve infinite values in any way. But now I really must be off; I have an assignation in the Seventh Dimension that I’d rather not miss. Your wallet, please!"
(1) You don't have to construe the gamble as some sort of coin flips. It could also be something like "the weather in Santa Clara, California in 20 September 2012 will be sunny" - i.e. a singular non-repeating event, in which case having 100 hundred people (as confused as me) will not help you.
A coin flip is not fundamentally a less singular non-repeating event than the weather at a specific location and specific time. There are no true repeating events on a macro scale if you specify location and time. The relevant difference is how confident you can be that past events are good predictors of the probability of future events. Pretty confident for a coin toss, less so for weather. Note however that if your probability estimates are sufficiently accurate / well-calibrated you can make money by betting on lots of dissimilar events. See for example how insurance companies, hedge funds, professional sports bettors, bookies and banks make much of their income.
(3) Besides, suppose you have a gamble Z with negative expectation with probability of a positive outcome 1-x, for a very small x. I claim that for small enough x, every one should take Z - despite the negative expectation.
'Small enough' here would have to be very much smaller than 1 in 100 for this argument to begin to apply. It would have to be 'so small that it won't happen before the heat death of the universe' scale. I'm still not sure the argument works even in that case.
I believe there is a sense in which small probabilities can be said to also have an associated uncertainty not directly captured by the simple real number representing your best guess probability. I was involved in a discussion on this point here recently.
'Small enough' here would have to be very much smaller than 1 in 100 for this argument to begin to apply. It would have to be 'so small that it won't happen before the heat death of the universe' scale. I'm still not sure the argument works even in that case.
How small should x be? And if the argument does hold, are you going to have two different criteria for rational behavior - one with events where probability of positive outcome is 1-x and one that isn't.
And also, from Nick Bostrom's piece (formatting will be messed up):
Mugger: Good. Now we will do some maths. Let us say that the 10 livres that you have in your wallet are worth to you the equivalent of one happy day. Let’s call this quantity of good 1 Util. So I ask you to give up 1 Util. In return, I could promise to perform the magic tomorrow that will give you an extra 10 quadrillion happy days, i.e. 10 quadrillion Utils. Since you say there is a 1 in 10 quadrillion probability that I will fulfil my promise, this would be a fair deal. The expected Utility for you would be zero. But I feel generous this evening, and I will make you a better deal: If you hand me your wallet, I will perform magic that will give you an extra 1,000 quadrillion happy days of life. ... Pascal hands over his wallet [to the Mugger].
Of course, by your reasoning, you would hand your wallet. Bravo.
In response to
Is Rationality Maximization of Expected Value?
Suppose someone offers you a (single trial) gamble C in which you stand to gain a nickel with probability 0.95 and stand to lose an arm and a leg with probability 0.05. Even though expectation is (-0.05arm -0.05leg + 0.95nickel), you should still take the gamble since the probability of winning on a single trial is very high - 0.95 to be exact.
Non-sarcastic version: Losing $100M is much worse than gaining $100K is good, regardless of utility of money being nonlinear. This is something you must consider, rather than looking at just the probabilities - so you shouldn't take gamble A. This is easier to see if you formulate the problems with gains and losses you can actually visualize.
Is the problem that 0.01 or 0.05 too high?
Take a smaller value then.
In fact, people take such gambles (with negative expectation but with high probability of winning) everyday.
They fly on airplanes and drive to work.
In response to
Is Rationality Maximization of Expected Value?
Thanks, I already knew about this.
Related is also Martingale gambling.
In response to
Is Rationality Maximization of Expected Value?
Gamblers are maximizing expected utility, not expected cash. That is all.
I think it's hard to enjoy gambling if you are sure you'll lose money, which is how I feel like. I may be over pessimistic.
Roulette gives you odds of 1.111 to 1 if you place on Red or Black with expectation -0.053 on the dollar. So I may be over-pessimistic. See the wiki entry.
In response to
Is Rationality Maximization of Expected Value?
The whole nonlinear utility thing makes this specific point wrong, but:
It seems like the main counter-intuitive part of expected utility theory (or counter-expected utility theory part of intuition) is just this type of question. See: Pascal's Mugging.
Humans tend to be loathe to trade of high probabilities of small benefits for low probabilities of big benefits in cases where linearity is very plausible, such as # of people saved.
But people seem to just as often make the opposite mistake about various scary risks.
Are people just bad at dealing with small probabilities?
What does that mean for coming to a reflective equilibrium about ethics?
The nonlinear utility of money?
Well, the point I was trying to make was supposed to be abstract and general. Nick Bostrom's Pascal's Mugging piece argues for a very similar (if not identical) point. Thanks for letting me know about this.
And yes, I'm bad at dealing with small probabilities. I feel that these evoke some philosophical questions about the nature of probability in general - or whatever we talk about when we talk about probabilities.
In response to
Is Rationality Maximization of Expected Value?
Does it make sense to speak of probabilities only when you have numerous enough trials?
No, probability theory also has non-frequency applications.
Can we speak of probabilities for singular, non-repeating events?
Yes. This is the core of a Bayesian approach to decision making. The usual interpretation is that the probabilities reflect your state of knowledge about events rather than frequencies of actual event outcomes. Try starting with the LW wiki article on Baesian probability and the blog posts linked therefrom.
Obviously, this needs more discussion but the kind of thought I was trying to motivate was the following:
How is that saying a non-repeating singular event has a very small probability of occurring different from saying it will not happen?
This was motivated by the lottery paradox. Questions like, when you buy a lottery ticket, you don't believe you will win, so why are you buying it?
Examples like these sort of pull my intuitions towards thinking no, it doesn't make sense to speak of probabilities for certain events.
Yeah, uhm, I figured I'd misunderstood that, because my second hypothesis was that someone was trolling us. Looking at the poster's previous comments I'm more inclined to think that he just missed the whole 'Bayes is god' meme.
Sorry that talking about money lead to confusion. I guess the point I was making was the following. See my respond to mattnewport, i.e.:
Suppose you have a gamble Z with negative expectation with probability of a positive outcome 1-x, for a very small x. I claim that for small enough x, every one should take Z - despite the negative expectation.
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I recently read an anecdote (so far unconfirmed) that Ataturk tried to ban the veil in Turkey, but got zero compliance from religious people, who simply ignored the law. Instead of cracking down, Ataturk decreed a second law: all prostitutes were required to wear a veil. The general custom of veil-wearing stopped immediately.
This might be the most impressive display of rationality I've ever heard of in a world leader.
As a Turk, I strongly believe that story is fictional.
Where and how was this ban issued? Can you give more details?
You may be hearing some fictional story based on his social reforms.
See here
And the veil, currently banned in public universities, is still very much a hot button issue. Also, a large segment of the Turkish population still wears the veil. The country is deeply divided over this issue.