Instead of charging $X, why not charge $2X with prob 1/2 and $0 with prob 1/2?
Given that people are often irrational (e.g., they buy lottery tickets), would you make more money this way?
There's a reason why people indulge "If I won the lottery..." fantasies.
http://www.jstor.org/pss/1914185
Unlike most people in this discussion, I like the idea and think it might work under certain circumstances (probably goods with low prices and high price inelasticity).
More generally, you could let them specify the pay probability p or price $Y >= $X. Given p, have them pay $X/p. Given $Y, make them pay with probability $X/$Y.
A: Buy a ticket
Lottery addicts certainly can't do math since they waste a lot of money on tickets. But a good many other people just buy one a week or just occasionally, eg when there's a particularly large jackpot on offer. This isn't so irrational; a regular player will make a few bucks back over the year on small prizes, and the risk/reward ratio improves for a big jackpot even though the odds don't.
It's worth recalling that while the odds on any individual ticket are awful, enough people are playing that people do win big prizes on a semi-regular basis. For small-spend players who can afford it, and may think of it as semi-charitable given the uses to which lottery profits are put, is it really any more irrational than certain kinds of insurance?
Some people get a kick out of the whole buy a ticket, wait for the numbers, get excited dreaming thing.
For some folks, it's entertainment.
This rant used to conclude with "therefore, buying lottery tickets is irrational; period", but someone on Less Wrong recently brought up a valid exception: it can be rational if you're currently deeply in debt. Utility functions are actually sigma-shaped: they flatten as you get more deeply negative as well. Being two trillion in debt isn't very much worse than being one trillion in debt. So when you're in debt, your utility function can get steeper before it gets shallower.
Actually, this is a big problem with conventional economics: it assumes that all variables are conventional. So income taxes don't affect people's willingness to work nearly as much as theory predicts, because their choice is usually between "Have a job" and "Don't have a job" and not "Do $30K of work per year" and "Do $40k of work per year". Unemployment exists because a business can't just drop wages to the equilibrium level, but rather workers make the choice between "I'm being paid fairly" and "I'm not being paid fairly" and act appropriately. Market niches are not instantly filled because entrepreneurship often has a binary outcome of "Get rich" and "Fail", and not a continuous distribution of outcomes.
s/conventional/continuous/. I can't type at night.
If you make people commit before revealing the amount, the issue is trust: how do customers know that the odds are actually 1/2 that they'll pay $0? How do they know you aren't just charging everyone $2X? You'd have to use some verifiably random bit, which gets complicated. People will smell a scam.
(Speaking of which: are there any commonly used publicly verifiable random numbers, like a die roll, but outside of any party's control? I presume there must be a need for them in a number of situations.)
There is an authoritative source of random numbers. In 1955, The RAND Corporation (a Cold War-era think tank) published a wonderful book entitled "A Million Random Digits with 100,000 Normal Deviates". Amazon.com has a copy of it, including some amusing reviews: http://www.amazon.com/Million-Random-Digits-Normal-Deviates/... .
For a good source of truly random numbers accessible from an API, I personally use http://random.org . They use atmospheric noise to create random numbers, which ultimately boils down to the true randomness that is Quantum Mechanics. It's not suitable for high-security applications -- I could do some network trickery and redirect your API request to my server that always returns 1.0 -- but it's better than the PRNGs on modern computers.
For example, I could tell my friend that I'll do his laundry for a month if Barack Obama's speech tomorrow contains an even number of words, and he'll do my laundry otherwise. This would appear to be truly random, but what if I secretly am friends with Barack Obama and can rig it in advance?
Is there some easy, practical way of producing a number whose randomness a large number of mutually distrustful parties can agree on? (I am thinking of something like asking each party to produce a number, and then publicly summing everyone's numbers together.)
Before we even begin delving into true RNGs, it is worthwhile to consider if we're over-engineering the problem. Is a PRNG "good enough" for the task at hand? In most cases, it is. A well-seeded PRNG, such as the ones present in modern-day Operating Systems, are suitable for most tasks not involving National Security.
However, if you have a genuine need for a true RNG, a computer algorithm certainly won't be generating it. Computers are by nature deterministic, and thus are very poor at behaving unpredictably. Thus, like with all good problems, we are forced to enter the realm of physics.
The fundamental problem is, who will be observing the source of randomness? To my knowledge, there isn't a physical source of randomness that is observable to everybody. (I did some quick googling to confirm my suspicions, but if you're planning on using my words to do anything useful, please confirm my statement!). Thus, as I see it, you have several options:
1) Your idea, where all parties derive a true random number from physics and publically sum everyone's numbers together. Your qualifiers (2) through (4) hold, but (1) is difficult. In this case, a certain number of rogue nodes will be able to manipulate the computation. I remember a Professor of mine, who does research in this field, stating the minimum number of truthful nodes needed in a distributed computation is (2/3)n + 1, but again confirm this number for yourself.
My point here is, using the public-summing method, you are faced with the difficult problem of determining if a node is attempting to manipulate the system. While a provable solution has been found, after decades of research across the globe, my Professor still is working on the problem, thus demonstrating it is not an easy task.
2) Have a single computer (say, random.org) that is observing a true RNG, and publicizes the result for all to see. Of course you'd use asymmetric crypto so that all parties can prove that they've received the correct number.
Again, your qualifiers (2) through (4) pass, but (1) presents a different problem. We're back to where we started: do you trust random.org?
However, it's an easier task to make an authority trustable than a peer in a distributed computation. We trust Verisign to be a certificate authority, because we believe their security practices are half-decent, and they can be audited. (Every year all CA's are independently audited by WebTrust.org using pretty strict criteria.)
The same could be done for random.org: why not require certification and frequent auditing of their RNG equipment?
The other nice thing about RNGs is that you can audit them yourself, given enough time and expertise. When viewed correctly, poor RNGs will have distinct patterns visible to the eye (example: http://www.cs.hku.hk/cisc/projects/va/index.htm ).
The crux of my argument is: I don't know of a true RNG, that upon observation by multiple parties, all will receive the same random number. My instinct and my research says that none exist, but I would love to be proven wrong. Crypto experts, correct me!
If you'd like more information, Wikipedia is always invaluable: http://en.wikipedia.org/wiki/Hardware_random_number_generato...
Just use next week's state lottery's numbers and similar things. That is fairly impartial and hard to influence. Though that might be too few bits and too much latency for your liking.
There's always going to be less people you can afford 2X than X.
On a practical note, what stops them getting a refund and repurchasing until they get it for free.
Your other observation is good, too, but I think this sentence gets to the heart of the problem. The double-or-nothing transaction is irreversible, which is what makes it a bad one.
People like commercial transactions that can be reversed. Ideally, both parties should walk away feeling like they got a fine deal, but that if the deal goes sour at any point in the near future (because a part was broken; because the customer had second thoughts) they would be both be willing to "undo" it without much harm done.
To take advantage of this, you might charge 1.1X with a 10% chance of getting 10X worth of similar goods. But that wouldn't be too exciting, either.
My service costs $X per month. So if you like it, you'll want to pay $X. ... But for a just this month, there is a 1/10 chance your $X will get you the service for the entire year!! ... and if you don't "win", you still get what you paid for.
I think this might encourage people who are on the edge to sign up now, and still feel good no matter what the outcome, plus they might tell their friends about it.
-- David Jones
Car dealerships seem to like this one, and Visa ran a similar contest here in Canada recently (called "Win What You Buy").
The success of this kind of promotion is most likely proportional to the price of your product.
Long story short: Make your new price $1.1X without fanfare, offer a 1 in 10 chance of getting it for free and you all win. They get their chance at a freebie at a cost they don't notice, you get your promotion vibe bumping up sales.
(1)I know that that's obvious, and it was dumb to run it - but we ran it anyway for the sake of completeness.
(0.5 * 2X) + (0.5 * 0) = X = (X * 1.0)
but perhaps you would be able to attract more customers. I doubt it. While it's certainly true that everybody loves a freebie, no one likes to pay twice for something.
Why not charge X with probability (1-p) and charge zero with probability p? The expected value would be reduced to (1-p) but perhaps you would attract people looking for freebies. More customers is always a good thing, right?
Problem solved.