There's no problem here at all; except perhaps the wildly absurd fantasy of "perfectly rational (and/or intelligent) markets," which the popularity of e.g. lotteries and casinos immediately disproves. Not much to see here.
That said, even for things that aren't publicly decided - the 'wisdom of the masses' [1] is very real and kind of weird phenomena. If you get people to try to guess how many beans are in a large jar, you'll get some people making really stupid guesses like 100, or a million. Yet when you average out the result over all answers, it tends to be extremely close to the right answer. At least unless you let people collaborate, then the effect disappears and we become stupid again. Kind of explains a lot of things.
[1] - https://en.wikipedia.org/wiki/Wisdom_of_the_crowd#Examples
The behaviour of customers of lotteries is indeed clearly not perfectly rational if you consider the monetary expected value of your wins. On the other hand, the customers' behaviour gets much more rational if you consider the customer's utility function: a huge lottery win has a disproportional benefit for the customer with respect to the customer's utility function (since the lottery win will hopefully change the customer's life). In other words: what the customer optimizes for is the expected value of his utility function.
What is the exact shape of the convex utility function you theorise these people have? And what data supports that?
Indemnity insurance and lotteries are actually symmetric to each other: indemnity insure is about preventing big losses in a rare event, and lotteries are about winning big in a rare event.
The central difference is that the behaviour of people with respect to wins and losses is not symmetric, that is why insurance policies and lotteries are marketed quite differently.
> What is the exact shape of the convex utility function you theorise these people have? And what data supports that?
I am very sure that big lottery companies have data available to answer this question, but they won't make the data available. So I honestly admit that I don't know the answer, but I can propose an experiment how one might get a partial answer:
Create some lottery offers, paying out, say, 1000, 10.000, 100.000, 1.000.000, 10.000.000 $ with probabilities such that the expected payout is 5 $ for each of them. Now sell/auction the lottery tickets and look what prices customers are willing to pay. This should give a rough idea how the utility function might look like.
The discussion was specifically about negative-EV lotteries since those are the ones that require convex utility.
No, 5$. The reason is that you sell/auction these lottery tickets. The price that the customer pays has to be subtracted from this payout (and this way the final payout very likely will become negative). But the price that the customer will pay in this experiment is not known beforehand, so the initial payout has to be positive.
On the other hand, maybe they're somewhat useful as a rough indicator.
Rich people can go to tennis matches, get a pilot's licence, and buy stock options. Poor people have much fewer alternatives.
Most players are fully aware of this, just as most beer-drinkers drink in moderation. (See Brenner, Brenner & Brown for more data.)