Preferably, let other people play the game first to gather the evidence at no cost to myself.
For the record, this is not permitted.
My take at it is basically this: average over all possible distributions
It's easy to say this but I don't think this works when you start doing the maths to get actual numbers out. Additionally, if you really take ALL possible distributions then you're already in trouble, because some of them are pretty weird - e.g. the Cauchy distribution doesn't have a mean or a variance.
distribution about which we initially don’t know anything and gradually build up evidence
I'd love to know if there are established formal approaches to this. The only parts of statistics that I'm familiar with assume known distributions and work from there. Anyone?
I'd love to know if there are established formal approaches to this.
You should probably look at Jaynes's book "Probability Theory: the Language of Science". In particular, I think that the discussion there dealing with the Widget Problem and with Laplace's Rule of Succession may be relevant to your question.
Suppose I tell you I have an infinite supply of unfair coins. I pick one randomly and flip it, recording the result. I've done this a total of 100 times and they all came out heads. I will pay you $1000 if the next throw is heads, and $10 if it's tails. Each unfair coin is entirely normal, whose "heads" follow a binomial distribution with an unknown p. This is all you know. How much would you pay to enter this game?
I suppose another way to phrase this question is "what is your best estimate of your expected winnings?", or, more generally, "how do you choose the maximum price you'll pay to play this game?"
Observe that the only fact you know about the distribution from which I'm drawing my coins is those 100 outcomes. Importantly, you don't know the distribution of each coin's p in my supply of unfair coins. Can you reasonably assume a specific distribution to make your calculation, and claim that it results in a better best estimate than any other distribution?
Most importantly, can one actually produce a "theoretically sound" expectation here? I.e. one that is calibrated so that if you pay your expected winnings every time and we perform this experiment lots of times then your average winnings will be zero - assuming I'm using the same source of unfair coins each time.
I suspect that the best one can do here is produce a range of values with confidence intervals. So you're 80% confident that the price you should pay to break even in the repeated game is between A80 and B80, 95% confident it's between A95 and B95, etc.
If this is really the best obtainable result, then what is a bayesianist to do with such a result to make their decision? Do you pick a price randomly from a specially crafted distribution, which is 95% likely to produce a value between A95..B95, etc? Or is there a more "bayesian" way?