I guess before this puzzle becomes stale I wanted to share my thoughts on it and see what people think. This puzzle is a bit of a joke, as I think most people got, which is that math seems to tell us to play forever even though that is, arguably, the one wrong answer.
Just because the math is wonky, doesn't mean that there isn't a reasonable approach to answering it. If Warren Buffet went crazy and decided to play this game with you, such that his entire multi-billion dollar fortune was at stake, where would you stop? I don't claim to have the math perfect but as I see it you have something like a 11% chance of taking all of his money assuming you wanted to shoot for that.
The more we ground this question in reality, the more problems it would have. I'm not talking about Canadian pennies having no value (they do) or limits on how much money anybody would actually pay out because that shouldn't change the approach you would have. Other complications I thought of:
- You could probably figure out some kind of insurance plan where a big bank pays you a fixed amount and they collect your actual winnings, up to whatever point they specify.
- Taxes. Probably not a huge issue in the scheme of things.
- If you play this game for charity, vast amounts of money won't lose their utility like they would for someone playing for themselves. I have no idea how to approach that so let's just say you are playing selfishly.
Like Gareth McCaughan suggested, money has decreasing utility the more of it you have, but I feel like that isn't quite the right perspective to have. If your husband has cancer and needs a $100,000 treatment, I imagine you aren't going to be weighing the utility of money beyond that point. I think the right approach is understanding what you stand to lose at every stage, and weighing it against what you stand to gain in terms of what you would actually buy.
I would suggest making a list of things that you can't really currently afford that would solve a big problem you have. Do you worry about your car breaking down? A newer car would would give you peace of mind. Don't like where you live? A down-payment on a new house would solve that. Hate your job? Maybe enough money would let you open that restaurant you always wanted. Alternately you could win enough money to retire right now but that can cause a whole world of new problems for you if you don't know how to keep yourself busy. Read up on stories of lottery winners if you want examples.
The opposite approach, which is asking yourself "what is something I'd like to have?", is where people make bad financial decisions. Wanting a huge mansion with a few Bugattis in the garage is fine but you should appreciate how far beyond the point of fixing a problem that goes. This undoubtedly ties in to why studies show that there is something of a threshold for where a person's income stops having a meaningful impact on their reported happiness.
Anyway, to tie this up, when you sum up things that fix problems you have, that is a reasonable upper-end stopping point for the game. You may want to stop earlier, depending on which items you may not be willing to risk, but going beyond that falls in the category of being needlessly greedy. I didn't spend long calculating it for myself, but personally I would stop in the range of 3-years worth of my current salary. My odds of getting there aren't great, though, I'm only looking at a 25% chance. What's interesting is how my answer has almost nothing to do with the actual odds of the game... so if the die were a D20 instead I would still stop at the same dollar amount.