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Sure. The question is more like, should you be willing to pay $1 million to play? And honestly, if you can't afford to burn $1 million, that would be a pretty b
by oconnor663 11y ago
Sure. The question is more like, should you be willing to pay $1 million to play? And honestly, if you can't afford to burn $1 million, that would be a pretty bad idea.
However! Suppose you were allowed to modify things ever so slightly. Suppose you get to say, "Sure, I'll pay $1 million to play. But instead of playing once and taking my winnings, I want to play a million times, and only take one millionth of whatever I win each time." You haven't done anything to change the average payout of the game. But you've made it a much better idea to play, by reducing the variance.
That's what it means to say the game has infinite expected value. You really should be willing to pay anything to play it, as long as you're allowed to repeat the game over and over.
Of course, if you're stuck with playing the game just once, then lots of games look like a bad idea despite having a positive expected value. The St. Petersburg game is hardly alone here. For example, suppose you have a 1-in-a-million chance of winning $10 billion, but it costs $1000 to play. Should you play? The expected value is good, but I don't think most people would touch it. Not much of a paradox there.
- grkvlt 11y agoActually, for the 1/1,000,000 chance of GBP 10,000,000,000 then GBP 1,000 seems OK? My EV is GBP 10,000 or 10.0 times my stake, which is _much_ better that buying 1,000 lottery tickets at GBP 1.00 each, for a 1/14,000 chance of GBP 10,000,000 (nominal; neglecting rollovers) and an EV of GBP 714.28 (approximately) which is an under unity gain (0.71 of the stake returned). However using the Kelly criterion for placing bets, I think it turns out that even though the return is high for the ten billion wager, you should only place the GBP 1,000 bet if you have a bankroll of 1,000,000,000 already ;( Corrections to my maths welcome!
- morgante 11y ago> For example, suppose you have a 1-in-a-million chance of winning $10 billion, but it costs $1000 to play. Should you play? I'd happily take those odds.
- dsjoerg 11y agoIt's not only the odds but the magnitude of the bet that matter. Because losing $1000 wouldn't bother you much, you're happy to do it. But scale the proposition up enough, perhaps 10^3, and you wouldn't be so relaxed about it. E.g. 1-in-a-million chance of winning $10 trillion, but it costs $1 million to play.