So one of the major issues I’ve identified with why our gut feelings don’t always match with good expected utility models is that we don’t live in a hypothetical universe. I typically use log utility of end state wealth to judge bets where I am fairly confident of my probability distributions as per Vaniver in another comment.
But there are reasons that even this doesn’t really match with our gut.
Our “gut” has evolved to like truly sure things, and we have sayings like “a bird in the hand is worth two in the bush” partly because we are not very good at mapping probability distributions, and because we can’t always trust everything we are told by outside parties.
When presented with a real life monty haul bet like this, except in very strange and arbitrary circumstances, we usually have reason to be more confident of our probability map on the sure bet than on the unsure one.
If someone has the $240 in cash in their hand, and is saying that if you take option B, they will hand it you right now and you can see it, you can usually be pretty sure that if you take option B you will get the money—there is no way they can deny you the money without it being obvious that they have plain and simply lied to you and are completely untrustworthy.
OTOH, if you take the uncertain option—how sure can you really be that the game is fair? How will the chance be determined? The person setting up the game understands this better than you, and may know tricks they are not telling you. If the real chance is much lower than promised, how will you be able to tell? If they have no intention of paying you for a “win”, how could you tell?
The more uncertainty is promised, the more uncertainty we will and should have in our trust and other unknown considerations. That’s a general rule of real life bets that’s summed up more perfectly than I ever could have in this famous quote from Guys and Dolls:
“One of these days in your travels, a guy is going to show you a brand new deck of cards on which the seal is not yet broken. Then this guy is going to offer to bet you that he can make the jack of spades jump out of this brand new deck of cards and squirt cider in your ear. But, son, do not accept this bet, because as sure as you stand there, you’re going to wind up with an ear full of cider.”
So for these reasons, this gamble, where the difference in expected value is fairly small compared to the value of the sure win—even though a log expected utility curve says to take the risk at almost any reasonable level of rich country wealth, unless you have a short term liquidity crunch—I’d probably take the 240. The only situations under which I would even consider taking the best are ones where I was very confident in my estimate of the probability distribution (we’re at a casino poker table and I have calculated the odds myself for example), and either already have nearly complete trust or don’t require significant trust in the other bettor/game master to make the numbers work.
In the hypothetical where we can assume complete trust and knowledge of the probability distribution, then yes I take the gamble. The reason my gut doesn’t like this, is because we almost never have that level of trust and knowledge in real life except in artificial circumstances.
I could pick the jack of spades out of a new deck of cards too; they tend to come pre-sorted. All it takes is studying another brand new deck of cards. (I’d have to do this studying before I would be able to pull this trick off, though.)
My guess is that you’d want to flip it backside-up, and from the then bottom, pull 10 cards and flip the top one of that one over. Then you’ll have either a Jack (if it starts with an ace, I assume this to be very likely, 50%?), a 9 (if it starts with a placeholder card, like a rule card, I presume this to be probable… like, 20%?), a Queen (if the ace is last, a la Jack Queen King Ace, 20% for this one as well) an 8 (two jokers at the front of the deck? 8%?), a 7 (2 jokers AND a rule card?! 1%) or 1% of me just being wrong entirely. It sounds like a weird distribution, but rule cards and jokers tend to be at the back, and an ace with fancy artwork tends to be at the front of the deck because it looks good.
A bit of googling reveals that a new deck usually starts with the Ace of Spades, so I’d guess that flipping the deck over, then drawing 10 cards from the bottom of the deck (what used to be the front) and then flipping the 10th card over will give you a Jack of Spades.
So one of the major issues I’ve identified with why our gut feelings don’t always match with good expected utility models is that we don’t live in a hypothetical universe. I typically use log utility of end state wealth to judge bets where I am fairly confident of my probability distributions as per Vaniver in another comment.
But there are reasons that even this doesn’t really match with our gut.
Our “gut” has evolved to like truly sure things, and we have sayings like “a bird in the hand is worth two in the bush” partly because we are not very good at mapping probability distributions, and because we can’t always trust everything we are told by outside parties.
When presented with a real life monty haul bet like this, except in very strange and arbitrary circumstances, we usually have reason to be more confident of our probability map on the sure bet than on the unsure one.
If someone has the $240 in cash in their hand, and is saying that if you take option B, they will hand it you right now and you can see it, you can usually be pretty sure that if you take option B you will get the money—there is no way they can deny you the money without it being obvious that they have plain and simply lied to you and are completely untrustworthy.
OTOH, if you take the uncertain option—how sure can you really be that the game is fair? How will the chance be determined? The person setting up the game understands this better than you, and may know tricks they are not telling you. If the real chance is much lower than promised, how will you be able to tell? If they have no intention of paying you for a “win”, how could you tell?
The more uncertainty is promised, the more uncertainty we will and should have in our trust and other unknown considerations. That’s a general rule of real life bets that’s summed up more perfectly than I ever could have in this famous quote from Guys and Dolls:
“One of these days in your travels, a guy is going to show you a brand new deck of cards on which the seal is not yet broken. Then this guy is going to offer to bet you that he can make the jack of spades jump out of this brand new deck of cards and squirt cider in your ear. But, son, do not accept this bet, because as sure as you stand there, you’re going to wind up with an ear full of cider.”
So for these reasons, this gamble, where the difference in expected value is fairly small compared to the value of the sure win—even though a log expected utility curve says to take the risk at almost any reasonable level of rich country wealth, unless you have a short term liquidity crunch—I’d probably take the 240. The only situations under which I would even consider taking the best are ones where I was very confident in my estimate of the probability distribution (we’re at a casino poker table and I have calculated the odds myself for example), and either already have nearly complete trust or don’t require significant trust in the other bettor/game master to make the numbers work.
In the hypothetical where we can assume complete trust and knowledge of the probability distribution, then yes I take the gamble. The reason my gut doesn’t like this, is because we almost never have that level of trust and knowledge in real life except in artificial circumstances.
I could pick the jack of spades out of a new deck of cards too; they tend to come pre-sorted. All it takes is studying another brand new deck of cards. (I’d have to do this studying before I would be able to pull this trick off, though.)
My guess is that you’d want to flip it backside-up, and from the then bottom, pull 10 cards and flip the top one of that one over. Then you’ll have either a Jack (if it starts with an ace, I assume this to be very likely, 50%?), a 9 (if it starts with a placeholder card, like a rule card, I presume this to be probable… like, 20%?), a Queen (if the ace is last, a la Jack Queen King Ace, 20% for this one as well) an 8 (two jokers at the front of the deck? 8%?), a 7 (2 jokers AND a rule card?! 1%) or 1% of me just being wrong entirely. It sounds like a weird distribution, but rule cards and jokers tend to be at the back, and an ace with fancy artwork tends to be at the front of the deck because it looks good.
A bit of googling reveals that a new deck usually starts with the Ace of Spades, so I’d guess that flipping the deck over, then drawing 10 cards from the bottom of the deck (what used to be the front) and then flipping the 10th card over will give you a Jack of Spades.