If C and E—and I’d say all 4 of them really, at least regarding a 98 0 1 0 1 solution—were inclined to be outraged as I suggest, and A knew this, they would walk away with more money. For me, that trumps any possible math and logic you could put forward.
And just in case A is stupid:
“But look, C and E, this is the optimal solution, if you don’t listen to me you’ll get less gold!”
“Nice try, smartass. Overboard you go.”
B watched on, starting to sweat...
EDIT: Ooops, I notice that I missed the fact that B doesn’t need to sweat since he just needs D. Still, my main point isn’t about B, but A.
Also I wanna make it 100% clear: I don’t claim that the proof is incorrect, given all the assumptions of the problem, including the ones about how the agents work. I’m just not impressed with the agents, with their ability to achieve their goals. Leave A unchanged and toss in 4 reasonably bright real humans as B C D E, at least some of them will leave with more money.
...because it’s very hot in Pirate Island’s shark-infested Pirate Bay, not out of any fear or distress at an outcome that put her in a better position than she had occupied before.
Whereas A had to build an outright majority to get his plan approved, B just had to convince one other pirate. D was the natural choice, both from a strict logico-mathematical view, and because D had just watched C and E team up to throw their former captain overboard. It wasn’t that they were against betraying their superiors for a slice of the treasure, it was that the slice wasn’t big enough! D wasn’t very bright—B knew from sharing a schooner with him these last few months—but team CE had been so obliging as to slice a big branch off D’s decision tree. What was left was a stump. D could take her offer of 1 coin, or be left to the mercy of the outrageously blood-thirsty team CE.
C and E watched on, dreaming of all the wonderful things they could do with their 0 coins.
[I think the “rationality = winning” story holds here (in the case where A’s proposal passes, not in this weird counterfactual cul-de-sac) but in a more subtle way. The 98 0 1 0 1 solution basically gives a value of the ranks, i.e., how much a pirate should be willing to pay to get into that rank at the time treasure will be divided. From this perspective, being A is highly valuable, and A should have been willing to pay, say, 43 coins for his ship, 2 coins for powder, 7 coins for wages to B, C, D, E, etc., to make it into that position. C, on the other hand, might turn down a promotion to second-in-command over B, unless it’s paired with a wage hike of one 1 coin; B would be surprisingly happy to be given such a demotion, if her pay remained unchanged. All the pirates can win even in a 98 0 1 0 1 solution, if they knew such a treasure would be found and planned accordingly.]
It seems to me that the extent to which B C D E will be able to get more money is to some extent dependent on their ability to plausibly precommit to rejecting an “unfair” deal… and possibly their ability to plausibly precommit to accepting a “fair” one.
Emphasis on “plausibly” and “PIRATES.”
At minimum, if they can plausibly precommit to things, I’d expect at the very least CDE to precommit to tossing A B overboard no matter what is offered and splitting the pot three ways. There are quite possibly better commitments to make even than this.
Heh, if I can’t convince you with “any possible math and logic”, let me try with this great video (also that one) that shows the consequences of “reasoning via outrage”.
Watched the first one. It was very different from the scenario we’re discussing. No one’s life was at stake. Also the shares were unequal from the start, so there was no fair scenario being denied, to get outraged about.
I’m not in favor of “reasoning via outrage” in general. I’m simply in favor of possessing a (known) inclination to turn down overly skewed deals (like humans generally have, usefully I might add); if I have it, and your life is at stake, you’d have to be suicidal to propose a 98 0 1 0 1 if I’m one of the people whose vote you need.
What makes it different from the video example is that, in the pirate example, if I turn down the deal the proponent loses far, far more than I do. Not just 98 coins to my 1, but their life, which should be many orders of magnitude more precious. So there’s clearly room for a more fair deal. The woman in that case wasn’t like my proposed E or C, she was like a significantly stupider version of A, wanting an unfairly good deal in a situation when there was no reason for her to believe her commitment could reliably prevail over the other players’ ability to do the same.
Watched the first one. It was very different from the scenario we’re discussing. No one’s life was at stake. Also the shares were unequal from the start, so there was no fair scenario being denied, to get outraged about.
A suggestion to randomize was made and denied. They fail at thinking. Especially Jo, who kept trying to convince herself and others that she didn’t care about money. Sour grapes—really pathetic.
Whenever I come across highly counterintuitive claims along these lines, I code them up and see how they perform over many iterations.
This is a lot trickier to do in this case compared to, say, the Monty Hall problem, but if you restricted it just to cases in which Pirate A retained 98 of the coins, you could demonstrate whether the [98, 0, 1, 0, 1] distribution was stable or not.
Also, I’d suggest thinking about this in a slightly different way to the way you’re thinking about it. The only pirate in the scenario who doesn’t have to worry about dying is pirate E, who can make any demands he likes from pirate D. What distribution would he suggest?
Edit: Rereading the wording of the scenario, pirate E can’t make any demands he likes from pirate D, and pirate D himself also doesn’t need to worry about dying.
Heh, if I can’t convince you with “any possible math and logic”, let me try with this great video (also that one) that shows the consequences of “reasoning via outrage”.
That shows one aspect of the consequences of reasoning via outrage. It doesn’t indicate that the strategy itself is bad. In a similar way the consequences of randomizing and defending Podunk 1⁄11 times is that 1⁄11 times (against a good player) you will end up on youtube losing Metropolis.
“I will stand firm on A until A becomes worth less than C is now, then I will accept B or C. You get more by accepting B or C now than you get by trying to get more.”
One rational course of action is to mutually commit to a random split, and follow through. What’s the rational course of action to respond to someone who makes that threat and is believed to follow through on it? If it is known that the other two participants are rational, why isn’t making a threat of that nature rational?
If C and E—and I’d say all 4 of them really, at least regarding a 98 0 1 0 1 solution—were inclined to be outraged as I suggest, and A knew this, they would walk away with more money. For me, that trumps any possible math and logic you could put forward.
And just in case A is stupid:
“But look, C and E, this is the optimal solution, if you don’t listen to me you’ll get less gold!”
“Nice try, smartass. Overboard you go.”
B watched on, starting to sweat...
EDIT: Ooops, I notice that I missed the fact that B doesn’t need to sweat since he just needs D. Still, my main point isn’t about B, but A.
Also I wanna make it 100% clear: I don’t claim that the proof is incorrect, given all the assumptions of the problem, including the ones about how the agents work. I’m just not impressed with the agents, with their ability to achieve their goals. Leave A unchanged and toss in 4 reasonably bright real humans as B C D E, at least some of them will leave with more money.
...because it’s very hot in Pirate Island’s shark-infested Pirate Bay, not out of any fear or distress at an outcome that put her in a better position than she had occupied before.
Whereas A had to build an outright majority to get his plan approved, B just had to convince one other pirate. D was the natural choice, both from a strict logico-mathematical view, and because D had just watched C and E team up to throw their former captain overboard. It wasn’t that they were against betraying their superiors for a slice of the treasure, it was that the slice wasn’t big enough! D wasn’t very bright—B knew from sharing a schooner with him these last few months—but team CE had been so obliging as to slice a big branch off D’s decision tree. What was left was a stump. D could take her offer of 1 coin, or be left to the mercy of the outrageously blood-thirsty team CE.
C and E watched on, dreaming of all the wonderful things they could do with their 0 coins.
[I think the “rationality = winning” story holds here (in the case where A’s proposal passes, not in this weird counterfactual cul-de-sac) but in a more subtle way. The 98 0 1 0 1 solution basically gives a value of the ranks, i.e., how much a pirate should be willing to pay to get into that rank at the time treasure will be divided. From this perspective, being A is highly valuable, and A should have been willing to pay, say, 43 coins for his ship, 2 coins for powder, 7 coins for wages to B, C, D, E, etc., to make it into that position. C, on the other hand, might turn down a promotion to second-in-command over B, unless it’s paired with a wage hike of one 1 coin; B would be surprisingly happy to be given such a demotion, if her pay remained unchanged. All the pirates can win even in a 98 0 1 0 1 solution, if they knew such a treasure would be found and planned accordingly.]
It seems to me that the extent to which B C D E will be able to get more money is to some extent dependent on their ability to plausibly precommit to rejecting an “unfair” deal… and possibly their ability to plausibly precommit to accepting a “fair” one.
Emphasis on “plausibly” and “PIRATES.”
At minimum, if they can plausibly precommit to things, I’d expect at the very least CDE to precommit to tossing A B overboard no matter what is offered and splitting the pot three ways. There are quite possibly better commitments to make even than this.
Heh, if I can’t convince you with “any possible math and logic”, let me try with this great video (also that one) that shows the consequences of “reasoning via outrage”.
Watched the first one. It was very different from the scenario we’re discussing. No one’s life was at stake. Also the shares were unequal from the start, so there was no fair scenario being denied, to get outraged about.
I’m not in favor of “reasoning via outrage” in general. I’m simply in favor of possessing a (known) inclination to turn down overly skewed deals (like humans generally have, usefully I might add); if I have it, and your life is at stake, you’d have to be suicidal to propose a 98 0 1 0 1 if I’m one of the people whose vote you need.
What makes it different from the video example is that, in the pirate example, if I turn down the deal the proponent loses far, far more than I do. Not just 98 coins to my 1, but their life, which should be many orders of magnitude more precious. So there’s clearly room for a more fair deal. The woman in that case wasn’t like my proposed E or C, she was like a significantly stupider version of A, wanting an unfairly good deal in a situation when there was no reason for her to believe her commitment could reliably prevail over the other players’ ability to do the same.
A suggestion to randomize was made and denied. They fail at thinking. Especially Jo, who kept trying to convince herself and others that she didn’t care about money. Sour grapes—really pathetic.
Whenever I come across highly counterintuitive claims along these lines, I code them up and see how they perform over many iterations.
This is a lot trickier to do in this case compared to, say, the Monty Hall problem, but if you restricted it just to cases in which Pirate A retained 98 of the coins, you could demonstrate whether the [98, 0, 1, 0, 1] distribution was stable or not.
Also, I’d suggest thinking about this in a slightly different way to the way you’re thinking about it. The only pirate in the scenario who doesn’t have to worry about dying is pirate E, who can make any demands he likes from pirate D. What distribution would he suggest?
Edit: Rereading the wording of the scenario, pirate E can’t make any demands he likes from pirate D, and pirate D himself also doesn’t need to worry about dying.
That shows one aspect of the consequences of reasoning via outrage. It doesn’t indicate that the strategy itself is bad. In a similar way the consequences of randomizing and defending Podunk 1⁄11 times is that 1⁄11 times (against a good player) you will end up on youtube losing Metropolis.
Whoever came up with that game show is a genius.
“I will stand firm on A until A becomes worth less than C is now, then I will accept B or C. You get more by accepting B or C now than you get by trying to get more.”
One rational course of action is to mutually commit to a random split, and follow through. What’s the rational course of action to respond to someone who makes that threat and is believed to follow through on it? If it is known that the other two participants are rational, why isn’t making a threat of that nature rational?