If I understand the proposal correctly, I think this bargaining solution can heavily favor someone with diminishing marginal utility vs someone with linear utility. For example suppose Alice and Bob each own 1 (infinitely divisible) unit of resource, and that’s the default point. Alice values all resources linearly, with utility function A(a,b)=a, where a is resources consumed by Alice, b is resources consumed by Bob. Bob’s utility function is B(a,b) = b if b<=1, else 1+(b-1)/9 if b<=1.9, else 1.1. Normalization causes Bob’s utility function to be multiplied by 10, and the bargaining solution ends up giving Bob 1.9 units of resources. Correct?
Here’s another strange property of the bargaining solution. (Again I would appreciate confirmation that I’m understanding it correctly.) Suppose Alice and Carol both have linear utility, and Alice owns a tiny bit more resources than Carol. Then Alice’s utility function gets normalized to have a higher marginal utility than Carol, and all the resources go to Alice. If Carol instead brought a tiny bit more resources to the table, then all the resources would go to Carol.
I find it puzzling that the post is getting upvoted so much with so little discussion. Is anyone else looking at whether the solution produces outcomes that make intuitive sense?
It’s even worse. Suppose Alice and Carol are as you say, with Alice having x more resources than Carol. Before the trade, Alice is given the chance to benefit them both; if she accepts the offer, she will gain y resources, and Carol will gain y+2x. This is a pareto improvement, and prima facie it seems she should do so.
But alas! If she does so, Carol will have x more resources that Alice does—so all the resources will go to Carol! So Alice definitely does not want to accept the offer—even though accepting Omega’s offer could leave them both better off post-trade.
Ideally the default point would be set before Omega’s offer. After all, that is a decision point of Alice’s, that Carol woul value being able to decide...
I’m certainly not looking at the solution; I have no idea how to work any of this maths, and am mostly skimming along with a bazillion other blogs anyway! I know I trust Lesswrong, so obviously if there’s a confident-sounding post here I assume it’s correct.
You are correct. But if both utilities are linear in resources, then the outcome space forms a line in the joint utility space. And when maximising utility versus a line, the generic outcome is to pick one end of the line or the other, never a midpoint.
This is a feature of any approach that maximises a summed utility: you (almost always) will pick the end point of any straight segment.
To me that suggests looking at approaches (like NBS) that don’t involve directly maximizing summed utility. The solution could still guarantee a Pareto optimal outcome, which means it ends up being maximal for some summed utility function, but you’re much more likely to end up in the middle of a line segment (which often seem to be the most intuitively acceptable outcome).
Why? If you can end up on either end of a long line segment, then you have a chance of winning a lot or losing a lot. But you shouldn’t be risk averse with your utility—risk aversion should already be included. So “towards the middle” is no better in expectation than “right end or left end”.
Maybe you’re thinking we shouldn’t be maximising expected utility? I’m actually quite sympathetic to that view...
And with complex real world valuations (eg anything with a diminishing marginal utility), then any Pareto line segments are likely to be short.
Nonlinear utility functions (as a function of resources) do not accurately model human risk aversion. That could imply that we should either change our (or they/their) risk aversion or not be maximising expected utility.
Nonlinear jumps in utility from different amounts of a resource seem common for humans at least at some points in time. Example: Either I have enough to pay off the loan shark, or he’ll break my legs.
If you have a line segment that crosses the quadrant of joint utility space that represents Pareto improvements over the status quo, then ending up on an end means one of the parties is made worse off. To generalize this observation, it’s hard to guarantee that no one is made worse off, unless the bargaining solution explicitly tries to do that. If you maximize summed utility, and your weights are not picked to ensure that the outcome is a Pareto improvement (which generally involves picking the Pareto improvement first and then working backwards to find the weights), then there will be situations where it makes one party worse off.
You talk about “in expectation” but I’m not sure that changes the picture. It seems like the same argument applies: you can’t guarantee that nobody is made worse off in expectation, unless you explicitly try to.
You can restrict the MWBS to only consider strict Pareto improvements over default, if you want. That’s another bargaining solution—call it PMWBS.
My (informal) argument was that in situations of uncertainty as to who you are facing, MWBS gives you a higher expected value than PMWBS (informally because you expect to gain more when the deal disadvantages your opponent, than you expect to lose when it disadvantages you). Since the expected value of PMWBS is evidently positive, that of MWBS must be too.
I think you may have to formalize this to figure out what you need to assume to make the argument work. Clearly MWBS doesn’t always give positive expected value in situations of uncertainty as to who you are facing. For example suppose Alice expects to face either Carol or Dave, with 50⁄50 probability. Carol has slightly more resources than Alice, and Dave has almost no resources. All three have linear utility. Under MWBS, Alice now has 50% probability of losing everything and 50% probability of gaining a small amount.
I was trying to assume maximum ignorance—maximum uncertainty as to who you might meet, and their abilities and values.
If you have a better idea as to what you face, then you can start shopping around bargaining solutions to get the one you want. And in your example, Alice would certainly prefer KSBS and NBS over PMWBS, which she would prefer over MWBS.
But if, for instance, Dave had slightly less resources that Alice, then it’s no longer true. And if any of them depart from (equal) linear utility in every single resource, then it’s likely no longer true either.
Indeed. That is discussed in the last part of the “properties” section.
I think the argument is that this possible outcome is acceptable to Alice because she expects an equal chance of encountering trade opportunities where she benefits from the bargain.
I see a similarity between this risk and the Newcomb problem, but I’m not sure what additional assumptions this brings into the theory. What knowledge of your trading partner’s decision mechanisms (source code) is necessary to commit to this agreement?
Seems correct. But high marginal utility in strategic places can work too. What if Alice had linear utility, up until 1.05, where she suddenly has utility 2? Then the split is Alice 1.05, Bob 0.95.
Diminishing marginal returns isn’t so much the issue, rather it’s the low utopia point (as a consequence of diminishing marginal returns).
If I understand the proposal correctly, I think this bargaining solution can heavily favor someone with diminishing marginal utility vs someone with linear utility. For example suppose Alice and Bob each own 1 (infinitely divisible) unit of resource, and that’s the default point. Alice values all resources linearly, with utility function A(a,b)=a, where a is resources consumed by Alice, b is resources consumed by Bob. Bob’s utility function is B(a,b) = b if b<=1, else 1+(b-1)/9 if b<=1.9, else 1.1. Normalization causes Bob’s utility function to be multiplied by 10, and the bargaining solution ends up giving Bob 1.9 units of resources. Correct?
Here’s another strange property of the bargaining solution. (Again I would appreciate confirmation that I’m understanding it correctly.) Suppose Alice and Carol both have linear utility, and Alice owns a tiny bit more resources than Carol. Then Alice’s utility function gets normalized to have a higher marginal utility than Carol, and all the resources go to Alice. If Carol instead brought a tiny bit more resources to the table, then all the resources would go to Carol.
I find it puzzling that the post is getting upvoted so much with so little discussion. Is anyone else looking at whether the solution produces outcomes that make intuitive sense?
It’s even worse. Suppose Alice and Carol are as you say, with Alice having x more resources than Carol. Before the trade, Alice is given the chance to benefit them both; if she accepts the offer, she will gain y resources, and Carol will gain y+2x. This is a pareto improvement, and prima facie it seems she should do so.
But alas! If she does so, Carol will have x more resources that Alice does—so all the resources will go to Carol! So Alice definitely does not want to accept the offer—even though accepting Omega’s offer could leave them both better off post-trade.
Ideally the default point would be set before Omega’s offer. After all, that is a decision point of Alice’s, that Carol woul value being able to decide...
I’m certainly not looking at the solution; I have no idea how to work any of this maths, and am mostly skimming along with a bazillion other blogs anyway! I know I trust Lesswrong, so obviously if there’s a confident-sounding post here I assume it’s correct.
[/confession]
You are correct. But if both utilities are linear in resources, then the outcome space forms a line in the joint utility space. And when maximising utility versus a line, the generic outcome is to pick one end of the line or the other, never a midpoint.
This is a feature of any approach that maximises a summed utility: you (almost always) will pick the end point of any straight segment.
It is the fundamental theorem of linear programming.
To me that suggests looking at approaches (like NBS) that don’t involve directly maximizing summed utility. The solution could still guarantee a Pareto optimal outcome, which means it ends up being maximal for some summed utility function, but you’re much more likely to end up in the middle of a line segment (which often seem to be the most intuitively acceptable outcome).
Why? If you can end up on either end of a long line segment, then you have a chance of winning a lot or losing a lot. But you shouldn’t be risk averse with your utility—risk aversion should already be included. So “towards the middle” is no better in expectation than “right end or left end”.
Maybe you’re thinking we shouldn’t be maximising expected utility? I’m actually quite sympathetic to that view...
And with complex real world valuations (eg anything with a diminishing marginal utility), then any Pareto line segments are likely to be short.
Nonlinear utility functions (as a function of resources) do not accurately model human risk aversion. That could imply that we should either change our (or they/their) risk aversion or not be maximising expected utility.
Nonlinear jumps in utility from different amounts of a resource seem common for humans at least at some points in time. Example: Either I have enough to pay off the loan shark, or he’ll break my legs.
Yep. Humans are not expected utility maximisers. But there’s strong arguments that an AI would be...
If you have a line segment that crosses the quadrant of joint utility space that represents Pareto improvements over the status quo, then ending up on an end means one of the parties is made worse off. To generalize this observation, it’s hard to guarantee that no one is made worse off, unless the bargaining solution explicitly tries to do that. If you maximize summed utility, and your weights are not picked to ensure that the outcome is a Pareto improvement (which generally involves picking the Pareto improvement first and then working backwards to find the weights), then there will be situations where it makes one party worse off.
You talk about “in expectation” but I’m not sure that changes the picture. It seems like the same argument applies: you can’t guarantee that nobody is made worse off in expectation, unless you explicitly try to.
You can restrict the MWBS to only consider strict Pareto improvements over default, if you want. That’s another bargaining solution—call it PMWBS.
My (informal) argument was that in situations of uncertainty as to who you are facing, MWBS gives you a higher expected value than PMWBS (informally because you expect to gain more when the deal disadvantages your opponent, than you expect to lose when it disadvantages you). Since the expected value of PMWBS is evidently positive, that of MWBS must be too.
I think you may have to formalize this to figure out what you need to assume to make the argument work. Clearly MWBS doesn’t always give positive expected value in situations of uncertainty as to who you are facing. For example suppose Alice expects to face either Carol or Dave, with 50⁄50 probability. Carol has slightly more resources than Alice, and Dave has almost no resources. All three have linear utility. Under MWBS, Alice now has 50% probability of losing everything and 50% probability of gaining a small amount.
I was trying to assume maximum ignorance—maximum uncertainty as to who you might meet, and their abilities and values.
If you have a better idea as to what you face, then you can start shopping around bargaining solutions to get the one you want. And in your example, Alice would certainly prefer KSBS and NBS over PMWBS, which she would prefer over MWBS.
But if, for instance, Dave had slightly less resources that Alice, then it’s no longer true. And if any of them depart from (equal) linear utility in every single resource, then it’s likely no longer true either.
Indeed. That is discussed in the last part of the “properties” section.
I think the argument is that this possible outcome is acceptable to Alice because she expects an equal chance of encountering trade opportunities where she benefits from the bargain.
I see a similarity between this risk and the Newcomb problem, but I’m not sure what additional assumptions this brings into the theory. What knowledge of your trading partner’s decision mechanisms (source code) is necessary to commit to this agreement?
Seems correct. But high marginal utility in strategic places can work too. What if Alice had linear utility, up until 1.05, where she suddenly has utility 2? Then the split is Alice 1.05, Bob 0.95.
Diminishing marginal returns isn’t so much the issue, rather it’s the low utopia point (as a consequence of diminishing marginal returns).