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.
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.