Gains from trade: Slug versus Galaxy—how much would I give up to control you?

Edit: Moved to main at ThrustVec­tor­ing’s sug­ges­tion.

A sug­ges­tion as to how to split the gains from trade in some situ­a­tions.

The prob­lem of Power

A year or so ago, peo­ple in the FHI em­barked on a grand pro­ject: to try and find out if there was a sin­gle way of re­solv­ing ne­go­ti­a­tions, or a sin­gle way of merg­ing com­pet­ing moral the­o­ries. This pro­ject made a lot of progress in find­ing out how hard this was, but very lit­tle in terms of solv­ing it. It seemed ev­i­dent that the cor­rect solu­tion was to weigh the differ­ent util­ity func­tions, and then for ev­ery­one max­imise the weighted sum, but all ways of weight­ing had their prob­lems (the weight­ing with the most good prop­er­ties was a very silly one: use the “min-max” weight­ing that sets your max­i­mal at­tain­able util­ity to 1 and your min­i­mal to 0).

One thing that we didn’t get close to ad­dress­ing is the con­cept of power. If two part­ners in the ne­go­ti­a­tion have very differ­ent lev­els of power, then ab­stractly com­par­ing their util­ities seems the wrong solu­tion (more to the point: it wouldn’t be ac­cepted by the pow­er­ful party).

The New Repub­lic spans the Galaxy, with Jedi knights, bat­tle fleets, armies, gen­eral cool­ness, and the man­u­fac­tur­ing and hu­man re­sources of countless sys­tems at its com­mand. The dull slug, ARthUr­pHilIpDenu, moves very slowly around a plant, and pos­si­bly owns one leaf (or not—he can’t pro­duce the pa­per­work). Both these en­tities have prefer­ences, but if they meet up, and their util­ities are nor­mal­ised ab­stractly, then ARthUr­pHilIpDenu’s prefer­ences will weigh in far too much: a size­able frac­tion of the galaxy’s pro­duc­tion will go to­wards satis­fy­ing the slug. Even if you think this is “fair”, con­sider that the New Repub­lic is the merg­ing of countless in­di­vi­d­ual prefer­ences, so it doesn’t make any sense that the two util­ities get weighted equally.

The de­fault point

After look­ing at var­i­ous black­mail situ­a­tions, it seems to me that it’s the con­cept of de­fault, or sta­tus quo, that most clearly differ­en­ti­ates be­tween a threat and an offer. I wouldn’t want you to make a cred­ible threat, be­cause this wors­ens the sta­tus quo, I would want you to make a cred­ible offer, be­cause this im­proves it. How this de­fault is es­tab­lished is an­other mat­ter—there may be some su­per-UDT ap­proach that solves it from first prin­ci­ples. Maybe there is some deep way of dis­t­in­guish­ing be­tween threats and promises in some other way, and the de­fault is sim­ply the point be­tween them.

In any case, with­out go­ing any fur­ther into it’s mean­ing or deriva­tion, I’m go­ing to as­sume that the prob­lem we’re work­ing on has a defini­tive de­fault/​dis­agree­ment/​threat point. I’ll use the de­fault point ter­minol­ogy, as that is closer to the con­cept I’m con­sid­er­ing.

Sim­ple trade prob­lems of­ten have a very clear de­fault point. Th­ese are my goods, those are your goods, the de­fault is we go home with what we started with. This is what I could build, that’s what you could build, the de­fault is that we both build purely for our­selves.

If we imag­ine ARthUr­pHilIpDenu and the New Repub­lic were at op­po­site ends of a reg­u­lated worm­hole, and they could only trade in safe and sim­ple goods, then we’ve got a pretty clear de­fault point.

Hav­ing a de­fault point opens up a whole host of new bar­gain­ing equil­ibriums, such as the Nash Bar­gain­ing Solu­tion (NBS) and the Kalai-Smorod­in­sky Bar­gain­ing Solu­tion (KSBS). But nei­ther of these are re­ally quite what we’d want: the KSBKS is all about fair­ness (which gen­er­ally re­duced ex­pected out­comes), while the NBS uses a product of util­ity val­ues, some­thing that makes no in­trin­sic sense at all (NBS has some nice prop­er­ties, like in­de­pen­dence of ir­rele­vant al­ter­na­tives, but this only mat­ters if the de­fault point is reached through a pro­cess that has the same prop­er­ties—and it can’t be).

What am I re­ally offer­ing you in trade?

When two agents meet, es­pe­cially if they are likely to meet more in the fu­ture (and most es­pe­cially if they don’t know the num­ber of times and the cir­cum­stances in which they will meet), they should merge their util­ity func­tions: fix a com­mon scale for their util­ity func­tions, add them to­gether, and then both pro­ceed to max­imise the sum.

This ex­plains what’s re­ally be­ing offered in a trade. Not a few wid­gets or stars, but the pos­si­bil­ity of copy­ing your util­ity func­tion into mine. But why would you want that? Be­cause that will change my de­ci­sions, into a di­rec­tion you find more pleas­ing. So what I’m ac­tu­ally offer­ing you, is ac­cess to my de­ci­sion points.

What is ac­tu­ally on offer in a trade, is ac­cess by one player’s util­ity func­tion to the other player’s de­ci­sion points.

This gives a novel way of nor­mal­is­ing util­ity func­tions. How much, pre­cisely, is ac­cess to my de­ci­sion points worth to you? If the de­fault point gives a nat­u­ral zero, then com­plete con­trol over the other player’s de­ci­sion points is a nat­u­ral one. “Power” is a neb­u­lous con­cept, and differ­ent play­ers may dis­agree as to how much power they each have. But power can only be ar­tic­u­lated through mak­ing de­ci­sions (if you can’t change any of your de­ci­sions, you have no power), and this nor­mal­i­sa­tion al­lows each player to spec­ify ex­actly how much they value the power/​de­ci­sion points of the other. Out­comes that in­volve one player con­trol­ling the other player’s de­ci­sion points can be des­ig­nated the “utopia” point for that first player. Th­ese are what would hap­pen if ev­ery­thing went ex­actly ac­cord­ing to what they wanted.

What does this mean for ARthUr­pHilIpDenu and the New Repub­lic? Well, the New Repub­lic stands to gain a leaf (maybe). From it’s per­spec­tive, the differ­ence be­tween de­fault (all the re­sources of the galaxy and no leaf) and utopia (all the re­sources of the galaxy plus one leaf) is tiny. And yet that tiny differ­ence will get nor­mal­ised to one: the New Repub­lic’s util­ity func­tion will get mul­ti­plied by a huge amount. It will weigh heav­ily in any sum.

What about ARthUr­pHilIpDenu? It stands to gain the re­sources of a galaxy. The differ­ence be­tween de­fault (a leaf) and utopia (all the re­sources of a galaxy ded­i­cated to mak­ing leaves) is uni­mag­in­ably hu­mon­gous. And yet that huge differ­ence will get nor­mal­ised to one: the ARthUr­pHilIpDenu’s util­ity func­tion will get di­vided by a huge amount. It will weigh very lit­tle in any sum.

Thus if we add the two nor­mal­ised util­ity func­tions, we get one that is nearly to­tally dom­i­nated by the New Repub­lic. Which is what we’d ex­pect, given the power differ­en­tial be­tween the two. So this bar­gain­ing sys­tem re­flects the rel­a­tive power of the play­ers. Another way of think­ing of this is that each player’s util­ity is nor­mal­ised tak­ing into ac­count how much they would give up to con­trol the other. I’m call­ing it the “Mu­tual Worth Bar­gain­ing Solu­tion” (MWBS), as it’s the worth to play­ers of the other player’s de­ci­sion points that are key. Also be­cause I couldn’t think of a bet­ter ti­tle.

Prop­er­ties of the Mu­tual Worth Bar­gain­ing Solution

How does the MWBS com­pare with the NBS and the KSBS? The NBS is quite differ­ent, be­cause it has no con­cept of rel­a­tive power, nor­mal­is­ing purely by the play­ers’ prefer­ences. In­deed, one player could have no con­trol at all, no de­ci­sion points, and the NBS would still be un­changed.

The KSBS is more similar to the MWBS: the utopia points of the KSBS are the same as those of the MWBS. If we set the de­fault point to (0,0) and the utopia points to (1,-) and (-,1), then the KSBS is given by the high­est h such that (h,h) is a pos­si­ble out­come. Whereas the MWBS is given by the out­come (x,y) such that x+y is high­est pos­si­ble.

Which is prefer­able? Ob­vi­ously, if they knew ex­actly what the out­comes and util­ities were on offer, then each player would have prefer­ences as to which sys­tem to use (the one that gives them more). But if they didn’t, if they had un­cer­tain­ties as to what play­ers and what prefer­ences they would face in the fu­ture, then MWBS gen­er­ally comes out on top (in ex­pec­ta­tion).

How so? Well, if a player doesn’t know what other play­ers they’ll meet, they don’t know in what way their de­ci­sion points will be rele­vant to the other, and vice versa. They don’t know what pieces of their util­ity will be rele­vant to the other, and vice versa. So they can ex­pect to face a wide va­ri­ety of nor­mal­ised situ­a­tions. To a first ap­prox­i­ma­tion, it isn’t too bad an idea to as­sume that one is equally likely to face a cer­tain situ­a­tion as it’s sym­met­ric com­ple­ment. Us­ing the KSBS, you’d ex­pect to get a util­ity of h (same in both case); un­der the MWBS, a util­ity of (x+y)/​2 (x in one case, y in the other). Since x+y ≥ h+h = 2h by the defi­ni­tion of the MWBS, it comes out ahead in ex­pec­ta­tion.

Another im­por­tant dis­tinc­tion be­tween the MWBS is that while the KSBS and the NBS only al­low Pareto im­prove­ments from the de­fault point, MWBS does al­low for some situ­a­tion where one player will lose from the deal. It is pos­si­ble, for in­stance, that (1/​2,-1/​4) is a pos­si­ble out­come (summed util­ity 14), and there are no bet­ter op­tions pos­si­ble. Doesn’t this go against the spirit of the de­fault point? Why would some­one go into a deal that leaves them poorer than be­fore?

First off all, that situ­a­tion will be rare. All MWBS must be in the tri­an­gle bounded by x<1, y<1 and x+y>0. The first bounds are defi­ni­tional: one can­not get more ex­pected util­ity that one’s utopia point. The last bound comes from the fact that the de­fault point is it­self an op­tion, with summed util­ity 0+0=0, so all summed util­ities must be above zero. Sprin­kle a few ran­dom out­come points into that tri­an­gle, and it very likely that the one with high­est summed util­ity will be a Pareto im­prove­ment over (0,0).

But the other rea­son to ac­cept the risk of los­ing, is be­cause of the op­por­tu­nity of gain. One could mod­ify the MWBS to only al­low Pareto im­prove­ments over the de­fault: but in ex­pec­ta­tion, this would perform worse. The player would be im­mune from los­ing 14 util­ity from (1/​2,-1/​4), but un­able to gain 12 from the (-1/​4,1/​2): the ar­gu­ment is the same as above. In ig­no­rance as to the other player’s prefer­ences, ac­cept­ing the pos­si­bil­ity of loss im­proves the ex­pected out­come.

It should be noted that the max­i­mum that a player could the­o­ret­i­cally lose by us­ing the MWBS is equal to the max­i­mum they could the­o­ret­i­cally win. So the New Repub­lic could lose at most a leaf, mean­ing that even pow­er­ful play­ers would not be re­luc­tant to trade. For less pow­er­ful play­ers, the po­ten­tial losses are higher, but so are the po­ten­tial re­wards.

Direc­tions of research

The MWBS is some­what un­der­de­vel­oped, and the ex­pla­na­tion here isn’t as clear as I’d have liked. How­ever, me and Miriam are about to have a baby, so I’m not ex­pect­ing to have any time at all soon, so I’m push­ing out the idea, un­pol­ished.

Some pos­si­ble routes for fur­ther re­search: what are the other prop­er­ties of MWBS? Are they prop­er­ties that make MWBS feel more or less likely or ac­cept­able? The NBS is equiv­a­lent with cer­tain prop­er­ties: what are the prop­er­ties that are nec­es­sary and suffi­cient for the MWBS (and can they sug­gest bet­ter Bar­gain­ing Solu­tions)? Can we re­place the de­fault point? Maybe we can get a zero by imag­in­ing what would hap­pen if the sec­ond player’s de­ci­sion nodes were un­der the con­trol of an anti-agent (an agent that’s the op­po­site of the first player), or a ran­domly se­lected agent?

The most im­por­tant re­search route is to analyse what hap­pens if sev­eral play­ers come to­gether at differ­ent times, and re­peat­edly nor­mal­ise their util­ities us­ing the MWBS: does it mat­ter the or­der in which they meet? I strongly feel that it’s ex­plor­ing this av­enue that will reach “the ul­ti­mate” bar­gain­ing solu­tion, if such a thing is to be found. Some solu­tion that is sta­ble un­der large num­bers of agents, who don’t know each other or how many they are, com­ing to­gether in a or­der they can’t pre­dict.