Conflict vs. mistake in non-zero-sum games

Sum­mary: Whether you be­have like a mis­take the­o­rist or a con­flict the­o­rist may de­pend more on your ne­go­ti­at­ing po­si­tion in a non-zero-sum game than on your wor­ld­view.

Dis­claimer: I don’t re­ally know game the­ory.

Plot the pay­offs in a non-zero-sum two-player game, and you’ll get a con­vex[1] set with the Pareto fron­tier on the top and right:

Payoff to player 2 vs. payoff to player 1

You can de­scribe this set with two pa­ram­e­ters: The sur­plus is how close the out­come is to the Pareto fron­tier, and the al­lo­ca­tion tells you how much the out­come fa­vors player 1 ver­sus player 2. In this illus­tra­tion, the level sets for sur­plus and al­lo­ca­tion are de­picted by con­cen­tric curves and ra­dial lines, re­spec­tively.

It’s tempt­ing to de­com­pose the game into two phases: A co­op­er­a­tive phase, where the play­ers co­or­di­nate to max­i­mize sur­plus; and a com­pet­i­tive phase, where the play­ers ne­go­ti­ate how the sur­plus is al­lo­cated.

Of course, in the usual for­mu­la­tion, both phases oc­cur si­mul­ta­neously. But this sug­gests a cou­ple of ne­go­ti­a­tion strate­gies where you try to make one phase hap­pen be­fore the other:

  1. “Let’s agree to max­i­mize sur­plus. Once we agree to that, we can talk about al­lo­ca­tion.”

  2. “Let’s agree on an al­lo­ca­tion. Once we do that, we can talk about max­i­miz­ing sur­plus.”

I’m go­ing to provoca­tively call the first strat­egy mis­take the­ory, and the sec­ond con­flict the­ory.

In­deed, the mis­take the­ory strat­egy pushes the ob­vi­ously good plan of mak­ing things bet­ter off for ev­ery­one. It can frame all op­po­si­tion as mak­ing the mis­take of leav­ing sur­plus on the table.

The con­flict the­ory strat­egy threat­ens to de­stroy sur­plus in or­der to get a more fa­vor­able al­lo­ca­tion. Its nar­ra­tive em­pha­sizes the fact that the play­ers can’t max­i­mize their re­wards si­mul­ta­neously.

Now I don’t have a good model of ne­go­ti­a­tion. But in­tu­itively, it seems that mis­take the­ory is a good strat­egy if you think you’ll be in a bet­ter ne­go­ti­at­ing po­si­tion once you move to the Pareto fron­tier. And con­flict the­ory is a good strat­egy if you think you’ll be in a worse ne­go­ti­at­ing po­si­tion at the Pareto fron­tier.

If you’re nat­u­rally a mis­take the­o­rist, this might make con­flict the­ory seem more ap­peal­ing. Imag­ine ne­go­ti­at­ing with a pa­per­clip max­i­mizer over the fate of billions of lives. Mu­tual co­op­er­a­tion is Pareto effi­cient, but un­ap­peal­ing. It’s more sen­si­ble to threaten defec­tion in or­der to save a few more hu­man lives, if you can get away with it.

It also makes mis­take the­ory seem un­sa­vory: Ap­par­ently mis­take the­ory is about post­pon­ing the al­lo­ca­tion ne­go­ti­a­tion un­til you’re in a com­fortable ne­go­ti­at­ing po­si­tion. (Or, some­what bet­ter: It’s about trick­ing the other play­ers into co­op­er­at­ing be­fore they can ex­tract con­ces­sions from you.)

This is kind of un­fair to mis­take the­ory, which is sup­posed to be about ed­u­cat­ing de­ci­sion-mak­ers on effi­cient poli­cies and build­ing in­sti­tu­tions to en­able co­op­er­a­tion. None of that is pre­sent in this model.

But I think it de­scribes some­thing im­por­tant about mis­take the­ory which is usu­ally rounded off to some­thing like “[mis­take the­o­rists have] be­come part of a class that’s more in­ter­ested in pro­tect­ing its own priv­ileges than in helping the poor or work­ing for the good of all”.

The rea­son I’m think­ing about this is that I want a the­ory of non-zero-sum games in­volv­ing coun­ter­fac­tual rea­son­ing and su­per­ra­tional­ity. It’s not clear to me what su­per­ra­tional agents “should” do in gen­eral non-zero-sum games.