Moloch games

tl;dr: This post sug­gests a di­rec­tion for mod­el­ling Molochs. The main thing this post does is to re­name the con­cept of “po­ten­tial games” (an ex­ist­ing con­cept in game the­ory) to “Moloch games” to sug­gest an in­ter­pre­ta­tion of this class of games. I also define “the prefer­ences of a Moloch” to gen­er­al­ize that no­tion (the prefer­ences may be in­tran­si­tive).

This post as­sumes that you have fa­mil­iar­ity with game the­ory and the con­cept of a Moloch.


What do group dy­nam­ics want? If a so­ciety/​group (the Moloch) wants things that are differ­ent from what the in­di­vi­d­u­als want, how can we as­sign prefer­ences or a util­ity func­tion to that so­ciety/​group that doesn’t model what would be good for the ag­gre­gated prefer­ence of the in­di­vi­d­u­als of the group, but that de­scribes what the group dy­nam­ics is ac­tu­ally “try­ing” to achieve (even against the in­ter­est of its in­di­vi­d­ual mem­bers)? Here is a sug­gested an­swer.

In­tu­ition. A Moloch game is a game such that there is a util­ity func­tion , called “the Moloch’s util­ity func­tion”, such that if the agents be­have in­di­vi­d­u­ally ra­tio­nally, then they col­lec­tively be­have as a “Moloch” that con­trols all play­ers si­mul­ta­neously and op­ti­mizes . In par­tic­u­lar, the Nash equil­ibria cor­re­spond to lo­cal op­tima of .

Not all games are Moloch games.

Defi­ni­tion. A game with finite num­ber of play­ers and for each player a strat­egy space is a car­di­nal Moloch game (in the game the­ory liter­a­ture, a car­di­nal po­ten­tial game), if there is a util­ity func­tion such that for all play­ers , and all strate­gies for the other play­ers,

In­tu­itively, if you take any strat­egy-pro­file for all the play­ers, and ad­just the strat­egy of one player, then the Moloch’s util­ity will in­crease/​de­crease by the same amount as the util­ity func­tion of that par­tic­u­lar player. Hence, in­tu­itively, ev­ery player be­haves always as if they are op­ti­miz­ing the Moloch’s util­ity func­tion.

The defi­ni­tion for an or­di­nal Moloch game re­places the con­di­tion with

In­tu­itively, rep­re­sents the Moloch’s prefer­ences or­di­nally but not car­di­nally. Ob­vi­ously, car­di­nal Moloch games are also or­di­nal Moloch games.


Ex­am­ple. The pris­on­ers dilemma:

Player 2
Co­op­er­ateDefec­t
Player 1Co­op­er­ate1, 1-1, 2
Defect2, −10 , 0

We will show that this is a car­di­nal Moloch game, by just com­put­ing the car­di­nal util­ity func­tion and show­ing that there are no in­con­sis­ten­cies:

How to com­pute the car­di­nal util­ity func­tion of the Moloch: Pick an ar­bi­trary strat­egy pro­file to have util­ity 0 (I take Defect, Defect). Then iter­a­tively com­pute the util­ity of rows and columns by just ap­ply­ing the con­straint that the defi­ni­tion gives: Com­pute the differ­ence in util­ity of the player whose row/​column you’re mov­ing along (i.e. player 2 for the rows, player 1 for the columns) of each cell in the row/​column from a cell of which you know the value of . In this case, we know . So com­pute as which equals . Similar for . For , there are two ways to com­pute it: Us­ing player 1′s util­ity func­tion and or player 2′s util­ity func­tion and . If these two give differ­ent an­swers, then the game is not a car­di­nal Moloch game.

Here is the car­di­nal util­ity func­tion of the Moloch for the pris­oner’s dilemma (The above al­gorithm gives a util­ity func­tion that is unique up to trans­la­tions) :

Co­op­er­ateDefec­t
Co­op­er­ate-3-1
Defect-10

In­tu­ition. In this case, even though all play­ers pre­fer Co­op­er­ate, Co­op­er­ate over Defect, Defect, the Moloch prefers the op­po­site. This cor­re­sponds to the fact that it is in­di­vi­d­u­ally ra­tio­nal for the play­ers to Defect. This Moloch util­ity func­tion cap­tures the “prefer­ences of the group dy­nam­ics” as op­posed to the prefer­ences of the in­di­vi­d­u­als. (It is ob­vi­ously very differ­ent from the no­tion of “ag­gre­gate prefer­ences” or “welfare”).


A Moloch game as­sumes in some sense that “the Moloch has tran­si­tive prefer­ences”. We can gen­er­al­ize to Molochs with pos­si­bly in­tran­si­tive prefer­ences (I don’t know of this be­ing defined this way in the liter­a­ture on po­ten­tial games):

Defi­ni­tion. Let be a game with a finite num­ber of play­ers, each of which has a prefer­ence re­la­tion over the strat­egy space (by de­fault de­rived from a util­ity func­tion ). Then the Moloch’s prefer­ences are defined as the prefer­ence re­la­tion satis­fy­ing for all play­ers , and all strat­egy pro­files for the other play­ers:

Ob­ser­va­tion. Th­ese prefer­ences are always in­com­plete (in­tu­itively, the Moloch doesn’t have an opinion on the com­par­i­son be­tween differ­ent play­ers chang­ing their strate­gies, be­cause it doesn’t have this in­for­ma­tion: play­ers in­di­vi­d­u­ally make choices given their op­tions). They may be ei­ther tran­si­tive or in­tran­si­tive. I’ll say a Moloch’s prefer­ences are ra­tio­nal if they are tran­si­tive (ne­glect­ing the usual re­quire­ment of com­plete­ness).

Just to show that the con­cepts are what they should be:

Lemma. Any game whose Moloch has tran­si­tive prefer­ences is an or­di­nal Moloch game. Any or­di­nal Moloch game has a Moloch with tran­si­tive prefer­ences.

Proof. For any tran­si­tive re­la­tion on a space there is a real-val­ued func­tion on it that is con­sis­tent with that re­la­tion. The other di­rec­tion fol­lows di­rectly.

In­tu­ition. If the Moloch has tran­si­tive prefer­ences, then the Moloch knows what it wants and the game will have a pure Nash equil­ibrium (there is a the­o­rem that for­mal­izes this). Con­versely, if the Moloch has in­tran­si­tive prefer­ences, then the Moloch doesn’t know what it wants and the game will tend to have cy­cles (not all of them will be­cause the play­ers might want to move out of them into a “tran­si­tive re­gion” of the Moloch’s prefer­ences).

I won’t show this here, but the liter­a­ture on po­ten­tial games (cf. the thing I am call­ing Moloch games), these are ex­am­ples:

Games with ra­tio­nal Molochs (i.e. Moloch games /​ po­ten­tial games):

  • Pri­soner’s dilemma

  • Bat­tle of the sexes

  • Co­or­di­na­tion game

  • Game of Chicken

Games with ir­ra­tional Molochs (i.e. not Moloch games /​ po­ten­tial games):

  • Match­ing pennies

  • Rock pa­per scissors


I prob­a­bly won’t spend much more time on this, but here is a sug­ges­tion for tak­ing this as a start­ing point to mod­el­ling Molochs:

  • Check if var­i­ous in­for­mal ideas about Molochs can be phrased in this lan­guage. Check if the lan­guage is satis­fy­ing to talk about ac­tual Molochs.

  • Look at the liter­a­ture on po­ten­tial games to see if it con­tains much in­sight. Make a dic­tio­nary of con­cepts named in the ter­minol­ogy of the on­tol­ogy we’re in­ter­ested in (similar to how I re­named “po­ten­tial game” to “Moloch game”) to make this liter­a­ture an “effi­ciently queryable database” for in­sights into Molochs.

  • I sus­pect that there might be ideas to be had about Moloch games that aren’t treated there, be­cause as far as I know, po­ten­tial games were de­vel­oped mostly as a trick to make com­pu­ta­tions eas­ier, not as a con­cep­tual tool for think­ing about Molochs, so­cietal in­ad­e­quacy and so forth. It’s plau­si­ble that cer­tain ob­vi­ous ques­tions haven’t been asked about them for this rea­son. Try to ac­tu­ally model Molochs this way and see if these defi­ni­tions al­low us to an­swer ques­tions we want to ask about them. Use this as a step­ping stone and see where it is un­satis­fy­ing. Build on top of that to push the anal­y­sis fur­ther.

Feel free to con­tact me if you want to think about this.


Some read­ing:

Flows and De­com­po­si­tions of Games: Har­monic and Po­ten­tial Games. In the lan­guage of this post: de­com­pos­ing a game into a “ra­tio­nal part” of the Moloch, and an ir­ra­tional de­vi­a­tion from it. Find­ing the “clos­est ra­tio­nal Moloch” of a game.

Some fur­ther ideas and ques­tions to ask:

  • Can real world so­cieties be de­com­posed into mul­ti­ple Molochs? In the style of the “Flows and De­com­po­si­tions of Games” pa­per, it wouldn’t have to be a de­com­po­si­tion in terms of sub­groups of play­ers, but of “as­pects of the game-the­o­retic in­ter­ac­tion”. (e.g. an in­di­vi­d­ual might si­mul­ta­neously be part of a “cap­i­tal­ism Moloch” and a “poli­tics Moloch”). Maybe Molochs can be ap­prox­i­mately de­com­posed.

  • Is there a no­tion of “Moloch game” for se­quen­tial games? Games with limited in­for­ma­tion? (The po­ten­tial game liter­a­ture prob­a­bly has asked analo­gous ques­tions).