Don’t Get Distracted by the Boilerplate

Author’s Note: Please don’t get scared off by the first sen­tence. I promise it’s not as bad as it sounds.

There’s a the­o­rem from the early days of group the­ory which says that any con­tin­u­ous, mono­tonic func­tion which does not de­pend on the or­der of its in­puts can be trans­formed to ad­di­tion. A good ex­am­ple is mul­ti­pli­ca­tion of pos­i­tive num­bers: f(x, y, z) = x*y*z. It’s con­tin­u­ous, it’s mono­tonic (in­creas­ing any of x, y, or z in­creases f), and we can change around the or­der of in­puts with­out chang­ing the re­sult. In this case, f is trans­formed to ad­di­tion us­ing a log­a­r­ithm: log(f(x, y, z)) = log(x) + log(y) + log(z).

Now, at first glance, we might say this is a very spe­cial­ized the­o­rem. “Con­tin­u­ous” and “mono­tonic” are very strong con­di­tions; they’re not both go­ing to ap­ply very of­ten. But if we ac­tu­ally look through the proof, it be­comes clear that these as­sump­tions aren’t as im­por­tant as they look. Weak­en­ing them does change the the­o­rem, but the core idea re­mains. For in­stance, if we re­move mono­ton­ic­ity, then our func­tion can still be writ­ten in terms of vec­tor ad­di­tion.

Many the­o­rems/​proofs con­tain pieces which are re­ally just there for mod­el­ling pur­poses. The cen­tral idea of the proof can ap­ply in many differ­ent set­tings, but we need to pick one of those set­tings in or­der to for­mal­ize it. This cre­ates some math­e­mat­i­cal boiler­plate. Typ­i­cally, we pick a set­ting which keeps the the­o­rem sim­ple—but that may in­volve stronger boiler­plate as­sump­tions than are strictly nec­es­sary for the main idea.

In such cases, we can usu­ally re­lax the boiler­plate as­sump­tions and end up with slightly weaker forms of the the­o­rem, which nonethe­less main­tain the cen­tral con­cepts.

Un­for­tu­nately, the boiler­plate oc­ca­sion­ally dis­tracts peo­ple who aren’t fa­mil­iar with the full idea un­der­ly­ing the proof. For some rea­son, I see this prob­lem most with the­o­rems in eco­nomics, game the­ory and de­ci­sion the­ory—the sort of the­o­rems which say “ei­ther X, or some­body is giv­ing away free money”. Peo­ple will come along and say “but wait, the the­o­rem as­sumes Y, which is com­pletely un­re­al­is­tic!” But re­ally, Y is of­ten just boiler­plate, and the core ideas still ap­ply even if Y is re­laxed to some­thing more re­al­is­tic. In fact, in many cases, the con­fu­sion is over the word­ing of the boiler­plate! Just be­cause we use the word “bet”, doesn’t mean peo­ple need to be at a cas­ino for the the­o­rem to ap­ply.

A few ex­am­ples:

  • “VNM util­ity the­o­rem is un­re­al­is­tic! It re­quires that we have prefer­ences over ev­ery pos­si­ble state of the uni­verse.” Re­sponse: Com­plete­ness is re­ally just there to keep the math clean. The core ideas of the proof still show that, if we don’t have a util­ity func­tion over some neigh­bor­hood of world-states, then we can be ex­ploited us­ing only those world-states.

  • “All these ra­tio­nal­ity the­o­rems are un­re­al­is­tic! They’re only rele­vant to wor­lds where evil agents are con­stantly run­ning around look­ing to ex­ploit us.” Re­sponse: We face trade-offs in the real world, and if we don’t choose be­tween the op­tions con­sis­tently, we’ll end up throw­ing away re­sources un­nec­es­sar­ily. Whether an evil per­son ma­nipu­lates us into it, or we stum­ble into it, isn’t re­ally rele­vant. The more situ­a­tions where we lo­cally vi­o­late VNM util­ity, the more situ­a­tions where we’ll lose re­sources.

  • “VNM util­ity the­o­rem is un­re­al­is­tic! It as­sumes we’re will­ing to ac­cept ei­ther a trade or its op­po­site (or both) - rather than just ig­nor­ing offers.” Re­sponse: We face trade-offs in the real world where “ig­nore” is not an op­tion, and if we don’t choose be­tween the op­tions con­sis­tently, we’ll end up throw­ing away re­sources un­nec­es­sar­ily.

  • “Dutch Book the­o­rems are un­re­al­is­tic! They as­sume we’re will­ing to ac­cept ei­ther a bet or it’s op­po­site, rather than ig­nor­ing both.” Re­sponse: same as pre­vi­ous. Alter­na­tively, we can build bid-ask spreads into the model, and most of the struc­ture re­mains.

  • “Dutch Book The­o­rems are un­re­al­is­tic! They as­sume we’re con­stantly mak­ing bets on ev­ery­thing pos­si­ble.” Re­sponse: ev­ery time we make a de­ci­sion un­der un­cer­tainty, we make a bet. Do so in­con­sis­tently, and we throw away re­sources un­nec­es­sar­ily.

In clos­ing, one im­por­tant note: I definitely do not want to claim that all ob­jec­tions to the use of VNM util­ity the­o­rem, Dutch Book the­o­rems, etc make this kind of mis­take.