Kelly and the Beast

Poor models

Con­sider two al­ter­na­tive ex­pla­na­tions to each of the fol­low­ing ques­tions:

  • Why do some birds have brightly-col­ored feathers? Be­cause (a) evolu­tion has found that they are bet­ter able to at­tract mates with such feathers or (b) that’s just how it is.

  • Why do some moths, af­ter a few gen­er­a­tions, change color to that of sur­round­ing man-made struc­tures? Be­cause (a) evolu­tion has found that the change in color helps the moths hide from preda­tors or (b) that’s just how it is.

  • Why do some cells com­mu­ni­cate pri­mar­ily via me­chan­i­cal im­pulses rather than elec­tro­chem­i­cal im­pulses? Be­cause: (a) for such cells, evolu­tion has found a trade-off be­tween en­ergy con­sump­tion, in­for­ma­tion trans­fer, and in­for­ma­tion pro­cess­ing such that me­chan­i­cal im­pulses are preferred or (b) that’s just how it is.

  • Why do some cells com­mu­ni­cate pri­mar­ily via elec­tro­chem­i­cal im­pulses rather than me­chan­i­cal im­pulses? Be­cause: (a) for such cells, evolu­tion has found a trade-off be­tween en­ergy con­sump­tion, in­for­ma­tion trans­fer, and in­for­ma­tion pro­cess­ing such that elec­tro­chem­i­cal im­pulses are preferred or (b) that’s just how it is.

Clearly the first set of ex­pla­na­tions are bet­ter, but I’d like to say a few things in defense of the sec­ond.

  • The prefer­ence of evolu­tion to­wards one against the other could very likely have noth­ing to do with mat­ing, preda­tors, en­ergy con­sump­tion, in­for­ma­tion trans­fer, or in­for­ma­tion pro­cess­ing. Those are the best the­o­rized guesses we have, and they have no ex­per­i­men­tal back­ing.

  • Evolu­tion works as a jus­tifi­ca­tion for con­tra­dic­tory phe­nom­ena.

  • The sec­ond set of ex­pla­na­tions are sim­pler.

  • The sec­ond set of ex­pla­na­tions have perfect sen­si­tivity, speci­fic­ity, pre­ci­sion, etc.

If that’s not enough to con­vince you, then I pro­pose as a mid­dle-ground an­other al­ter­na­tive ex­pla­na­tion for any situ­a­tion where evolu­tion alone might be used as such: “I don’t have a clue.” It’s more hon­est, more in­for­ma­tive, and it does more to get peo­ple to ac­tu­ally in­ves­ti­gate open ques­tions, as op­posed to pre­tend­ing those ques­tions have been ad­dressed in any mean­ingful way.

Less poor models

When peo­ple use evolu­tion as a jus­tifi­ca­tion for a phe­nomenon, what they tend to imag­ine is this:

  • Changes are grad­ual.

  • Changes oc­cur on the time scale of gen­er­a­tions, not in­di­vi­d­u­als.

  • Du­pli­ca­tion and ter­mi­na­tion are, re­spec­tively, pos­i­tively and nega­tively cor­re­lated with the chang­ing thing.

If you agree, then I’m sure the fol­low­ing ques­tions re­gard­ing stan­dard evopsych ex­pla­na­tions of so­cial sig­nal­ing phe­nomenon X should be easy to an­swer:

  • What in­di­ca­tion is there that the change in adop­tion of X was grad­ual?

  • What in­di­ca­tion is there that change in adop­tion of X hap­pens on the time scale of gen­er­a­tions and not in­di­vi­d­u­als (i.e., that in­di­vi­d­u­als have lit­tle in­fluence in their own lo­cal adop­tion of X)?

  • What con­sti­tutes du­pli­ca­tion and ter­mi­na­tion? Is the hy­poth­e­sized chain of cor­re­la­tion short enough or re­li­able enough to be con­vinc­ing?

If you agreed with the de­com­po­si­tion of “evolu­tion” and dis­agreed with any of the sub­se­quent ques­tions, then your model of evolu­tion might not be con­sis­tent, or you may have a prefer­ence for un­jus­tified ex­pla­na­tions. In con­ver­sa­tion, this isn’t re­ally an is­sue, but per­haps there are some down­sides to us­ing in­con­sis­tent mod­els for your per­sonal wor­ld­views.

Op­ti­mal models

In March 1956, John Kelly de­scribed an equa­tion for bet­ting op­ti­mally on a coin-toss game weighted in your fa­vor. If you were ex­pected to gain money on av­er­age, and if you could place bets re­peat­edly, then the Kelly bet let you grow your prin­ci­pal at the great­est pos­si­ble rate.

You can read the pa­per here: http://​​www.her­rold.com/​​bro­ker­age/​​kelly.pdf. It’s im­por­tant for you to be able to read pa­pers like this…

The ar­gu­ment goes like this. Given a coin that lands on heads with prob­a­bil­ity p and tails with prob­a­bil­ity q=1-p, I let you bet k (a frac­tion of your to­tal) that the coin will land heads. If you win, I give you b*k. If you lose, I take your k. After n rounds, you will have won on av­er­age p*n times, and you will have lost q*n times. Your new to­tal will look like this:

Your bet is op­ti­mized when the gra­di­ent of this value with re­spect to k is zero and de­creas­ing in both di­rec­tions away.

You can check eas­ily that the equa­tion is always con­cave down when the odds are in your fa­vor and when k is be­tween 0 and 1. Note that there is alway one lo­cal max­i­mum: the value of k found above. There is also one un­defined value, k=1 (all-in ev­ery time), which if you plug into the origi­nal equa­tion re­sults in you be­ing broke.

The Kelly bet makes one key as­sump­tion: chance is nei­ther with you nor against you. If you play n games, you will win n*p of them, and you will lose n*q. With this as­sump­tion, which of­ten al­igns closely with re­al­ity, your prin­ci­pal will grow fairly re­li­ably, and it will grow ex­po­nen­tially. More­over, you will never go broke with a Kelly bet.

There is a sec­ond an­swer though that doesn’t make this as­sump­tion: go all-in ev­ery time. Your ex­pected win­nings, summed over all pos­si­ble coin con­figu­ra­tions, will be:

If you run the num­bers, you’ll see that this sec­ond strat­egy of­ten beats the Kelly bet on av­er­age, though most out­comes re­sult in you be­ing broke.

So I’ll offer you a choice. We’ll play the coin game with a fair coin. You get 3U (util­ity) for ev­ery 1U you bet if you win, and you lose your 1U oth­er­wise. You can play the game with any amount of util­ity for up to, say, a trillion rounds. Would you use Kelly’s strat­egy, with which your util­ity would al­most cer­tainly grow ex­po­nen­tially to be far larger than your ini­tial sum, or would you use the sec­ond strat­egy, which performs far, far bet­ter on av­er­age, though through which you’ll al­most cer­tainly end up turn­ing the world into a per­ma­nent hell?

This as­sumes noth­ing about the util­ity func­tion other than that util­ity can re­li­ably be in­creased and de­creased by spe­cific quan­tities. If you pre­fer the Kelly bet, then you’re not op­ti­miz­ing for any util­ity func­tion on av­er­age, and so you’re not op­ti­miz­ing for any util­ity func­tion at all.