# A semi-technical question about prediction markets and private info

There ex­ists a 6-sided die that is weighted such that one of the 6 num­bers has a 50% chance to come up and all the other num­bers have a 1 in 10 chance. No­body knows for cer­tain which num­ber the die is bi­ased in fa­vor of, but some peo­ple have had a chance to roll the die and see the re­sult.

You get a chance to roll the die ex­actly once, with no­body else watch­ing. It comes up 6. Run­ning a quick Bayes’s The­o­rem calcu­la­tion, you now think there’s a 50% chance that the die is bi­ased in fa­vor of 6 and a 10% chance for the num­bers 1 through 5.

You then dis­cover that there’s a pre­dic­tion mar­ket about the die. The pre­dic­tion mar­ket says there’s a 50% chance that “3” is the num­ber the die is bi­ased in fa­vor of, and each other num­ber is given 10% prob­a­bil­ity.

How do you up­date based on what you’ve learned? Do you make any bets?

I think I know the an­swer for this toy prob­lem, but I’m not sure if I’m right or how it gen­er­al­izes to real life...

• Miss­ing in­for­ma­tion—how many peo­ple have rol­led the die how many times be­fore par­ti­ci­pat­ing in the mar­ket? If you don’t ex­pect that there’s pri­vate in­for­ma­tion that the mar­ket can make pub­lic, you shouldn’t ex­pect it to in­di­cate truth.

Also miss­ing—what are the ac­tual con­tracts? Is this a wa­ger on the next roll of the die? Mean­ing 50% is pay­ing 1:1, 10% pay­ing 9:1 (\$100 bet on 3 pays \$100, \$100 bet on 6 pays \$900 if it wins)?

In that case, if the mar­ket is one per­son who saw one roll, I bet 5 con­tracts on 6, and one on each of 1,2,4,5. If the mar­ket is hun­dreds of peo­ple, even if they’ve each only seen it once (in­de­pen­dently; hun­dreds of rolls with one ob­server each, not 100s of ob­servers of one roll), then the mar­ket has likely already worked out the cor­rect odds, so my ob­ser­va­tion doesn’t add much.

• Note: if the mar­ket is hun­dreds of peo­ple, and it’s a mar­ket on which face comes up 50% of the time (not on which face will come up on a spe­cific roll), and it only gets 50% odds on that be­ing a par­tic­u­lar num­ber, then some­thing un­usual is hap­pen­ing. An effi­cient mar­ket un­der these cir­cum­stances should be very con­fi­dent.

(I haven’t done any ex­plicit calcu­la­tions, but I’m rea­son­ably con­fi­dent.)

• Un­less only one roll of the die was seen by the hun­dreds of peo­ple, and it came up “3”.

• Ah, sure. It would have been bet­ter of me to say “if the mar­ket has col­lec­tively ob­served hun­dreds of rolls”.

• I’d bet on 6. I have in­for­ma­tion that the mar­ket doesn’t have, and my in­for­ma­tion points to 6 as the an­swer, so the mar­ket is un­der­pric­ing 6 (com­pared to how it would price 6 if it had all the in­for­ma­tion).

Another way to think of it: sup­pose that, in­stead of a mar­ket, there was just a sin­gle per­son look­ing at all the die rolls and up­dat­ing us­ing Bayes’s Rule. There have been n rolls and that per­son has as­signed the ap­pro­pri­ate prob­a­bil­ities to each of the pos­si­ble die weight­ings. Then the n+1th roll is a 6. The per­son then up­dates their prob­a­bil­ity as­sign­ments to give 6 a higher chance of be­ing the fa­vored side.

If the pre­dic­tion mar­ket is effi­cient, then it should be analo­gous to this situ­a­tion. The mar­ket price re­flects the first n rolls, and now I know that the n+1th roll was a 6, so I get to profit (in ex­pec­ta­tion) by up­dat­ing the mar­ket’s prob­a­bil­ities to take that new piece of in­for­ma­tion into ac­count.

It may be pos­si­ble to give a more pre­cise an­swer, but this is what I have for now.

• I agree. This should be time-sym­met­ri­cal; your ac­tions should be the same whether you rol­led first then saw mar­ket prices or vice-versa. If you ob­serve then roll, you have gained in­for­ma­tion mak­ing 6 more prob­a­ble at least a lit­tle bit, and you must trade in the di­rec­tion of your in­for­ma­tion (go­ing long 6 or short the oth­ers), which is +EV.

The real ques­tion is how much should you trade, since you can­not af­ford to bet in­definite amounts of money on tiny +EV op­por­tu­ni­ties. For a pre­dic­tion mar­ket that is a bit tricky, since they won’t typ­i­cally be on prob­lems where you can read off the pos­te­rior dis­tri­bu­tion from a few sum­mary prices (if it’s 50/​10/​10/​10/​10/​10%, does that mean that the mar­ket is highly cer­tain that it’s a prob­lem which sim­ply has those true fre­quen­cies, or could it mean that it’s a de­ter­minis­tic prob­lem but that the mar­ket is cur­rently highly un­cer­tain? in the former case, you have al­most no edge and want to bet lit­tle and in the lat­ter you want to bet a lot), and you have to start look­ing at things like volatility of the prices to gauge how much in­for­ma­tion you have rel­a­tive to it and so how much of an edge and how much you can af­ford to bet.

• It may be pos­si­ble to give a more pre­cise an­swer, but this is what I have for now.

AlexMen­nen and Os­car_Cun­ning­ham have run the num­bers and got­ten that more pre­cise an­swer. I did some calcu­la­tions my­self and agree with them. If the mar­ket has been effi­ciently in­cor­po­rat­ing in­for­ma­tion, then the prior die rolls in­cluded k+1 rolls of 3, and k rolls of each of the other num­bers (this gives 1:1:5:1:1:1 odds re­gard­less of k). My roll brings it up to k+1 6′s, so the odds should now be 1:1:5:1:1:5 (i.e., 114 for most num­bers and 514 for 3 and 6).

This is as­sum­ing that the mar­ket is ba­si­cally just do­ing Bayesian up­dat­ing; it’s pos­si­ble that there are some more com­pli­cated things hap­pen­ing with the mar­ket which make it a bad idea to make this as­sump­tion.

• Let’s as­sume pre­dic­tion mar­kets are effi­cient and you didn’t already pos­sess any rele­vant in­for­ma­tion that you weren’t trad­ing on be­fore­hand. Then you should treat the mar­ket odds as a prior and your die roll as ev­i­dence, in ex­actly the way you always do Bayesian up­dates. In this case, that means it looks like that gives you a pos­te­rior prob­a­bil­ity of 514 each for the die be­ing weighted in fa­vor of 3 or 6, and 114 for each of the other pos­si­bil­ities. Con­trary to what other com­menters were say­ing, it doesn’t mat­ter what in­for­ma­tion led to the mar­ket odds un­der these as­sump­tions.

• I don’t know if there are any eco­nomic the­o­rems about how mar­kets in­cor­po­rate in­for­ma­tion from their par­ti­ci­pants, but if we as­sume that they in­cor­po­rate all of the in­for­ma­tion then it must be that the mar­ket par­ti­ci­pants had seen ev­ery num­ber equally of­ten ex­cept for see­ing one ex­tra three. So you should up­date as though you had seen an ex­tra three, to get odds of 1:1:5:1:1:5. Then it would be worth plac­ing a bet on six and against each other num­ber.

• The mar­ket prices re­flect what one would see if one per­son had rol­led the die once and had the re­sult be a 3. Get­ting the re­sult any other way seems pretty hard.

I would there­fore as­sume that the mar­ket prob­a­bly got that way be­cause one per­son rol­led the die once, and got a 3, so now I have two rolls, 3 and 6, and I should up­date ac­cord­ingly.

If the pre­dic­tion mar­ket let me bet non-triv­ial amounts with­out mov­ing the prices, then some­thing strange is go­ing on, and “it’s time for some game the­ory.”

• The num­bers I picked are kind of ar­bi­trary—yes, they are what you would get from some­one rol­ling the die once and get­ting a 3. The ba­sic con­cern is that pre­dic­tion mar­kets might be vuln­er­a­ble to in­for­ma­tion cas­cades—you think the pre­dic­tion mar­ket knows bet­ter than you, so you don’t go and con­tra­dict it even though your per­sonal anal­y­sis of the situ­a­tion gives a differ­ent an­swer.

• I think what Un­named says is the most im­por­tant ob­ser­va­tion:

I’d bet on 6. I have in­for­ma­tion that the mar­ket doesn’t have, and my in­for­ma­tion points to 6 as the an­swer, so the mar­ket is un­der­pric­ing 6 (com­pared to how it would price 6 if it had all the in­for­ma­tion).

Any time you have pri­vate in­for­ma­tion that the mar­ket doesn’t have you should bet to move the mar­ket in the di­rec­tion of your in­for­ma­tion. The difficult ques­tion is how much you should bet.

• What in­for­ma­tion do they have?

This is the gen­eral prob­lem of a mix­ture of ex­perts when all you have are the pre­dic­tions but not the in­for­ma­tion on which the pre­dic­tions are based (at least for the mar­ket). I don’t think there is a real an­swer to that un­til you in­put more in­for­ma­tion into the sys­tem.

We want P(side | I_them, I_me)

We have P(side|I_them), P(side|I_me)

The lat­ter don’t give the former.

• Any time you have pri­vate in­for­ma­tion that the mar­ket doesn’t have you should bet to move the mar­ket in the di­rec­tion of your in­for­ma­tion. The difficult ques­tion is how much you should bet.

I don’t think that’s true. If it would be true I don’t think most mu­tual fund man­agers would un­der­perform the SAP 500. A mu­tual manger might get some su­perfi­cial in­for­ma­tion about stocks he in­vests in that take ac­tual re­search.

• Mu­tual fund man­agers have in­cen­tives other than max­i­miz­ing ex­pec­ta­tion of price. OPM.

Also, they likely over­es­ti­mate the value of their priv­ileged in­for­ma­tion.

• This is an in­ter­est­ing puz­zle. I catch my­self fight­ing the hy­po­thet­i­cal a lot.

I think it hinges on what would be the right move if you saw a six, and the mar­ket also had six as the fa­vored op­tion. In that situ­a­tion, it would be ap­pro­pri­ate to bet on the six which would move it past the 50% equil­ibrium, be­cause you have the in­for­ma­tion from the mar­ket and the in­for­ma­tion from the die. I think maybe your equil­ibrium price can only ex­ist if there is only one par­ti­ci­pant cur­rently offer­ing all of those bets, and they saw a six (so it’s not re­ally a true mar­ket yet, or there is only one in­formed par­ti­ci­pant and many un­in­formed). In that case, you hav­ing seen a six would im­ply a prob­a­bil­ity of higher than 50% that it is the weighted side. Given that think­ing, if you see that pre­dic­tion mar­ket fa­vor­ing a differ­ent num­ber (“3”), you should in­deed bet against it, be­cause very lit­tle in­for­ma­tion is con­tained in the mar­ket (one die throw worth).

The mar­ket show­ing a 50% price for one num­ber and 10% for the rest is an un­sta­ble equil­ibrium. If you started out with a mar­ket-maker with no in­for­ma­tion offer­ing 16 for all sides and there were many par­ti­ci­pants who only saw a sin­gle die, the bet­ting would in­crease on the cor­rect side. At each price that fa­vors that side, ev­ery per­son who again sees that side would have both the guess from the mar­ket and the in­for­ma­tion from their roll, then they would use that in­for­ma­tion to es­ti­mate a slightly greater prob­a­bil­ity and the price would shift fur­ther in the cor­rect di­rec­tion. It would blow past 50% prob­a­bil­ity with­out even paus­ing.

Those don’t seem like very satis­fac­tory an­swers though.

• Has any­one rol­led the die more than once? If not, it’s hard to see how it could con­verge on that out­come un­less ev­ery­body that’s bet­ting saw a 3 (even a sin­gle per­son see­ing differ­ently should drive the price down­ward). There­fore, it de­pends on how many peo­ple saw rolls, and you should up­date as if you’ve seen as many 3s as other peo­ple have bet.

You should bet on six if your prob­a­bil­ity is still higher than 10%.

If the pre­dic­tion mar­ket caused oth­ers to up­date pre­vi­ously then it’s more com­pli­cated. Prob­a­bly you should as­sume it re­flects all available in­for­ma­tion, and there­fore ex­actly one 3 was seen. Ul­ti­mately there’s no good an­swer be­cause there’s Knigh­tian un­cer­tainty in mar­kets.