As I mentioned in my previous post, I recently presented my Hayekian-inspired “truth markets” idea (see here and here) at the annual Loyola ConLaw Colloquium. After my talk, my colleague and friend David Schraub made an insightful observation. According to Schraub, the price of a belief contract won’t reflect the truth value of the conspiracy theory or fake news story being bet on and thus won’t converge on the “right” answer; instead, the price will track underlying beliefs among bettors about the beliefs of others. For Schraub, if we are placing bets not on their own beliefs (first-order bets) but rather on their beliefs about what others think (second-order bets), then all our bets are effectively “arbitrary” (Schraub’s term, not mine).
To illustrate this point, Professor Schraub referred to a multi-player number game known as “guess 2/3 of the average“. In this game, each player secretly chooses a real number between 1 to 100. The winner is the player who chooses the number that is closest to 2/3 of the average of the sum of everyone’s choices. Given this set-up, Schraub posed the following question to me: How is a someone’s belief that a player in this game will pick the number ‘31’ or ‘57’ and different from my proposed market in belief contracts?
As an historical aside, Schraub’s critique also reminds me of John Maynard Keynes’ psychological critique of stock market investors in Chapter 12 of his classic work on The General Theory of Employment, Interest and Money:
… professional investment may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole; so that each competitor has to pick, not those faces which he himself finds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view.
For Keynes, the optimal strategy in this beauty contest game is not pick the six faces that you, the contestant, may personally find the most attractive. To win, you must pick the faces that you think other people will find most attractive:
It is not a case of choosing those which, to the best of one’s judgment, are really the prettiest, nor even those which average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practise the fourth, fifth and higher degrees.
In reply, I am tempted to cite Hayek and be done with it. After all, Keynes is wrong: in the real world stock market prices do, in fact, reflect the underlying financial value of the companies whose stocks are being traded. (See here for example.) Also, contra Schraub, his artificial number game example does indeed have a rational or game-theoretic solution; see here and here. But that said, one could argue in counter-reply that my truth market idea is qualitatively different from a traditional market, since people would be trading “belief contracts”, i.e. people would placing bets on whether a conspiracy is true or false. I will therefore give Schraub’s Keynesian-inspired critique further thought and report back soon.
prior probability 
This is very interesting — I did not actually realize I was channeling Keynes (good company to be in!). Three further notes:
1) The possibility that my “arbitrary” game has an equilibrium doesn’t resolve the problem that the market won’t be pegged to the truth/falsity of the conspiracy unless the equilibrium is one tied to the putative value of the underlying asset (here, the “actual” truth or falsity of the conspiracy). In the Guess 2/3 game, the equilibrium is to guess zero, which almost certainly would *not* be 2/3 of the average in a world where people were naively just picking their favorite number. So there might be a strategy to win your betting market, but it’s not a strategy that necessarily has anything to do with being more accurate about whether a conspiracy is true or false.
2) I actually don’t think your market entails making second-order bets about beliefs; I think it is in fact infinitely recursive (and that’s why it is arbitrary). No player has any reason to “buy” a contract based on their actual belief in the truth/falsity of the conspiracy aside from a belief that other market participants will make their choices based on actual belief; but since the other market participants also have no reason to buy based on actual belief, nobody’s purchasing decisions are pegged to beliefs about the truth/falsity of the conspiracy even on a second-order basis.
3) One reason stock market prices might reflect underlying valuations is, paradoxically enough, that it is possible to be right, and be rewarded for being right, even if “the market” disagrees with you. Suppose there is a company that is doing very well, but for whatever reason “the market” stubbornly refuses to buy its stock. Congratulations — you buy up shares at $1 a pop and enjoy getting rich off fat dividend checks (the opposite is true for an irrationally high-priced stock — if it goes insolvent, everyone loses their money no matter how stubbornly people kept buying the stock). The reason we don’t see that, of course, is precisely that the stock market offers a mechanism to punish people for being “wrong” that is not fully endogenous to the market itself. The truth market doesn’t have that — if people just keep on stubbornly buying “This is a conspiracy” for something that is true, there is not even in concept any mechanism that punishes them for being wrong (or in the opposite case, rewards them for being right), because the pricing mechanism is entirely self-referential. That’s what causes the bet price to be fundamentally decoupled from the “underlying value of the asset” and makes the price ultimately arbitrary.
Thanks for the further feedback! All three points are well-taken. I will report back soon …
Hi again! I had a chance to read some of the beauty contest literature (see, for example, my follow-up blog post from Nov. 17) on my train ride to the Southern Econ Assoc conference this weekend, and I also had some time early this morning (Nov. 19) to mull over your excellent points.
I will begin by restating the research problem we are both trying to solve. To the point, what do my beliefs (b-1) about other people’s beliefs (b-2) tell us about the truth values of those third-party beliefs b-2? Your position is that there is no logical relation between b-1 and b-2, but I am not so sure. Although I understand your reasoning (especially your three points above), I want to think about this question more.
In the meantime, I want to make two additional points for us to think about. One is as follows: What is the relation between Keynes’ “beauty contest” game and the “guess the average” game? Are they essentially the same, as Richard Thaler claims? (See the Thaler link in my 11/17 blog post.) Or is the beauty contest game subtly different than the guessing game? (For the latter view, see item #4 on page 4 of the ZBW link in my 11?17 blog post.)
Either way, my other point is that Keynes’s beauty contest game is, in principle, possible to win. In addition, some people are, in fact, able to win the guessing game without having to guess zero, but the winning number usually moves toward zero the more times the game is played.
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“What do my beliefs (b-1) about other people’s beliefs (b-2) tell us about the truth values of those third-party beliefs b-2? Your position is that there is no logical relation between b-1 and b-2….”
Again, just to reiterate: I think the core problem in your market set up is that it actually never relates to my beliefs (b-1) about other people’s beliefs [about the truth/falsity of the conspiracy] (b-2). There might be a relation between b-1 and b-2, but your market won’t find the relation, because it’s not set up in a way that actually encourages me or anyone else to make choices EITHER based on my own first-order beliefs or my beliefs about the beliefs of others.
I should not “buy” based on my belief about the conspiracy; I also should not “buy” based on my best assessment of other peoples’ beliefs about the conspiracy. Nobody, even in an attenuated, second- or third-order fashion, will be buying contracts based on belief about the conspiracy (or belief about belief, or belief about belief about belief …). And if no market participant logically should be making a decision that at some level ties to somebody’s perceived belief about the conspiracy (whether their own first-order belief, or their perception of others’ beliefs, or their perception of the perception of others’ beliefs ….), then I shouldn’t make a choice on that basis either, which means the basis for the choice is arbitrary.
To clarify: I am proposing a “truth market” or “fake news futures” to allow people to buy and sell “belief contracts”…
You: “I should not ‘buy’ [a belief contract] based on my belief about the conspiracy; I also should not ‘buy’ based on my best assessment of other peoples’ beliefs about the conspiracy.”
Me: Then what am I “buying”?
[You: “Then what am I ‘buying’?”]
Me: Exactly!
Touche’
It might be time to hit “pause” on this discussion, but let me first make two quick points:
1. Because of your feedback, I have now become more fascinated by the theoretical “beauty contest” question (i.e. the ontology and epistemology of beliefs about beliefs) than about possible applications of this theoretical problem.
2. That said, however we solve these ontological and epistemological problems, I am still inclined to believe that, with a sufficiently large and diverse population of bettors, my proposal for a truth market in belief contracts would work just as well (or just as bad, depending on one’s priors!) as more familiar markets for goods and services.
That is all, for now …
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