Suppose I’m trying to infer probabilities about some set of events by looking at betting markets. My idea was to visualise the possible probability assignments as a high-dimensional space, and then for each bet being offered remove the part of that space for which the bet has positive expected value. The region remaining after doing this for all bets on offer should contain the probability assignment representing the “market’s beliefs”.
My question is about the situation where there is no remaining region. In this situation for every probability assignment there’s some bet with a positive expectation. Is it a theorem that there is always an arbitrage in this case? In other words, can one switch the quantifiers from “for all probability assignments there exists a positive expectation bet” to “there exists a bet such that for all probability assignments the bet has positive expectation”?
Yes, I think you can. If there’s a bunch of linear functions F_i defined on a simplex, and for any point P in the simplex there’s at least one i such that F_i(P) > 0, then some linear combination of F_i with non-negative coefficients will be positive everywhere on the simplex.
Unfortunately I couldn’t come up with a simple proof yet. Here’s how a not so simple proof could work: consider the function G(P) = max F_i(P). Let Q be the point where G reaches minimum. Q exists because the simplex is compact, and G(Q) > 0 by assumption. Then you can take a linear combination of those F_i whose value at Q coincides with G. There are two cases: 1) Q is in the interior of the simplex, in this case you can make the linear combination come out as a positive constant; 2) Q is on one of the faces (or edges, etc), in this case you can recurse to that face which is itself a simplex. Eventually you get a function that’s a positive constant on that face and greater everywhere else.
You should be able to get it as a corollary of the lemma that given two disjoint convex subsets U and V of R^n (which are non-zero distance apart), there exists an affine function f on R^n such that f(u) > 0 for all u in V and f(v) < 0 for all v in V.
Our two convex sets being (1) the image of the simplex under the F_i : i = 1 … n and (2) the “negative quadrant” of R^n (i.e. the set of points all of whose co-ordinates are non-positive.)
I’m confused what the word “fairly” means in this sentence.
Do you mean that they make a zero-expected-value bet, e.g., 1:1 odds for a fair coin? (Then “fairly” is too strong; non-degenerate odds (i.e., not zero on either side) is the actual required condition.)
Do you mean that they bet without fraud, such that one will get a positive payout in one outcome and the other will in the other? (Then I think “fairly” is redundant, because I would say they haven’t actually bet on the outcome of the coin if the payouts don’t correspond to coin outcomes.)
If I’m understanding properly, you’re trying to use the set of bets offered as evidence to infer the common beliefs of the market that’s offering them. Yet from a Bayesian perspective, it seems like you’re assigning P( X offers bet B | bet B has positive expectation ) = 0. While that’s literally the statement of the Efficient Markets Hypothesis, presumably you—as a Bayesian—don’t actually believe the probability to be literally 0.
Getting this right and generalizing a bit (presumably, you think that P( X offers B | B has expectation +epsilon ) < P( X offers B | B has expectation +BIG_E )), should make the market evidence more informative (and cases of arbitrage less divide-by-zero, break-your-math confusing).
The efficient markets hypothesis is that one should expect ‘no remaining region’ to be the default. While betting may not be as competed as finance, there are still hedge funds etc doing betting.
Also I would suggest thinking about expected utility greater than some positive threshold to take into account transaction costs. I suppose that this would make a good deal of difference to how many such regions you could expect there to be.
Suppose I’m trying to infer probabilities about some set of events by looking at betting markets. My idea was to visualise the possible probability assignments as a high-dimensional space, and then for each bet being offered remove the part of that space for which the bet has positive expected value. The region remaining after doing this for all bets on offer should contain the probability assignment representing the “market’s beliefs”.
My question is about the situation where there is no remaining region. In this situation for every probability assignment there’s some bet with a positive expectation. Is it a theorem that there is always an arbitrage in this case? In other words, can one switch the quantifiers from “for all probability assignments there exists a positive expectation bet” to “there exists a bet such that for all probability assignments the bet has positive expectation”?
Yes, I think you can. If there’s a bunch of linear functions F_i defined on a simplex, and for any point P in the simplex there’s at least one i such that F_i(P) > 0, then some linear combination of F_i with non-negative coefficients will be positive everywhere on the simplex.
Unfortunately I couldn’t come up with a simple proof yet. Here’s how a not so simple proof could work: consider the function G(P) = max F_i(P). Let Q be the point where G reaches minimum. Q exists because the simplex is compact, and G(Q) > 0 by assumption. Then you can take a linear combination of those F_i whose value at Q coincides with G. There are two cases: 1) Q is in the interior of the simplex, in this case you can make the linear combination come out as a positive constant; 2) Q is on one of the faces (or edges, etc), in this case you can recurse to that face which is itself a simplex. Eventually you get a function that’s a positive constant on that face and greater everywhere else.
Does that make sense?
You should be able to get it as a corollary of the lemma that given two disjoint convex subsets U and V of R^n (which are non-zero distance apart), there exists an affine function f on R^n such that f(u) > 0 for all u in V and f(v) < 0 for all v in V.
Our two convex sets being (1) the image of the simplex under the F_i : i = 1 … n and (2) the “negative quadrant” of R^n (i.e. the set of points all of whose co-ordinates are non-positive.)
Yeah, I think that works. Nice!
I was trying to construct a proof along similar lines, so thank you for beating me to it!
Note that 2 is actually a case of 1, since you can think of the “walls” of the simplex as being bets that the universe offers you (at zero odds).
If Alice and Bob bet fairly on the outcome of a coin, there is no arbitrage.
I’m confused what the word “fairly” means in this sentence.
Do you mean that they make a zero-expected-value bet, e.g., 1:1 odds for a fair coin? (Then “fairly” is too strong; non-degenerate odds (i.e., not zero on either side) is the actual required condition.)
Do you mean that they bet without fraud, such that one will get a positive payout in one outcome and the other will in the other? (Then I think “fairly” is redundant, because I would say they haven’t actually bet on the outcome of the coin if the payouts don’t correspond to coin outcomes.)
(This comment isn’t an answer to your question.)
If I’m understanding properly, you’re trying to use the set of bets offered as evidence to infer the common beliefs of the market that’s offering them. Yet from a Bayesian perspective, it seems like you’re assigning P( X offers bet B | bet B has positive expectation ) = 0. While that’s literally the statement of the Efficient Markets Hypothesis, presumably you—as a Bayesian—don’t actually believe the probability to be literally 0.
Getting this right and generalizing a bit (presumably, you think that P( X offers B | B has expectation +epsilon ) < P( X offers B | B has expectation +BIG_E )), should make the market evidence more informative (and cases of arbitrage less divide-by-zero, break-your-math confusing).
The efficient markets hypothesis is that one should expect ‘no remaining region’ to be the default. While betting may not be as competed as finance, there are still hedge funds etc doing betting.
Also I would suggest thinking about expected utility greater than some positive threshold to take into account transaction costs. I suppose that this would make a good deal of difference to how many such regions you could expect there to be.