Well, if we knew we were looking for a proof of A and not a disproof, things would be easy: We could just print shares in A alone. These could only be redeemed for $1 if it were possible to prove A. But having a bunch of shares of A and ¬A hanging around doesn’t mean that they only have value if a proof or disproof is found. It means that they have value right away because they can be cancelled with each other to yield $1. And once all those shares have been cancelled, there’s no value for traders in continuing to search for a proof or disproof.
why is this a problem?
how does the attacker acquire the shares in A and not-A? they have to buy one of each, right? so presumably price(A) + price(not-A) = $1 +/- some transaction costs.
this seems like a desirable property of the market (if we accept the l.e.m.), since it ensures that the price of A (which we can interpret as the probability that A is true) is equal to $1 - price(not-A).
if we turn dollars into A + not-A, and allow them to trade on the open market, why is anyone incentivized to turn them back into dollars, unless they are closing out a position after market moves?
It’s not a huge problem, but it is a minor problem, at least for people trying to set bounties on a problem.
In the intuitionist version, the prices of a statement and its negation can sum to at most $1, but less is possible. (As you say, they sum to exactly $1 if one assumes LEM.) [1] If traders know that a statement is undecidable, for example, then this sum should be $0 and not $1. They don’t expect to be able to redeem shares in either the statement or its negation.
Imagine that I am trying to offer a bounty on a problem with statement , which can be resolved both by proving or by disproving the statement. I can sell a pair for some price . If we assume LEM:
If I set the price below $1, then people can get free money without proving the theorem by cancelling shares.
If I set the price at $1, then the deal I’m offering people who try to prove my theorem is that they can pay $1 to put in a bunch of effort to prove a theorem, from which they’ll then earn $1, for a profit of $0.
Even if we account for transaction costs, cancelling shares is usually going to be much easier than proving theorems, so it will become profitable first.
Therefore, I have to set equal to $1 as the least of two evils. Provers can still make a profit by proving theorems if the market is trading at an intermediate price like $0.5, because they can use their inside knowledge from proving the theorem to correct the market. But this is not relevant in cases where “everyone knows” that the theorem should be true, but the hard part is to find a proof.
On the other hand, if cancellation is no longer possible, then the best option is simply to set less than $1. In a world where (or ) is provable, I do lose money. But that can only happen with a proof of (or ), which is exactly the thing I was trying to bounty.
Because an irreversible conversion from $1 to is possible, we have . It’s always possible to create opposing shares in a statement. The reverse is not always possible (though it’s possible for “oracle” statements that point to real-world events, i.e. the kind of statements that would be traded on non-mathematical prediction markets).
why is this a problem?
how does the attacker acquire the shares in A and not-A? they have to buy one of each, right? so presumably price(A) + price(not-A) = $1 +/- some transaction costs.
this seems like a desirable property of the market (if we accept the l.e.m.), since it ensures that the price of A (which we can interpret as the probability that A is true) is equal to $1 - price(not-A).
if we turn dollars into A + not-A, and allow them to trade on the open market, why is anyone incentivized to turn them back into dollars, unless they are closing out a position after market moves?
It’s not a huge problem, but it is a minor problem, at least for people trying to set bounties on a problem.
In the intuitionist version, the prices of a statement and its negation can sum to at most $1, but less is possible. (As you say, they sum to exactly $1 if one assumes LEM.) [1] If traders know that a statement is undecidable, for example, then this sum should be $0 and not $1. They don’t expect to be able to redeem shares in either the statement or its negation.
Imagine that I am trying to offer a bounty on a problem with statement
If I set the price
If I set the price
Even if we account for transaction costs, cancelling shares is usually going to be much easier than proving theorems, so it will become profitable first.
Therefore, I have to set
On the other hand, if cancellation is no longer possible, then the best option is simply to set
Because an irreversible conversion from $1 to