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Extremely Simple Carroll Mechanism
Our goal is to create a cost function for the where when two shares (A and B) have been sold in large quantities, the cost of shares in the market that links the two (which we call A.B’s) cost is low.
The idea here is to make an incredibly simple mechanism where staking A.B (the intermediate market) competes with A•B (the state of the system having many shares of A and B sold).
the two simplest options you might imagine coming up with:
We could try something like these, as these are each a trivial instance of a function where as A and B rise C(A.B) will fall.
The obvious issue with these most trivial approaches is that they’re sensitive only to the magnitude of A and B, not to the relative magnitude between them. If we wanted something that accounted for that, such that we respect the rule A•B → ¬A.B, we could use a function like this:
Meaning our cost function for A.B would look like this:
Here’s a graph of that function showing how C(A.B) changes as A increases. Behind it we have a shadow of the typical cost function as A increases.
And here’s how the shape of the cost function changes as more of the competing token is sold. You can see the cost is maximized whenever both A and B have been sold in similar and large quantities.
X is A.B, the number of total shares sold in the relevance market (while we typically denote this A.B, it’s not possible to have multi-letter variables in desmos so this is what we use instead).
You can see X essentially just moves the floor of the function.
You can see that as X (A.B) rises, essentially it raises the floor of the function, meaning that buying X shares causes the cost of X shares to rise as well, in all the places where it’s not already maxed out due to competition between shares.
Since price is the derivative of cost, our corresponding price function comes out to:
And this would imply that our resultant pseudo-LMSR for the Carroll Markets would be:
Now our B quantity is being modified by our new price function for the relevance market.
An extremely simple carroll mechanism.
What’s left to think about is the problem of a reentry attack, where player can buy A.B and then buy A, then sell A.B then sell A for a profit.
Finally, this means that the last thing to play with is the addition of epistemic leverage. Epistemic leverage will be interesting because it essentially acts as a modification of A, B, and A.B, but where it is sensitive to the market price of the counterposition. So, this is where we get our propagation through the network while still keeping it game theoretic.
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