Think of a Priceline-like scenario: The hotel owner has posted some prices (hidden from the final consumer?). The customer bids a price, which is lower than the lowest price accepted by the hotel, by say $1. In that case, it makes sense to use a “subsidizing” automatic market maker to bridge the gap between bid-ask, and get the transaction to happen. Since the market owner is benefiting by the transaction (e.g., charges a fee equal to x% of the price), then it makes sense to provide the funds that will allow the two parties to transact. Same thing for labor market (e.g. on oDesk I want to hire people at $10/hr, the worker asks for $10.50/hr: the market owner subsidizes the difference, since they make 10% of the overall volume)

There are also the network effects that generate additional benefits when the market is moving. In principle even governments would be interested in doing so, as all transactions generate tax revenue, especially in countries with VAT.

I am not sure if this makes sense. I see strategic considerations here (what to bid in order to take full advantage of the subsidy), the markets do not have any external ground truth (on the other hand, stock markets do not have one either), and other objections.

Any thoughts? I guess there should be other people that have thought about these. They seem pretty straightforward ideas.

Thank you

You are correct to note that if we restrict ourselves to only the positive orthant, the sum of prices never quite reaches 1. Instead, for small alpha, we get values that sum very very very close to one (within machine precision) – this is because
a*log(exp(1/a)+n-1) ~ a*log(exp(1/a)) = 1 for small a.
The key here is a comparison between n-1 and exp(1/a), where for the values we consider exp(1/a) >>> n-1. Since small alpha are appropriate for this market maker, the bounds we have are still good in practice. This is something we will be clarifying in the journal version.

Update: Actually I spoke too soon: Abe has now discovered some generalizations.

Set r_j = q_j / b(q)
Set k = sum_j exp(r_j)
Set s_j = exp(r_j) / k

p_i(q) = s_i + a * H(s)
sum_i p_i(q) = 1 + a * n * H(s)

where H(s) is the entropy of the distribution s. This seems a much more natural description of what the algorithm is doing, especially if the probability distribution s can be given a physical interpretation (perhaps races between exponential variables). In addition many of the properties proved in the paper, cast in this light, rely only on the fact that s is a distribution and ignore its relation to q. I’m going to try to produce a cost function for which treats s simply as an unspecified function of q – it may be that this can be generalised to a class of market makers.

I would be interested to hear your thoughts on this if you have a moment.