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Price Discovery in Waiting List

In economics and finance, the price discovery in waiting lists is the process of agent assignment to a waiting list, using the waiting time as the price.

Overview
Price discovery in waiting list is different then classic price discovery with prices and sellers. In this case the price is the waiting time and no seller. The target is to determine the optimum waiting time that the agent still enters to the list. This is an infinite process which is changes over the time when buyer enter or exit from a list.

Example
Waiting list for a single item type. Items arrive at Poisson rate 1.

Agents arrive at Poisson rate 2. Each agent ha quasilinear utility which depends on its private utility and the waiting list length. .  i.i.d is the agent private utility and   is the waiting list length when the agent arrives.

When an agent arrives, he checks the waiting list length, calculate  and the decides if join or not.

To maximize the allocative efficiency, we would like to server only half of the agents, the agents with, so the market price.

Target
Present a mechanism which allocate agents to waiting list in efficient way under fluctuating price with bonded loss by the adjustment size

Items
Finite number of items                               arrives according to Poisson process, total rate. The probability of item to arrive is. Each item type has a separate waiting list.

Agents
Uncountably many or finitely many types of agents which arrive according to Poisson process with total rate. Agent type  drawn i.i.d according to a distribution

Quasi-Linear Utility
Each agent type has a utility value for each item type. The value depends also the writing list length of the item. .

The value of the empty set  for any agent type.

is bounded;  is smooth, strictly increasing and convex or concave

Assignment
     is the waiting list that the agent who arrived at time   joined. The agent can choose not to join any waiting list (select )

Selected by.

When  is the waiting list length.

Allocative Efficiency Bound
This mechanism promise the following bound over its assignments:

Note: If the waiting time cost is linear method then

Relation to static assignment
Let  be the optimal allocative efficiency in the corresponding static assignment problem

Subject to

Then

The SGD problem
denote optimal static prices

The current price, is changed when an agent join or left the waiting list

is the maximal adjustment size

The expected price changed is:

This is a sub-gradient of the dual objective

Optimal Adjustment Size and Price Rigidity
The problem can be considered as a problem with a planner who can set prices, but doesn’t know the distribution of the buyers. This distribution can be learned from over time from past agents’ choices in a finite horizon.

The SGD pricing heuristic can be to increase price of item  by  when agent chooses   and reduce it by   at rate proportional to supply.

The allocative efficiency of SGD pricing with adjustment  is at least

Note:  is the minimal possible loss