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About

This is an implementation of the algorithms developed in the working paper (BGKL) by Elizabeth Baldwin, Paul Goldberg, Paul Klemperer and Edwin Lock available on the ArXiv here. The algorithms solve the Product-Mix Auction (originally developed by Paul Klemperer, see original paper) that uses the tropical bidding language with positive and negative bids; that is, they find a competitive (Walrasian) equilibrium.

Installation instructions for this implementation can be found here.

As described in BGKL, computing a competitive equilibrium can be separated into two parts:

  1. Find the component-wise minimal market-clearing price using a steepest descent approach. Both long-step methods described in the paper are implemented.

  2. Find an allocation of the supply (=target) bundle among the various bidders so that each bidder receives a bundle they demand at the market-clearing price. This is implemented according to algorithm described in the BGKL paper.

A note on the encoding of prices and bundles

For an auction with n goods, prices and bundles of goods are represented as (n+1)-dimensional vectors, where the i-th entry corresponds to the i-th good and the 0-th entry corresponds to a notional 'reject' good that is useful for technical reasons (see BGKL). In particular, every price vector has an 0-th entry of value 0. Moreover, an allocation of the target bundle among the bidders consists of a list containing a bundle vector for each bidder, and each vector's 0-th entry denotes how many notional 'reject' goods the bidder receives.

Example Usage

Initial

Activate the virtual environment and launch an interactive Python shell:

$ cd ..path..to../product-mix
$ source venv/bin/activate
$ python

Import the product-mix package

>>> import productmix as pm

Solving the Product-Mix auction

Load an allocation problem from a file

>>> alloc = pm.load_from_json('examples/example2.json')

Find a market-clearing price using unit step steepest descent

>>> prices = pm.min_up(alloc, long_step_routine="")

Find market-clearing prices using long step steepest descent

>>> prices = pm.min_up(alloc, long_step_routine="demandchange")

or

>>> prices = pm.min_up(alloc, long_step_routine="binarysearch")

Print and set market-clearing prices in allocation problem object

>>> print(prices)
[0. 2. 4.]
>>> alloc.prices = prices

Compute a valid allocation. This outputs a list of bundles.

Note that running the pm.allocate(alloc) method has the side effect that all bids in the allocation problem instance alloc are deleted!

>>> allocation = pm.allocate(alloc)
>>> print(allocation)
[array([3., 3., 0.]), array([2., 4., 1.])]

Hence the first bidder is allocated 3 items of good 1 and no items of good 2, while the second bidder is allocated 4 items of good 1 and one 1 item of good 2.

Miscellaneous methods

Reload the allocation problem from a file

>>> alloc = pm.load_from_json('examples/example2.json')

Check validity of bid lists

>>> pm.is_valid(alloc)
True

Define some price vector p and compute market-clearing vector prices

>>> import numpy as np
>>> p = np.array([0,1,1])
>>> prices = pm.min_up(alloc)  # Uses 'binary search' long step technique by default

Compute the Lyapunov function at price vectors p and prices

>>> pm.lyapunov(alloc, p)
49.0
>>> pm.lyapunov(alloc, prices)
39.0

Get a demanded bundle at p and prices (not necessarily unique!)

>>> pm.demanded_bundle(alloc, p)
array([0., 7., 6.])
>>> pm.demanded_bundle(alloc, prices)
array([5., 7., 1.])

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An implementation of the Product-Mix auction

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