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cs234r
Markets for Networks and Crowds
BRENDAN LUCIER, MICROSOFT RESEARCH NE
NICOLE IMMORLICA, MICROSOFT RESEARCH NE
Lecture 4:
Richard Cole and Lisa Fleischer. Fast-Converging
Tatonnement Algorithms for One-Time and
Ongoing Market Problems.
Preliminary version in ACM Symposium on the
Theory of Computing, 2008.
Market Convergence
Question: how do prices converge?
Markets seem to clear quickly and efficiently in
practice. Can we reverse-engineer the process?
One answer: centralized tâtonnement.
This paper: is there a natural non-centralized
process? Can we avoid the “invisible auctioneer?”
Model
Fisher Market:
n divisible goods, each from a different seller.
• WLOG: one unit of each good
m buyers, each with
• a budget 𝐵𝑗
• a valuation function 𝑣𝑗 over baskets of goods
𝑣𝑗 (𝑥1𝑗 , … , 𝑥𝑛𝑗 ): buyer 𝑗’s value when obtaining
𝑥𝑖𝑗 quantity of good 𝑖.
Model
Single-shot Market:
𝐷𝑗 (𝑝1 , … , 𝑝𝑛 ): bundle of goods most preferred by
buyer 𝑗, subject to ∑𝑝𝑖 𝑥𝑖𝑗 ≤ 𝐵𝑗 .
Assumption: gross substitutes condition
(if some prices ↑, demand for other goods not ↓).
Price equilibrium:
Prices (𝑝1 , … , 𝑝𝑛 ) such that, when each buyer
takes 𝐷𝑖 𝑝1 , … , 𝑝𝑛 , all goods are sold.
Model:
Ongoing Market
Each seller has a warehouse that holds the
current stock of their good.
• 𝑠𝑖 : amount of 𝑖 currently in the warehouse
Each round, seller 𝑖:
• gets one additional unit of good 𝑖.
• sets price 𝑝𝑖 .
Buyers buy 𝐷𝑗 (𝒑). Any unsold quantity of good 𝑖
stays in the warehouse until the next round.
Example:
Good 1
Good 2
Current Stock
Desired level
Example:
Good 2
Good 1
One unit
One unit
Example:
Good 1
Good 2
Price: 𝑝1 = $2
Price: 𝑝2 = $7
Example:
Good 1
Good 2
Model:
Stability:
A vector of prices 𝒑 forms an equilibrium if
warehouse quantities are stable over time, and
are equal to an “ideal quantity” 𝑠𝑖𝐹
Note: an equilibrium of the one-shot market is
also an equilibrium in the ongoing market.
Goal:
A local price update rule that converges quickly
to equilibrium prices.
Local: price updates for good i depend only on
current supply, past prices, and past demand.
Quickly: depends on market parameters.
Update Rule:
Update prices multiplicatively, in response to
over-demand / under-demand.
𝑝𝑖 ← 𝑝𝑖 (1 + 𝜆 ⋅ min 1, 𝑥𝑖 − 1 + 𝜅 𝑠𝑖 − 𝑠𝑖𝐹
Total demand
last period for
good 𝑖.
The demand needed to
reduce the warehouse
gap by a factor of 𝜅.
Parameters 𝜆, 𝜅 ≤ 1 are intended to dampen
price changes, to prevent overreaction.
Result (informal):
Theorem: If 𝜆 and 𝜅 are sufficiently small,
relative to the max rate of change in demand
with respect to changes in price and budgets,
THEN
a 𝛿-approximate equilibrium is reached in
1 1 1
POLY log
, ,
𝛿 𝜆 𝜅
rounds.
Analysis Idea:
Price rule updates prices in the “right” direction,
but slowly enough to avoid oscillation.
𝜆 and 𝜅: set to keep prices in a range where
Δ excess demand = Θ Δ pi
Each step, update to 𝑝𝑖 is large enough to make
constant-factor progress toward equilibrium.
Informal Discussion:
Motivation: Since price updates depend only on
seller-specific feedback, tâtonnement occurs
without a centralized market-maker.
Questions:
• Is this seller behavior justified?
• Other models of repeated markets?