price differentiation in the kelly mechanism
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Price Differentiation in the Kelly Mechanism. Richard Ma Advanced Digital Sciences Center, Illinois at Singapore School of Computing, National University of Singapore. Joint work with Dah Ming Chiu, John Lui (Chinese University of Hong Kong) - PowerPoint PPT PresentationTRANSCRIPT
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Price Differentiation in the Kelly Mechanism
Richard MaAdvanced Digital Sciences Center, Illinois at SingaporeSchool of Computing, National University of Singapore
Joint work with Dah Ming Chiu, John Lui (Chinese University of Hong Kong)
Vishal Misra, Dan Rubenstein (Columbia University)
W-PIN 2012
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A Resource Allocation Problem One divisible resource with capacity
E.g., bandwidth , CPU cycles users compete for the resource : user ‘s valuation (or monetary utility)
on amount of resource Increasing, Concave and Differentiable
A social welfare maximization problem: Maximize Subject to and
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The Kelly Mechanism Each user submits a bid , which is the
willingness to pay (for unknown amount of resource)
Resource is allocated proportionally by
The utility of each user is
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Properties of Kelly Mechanism Equal price (per unit resource) Price-taking assumption: Given a price ,
each user maximizes
First-order condition:
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Properties of Kelly Mechanism [Kelly ‘98] Under the price-taking
assumption, there exists a unique competitive equilibrium under which the network “clears the market”: the social welfare is maximized
It works when the number of users is big, where each user’s strategy does not move the market price much.
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Non-cooperative Game Without the price-taking assumption,
Kelly mechanism creates a non-cooperative game User ’s strategy: User ’s objective: Maximize
[Hajek et al. 02] There exists a unique Nash equilibrium for the game.
[Johari et al. 04] Efficiency loss from the Nash equilibrium could be as big as 25% of the social optimum (or PoA ).
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Price Differentiation Each user buys “tickets” for bidding Allocation is proportional to # of
“tickets”
User pays price for each “ticket”
Given a fixed price vector User uses a strategy to maximize
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A Generalized Mechanism Price differentiation: per unit resource
price for user is
If the price vector , the special case is the Kelly mechanism
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Properties Theorem 1: Under any price vector ,
there exists a unique Nash equilibrium. Theorem 2: For any allocation vector ,
there exists a vector such that is the allocation of the unique Nash equilibrium.
Theorem 3: For any two price vectors , with , the Nash equilibrium satisfies
and .
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Properties Theorem 4: If any user gets zero
under , then the equilibrium does not change if we further increase unilaterally.
Theorem 5: There is a connected set of price vectors that maps to the set of all resource allocations continuously and bijectively.
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Mapping from to
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Valuation Revelation We want to find the vector that
achieves the social welfare as a Nash equilibrium
Problem: we still don’t know the valuation
Theorem 6: In equilibrium, we have
In theory, we can recover the (shape of) valuation functions.
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Unsolved Problem (Future work) Theorem 7: achieve social optimum iff
for all users and . The above provides some hint about
how to adjust the prices between a pair of users.
Question: how can we utilize the above result to maximize social welfare? Feedback control? Convergence?
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