Algorithmic Game Theory and Internet Computing

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Algorithmic Game Theoryand Internet Computing
New Market Models and Algorithms

• Vijay V. Vazirani
• Georgia Tech

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How do we salvage the situation??

• Algorithmic ratification of the
• invisible hand of the market

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What is the right model??
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Linear Fisher Market

• DPSV, 2002
• First polynomial time algorithm
• Extend to separable, plc utilities??

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What makes linear utilities easy?

• Weak gross substitutability
• Increasing price of one good cannot
• decrease demand of another.
• Piecewise-linear, concave utilities do not
• satisfy this.

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Piecewise linear, concave
utility
amount of j
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Differentiate
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rate
amount of j
money spent on j
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Spending constraint utility function
rate utility/unit amount of j
rate
20
40
60
money spent on j
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• Theorem (V., 2002)
• Spending constraint utilities
• 1). Satisfy weak gross substitutability
• 2). Polynomial time algorithm for
• computing equilibrium
• 3). Equilibrium is rational.

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An unexpected fallout!!

• Has applications to
• Googles AdWords Market!

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Application to Adwords market
rate utility/click
rate
money spent on keyword j
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Is there a convex program for this model?

• We believe the answer to this question should be
  yes. In our experience, non-trivial polynomial
  time algorithms for problems are rare and happen
  for a good reason a deep mathematical structure
  intimately connected to the problem.

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Devanurs program for linear Fisher
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C. P. for spending constraint!
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Spending constraint market
Fisher market with plc utilities
EG convex program Devanurs program
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Price discrimination markets

• Business charges different prices from different
• customers for essentially same goods or
  services.
• Goel V., 2009
• Perfect price discrimination market.
• Business charges each consumer what
• they are willing and able to pay.

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plc utilities
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• Middleman buys all goods and sells to buyers,
• charging according to utility accrued.
• Given p, each buyer picks rate for accruing
• utility.

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• Middleman buys all goods and sells to buyers,
• charging according to utility accrued.
• Given p, each buyer picks rate for accruing
• utility.
• Equilibrium is captured by a
• rational convex program!

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Generalization of EG program works!
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• V., 2010 Generalize to
• Continuously differentiable, quasiconcave
• (non-separable) utilities, satisfying
  non-satiation.

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• V., 2010 Generalize to
• Continuously differentiable, quasiconcave
• (non-separable) utilities, satisfying
  non-satiation.
• Compare with Arrow-Debreu utilities!!
• continuous, quasiconcave, satisfying
  non-satiation.

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Spending constraint market
Price discrimination market (plc utilities)
EG convex program Devanurs program
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Eisenberg-Gale Markets Jain V., 2007
(Proportional Fairness) (Kelly, 1997)
Price disc. market
Spending constraint market
Nash Bargaining V., 2008
EG convex program Devanurs program
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A combinatorial market
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A combinatorial market
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A combinatorial market
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A combinatorial market

• Given
• Network G (V,E) (directed or undirected)
• Capacities on edges c(e)
• Agents source-sink pairs
• with money m(1), m(k)
• Find equilibrium flows and edge prices

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Equilibrium

• Flows and edge prices
• f(i) flow of agent i
• p(e) price/unit flow of edge e
• Satisfying
• p(e)gt0 only if e is saturated
• flows go on cheapest paths
• money of each agent is fully spent

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Kellys resource allocation model, 1997
Mathematical framework for understanding TCP
congestion control
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• Van Jacobson, 1988 AIMD protocol
• (Additive Increase Multiplicative Decrease)

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• Van Jacobson, 1988 AIMD protocol
• (Additive Increase Multiplicative Decrease)
• Why does it work so well?

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• Van Jacobson, 1988 AIMD protocol
• (Additive Increase Multiplicative Decrease)
• Why does it work so well?
• Kelly, 1977 Highly successful theory

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TCP Congestion Control

• f(i) source rate
• prob. of packet loss (in TCP Reno)
• queueing delay (in TCP Vegas)

p(e)
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TCP Congestion Control

• f(i) source rate
• prob. of packet loss (in TCP Reno)
• queueing delay (in TCP Vegas)
• Low Lapsley, 1999
• AIMD RED converges to equilibrium in limit

p(e)
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TCP Congestion Control

• f(i) source rate
• prob. of packet loss (in TCP Reno)
• queueing delay (in TCP Vegas)
• Kelly Equilibrium flows are proportionally
  fair
• only way of adding 5 flow to
  someone
• is to decrease total of 5 flow from
  rest.

p(e)
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• Kelly V., 2002 Kellys model is a
• generalization of Fishers model.

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• Kelly V., 2002 Kellys model is a
• generalization of Fishers model.
• Find combinatorial poly time algorithms!

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• Kelly V., 2002 Kellys model is a
• generalization of Fishers model.
• Find combinatorial poly time algorithms!
• (May lead to new insights for
• TCP congestion control
  protocol)

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Jain V., 2005

• Strongly polynomial combinatorial algorithm
• for single-source multiple-sink market

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Single-source multiple-sink market

• Given
• Network G (V,E), s source
• Capacities on edges c(e)
• Agents sinks
• with money
• Find equilibrium flows and edge prices

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Equilibrium

• Flows and edge prices
• f(i) flow of agent i
• p(e) price/unit flow of edge e
• Satisfying
• p(e)gt0 only if e is saturated
• flows go on cheapest paths
• money of each agent is fully spent

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5
5
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30
10
40
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Jain V., 2005

• Strongly polynomial combinatorial algorithm
• for single-source multiple-sink market
• Ascending price auction
• Buyers sinks (fixed budgets, maximize flow)
• Sellers edges (maximize price)

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Auction of k identical goods

• p 0
• while there are gtk buyers
• raise p
• end
• sell to remaining k buyers at price p

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Find equilibrium prices and flows
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Find equilibrium prices and flows
cap(e)
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min-cut separating from all the sinks
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60
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Throughout the algorithm
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sink demands flow
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Auction of edges in cut

• p 0
• while the cut is over-saturated
• raise p
• end
• assign price p to all edges in the cut

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60
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60
50
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nested cuts
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• Flow and prices will
• Saturate all red cuts
• Use up sinks money
• Send flow on cheapest paths

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• Exercise Find the red cuts efficiently!

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Convex Program for Kellys Model
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JV Algorithm

• primal-dual alg. for nonlinear convex program
• primal variables flows
• dual variables prices of edges
• algorithm primal dual improvements

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Rational!!
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Other resource allocation markets

• k source-sink pairs (directed/undirected)

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Other resource allocation markets

• k source-sink pairs (directed/undirected)
• Branchings rooted at sources (agents)
• Spanning trees
• Network coding

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Branching market (for broadcasting)
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Branching market (for broadcasting)
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Branching market (for broadcasting)
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Branching market (for broadcasting)
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Branching market (for broadcasting)

• Given Network G (V, E), directed
• edge capacities
• sources,
• money of each source
• Find edge prices and a packing
• of branchings rooted at sources
  s.t.
• p(e) gt 0 gt e is saturated
• each branching is cheapest possible
• money of each source fully used.

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Eisenberg-Gale-type program for branching
market
s.t. packing of branchings
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Eisenberg-Gale-Type Convex Program
s.t. packing constraints
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Eisenberg-Gale Market

• A market whose equilibrium is captured
• as an optimal solution to an
• Eisenberg-Gale-type program

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Other resource allocation markets

• k source-sink pairs (directed/undirected)
• Branchings rooted at sources (agents)
• Spanning trees
• Network coding

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Irrational for 2 sources 3 sinks
1
1
1
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Irrational for 2 sources 3 sinks
Equilibrium prices
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Max-flow min-cut theorem!
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• Theorem Strongly polynomial algs for
• following markets
• 2 source-sink pairs, undirected (Hu, 1963)
• spanning tree (Nash-William Tutte, 1961)
• 2 sources branching (Edmonds, 1967 JV, 2005)
• 3 sources branching irrational

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• Theorem Strongly polynomial algs for
• following markets
• 2 source-sink pairs, undirected (Hu, 1963)
• spanning tree (Nash-William Tutte, 1961)
• 2 sources branching (Edmonds, 1967 JV, 2005)
• 3 sources branching irrational
• Open (no max-min theorems)
• 2 source-sink pairs, directed
• 2 sources, network coding

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Chakrabarty, Devanur V., 2006

• EG2 Eisenberg-Gale markets with 2 agents
• Theorem EG2 markets are rational.

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Chakrabarty, Devanur V., 2006

• EG2 Eisenberg-Gale markets with 2 agents
• Theorem EG2 markets are rational.
• Combinatorial EG2 markets polytope
• of feasible utilities can be described via
• combinatorial LP.
• Theorem Strongly poly alg for Comb EG2.

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2 source-sink market in directed graphs
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2
1
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Polytope of feasible flows
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LPs corresponding to facets
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30
60
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Polytope of feasible flows
(1, 2)
2
1
(0, 1)
0
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• Find the two (one) facets
• Exponentially many facets!
• Binary search on

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5
10
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• Find relative prices of
• two facets, say
• Compute duals

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• Find relative prices of
• two facets, say
• Compute duals
• Compute

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5, each
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10/2 5, each
10, each
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10
5
30
15
60
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10
5
30
15
60
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10
5
3015x2
15
6020x3
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3-source branching
Single-source
SUA
2 s-s undir
Comb EG2
2 s-s dir
Rational
Fisher
EG2
EG
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