Algorithmic Game Theory and Internet Computing Vijay V. Vazirani 3) New Market Models, Resource Allocation Markets

Fisher’s Model n buyers, with specified money, m(i) for buyer i k goods (unit amount of each good) Linear utilities: is utility derived by i on obtaining one unit of j Total utility of i,

Fisher’s Model n buyers, with specified money, m(i) k goods (each unit amount, w.l.o.g.) Linear utilities: is utility derived by i on obtaining one unit of j Total utility of i, Find prices s.t. market clears

Eisenberg-Gale Program, 1959

Via KKT Conditions can establish: Optimal solution gives equilibrium allocations Lagrange variables give prices of goods

Equilibrium exists (under mild conditions) Equilibrium utilities and prices are unique Eisenberg-Gale program helps establish:

Equilibrium exists (under mild conditions) Equilibrium utilities and prices are unique Rational!! Eisenberg-Gale program helps establish:

Kelly’s resource allocation model, 1997 Mathematical framework for understanding TCP congestion control

Kelly’s model Given: network G = (V,E) (directed or undirected) capacities on edges source-sink pairs (agents) m(i): money agent i is willing to pay

Kelly’s model Network determines: f(i): flow of agent i Assume utility u(i) = m(i) log f(i) Total utility is additive

Convex Program for Kelly’s Model

Kelly’s model Lagrange variables: p(e): price/unit flow

Kelly’s model Optimum flow and edge prices are in equilibrium: 1). p(e)>0 only if e is saturated 2) flows go on cheapest paths 3) money of each agent is fully used Let rate(i) = cost of cheapest path for i m(i) = f(i) rate(i)

Kelly’s model Optimum flow and edge prices are in equilibrium: 1). p(e)>0 only if e is saturated 2) flows go on cheapest paths 3) money of each agent is fully used Let rate(i) = cost of cheapest path for i f(i)’s and rate(i)’s are unique!

TCP Congestion Control f(i): source rate prob. of packet loss (in TCP Reno) queueing delay (in TCP Vegas) p(e):

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’s dollar is to decrease 5% flow from someone else’s dollar. p(e):

TCP Congestion Control f(i): source rate prob. of packet loss (in TCP Reno) queueing delay (in TCP Vegas) Low, Doyle, Paganini: continuous time algs. for computing equilibria (not poly time). p(e):

TCP Congestion Control f(i): source rate prob. of packet loss (in TCP Reno) queueing delay (in TCP Vegas) Low, Doyle, Paganini: continuous time algs. for computing equilibria (not poly time). AIMD + RED converges to equilibrium primal-dual (source-link) alg. p(e):

TCP Congestion Control f(i): source rate prob. of packet loss (in TCP Reno) queueing delay (in TCP Vegas) Low, Doyle, Paganini: continuous time algs. for computing equilibria (not poly time). FAST: for high speed networks with large bandwidth p(e):

Combinatorial Algorithms Devanur, Papadimitriou, Saberi & V., 2002: for Fisher’s linear utilities case Kelly & V., 2002: Kelly’s model is a generalization of Fisher’s model. Find combinatorial poly time algorithms!

Irrational for 2 sources & 3 sinks $1

Irrational for 2 sources & 3 sinks Equilibrium prices

1 source & multiple sinks 2 source-sink pairs

$5

$10 $40 $30

Jain & V., 2005: strongly poly alg Primal-dual algorithm  Usual: linear programs & LP-duality  This: convex programs & KKT conditions Ascending price auction  Buyers: sinks (fixed budgets, maximize flow)  Sellers: edges (maximize price)

rate(i): cost of cheapest path

Capacity of edge =

min s-t cut

nested cuts

Find s-t max flow Flow and prices will:  Saturate all red cuts  Use up sinks’ money  Send flow on cheapest paths

$10 $40 $30

Rational!!

Max-flow min-cut theorem

Other resource allocation markets 2 source-sink pairs (directed/undirected) Branchings rooted at sources (agents)

Branching market (for broadcasting) Given: Network G = (V, E)  edge capacities  sources,  money of each source Find: edge prices and a packing of branchings rooted at sources s.t.  p(e) > 0 => e is saturated  each branching is cheapest possible  money of each source fully used.

Eisenberg-Gale-type program for branching market s.t. packing of branchings

Other resource allocation markets 2 source-sink pairs (directed/undirected) Branchings rooted at sources (agents) Spanning trees Network coding

Eisenberg-Gale-Type Convex Program s.t. packing constraints

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

Megiddo, 1974: Let T = set of sinks (agents) For define v(S) to be the max-flow possible from s to sinks in S. Then v is a submodular function, i.e., for

Simpler convex program for single-source market

Submodular Utility Allocation Market Any market which has simpler program and v is submodular

Submodular Utility Allocation Market Any market which has simpler program and v is submodular Theorem: Strongly polynomial algorithm for SUA markets.

Submodular Utility Allocation Market Any market which has simpler program and v is submodular Theorem: Strongly polynomial algorithm for SUA markets. Corollary: Rational!!

Theorem: Following markets are SUA:  2 source-sink pairs, undirected (Hu, 1963)  spanning tree (Nash-William & Tutte, 1961)  2 sources branching (Edmonds, JV, 2005) 3 sources branching: irrational

Theorem: Following markets are SUA:  2 source-sink pairs, undirected (Hu, 1963)  spanning tree (Nash-William & Tutte, 1961)  2 sources branching (Edmonds, JV, 2005) 3 sources branching: irrational Open (no max-min thoerems):  2 source-sink pairs, directed  2 sources, network coding

Theorem: Following markets are SUA:  2 source-sink pairs, undirected (Hu, 1963)  spanning tree (Nash-William & Tutte, 1961)  2 sources branching (Edmonds, JV) 3 sources branching: irrational Open (no max-min thoerems):  2 source-sink pairs, directed  2 sources, network coding Chakrabarty, Devanur & V., 2006

EG[2]: Eisenberg-Gale markets with 2 agents Theorem: EG[2] markets are rational.

EG[2]: Eisenberg-Gale markets with 2 agents Theorem: EG[2] markets are rational. Combinatorial EG[2] markets: polytope of feasible utilities can be described via combinatorial LP. Theorem: Strongly poly alg for Comb EG[2]. Using Tardos, 1986.

2 source-sink market in directed graphs

2 1

Polytope of feasible flows

LP’s corresponding to facets

$30 $60

For any m(1), m(2), need to ‘‘price’’ at most two facets, with prices say

$10 $5

For any m(1), m(2), need to ‘‘price’’ at most two facets, with prices say Exponentially many facets!  Binary search on

For any m(1), m(2), need to ‘‘price’’ at most two facets, with prices say Exponentially many facets!  Binary search on Compute duals

For any m(1), m(2), need to ‘‘price’’ at most two facets, with prices say Exponentially many facets!  Binary search on Compute duals Compute

$5, each

10/2 = $5, each $10, each

$30 $60 $5 $10 $15

$30 $60 $5 $10 $15

$30=$15x2 $60=$20x3 $5 $10 $15

EG Rational Comb EG[2] SUA EG[2] 3-source branching Fisher 2 s-s undir 2 s-s dir Single-source

Observe: Equilibrium is always an s-t max-flow

Efficiency of Markets ‘‘price of capitalism’’ Agents:  different abilities to control prices  idiosyncratic ways of utilizing resources Q: Overall output of market when forced to operate at equilibrium.

Efficiency

Rich classification!

MarketEfficiency Single-source1 3-source branching k source-sink undirected 2 source-sink directedarbitrarily small

Other properties: Fairness (max-min + min-max fair) Competition monotonicity

Open issues Strongly poly algs for approximating  nonlinear convex programs  equilibria Insights into congestion control protocols?

The End