Algorithmic Game Theoryand 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,

i ij ijj

U u xiju

]1,0[

x

xuuij

ijj iji

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

i ij ijj

U u xiju

xuu ijj iji

Eisenberg-Gale Program, 1959

max ( ) log

. .

:

: 1

: 0

ii

i ij ijj

iji

ij

m i u

s t

i u

j

ij

u xx

x

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

)1(m

t1

s2

t2)(ecs1

)2(m

Kelly’s model

Network determines:

f(i): flow of agent i

Assume utility u(i) = m(i) log f(i)

Total utility is additive

t1

s2

t2s1

Convex Program for Kelly’s Model

0:,

)()(:

)(:..

)(log)(max

f

f

p

i

p

p

i

i

pi

eceflowe

ifits

ifim

Kelly’s model

t1

s2

t2)(eps1

Lagrange variables:

p(e): price/unit flow

Kelly’s model

t1

s2

t2)(eps1

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

t1

s2

t2)(eps1

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

s1 t1

1

s2

t2

1

t21 2

$1 $1

$1

Irrational for 2 sources & 3 sinks

s1 t1

1

s2

t2

1

t2

313

3

Equilibrium prices

1 source & multiple sinks 2 source-sink pairs

s

t1

t2

2

2

110$

10$

s

t1

t2

2

2

1 10$

10$

$5

$5

s

t1

t2

2

2

1 10$

10$

120$

s

t1

t2

2

2

1 120$

10$

$10

$40

$30

Jain & V., 2005: strongly poly alg

Primal-dual algorithmUsual: linear programs & LP-dualityThis: convex programs & KKT conditions

Ascending price auctionBuyers: sinks (fixed budgets, maximize flow)Sellers: edges (maximize price)

st1

t2

t3

t4

rate(i): cost of cheapest path ts i

st1

t2

t3

t4

t

st1

t2

t3

t4

t

Capacity of edge =tt i

)()(irate

im

st1

t2

t3

t4

t

min s-t cut

st1

t2

t3

t4

t

p

st1

t2

t3

t4

t

p

st1

t2 t3

t4

t

pp 0

prate0

)2(

st1

t2 t3

t4

t

p0

p prate0

)2(

st1

t2 t3

t4

t

p0

p1

prate0

)2(

ppraterate10

)3()1(

st1

t2 t3

t4

t

p0

p1

p

st1

t2 t3

t4

t

p0

p1 p

2 nested cuts

st1

t2 t3

t4

t

p0

p1 p

2

prate0

)2(

ppraterate10

)3()1(

ppprate210

)4(

Find s-t max flow

Flow and prices will:

Saturate all red cutsUse up sinks’ moneySend flow on cheapest paths

s

t1

t2

2

2

1120$

10$

a

b

s

t1

t2

2

2

1 p120

p10

a

b

t

p

s

t1

t2

2

2

1 10120

11010

a

b

t

10p

s

t1

t2

2

2

1 p10120

1

a

b

t

100p p

s

t1

t2

2

2

1 33010

120

1

a

b

t

100p 30p

s

t1

t2

2

2

1 120$

10$

$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.

S V

Eisenberg-Gale-type program for branching market

max ( ) log ii Sm i b

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

max ( ) log iim i u

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

TS

)()()()(,

,

AvtAvBvtBvAt

TBA

Simpler convex program for single-source market

0)(:

)()(:

..)(log)(max

ifi

SvifTS

tsifim

Si

i

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, 1967 + 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, 1967 + JV, 2005)

3 sources branching: irrational

Open (no max-min thoerems):2 source-sink pairs, directed2 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, 1967 + JV)

3 sources branching: irrational

Open (no max-min thoerems):2 source-sink pairs, directed2 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

2s

1s 1t

2t

2

1

1s

2t

2s

1t

1s

2t

2s

1t

1f

2fPolytope of feasible flows

1 4f

1 2 5f f

1 22 8f f

LP’s corresponding to facets

1 2max. .

f fs tcapacity

1 1

2 2

min ( )

. .( ) 1( )

ee

c e x

s td s td s t

1 22 8f f

1s

2s

2t

1t

1ex

1 2 5f f

2s

t1

t2

1s

1/ 2ex

1ex

$30

$60

1s 1t

2t

2s

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

$10

$5

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

Exponentially many facets! Binary search on

1 2,p p

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

Exponentially many facets! Binary search on

Compute duals

1 2,p p

1 2,x x

1s

2s

2t

1t

1ex

2s

t1

t2

1s

1/ 2ex

1ex

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

1 1 2 2( ) ( ) ( )p e p x e p x e

1 2,p p

1 2,x x

1s

2s

2t

1t

$5, each

2s

t1

t2

1s

10/2 = $5, each

$10, each

$30

$60

$5$10

$15

1s

2s

1t

2t

$30

$60

$5$10

$15

1s

2s

2t

1t

$30=$15x2

$60=$20x3

$5$10

$15

1s

2s

2t

1t

EG

Rational

Comb EG[2]

SUA

EG[2]

3-source branching

Fisher

2 s-s undir

2 s-s dir

Single-source

st1

t2

t3

t4

t

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

( )( ) minmax ( )I

equilibrium utility Ieff Mutility I

Efficiency

Rich classification!

( )( ) minmax ( )I

equilibrium utility Ieff Mutility I

1/(2 1)k

Market EfficiencySingle-source 1

3-source branching

k source-sink undirected

2 source-sink directed arbitrarily small

1/ 2

. . 1/( 1)l b k

Other properties:

Fairness (max-min + min-max fair) Competition monotonicity

Open issues

Strongly poly algs for approximatingnonlinear convex programsequilibria

Insights into congestion control protocols?

The End