algorithmic game theory and internet computing vijay v. vazirani 3) new market models, resource...

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