Algorithmic Game Theoryand Internet Computing

Vijay V. Vazirani

Georgia Tech

Algorithms for the Linear Case,

and Beyond …

Irving Fisher, 1891

Defined a fundamental

market model

Special case of Walras’

model

Several buyers with different utility functions and moneys.

Find equilibrium prices!!

1p 2p3p

Linear Fisher Market

Assume:Buyer i’s total utility,

mi : money of buyer i.

One unit of each good j.

Find market clearing prices!

i ij ijj G

v u x

Eisenberg-Gale Program, 1959

max log

. .

:

: 1

: 0

i ii

i ij ijj

iji

ij

m v

s t

i v

j

ij

u xx

x

Eisenberg-Gale Program, 1959

max log

. .

:

: 1

: 0

i ii

i ij ijj

iji

ij

m v

s t

i v

j

ij

u xx

x

prices pj

Convex programs thatcapture market equilibria

Underly all “efficient” markets (so far).

Rational convex programs for many markets!

Algorithms: combinatorial, for rational (primal-dual) continuous (ellipsoid/interior point)

Combinatorial algorithms for rational convex programs

Natural extension of field of

combinatorial optimization!

Combinatorial algorithms for rational convex programs

Natural extension of field of

combinatorial optimization!

Central aspect of C.O. – efficient algorithms

for solving integral LP’s, e.g. matching, flow.

Combinatorial Algorithm for Linear Case of Fisher’s Model

Devanur, Papadimitriou, Saberi & V., 2002

By extending the primal-dual paradigm to the setting of convex programs & KKT conditions

Combinatorial algorithms

Yield deep structural insights.

Preferable for applications.

Auction for Google’s TV ads

N. Nisan et. al, 2009:

Used market equilibrium based approach.

Combinatorial algorithms for linear case

provided “inspiration”.

Primal-Dual Paradigm

Highly successful algorithm design

technique from exact and

approximation algorithms

Exact Algorithms for Cornerstone Problems in P:

Matching (general graph) Network flow Shortest paths Minimum spanning tree Minimum branching

Approximation Algorithms

set cover facility location

Steiner tree k-median

Steiner network multicut

k-MST feedback vertex set

scheduling . . .

Yin & Yang

An easier question

Given prices p, are they equilibrium prices?

If so, find equilibrium allocations.

An easier question

Given prices p, are they equilibrium prices?

If so, find equilibrium allocations.

Equilibrium prices are unique!

At prices p, buyer i’s most

desirable goods, Si =

Any goods from Si worth m(i)

constitute i’s optimal bundle

arg max ijj

j

u

p

Bang-per-buck

m(1)

m(2)

m(3)

m(4)

p(1)

p(2)

p(3)

p(4)

For each buyer, most desirable goods, i.e.

arg max iji j

j

uS

p

Network N(p)

m(1)

m(2)

m(3)

m(4)

p(1)

p(2)

p(3)

p(4)

infinite capacities

st

Max flow in N(p)

m(1)

m(2)

m(3)

m(4)

p(1)

p(2)

p(3)

p(4)

p: equilibrium prices iff both cuts saturated

Idea of algorithm

“primal” variables: allocations

“dual” variables: prices of goods

Approach equilibrium prices from below:start with very low prices; buyers have surplus money iteratively keep raising prices and decreasing surplus

An important consideration

The price of a good never exceeds

its equilibrium price

Invariant: s is a min-cut

Invariant: s is a min-cut in N(p)

m(1)

m(2)

m(3)

m(4)

p(1)

p(2)

p(3)

p(4)

p: low prices

s

Idea of algorithm

Iterations:

execute primal & dual improvements

Allocations Prices

How is primal-dual paradigm

adapted to nonlinear setting?

Fundamental difference betweenLP’s and convex programs

Complementary slackness conditions:

involve primal or dual variables, not both.

KKT conditions: involve primal and dual

variables simultaneously.

KKT conditions

1. : 0

2. : 0 1

3. , :( )

4. , : 0( )

j

j iji

ij i

j

ij iij

j

j p

j p x

u vi j

p m i

u vi j x

p m i

KKT conditions

1. : 0

2. : 0 1

3. , :( )

4. , : 0( ) ( )

j

j iji

ij i

j

ij ijij jiij

j

j p

j p x

u vi j

p m i

u xu vi j x

p m i m i

Primal-dual algorithms so far(i.e., LP-based)

Raise dual variables greedily. (Lot of effort spent

on designing more sophisticated dual processes.)

Primal-dual algorithms so far

Raise dual variables greedily. (Lot of effort spent

on designing more sophisticated dual processes.)

Only exception: Edmonds, 1965: algorithm

for max weight matching.

Primal-dual algorithms so far

Raise dual variables greedily. (Lot of effort spent

on designing more sophisticated dual processes.)

Only exception: Edmonds, 1965: algorithm

for max weight matching.

Otherwise primal objects go tight and loose.

Difficult to account for these reversals --

in the running time.

Our algorithm

Dual variables (prices) are raised greedily

Yet, primal objects go tight and looseBecause of enhanced KKT conditions

Our algorithm

Dual variables (prices) are raised greedily

Yet, primal objects go tight and looseBecause of enhanced KKT conditions

New algorithmic ideas needed!

Key Algorithmic Idea

Dual variables (prices) are raised greedily

Yet, primal objects go tight and looseBecause of enhanced KKT conditions

Balanced Flows: For limiting no. of such events

Max-flow in N

m p

W.r.t. a max-flow f, surplus(i) = m(i) – f(i,t)

i

t s

Max-flow in N

m p

surplus vector = vector of surpluses w.r.t. f

i

Obvious potential function

Total surplus money = l1 norm of surplus vector

Reduce l1 norm of surplus vector by

inverse polynomial fraction in each iteration

Balanced flow

A max-flow that

minimizes l2 norm of surplus vector.

Makes surpluses as equal as possible.

Balanced flow

A max-flow that

minimizes l2 norm of surplus vector.

Makes surpluses as equal as possible.

All balanced flows have same surplus vector.

Our algorithm

Reduces l2 norm of surplus vector by

inverse polynomial fraction in each iteration.

s1

s2

(1, 0)

(0, 1)

Property 1

f: max-flow in N.

R: residual graph w.r.t. f.

If surplus (i) < surplus(j) then there is no

path from i to j in R.

Property 1

i

j

R:

surplus(i) < surplus(j)

Property 1

i

surplus(i) < surplus(j)

j

R:

Property 1

i

Circulation gives a more balanced flow.

j

R:

Property 1

Theorem: A max-flow is balanced iff

it satisfies Property 1.

Construct N’(I, J)

Raise prices in J

New edge enters N

Stop when Invariant is threatened

Algorithm for an iteration

Network N(p)

m p

buyers goods

bang-per-buck edges

Construct N’(I, J)Find a balanced flow in N(p)

Let d = max surplus w.r.t. balanced flowI = buyers with surplus dJ = goods desired by I

Raise prices in J

New edge enters N

Stop when Invariant is threatened

Network N(p)

N’(I, J)I J

N - N’

Network N(p)

N’(I, J)I J

N - N’

Construct N’(I, J)

Raise prices in J N’ is decoupled from N - N’

New edge enters N

Stop when Invariant is threatened

Network N(p)

N’(I, J)I J

N - N’

Network N(p)

N’(I, J)I J

N - N’

By Property 1, this edge did not carry any flow.Hence Invariant is not violated by its removal.

Raise prices in J

proportionately, so that

edges in N’ don’t change.

p . x, for each p in J initialize x = 1raise x

Construct N’(I, J)

Raise prices in J

New edge enters N

Stop when Invariant is threatened

Network N(p)

N’(I, J)I J

N - N’

Construct N’(I, J)

Raise prices in J

New edge enters NRecompute balanced flowBuyers in N - N’ having residual paths to N’

Move to N’

Stop when Invariant is threatened

Network N(p)

N’(I, J)I J

N - N’

Network N(p)

N’(I, J)I J

N - N’

Network N(p)

N’(I, J)I J

N - N’

Construct N’(I, J)

Raise prices in J

New edge enters NRecompute balanced flowBuyers moved to N’ will have

sufficiently large surplus

Stop when Invariant is threatened

Construct N’(I, J)

Raise prices in J

New edge enters N

Stop when Invariant is threatened

Algorithm for an iteration

Tight set: p(S) = m(T)

N’(I, J)

T S

N - N’

Surplus of buyers in T drops to 0

Surplus of buyers in T drops to 0

l1 norm of surplus vector drops by 1/n fraction

after the iteration.

Assume k sub-iterations.

Let d0 = d. At the end of lth sub-iteration,

dl = min {surplus(i) | i is in I}. So, dk = 0.

Network N(p)

N’(I, J)I J

N - N’

Some i in old I will achieve minimum.

Its surplus must drop by at least (dl-1 – dl).

Therefore, decrease in l1 norm

in sub-iteration l is at least (dl-1 – dl)

Therefore, decrease in iteration is at least d

Network N(p)

N’(I, J)I J

N - N’

Assume k sub-iterations.

Let d0 = d. At the end of lth sub-iteration,

dl = min {surplus(i) | i is in I}. So, dk = 0.

Decrease in l1 norm in sub-iteration l

is at least (dl-1 – dl)

Decrease in l22 norm in sub-iteration l

is at least (dl-1 – dl)2

Our algorithm

Reduces l2 norm of surplus vector by

1/n2 fraction in each iteration

Open question

Can define balanced flow without l2 normBalanced flow = lexicographically smallest flow

Q: Can we dispense with l2 norm in proof?

Open question

Can define balanced flow without l2 normBalanced flow = lexicographically smallest flow

Q: Can we dispense with l2 norm in proof?

V, 2008: Family of examples s.t. l1 norm of surplus vector decreases by inverse exponential fraction in an iteration!

s1

s2

(1, 0)

(0, 1)

KKT conditions were relaxed

e(i): money currently spent by i

w.r.t. a balanced flow in N

surplus money of i γi =mi −e(i)

Relaxed KKT conditions

1. : 0

2. : 0 1

3. , :( )

4. , : 0( )

j

j iji

ij i

j

ij iij

j

j p

j p x

u vi j

p m i

u vi j x

p m i

e(i)

e(i)

Potential function

2 2 21 2 ... nγ γ γ

Algorithm drops potential by an inverse polynomial

factor in each iteration (strongly polynomial time).

Potential function

2 2 21 2 ... nγ γ γ

Algorithm drops potential by an inverse polynomial

factor in each iteration (strongly polynomial time).

= poly mii∑( )

Second point of departure

KKT conditions are satisfied via a

continuous process Normally: in discrete steps

utility

Piecewise linear, concave

amount of j

Additively separable over goods

Long-standing open problem

Complexity of finding an equilibrium for

Fisher and Arrow-Debreu models under

separable, plc utilities?

How do we build on solution to the linear case?

utility

amount of j

Generalize EG program to

piecewise-linear, concave utilities?

ijkl

ijkuutility/unit of j

,

,

max log

. .

:

: 1

:

: 0

i ii

i ijk ijkj k

iji k

ijk ijk

ijk

m v

s t

i v

j

ijk

ijk

u xx

x l

x

Generalization of EG program

,

,

max log

. .

:

: 1

:

: 0

i ii

i ijk ijkj k

iji k

ijk ijk

ijk

m v

s t

i v

j

ijk

ijk

u xx

x l

x

Generalization of EG program

V. & Yannakakis, 2007: Equilibrium is rational for Fisher and Arrow-Debreu models under separable, plc utilities.

Given prices p, are they equilibrium prices?

Build on combinatorial insights

utility

amount of j

Case 1

ijx

fully allocated

partially allocated

utility

amount of j

Case 2: no p.a. segment

ijx

fully allocated

p full & partial segments

Network N(p)

Theorem: p equilibrium prices iff

max-flow in N(p) = unspent money.

Network N(p)

partially allocated segments

s

t

q1

q2

q3

q4

m '1

m '2

m '3

m '4

LP for max-flow in N(p); variables = fe’s

Next, let p be variables!

“Guess” full & partial segments – gives N(p)

Write max-flow LP -- it is still linear! variables = fe’s & pj’s

Rationality proof

If “guess” is correct,

at optimality, pj’s are equilibrium prices.

Hence rational!

Rationality proof

If “guess” is correct,

at optimality, pj’s are equilibrium prices.

Hence rational!

In P??

NP-hardness does not apply

Megiddo, 1988:Equilibrium NP-hard => NP = co-NP

Papadimitriou, 1991: PPAD2-player Nash equilibrium is PPAD-completeRational

Etessami & Yannakakis, 2007: FIXP3-player Nash equilibrium is FIXP-completeIrrational

Markets with piecewise-linear, concave utilities

Chen, Dai, Du, Teng, 2009: PPAD-hardness for Arrow-Debreu model

Markets with piecewise-linear, concave utilities

Chen, Dai, Du, Teng, 2009: PPAD-hardness for Arrow-Debreu model

Chen & Teng, 2009: PPAD-hardness for Fisher’s model

V. & Yannakakis, 2009:PPAD-hardness for Fisher’s model

Markets with piecewise-linear, concave utilities

V, & Yannakakis, 2009: Membership in PPAD for both models,

How do we salvage the situation??

Algorithmic ratification of the

“invisible hand of the market”

Is PPAD really hard??

What is the “right” model??

Open

Can Fisher’s linear case

be captured via an LP?