algorithmic game theory and internet computing

Algorithmic Game Theoryand Internet Computing

Vijay V. Vazirani

Nash Bargaining via

Flexible Budget Markets

The new platform for computing

Internet

Massive computational power available

Sellers (programs) can negotiate with

individual buyers!

Internet



individual buyers!

Back to bargaining!

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individual buyers!

Algorithmic Game Theory

Bargaining and Game Theory

Nash (1950): First formalization of bargaining.

von Neumann & Morgenstern (1947):

Theory of Games and Economic Behavior

Game Theory: Studies solution concepts for

negotiating in situations of conflict of interest.

Bargaining and Game Theory

Nash (1950): First formalization of bargaining.

von Neumann & Morgenstern (1947):

Theory of Games and Economic Behavior

Game Theory: Studies solution concepts for

negotiating in situations of conflict of interest.

Theory of Bargaining: Central!

Nash bargaining

Captures the main idea that both players

gain if they agree on a solution.

Else, they go back to status quo.

Example

Two players, 1 and 2, have vacation homes:

1: in the mountains

2: on the beach

Consider all possible ways of sharing.

Utilities derived jointly

1v

2v

S : convex + compact

feasible set

Disagreement point = status quo utilities

1v

2v

1c

2c

S

Disagreement point = 1 2( , )c c

Nash bargaining problem = (S, c)

1v

2v

1c

2c

S

Disagreement point = 1 2( , )c c

Nash bargaining

Q: Which solution is the “right” one?

Solution must satisfy 4 axioms:

Paretto optimality

Invariance under affine transforms

Symmetry

Independence of irrelevant alternatives

1v

2v

1c

2c

v

S

( , ),

& ( , )

v N S c

T S v T v N T c

1v

2v

1c

2c

v

S

T

( , ),

& ( , )

v N S c

T S v T v N T c

Thm: Unique solution satisfying 4 axioms

1 2( , ) 1 1 2 2( , ) max {( )( )}v v SN S c v c v c

1v

2v

1c

2c

S

Generalizes to n-players

Theorem: Unique solution

1 1( , ) max {( ) ... ( )}v S n nN S c v c v c

Generalizes to n-players

Theorem: Unique solution

1 1( , ) max {( ) ... ( )}v S n nN S c v c v c

(S, c) is feasible if S contains a point that makes each i strictly happier than ci

Bargaining theory studies promise problem

Restrict to instances (S, c) which are feasible.

Linear Nash Bargaining (LNB)

Feasible set is a polytope defined by

linear packing constraints

Nash bargaining solution is

optimal solution to convex program:

max log( )

. .

i ii

v c

s t

packing constraints

Q: Compute solution combinatoriallyin polynomial time?

Study promise problem?

Decision problem reduces to promise problem

Therefore, study decision and search problems.

Linear utilities

B: n players with disagreement points, ci

G: g goods, unit amount each

S = utility vectors obtained by distributing

goods among players

0i ij ij ijj G

v u x x

e.g., ci = i’s utility for initial endowment




goods among players

0i ij ij ijj G

v u x x

Convex program giving NB solution

max log( )

. .

:

: 1

: 0

i ii

i ij ijj

iji

ij

v c

s t

i v

j

ij

u xx

x

Theorem

If instance is feasible,

Nash bargaining solution is rational! Polynomially many bits in size of instance

Theorem

If instance is feasible,

Nash bargaining solution is rational! Polynomially many bits in size of instance

Decision and search problems

can be solved in polynomial time.

Resource Allocation Nash Bargaining Problems

Players use “goods” to build “objects”

Player’s utility = number of objects

Bound on amount of goods available

2s

1s

2t

1t

( )cap e

Goods = edges Objects = flow paths

2s

1s

2t

1t

( )cap e

Given disagreement point, find NB soln.

Theorem:

Strongly polynomial, combinatorial algorithm

for single-source multiple-sink case.Solution is again rational.

Insights into game-theoretic properties of Nash bargaining problems

Chakrabarty, Goel, V. , Wang & Yu:

Efficiency (Price of bargaining)Fairness Full competitiveness

Linear utilities




goods among players

0i ij ij ijj G

v u x x

Game plan

Use KKT conditions to

transform Nash bargaining problem to

computing the equilibrium in a certain market.

Find equilibrium using primal-dual paradigm.

Crown jewel of mathematicaleconomics for over a century!

General Equilibrium Theory

A central tenet

Prices are such that demand equals supply, i.e.,

equilibrium prices.

A central tenet

Prices are such that demand equals supply, i.e.,

equilibrium prices.

Easy if only one good

Supply-demand curves

Irving Fisher, 1891

Defined a fundamental

market model

Fisher’s Model

B = n buyers, money mi for buyer i

G = g goods, w.l.o.g. unit amount of each good : utility derived by i

on obtaining one unit of j Total utility of i,

i ij ijj

U u xiju

[0,1]

i ij ijj

ij

v u xx

Fisher’s Model

B = n buyers, money mi for buyer i

G = g goods, w.l.o.g. unit amount of each good : utility derived by i

on obtaining one unit of j Total utility of i,

Find market clearing prices.

i ij ijj

U u xiju

[0,1]

i ij ijj

ij

v u xx

An almost entirely non-algorithmic theory!

General Equilibrium Theory

Flexible budget market,only difference:

Buyers don’t spend a fixed amount of money.

Instead, they know how much utility they desire.

At any given prices, they spend just enough

money to accrue utility desired.

Most cost-effective goods

At prices p, for buyer i: Si =

Define

arg min jj

ij

p

u

( ) min jj

ij

pcost i

u

Flexible budget market

Agent i wants utility

At prices p, must spend to get utility

ic

ic. ( )ic cost i




Define

Find market clearing prices.

ic

ic

1 . ( )i im c cost i

. ( )ic cost i




Define

Find market clearing prices -- may not exist!!

ic

ic


. ( )ic cost i




Define

Find market clearing prices -- may not exist!!

feasible/infeasible

ic

ic


. ( )ic cost i

Theorem: Nash Bargaining for linear utilities

reduces to

Equilibrium for flexible budget markets

Theorem: Nash Bargaining for linear utilitiesreduces to

Equilibrium for flexible budget markets

(S(u), c) M(u, c)

(S, c) is feasible iff M is feasible.

If feasible, x is Nash bargaining solution iff x is equilibrium allocation.

Primal-Dual Paradigm

Usual framework: LP-duality theory


Usual framework: LP-duality theory

Extension to convex programs and

KKT conditions.

Yin & Yang

Combinatorial Algorithm for Linear Case of Fisher’s Model

Devanur, Papadimitriou, Saberi & V., 2002

Using primal-dual paradigm

Combinatorial Algorithm for Linear Case of Fisher’s Model

Devanur, Papadimitriou, Saberi & V., 2002

Using primal-dual paradigm

Solves Eisenberg-Gale convex program

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

Why remarkable?

Equilibrium simultaneously optimizes

for all agents.

How is this done via a single objective function?

Idea of algorithm

primal variables: allocations dual variables: prices of goods iterations:

execute primal & dual improvements

Allocations Prices (Money)


Main differences:

mi ’s change as prices change.

problem is not total.



?

InfeasibleFeasible

Decision

Search

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.

For each i, 1 . ( )i im c cost i

m(1)

m(2)

m(3)

m(4)

p(1)

p(2)

p(3)

p(4)

For each i, most cost-effective goods

arg min ji j

ij

pS

u

Network N(p)

m(1)

m(2)

m(3)

m(4)

p(1)

p(2)

p(3)

p(4)

infinite capacities

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

Two important considerations

The price of a good never exceeds

its equilibrium priceInvariant: s is a min-cut

Max flow

m(1)

m(2)

m(3)

m(4)

p(1)

p(2)

p(3)

p(4)

p: low prices

Two important considerations

The price of a good never exceeds

its equilibrium priceInvariant: s is a min-cut

Rapid progress is madeBalanced flows

Max-flow in N

m p

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

i

Balanced flow

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

A max-flow that

minimizes l2 norm of surplus vector.



?

InfeasibleFeasible

Decision

Search

Balanced flow helps Decision as well!

Proof of infeasibility: dual solution to

max

. .

:

: 1

, : 0

ij ij ij G

iji B

ij

t

s t

i u x c t

j x

i j x

Theorem: Algorithm runs in polynomial time.

Theorem: Algorithm runs in polynomial time.

Q: Find strongly polynomial algorithm!

Nonlinear programs with rational solutions!

Open

Nonlinear programs with rational solutions!

Solvable combinatorially!!

Open


Combinatorial Optimization (1960’s & 70’s):

Integral optimal solutions to LP’s




Approximation Algorithms (1990’s):

Near-optimal integral solutions to LP’s






Algorithmic Game Theory (New Millennium):

Rational solutions to nonlinear convex programs






Algorithmic Game Theory (New Millennium):

Rational solutions to nonlinear convex programs

Approximation algorithms for convex programs?!

Open

Can Nash bargaining problem

for linear utilities case

be captured via an LP?

algorithmic game theory and internet computing

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