Nash Welfare, Market Equilibrium, and Stable Polynomials

IntroductionSpending-Restricted Outcome

Approximation Algorithm

Nash Welfare, Market Equilibrium,and Stable Polynomials

STOC 2019 tutorial23 June 2019

Nima Anari Stanford UniversityJugal Garg University of Illinois at Urbana ChampaignVasilis Gkatzelis Drexel University

Nima Anari, Jugal Garg, and Vasilis Gkatzelis Nash Welfare, Market Equilibrium, and Stable Polynomials



Resource AllocationNash Social WelfareApproximation Guarantee

Overview

First Section (9-10am)“Approximating the Nash Social Welfare with Indivisible Items”Vasilis Gkatzelis

Second Section (10-11am)“NSW Beyond Symmetric Agents with Additive Valuations”Jugal Garg

Coffee Break (11-11:20pm)

Third Section (11:20-12:20pm)“Nash Social Welfare and Stable Polynomials”Nima Anari





Resource Allocation

Distribute a collection of items among a set of agents

Each agent has additive valuations

i

5

4

3

2

1

[vi1, vi2, vi3, vi4, vi5][15, 2, 4, 5, 3]

vi ({1}) = 15

vi ({2}) = 2

vi ({2, 3}) = 6

vi ({2, 3, 4}) = 11

vi ({2, 3, 4, 5}) = 14

Items





Resource Allocation



i

5

4

3

2

1

[vi1, vi2, vi3, vi4, vi5][15, 2, 4, 5, 3]

vi ({1}) = 15

vi ({2}) = 2

vi ({2, 3}) = 6

vi ({2, 3, 4}) = 11

vi ({2, 3, 4, 5}) = 14

Items





Resource Allocation



i

5

4

3

2

1

[vi1, vi2, vi3, vi4, vi5][15, 2, 4, 5, 3]

vi ({1}) = 15

vi ({2}) = 2

vi ({2, 3}) = 6

vi ({2, 3, 4}) = 11

vi ({2, 3, 4, 5}) = 14

Items





Resource Allocation



i

5

4

3

2

1

[vi1, vi2, vi3, vi4, vi5][15, 2, 4, 5, 3]

vi ({1}) = 15

vi ({2}) = 2

vi ({2, 3}) = 6

vi ({2, 3, 4}) = 11

vi ({2, 3, 4, 5}) = 14

Items





Resource Allocation



i

5

4

3

2

1

[vi1, vi2, vi3, vi4, vi5][15, 2, 4, 5, 3]

vi ({1}) = 15

vi ({2}) = 2

vi ({2, 3}) = 6

vi ({2, 3, 4}) = 11

vi ({2, 3, 4, 5}) = 14

Items





Resource Allocation



i

5

4

3

2

1

[vi1, vi2, vi3, vi4, vi5][15, 2, 4, 5, 3]

vi ({1}) = 15

vi ({2}) = 2

vi ({2, 3}) = 6

vi ({2, 3, 4}) = 11

vi ({2, 3, 4, 5}) = 14

Items





Resource Allocation



i

5

4

3

2

1

[vi1, vi2, vi3, vi4, vi5][15, 2, 4, 5, 3]

vi ({1}) = 15

vi ({2}) = 2

vi ({2, 3}) = 6

vi ({2, 3, 4}) = 11

vi ({2, 3, 4, 5}) = 14

Items





Resource Allocation



i

5

4

3

2

1

[vi1, vi2, vi3, vi4, vi5][15, 2, 4, 5, 3]

vi ({1}) = 15

vi ({2}) = 2

vi ({2, 3}) = 6

vi ({2, 3, 4}) = 11

vi ({2, 3, 4, 5}) = 14

Items





Resource Allocation



i

5

4

3

2

1

[vi1, vi2, vi3, vi4, vi5][15, 2, 4, 5, 3]

vi ({1}) = 15

vi ({2}) = 2

vi ({2, 3}) = 6

vi ({2, 3, 4}) = 11

vi ({2, 3, 4, 5}) = 14

Items





Setting

Set N of n agents and set M of m indivisible items

For each agent i and item j : xij ∈ {0, 1}For each agent i : vi(x) =

∑j∈M xijvij

4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[1,0,0,0,0]

Agents Items





Setting



∑j∈M xijvij

4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[1,0,0,0,0]

Agents Items





Setting



∑j∈M xijvij

4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[1,0,0,0,0]

Agents Items





Setting



∑j∈M xijvij

4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[1,0,0,0,0]

Agents Items





Utilitarian Social Welfare

Maximize the utilitarian social welfare: maxx

∑i∈N

vi (x)

2

1

8

7

6

5

4

3

2

1

[1,1,1,1,7,7,7,7]

[1,1,1,1,1,1,1,1]

Agents Items







∑i∈N

vi (x)

2

1

8

7

6

5

4

3

2

1

[1,1,1,1,7,7,7,7]

[1,1,1,1,1,1,1,1]

Agents Items







∑i∈N

vi (x)

2

1

8

7

6

5

4

3

2

1

[1,1,1,1,7,7,7,7]

[1,1,1,1,1,1,1,1]

Agents Items





Egalitarian Social Welfare [BS’06,AS’10,F’08,...]

Maximize the egalitarian social welfare: maxx

{mini∈N

vi (x)

}

2

1

8

7

6

5

4

3

2

1

[1,1,1,1,7,7,7,7]

[1,1,1,1,1,1,1,1]

Agents Items







{mini∈N

vi (x)

}

2

1

8

7

6

5

4

3

2

1

[1,1,1,1,7,7,7,7]

[1,1,1,1,1,1,1,1]

Agents Items







{mini∈N

vi (x)

}

2

1

8

7

6

5

4

3

2

1

[1,1,1,1,7,7,7,7]

[1,1,1,1,1,1,1,1]

Agents Items





Nash Social Welfare

Maximize the Nash social welfare: maxx

(∏i∈N

vi (x)

)1/n

2

1

8

7

6

5

4

3

2

1

[1,1,1,1,7,7,7,7]

[1,1,1,1,1,1,1,1]

Agents Items





Nash Social Welfare


(∏i∈N

vi (x)

)1/n

2

1

8

7

6

5

4

3

2

1

[1,1,1,1,7,7,7,7]

[1,1,1,1,1,1,1,1]

Agents Items





Nash Social Welfare


(∏i∈N

vi (x)

)1/n

2

1

8

7

6

5

4

3

2

1

[1,1,1,1,7,7,7,7]

[1,1,1,1,1,1,1,1]

Agents Items





Nash Social Welfare


(∏i∈N

vi (x)

)1/n

2

1

8

7

6

5

4

3

2

1

[1,1,1,1,7,7,7,7]

[1,1,1,1,1,1,1,1]

Agents Items





Nash Social Welfare


(∏i∈N

vi (x)

)1/n

2

1

8

7

6

5

4

3

2

1

[1,1,1,1,7,7,7,7]

[1,1,1,1,1,1,1,1]

Agents Items





Nash Social Welfare


(∏i∈N

vi (x)

)1/n

2

1

8

7

6

5

4

3

2

1

[1,1,1,1,7,7,7,7]

[1,1,1,1,1,1,1,1]

Agents Items





Nash Social Welfare

The Nash SW objective satisfies highly desired properties:

Scale-independenceUsing v ′

ij = αivij for any αi > 0 does not affect the outcomeAvoids interpersonal comparability of individual’s preferences

Strikes a balance between fairness and efficiency

maxx

(1

n

∑i

[vi (x)]p

)1/p

Discovered by different communities:

Nash Bargaining [Nash ’50]

Proportional Fairness [Kelly ’97]

Competitive Equilibrium from Equal Incomes [Varian ’74]





Nash Social Welfare

The Nash SW objective satisfies highly desired properties:

Scale-independenceUsing v ′

ij = αivij for any αi > 0 does not affect the outcomeAvoids interpersonal comparability of individual’s preferences

Strikes a balance between fairness and efficiency

maxx

(1

n

∑i

[vi (x)]p

)1/p

Discovered by different communities:

Nash Bargaining [Nash ’50]

Proportional Fairness [Kelly ’97]

Competitive Equilibrium from Equal Incomes [Varian ’74]





Approximation Guarantee

Let x∗ be the integral allocation maximizing the Nash SW

Goal: Design algorithm computing an integral allocation x :(∏i∈N

vi (x)

)1/n

≥ 1

ρ·

(∏i∈N

vi (x∗)

)1/n

The first known algorithm achieved ρ ∈ Θ(m) [NR’14]

The problem is NP-hard even for two identical agents

In fact, this problem is APX-hard [L’15]

Theorem (CG’15, CDGJMVY’17)

There exists a poly-time algorithm that achieves ρ = 2








vi (x)

)1/n

≥ 1

ρ·

(∏i∈N

vi (x∗)

)1/n













vi (x)

)1/n

≥ 1

ρ·

(∏i∈N

vi (x∗)

)1/n













vi (x)

)1/n

≥ 1

ρ·

(∏i∈N

vi (x∗)

)1/n









Program FormulationMarket EquilibriumSpending-Restricted Outcome

Program Formulation

This problem can be expressed as an integer program (IP):

maximize:

(∏i∈N

ui

)1/n

subject to:∑j∈M

xijvij = ui , ∀i ∈ N

∑i∈N

xij ≤ 1, ∀j ∈ M

xij ∈ {0, 1}, ∀i ∈ N, j ∈ M

Observation

The integrality gap of the integer program IP is unbounded!





Program Formulation


maximize:∑i∈N

log ui

subject to:∑j∈M


∑i∈N

xij ≤ 1, ∀j ∈ M

xij ∈ {0, 1}, ∀i ∈ N, j ∈ M

Observation






Program Formulation


maximize:∑i∈N

log ui

subject to:∑j∈M


∑i∈N

xij ≤ 1, ∀j ∈ M

xij ≥ 0, ∀i ∈ N, j ∈ M

Observation






Program Formulation

The relaxation of IP is equivalent to the Eisenberg-Gale program:

maximize:∑i∈N

log ui

subject to:∑j∈M


∑i∈N

xij ≤ 1, ∀j ∈ M

xij ≥ 0, ∀i ∈ N, j ∈ M

Observation






Program Formulation

The relaxation of IP is equivalent to the Eisenberg-Gale program:

maximize:∑i∈N

log ui

subject to:∑j∈M


∑i∈N

xij ≤ 1, ∀j ∈ M

xij ≥ 0, ∀i ∈ N, j ∈ M

Observation






Integrality Gap

Observation


n

...

2

1

m

...

3

2

1

[V, 1, . . . , 1]

[V, 1, . . . , 1]

[V, 1, . . . , 1]

Agents Items

1/n

1/n

1/n





Integrality Gap

Observation


n

...

2

1

m

...

3

2

1

[V, 1, . . . , 1]

[V, 1, . . . , 1]

[V, 1, . . . , 1]

Agents Items

1/n

1/n

1/n





Integrality Gap

Observation


n

...

2

1

m

...

3

2

1

[V, 1, . . . , 1]

[V, 1, . . . , 1]

[V, 1, . . . , 1]

Agents Items

1/n

1/n

1/n





Integrality Gap

Observation


n

...

2

1

m

...

3

2

1

[V, 1, . . . , 1]

[V, 1, . . . , 1]

[V, 1, . . . , 1]

Agents Items





Integrality Gap

Observation


n

...

2

1

m

...

3

2

1

[V, 1, . . . , 1]

[V, 1, . . . , 1]

[V, 1, . . . , 1]

Agents Items





Market Equilibrium Interpretation

Each agent is allocated a budget of $1

4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

$0.2

$0.2

$0.2

$0.4

$3

Agents Items

$1

$1

$1

$0.4

$0.2

$0.2

$0.2






Each agent is allocated a budget of $1 and item j has price pj

4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

$0.2

$0.2

$0.2

$0.4

$3

Agents Items

$1

$1

$1

$0.4

$0.2

$0.2

$0.2







4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

$0.2

$0.2

$0.2

$0.4

$3

Agents Items

$1

$1

$1

$0.4

$0.2

$0.2

$0.2







4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

$0.2

$0.2

$0.2

$0.4

$3

Agents Items

$1

$1

$1

$0.4

$0.2

$0.2

$0.2







4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

$0.2

$0.2

$0.2

$0.4

$3

Agents Items

$1

$1

$1

$0.4

$0.2

$0.2

$0.2







4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

$0.2

$0.2

$0.2

$0.4

$3

Agents Items

$1

$1

$1

$0.4

$0.2

$0.2

$0.2







4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

$0.2

$0.2

$0.2

$0.4

$3

Agents Items

$1

$1

$1

$0.4

$0.2

$0.2

$0.2







4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

$0.2

$0.2

$0.2

$0.4

$3

Agents Items

$1

$1

$1

$0.4

$0.2

$0.2

$0.2







4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

$0.2

$0.2

$0.2

$0.4

$3

Agents Items

$1

$1

$1

$0.4

$0.2

$0.2

$0.2







4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

$0.2

$0.2

$0.2

$0.4

$3

Agents Items

$1

$1

$1

$0.4

$0.2

$0.2

$0.2





Spending-Restricted Outcome

Spending-Restricted outcome: at most $1 spent on any item

4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

$0.2

$0.2

$0.2

$0.4

$3

Agents Items

$1

$1

$1

$0.4

$0.2

$0.2

$0.2







4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

$0.2

$0.2

$0.2

$0.4

$3

Agents Items

$1

$1

$1

$0.4

$0.2

$0.2

$0.2







4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

$2/3

$2/3

$2/3

$4/3

$10

Agents Items

$1

$1

$2/3

$1/3

$2/3

$1/3







4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

$2/3

$2/3

$2/3

$4/3

$10

Agents Items

$1

$1

$2/3

$1/3

$2/3

$1/3







4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

$2/3

$2/3

$2/3

$4/3

$10

Agents Items

$1

$1

$2/3

$1/3

$2/3

$1/3







4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

$2/3

$2/3

$2/3

$4/3

$10

Agents Items

$1

$1

$2/3

$1/3

$2/3

$1/3







4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

$2/3

$2/3

$2/3

$4/3

$10

Agents Items

$1

$1

$2/3

$1/3

$2/3

$1/3







4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

$2/3

$2/3

$2/3

$4/3

$10

Agents Items

$1

$1

$2/3

$1/3

$2/3

$1/3







4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

$2/3

$2/3

$2/3

$4/3

$10

Agents Items

$1

$1

$2/3

$1/3

$2/3

$1/3







4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

$2/3

$2/3

$2/3

$4/3

$10

Agents Items

$1

$1

$2/3

$1/3

$2/3

$1/3







4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

$2/3

$2/3

$2/3

$4/3

$10

Agents Items

$1

$1

$2/3

$1/3

$2/3

$1/3




Computing the SR outcomeUpper BoundSRR Algorithm

Main Technical Contributions

The main technical contributions in the rest of the tutorial are:

1 SR outcome is computable in poly-time

2 SR outcome implies a better upper bound for OPT

3 SR outcome reveals useful information for rounding





1. Computing the SR outcome [CG’15]

Expressing the SR outcome via a convex program is not trivial:

Spending constraint combines primal and dual variables

Computed via complicated primal-dual algorithm in [CG’15]

maximize:∑i∈N

log ui

subject to:∑j∈M


∑i∈N

xij = 1, ∀j ∈ M

xij ≥ 0, ∀i ∈ N, j ∈ M









maximize:∑i∈N

log ui

subject to:∑j∈M


∑i∈N

xijpj = min{1,pj} ∀j ∈ M

xij ≥ 0, ∀i ∈ N, j ∈ M









maximize:∑i∈N

log ui

subject to:∑j∈M


∑i∈N

xijpj = min{1,pj} ∀j ∈ M

xij ≥ 0, ∀i ∈ N, j ∈ M





1. Computing the SR outcome [CDGJMVY’17]

An alternative “integer” program for the optimal NSW:

Let bij be the amount that agent i spends on item j

Let qj be the total amount spent on item j across all agents

max (∏

i ui )1/n s.t.

∀i , ui =∑

j xijvij

∀j ,∑

i xij = 1

∀i , j , xij ∈ {0, 1}.

max

(∏i

∏j v

bijij∏

j qqjj

)1/n

s.t.

∀j ,∑

i bij = qj

∀i ,∑

j bij = 1

∀i , j , qj ≤ 1, bij ∈ {0, qj}





1. Computing the SR outcome [CDGJMVY’17]

Solving the relaxation of this program yields the SR outcome!

Let bij be the amount that agent i spends on item j

Let qj be the total amount spent on item j across all agents

max (∏

i ui )1/n s.t.

∀i , ui =∑

j xijvij

∀j ,∑

i xij = 1

∀i , j , xij ∈ {0, 1}.

max

(∏i

∏j v

bijij∏

j qqjj

)1/n

s.t.

∀j ,∑

i bij = qj

∀i ,∑

j bij = 1

∀i , j , qj ≤ 1, bij ∈ [0, qj ]





2. Upper Bound [CG’15]

4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

[2, 43 ,23 ,

23 ,

23 ]

[10, 0, 23 ,23 ,

23 ]

[10, 43 , 0, 0, 0]

[10, 0, 0, 0, 0]

$2/3

$2/3

$2/3

$4/3

$10

H

L

$1

$1

$2/3

$1/3

$1/3

$2/3

Theorem

For SR prices p and scaled vi :∏

i∈N vi (x∗) ≤

∏j∈H pj






4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

[2, 43 ,23 ,

23 ,

23 ]

[10, 0, 23 ,23 ,

23 ]

[10, 43 , 0, 0, 0]

[10, 0, 0, 0, 0]

$2/3

$2/3

$2/3

$4/3

$10

H

L

$1

$1

$2/3

$1/3

$1/3

$2/3

Theorem


i∈N vi (x∗) ≤

∏j∈H pj






4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

[2, 43 ,23 ,

23 ,

23 ]

[10, 0, 23 ,23 ,

23 ]

[10, 43 , 0, 0, 0]

[10, 0, 0, 0, 0]

$2/3

$2/3

$2/3

$4/3

$10

H

L

$1

$1

$2/3

$1/3

$1/3

$2/3

Theorem


i∈N vi (x∗) ≤

∏j∈H pj






4

3

2

1

5

4

3

2

1

[3,2,1,1,1]

[15,0,1,1,1]

[15,2,0,0,0]

[15,0,0,0,0]

[2, 43 ,23 ,

23 ,

23 ]

[10, 0, 23 ,23 ,

23 ]

[10, 43 , 0, 0, 0]

[10, 0, 0, 0, 0]

$2/3

$2/3

$2/3

$4/3

$10

H

L

$1

$1

$2/3

$1/3

$1/3

$2/3

Theorem


i∈N vi (x∗) ≤

∏j∈H pj






4

3

2

1

5

4

3

2

1

[10, 43 ,23 ,

23 ,

23 ]

[10, 43 ,23 ,

23 ,

23 ]

[10, 43 ,23 ,

23 ,

23 ]

[10, 43 ,23 ,

23 ,

23 ]

$2/3

$2/3

$2/3

$4/3

$10

H

L

$1

$1

$2/3

$1/3

$1/3

$2/3

Theorem


i∈N vi (x∗) ≤

∏j∈H pj





2. Upper Bound [CDGJMVY’17]

New program’s optimal value is equal to previous upper bound!

Normalizing so that vij = pj when bij > 0 gives∏i

∏j v

bijij∏

j qqjj

=

∏j p

∑i bij

j∏j q

qjj

=∏j

(pjqj

)qj

Then, observing that qj = pj if j ∈ L, and qj = 1 if j ∈ H:

∏j

(pjqj

)qj

=∏j∈L

1qj ·∏j∈H

pj =∏j∈H

pj





2. Upper Bound [CDGJMVY’17]

New program’s optimal value is equal to previous upper bound!

Normalizing so that vij = pj when bij > 0 gives∏i

∏j v

bijij∏

j qqjj

=

∏j p

∑i bij

j∏j q

qjj

=∏j

(pjqj

)qj

Then, observing that qj = pj if j ∈ L, and qj = 1 if j ∈ H:

∏j

(pjqj

)qj

=∏j∈L

1qj ·∏j∈H

pj =∏j∈H

pj





3. Spending Restricted Rounding Algorithm

a

a a a

a a a a

$1.3 $0.3

$1.5 $0.8 $0.9

The SRR algorithm:

1. Allocate leaf-items

2. Allocate low price items (pj ≤ 1/2)

3. Match remaining items to agents:

-Let vi (xp) be i ’s current value

-Change vij to log[vi (xp) + vij ]

-Add dummy items of value log[vi (xp)]

-Run maximum weight matching alg.

Theorem

The allocation x that the SRR algorithm computes satisfies:(∏i∈N

vi (x∗)

)1/n

≤ 2 ·

(∏i∈N

vi (x)

)1/n






a

a a a

a a a a

$1.3 $0.3

$1.5 $0.8 $0.9

The SRR algorithm:








Theorem


vi (x∗)

)1/n

≤ 2 ·

(∏i∈N

vi (x)

)1/n






a

a a a

a a a a

$1.3 $0.3

$1.5 $0.8 $0.9

The SRR algorithm:








Theorem


vi (x∗)

)1/n

≤ 2 ·

(∏i∈N

vi (x)

)1/n






a

a a a

a a a a

$1.3 $0.3

$1.5 $0.8 $0.9

The SRR algorithm:








Theorem


vi (x∗)

)1/n

≤ 2 ·

(∏i∈N

vi (x)

)1/n






a

a a a

a a a a

$1.3 $0.3

$1.5 $0.8 $0.9

The SRR algorithm:








Theorem


vi (x∗)

)1/n

≤ 2 ·

(∏i∈N

vi (x)

)1/n






a

a a a

a a a a

$1.3 $0.3

$1.5 $0.8 $0.9

The SRR algorithm:








Theorem


vi (x∗)

)1/n

≤ 2 ·

(∏i∈N

vi (x)

)1/n






a

a a a

a a a a

$1.3 $0.3

$1.5 $0.8 $0.9

The SRR algorithm:








Theorem


vi (x∗)

)1/n

≤ 2 ·

(∏i∈N

vi (x)

)1/n






a

a a a

a a a a

$1.3 $0.3

$1.5 $0.8 $0.9

The SRR algorithm:








Theorem


vi (x∗)

)1/n

≤ 2 ·

(∏i∈N

vi (x)

)1/n






a

a a a

a a a a

$1.3 $0.3

$1.5 $0.8 $0.9

The SRR algorithm:








Theorem


vi (x∗)

)1/n

≤ 2 ·

(∏i∈N

vi (x)

)1/n






a

a a a

a a a a

$1.3 $0.3

$1.5 $0.8 $0.9

The SRR algorithm:








Theorem


vi (x∗)

)1/n

≤ 2 ·

(∏i∈N

vi (x)

)1/n






a

a a a

a a a a

$1.3 $0.3

$1.5 $0.8 $0.9

The SRR algorithm:








Theorem


vi (x∗)

)1/n

≤ 2 ·

(∏i∈N

vi (x)

)1/n






a

a a a

a a a a

$1.3 $0.3

$1.5 $0.8 $0.9

The SRR algorithm:








Theorem


vi (x∗)

)1/n

≤ 2 ·

(∏i∈N

vi (x)

)1/n






a

a a a

a a a a

$1.3 $0.3

$1.5 $0.8 $0.9

The SRR algorithm:








Theorem


vi (x∗)

)1/n

≤ 2 ·

(∏i∈N

vi (x)

)1/n






a

a a a

a a a a

$1.3 $0.3

$1.5 $0.8 $0.9

The SRR algorithm:








Theorem


vi (x∗)

)1/n

≤ 2 ·

(∏i∈N

vi (x)

)1/n






a

a a a

a a a a

$1.3 $0.3

$1.5 $0.8 $0.9

The SRR algorithm:








Theorem


vi (x∗)

)1/n

≤ 2 ·

(∏i∈N

vi (x)

)1/n






a

a a a

a a a a

$1.3 $0.3

$1.5 $0.8 $0.9

The SRR algorithm:








Theorem


vi (x∗)

)1/n

≤ 2 ·

(∏i∈N

vi (x)

)1/n





Overview

First Section (9-10am)“Approximating the Nash Social Welfare with Indivisible Items”Vasilis Gkatzelis

Second Section (10-11am)“NSW Beyond Symmetric Agents with Additive Valuations”Jugal Garg

Coffee Break (11-11:20pm)

Third Section (11:20-12:20pm)“Nash Social Welfare and Stable Polynomials”Nima Anari





Thank you!

THANK YOU!


Documents

Nash Welfare, Market Equilibrium, and Stable Polynomials