The Power of Small Coalitions in Cost Sharing Florian Schoppmann - - PowerPoint PPT Presentation

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The Power of Small Coalitions in Cost Sharing Florian Schoppmann - - PowerPoint PPT Presentation

Mar 18, 2024 315 likes •749 views

The Power of Small Coalitions in Cost Sharing Florian Schoppmann International Graduate School Dynamic Intelligent Systems University of Paderborn, Germany Cost Sharing A public excludable good (service) is to be made available to n players

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SLIDE 1

The Power of Small Coalitions in Cost Sharing

Florian Schoppmann International Graduate School Dynamic Intelligent Systems University of Paderborn, Germany

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SLIDE 2

Cost Sharing

  • A public excludable good (service) is to be

made available to n players

  • Player i is characterized by his valuation vi
  • Service cost C(q) depends on allocation
  • Task: Determine allocation and cost shares
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SLIDE 3

Example: Public Infrastructure Project

  • Steiner tree problem
  • Service = Connectivity

to new power plant

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SLIDE 4

Cost-Sharing Mechanism

  • Protocol: elicit reports first, then serve
  • Mathematically:

Mechanism M = (q, x) n players b !(!) !(!)

∈ {!, "}! ∈ R! ∈ R!

≥!

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SLIDE 5

What is the problem?

  • Valuations vi are private
  • A mechanism should elicit truthful bids bi
  • Budget-Balance, Efficiency, Polynomial time
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SLIDE 6

What is the problem?

  • Valuations vi are private
  • A mechanism should elicit truthful bids bi
  • Budget-Balance, Efficiency, Polynomial time
  • Recover cost
  • Bounded surplus
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SLIDE 7

What is the problem?

  • Valuations vi are private
  • A mechanism should elicit truthful bids bi
  • Budget-Balance, Efficiency, Polynomial time

Trade off service cost and excluded valuations

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SLIDE 8

Preliminaries

  • Quasi-linear utilities, ui(b|vi) = qi(b)· vi – xi(b)
  • Requirements:
  • Non-negative cost shares: xi(b) ≥ 0
  • Individually rational: If bi = vi then ui(b|vi) ≥ 0
  • Player sovereignty: If bi = b∞ then qi(b) = 1
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SLIDE 9

Strategy-Proofness

  • Unilateral deviation is never successful
  • Truth is always a Nash Equilibrium

(whatever the true valuations v)

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SLIDE 10

Strategy-Proofness

  • Unilateral deviation is never successful
  • Truth is always a Nash Equilibrium

(whatever the true valuations v)

b1 q1 = 0 x1 = 0 Threshold

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SLIDE 11

Strategy-Proofness

  • Unilateral deviation is never successful
  • Truth is always a Nash Equilibrium

(whatever the true valuations v)

b1 q1 = 0 x1 = 0 Threshold q1 = 1 x1 =

  • r
  • r
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SLIDE 12

Strategy-Proofness

  • Unilateral deviation is never successful
  • Truth is always a Nash Equilibrium

(whatever the true valuations v)

b1 Threshold q1 = 1 x1 =

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SLIDE 13

Coalitional Variants of Strategy-Proofness

  • Joint deviation is never successful
  • Weak notion of successful – strong collusion

resistance:

  • All players gain utility (weakly group-strategyproof)
  • Somebody better, nobody worse off (group-strategyproof)
  • Sum of utilities improve (“ultimately GSP”)
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SLIDE 14

Moulin Mechanisms

(Moulin, 1999)

  • Group-strategyproof
  • Cross-Monotonicity
  • Idea: Largest feasible set

bi 1 2 3 4

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SLIDE 15

Moulin Mechanisms

(Moulin, 1999)

  • Group-strategyproof
  • Cross-Monotonicity
  • Idea: Largest feasible set

bi 1 2 3 4

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SLIDE 16

Moulin Mechanisms

(Moulin, 1999)

  • Group-strategyproof
  • Cross-Monotonicity
  • Idea: Largest feasible set

bi 1 2 3 4

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SLIDE 17

Moulin Mechanisms

(Moulin, 1999)

  • Group-strategyproof
  • Cross-Monotonicity
  • Idea: Largest feasible set

bi 1 2 3 4

+ Universal technique − Poor BB and EFF

sometimes inevitable

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SLIDE 18

Implications of SP, WGSP, GSP, etc.

Transfers Communication Service Money Each with all None None

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SLIDE 19

Implications of SP, WGSP, GSP, etc.

Transfers Communication Service Money Each with all None None SP

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SLIDE 20

GSP

Implications of SP, WGSP, GSP, etc.

Transfers Communication Service Money Each with all None None SP

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SLIDE 21

GSP

Implications of SP, WGSP, GSP, etc.

Transfers Communication Service Money Each with all None None WGSP SP

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SLIDE 22

GSP

Implications of SP, WGSP, GSP, etc.

“ultimate” GSP Transfers Communication Service Money Each with all None None WGSP SP

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SLIDE 23

GSP

Implications of SP, WGSP, GSP, etc.

“ultimate” GSP Transfers Communication Service Money Each with all None None WGSP k-GSP SP

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SLIDE 24

GSP

Implications of SP, WGSP, GSP, etc.

“ultimate” GSP Transfers Communication Service Money Each with all None None WGSP k-GSP SP All these notions imply perfect information!

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SLIDE 25

Related Work

  • Effective pairwise strategyproof (Serizawa, 2006)
  • Weak utility non-bossy (Mutuswami, 2005)
  • Bribe-proof (Schummer, 2000)
  • Cost-Sharing: Assuming monetary transfers is too strong
  • k-strong equilibria (Andelman et al., 2007)
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SLIDE 26
  • Example with 3 players
  • Set thresholds
  • If b = (1, 1, 1) then serve all, else do not serve indifferents

k-GSP ≠ GSP

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SLIDE 27
  • Example with 3 players
  • Set thresholds
  • If b = (1, 1, 1) then serve all, else do not serve indifferents

k-GSP ≠ GSP

bi 1 2 3

1.5 1

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SLIDE 28
  • Example with 3 players
  • Set thresholds
  • If b = (1, 1, 1) then serve all, else do not serve indifferents

k-GSP ≠ GSP

bi 1 2 3

1.5 1

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SLIDE 29

Main Result: 2-GSP + Separability ⇔ GSP

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Technical Lemma

2-GSP ⇒ no coalition K with ∀i ∈ K: bi ∈ { –1, b∞ } is ever successful

  • Let p be last in with up(bp) < up(bp – 1)
  • Then: p gains service, loses utility: ui(bp) < 0
  • up(b|bp) ≤ up(bp|bp), i.e., xp(b) ≥ xp(bp) > vp
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SLIDE 31

Technical Lemma

2-GSP ⇒ no coalition K with ∀i ∈ K: bi ∈ { –1, b∞ } is ever successful

  • Let p be last in with up(bp) < up(bp – 1)
  • Then: p gains service, loses utility: ui(bp) < 0
  • up(b|bp) ≤ up(bp|bp), i.e., xp(b) ≥ xp(bp) > vp

1 … m m + 1 … k n

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SLIDE 32

Technical Lemma

2-GSP ⇒ no coalition K with ∀i ∈ K: bi ∈ { –1, b∞ } is ever successful

  • Let p be last in with up(bp) < up(bp – 1)
  • Then: p gains service, loses utility: ui(bp) < 0
  • up(b|bp) ≤ up(bp|bp), i.e., xp(b) ≥ xp(bp) > vp

1 … m m + 1 … k n b1 := (b1, v2, …, vn)

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SLIDE 33

Technical Lemma

2-GSP ⇒ no coalition K with ∀i ∈ K: bi ∈ { –1, b∞ } is ever successful

  • Let p be last in with up(bp) < up(bp – 1)
  • Then: p gains service, loses utility: ui(bp) < 0
  • up(b|bp) ≤ up(bp|bp), i.e., xp(b) ≥ xp(bp) > vp

bi := (b1, …, bi, vi + 1, …, vn) 1 … m m + 1 … k n

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SLIDE 34

Technical Lemma

2-GSP ⇒ no coalition K with ∀i ∈ K: bi ∈ { –1, b∞ } is ever successful

  • Let p be last in with up(bp) < up(bp – 1)

bi := (b1, …, bi, vi + 1, …, vn) 1 … m m + 1 … k n

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SLIDE 35

Technical Lemma

2-GSP ⇒ no coalition K with ∀i ∈ K: bi ∈ { –1, b∞ } is ever successful

  • Let p be last in with up(bp) < up(bp – 1)

bi := (b1, …, bi, vi + 1, …, vn) 1 … m m + 1 … k n = 0

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SLIDE 36

Technical Lemma

2-GSP ⇒ no coalition K with ∀i ∈ K: bi ∈ { –1, b∞ } is ever successful

  • Let p be last in with up(bp) < up(bp – 1)
  • Then: p gains service, loses utility: ui(bp) < 0

bi := (b1, …, bi, vi + 1, …, vn) 1 … m m + 1 … k n = 0

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SLIDE 37

Technical Lemma

2-GSP ⇒ no coalition K with ∀i ∈ K: bi ∈ { –1, b∞ } is ever successful

  • Let p be last in with up(bp) < up(bp – 1)
  • Then: p gains service, loses utility: ui(bp) < 0
  • up(b|bp) ≤ up(bp|bp), i.e., xp(b) ≥ xp(bp) > vp

bi := (b1, …, bi, vi + 1, …, vn) 1 … m m + 1 … k n = 0

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SLIDE 38

Technical Lemma

2-GSP ⇒ no coalition K with ∀i ∈ K: bi ∈ { –1, b∞ } is ever successful

  • Let p be last in with up(bp) < up(bp – 1)
  • Then: p gains service, loses utility: ui(bp) < 0
  • up(b|bp) ≤ up(bp|bp), i.e., xp(b) ≥ xp(bp) > vp

bi := (b1, …, bi, vi + 1, …, vn) 1 … m m + 1 … k n = 0

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SLIDE 39

2-GSP + Separability ⇒ GSP

  • Assume (k – 1)-GSP & ∃ successful K of size k
  • By Lemma: k < n and, w.l.o.g., un(b) < un(v)
  • Utilities are stationary in phase 2
  • Not everything can happen in phase 1

1 … … k n qi(b) > qi(v) and xi(b) = vi ui(b) > ui(v) or Mi(b) = Mi(v)

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SLIDE 40

Other results

GSP 2-GSP WGSP SP separable upper- continuous (outcome) non-bossy weakly utility non-bossy 2-WGSP

+

Mutuswami, 2005

+ +

Implications by definition This work

+ +

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SLIDE 41

Summary

  • Motivation
  • Weakening GSP in a way orthogonal to the known

relaxations GSP → WGSP

  • Communication is not unlimited
  • Results

− 2-GSP does not really allow for better performance + Characterization + Verifying GSP should become easier