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

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

Example: Public Infrastructure Project • Steiner tree problem • Service = Connectivity to new power plant

Cost-Sharing Mechanism • Protocol: elicit reports first, then serve • Mathematically: ! ( ! ) ∈ { ! , " } ! Mechanism b M = ( q , x ) ∈ R ! ! ( ! ) n players ∈ R ! ≥ !

What is the problem? • Valuations v i are private • A mechanism should elicit truthful bids b i • Budget-Balance, Efficiency, Polynomial time

What is the problem? • Valuations v i are private • A mechanism should elicit truthful bids b i • Budget-Balance, Efficiency, Polynomial time • Recover cost • Bounded surplus

What is the problem? • Valuations v i are private • A mechanism should elicit truthful bids b i • Budget-Balance, Efficiency, Polynomial time Trade off service cost and excluded valuations

Preliminaries • Quasi-linear utilities, u i ( b |v i ) = q i ( b )· v i – x i ( b ) • Requirements: • Non-negative cost shares: x i ( b ) ≥ 0 • Individually rational: If b i = v i then u i ( b |v i ) ≥ 0 • Player sovereignty: If b i = b ∞ then q i ( b ) = 1

Strategy-Proofness • Unilateral deviation is never successful • Truth is always a Nash Equilibrium (whatever the true valuations v )

Strategy-Proofness • Unilateral deviation is never successful • Truth is always a Nash Equilibrium (whatever the true valuations v ) Threshold b 1 q 1 = 0 x 1 = 0

Strategy-Proofness • Unilateral deviation is never successful • Truth is always a Nash Equilibrium (whatever the true valuations v ) Threshold b 1 q 1 = 0 q 1 = 1 or x 1 = 0 x 1 = or

Strategy-Proofness • Unilateral deviation is never successful • Truth is always a Nash Equilibrium (whatever the true valuations v ) Threshold b 1 q 1 = 1 x 1 =

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”)

Moulin Mechanisms (Moulin, 1999) • Group-strategyproof b i • Cross-Monotonicity • Idea: Largest feasible set 1 2 3 4

Moulin Mechanisms (Moulin, 1999) • Group-strategyproof b i • Cross-Monotonicity • Idea: Largest feasible set 1 2 3 4

Moulin Mechanisms (Moulin, 1999) • Group-strategyproof b i • Cross-Monotonicity • Idea: Largest feasible set 1 2 3 4

Moulin Mechanisms (Moulin, 1999) • Group-strategyproof b i • Cross-Monotonicity • Idea: Largest feasible set 1 2 3 4 + Universal technique − Poor BB and EFF sometimes inevitable

Implications of SP, WGSP, GSP, etc. Communication Each with all None Transfers None Service Money

Implications of SP, WGSP, GSP, etc. Communication Each with all SP None Transfers None Service Money

Implications of SP, WGSP, GSP, etc. Communication GSP Each with all SP None Transfers None Service Money

Implications of SP, WGSP, GSP, etc. Communication WGSP GSP Each with all SP None Transfers None Service Money

Implications of SP, WGSP, GSP, etc. Communication WGSP GSP “ultimate” GSP Each with all SP None Transfers None Service Money

Implications of SP, WGSP, GSP, etc. Communication WGSP GSP “ultimate” GSP Each with all k -GSP SP None Transfers None Service Money

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

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)

k -GSP ≠ GSP • Example with 3 players • Set thresholds • If b = (1, 1, 1) then serve all, else do not serve indifferents

k -GSP ≠ GSP • Example with 3 players • Set thresholds • If b = (1, 1, 1) then serve all, else do not serve indifferents b i 1.5 1 1 2 3

k -GSP ≠ GSP • Example with 3 players • Set thresholds • If b = (1, 1, 1) then serve all, else do not serve indifferents b i 1.5 1 1 2 3

Main Result: 2-GSP + Separability ⇔ GSP

Technical Lemma 2-GSP ⇒ no coalition K with ∀ i ∈ K : b i ∈ { – 1, b ∞ } is ever successful • Let p be last in with u p ( b p ) < u p ( b p – 1 ) • Then: p gains service, loses utility: u i ( b p ) < 0 • u p ( b |b p ) ≤ u p ( b p |b p ) , i.e., x p ( b ) ≥ x p ( b p ) > v p

Technical Lemma 2-GSP ⇒ no coalition K with ∀ i ∈ K : b i ∈ { – 1, b ∞ } is ever successful 1 … m m + 1 … k n • Let p be last in with u p ( b p ) < u p ( b p – 1 ) • Then: p gains service, loses utility: u i ( b p ) < 0 • u p ( b |b p ) ≤ u p ( b p |b p ) , i.e., x p ( b ) ≥ x p ( b p ) > v p

Technical Lemma 2-GSP ⇒ no coalition K with ∀ i ∈ K : b i ∈ { – 1, b ∞ } is ever successful b 1 := ( b 1 , v 2 , …, v n ) 1 … m m + 1 … k n • Let p be last in with u p ( b p ) < u p ( b p – 1 ) • Then: p gains service, loses utility: u i ( b p ) < 0 • u p ( b |b p ) ≤ u p ( b p |b p ) , i.e., x p ( b ) ≥ x p ( b p ) > v p

Technical Lemma 2-GSP ⇒ no coalition K with ∀ i ∈ K : b i ∈ { – 1, b ∞ } is ever successful b i := ( b 1 , …, b i , v i + 1 , …, v n ) 1 … m m + 1 … k n • Let p be last in with u p ( b p ) < u p ( b p – 1 ) • Then: p gains service, loses utility: u i ( b p ) < 0 • u p ( b |b p ) ≤ u p ( b p |b p ) , i.e., x p ( b ) ≥ x p ( b p ) > v p

Technical Lemma 2-GSP ⇒ no coalition K with ∀ i ∈ K : b i ∈ { – 1, b ∞ } is ever successful b i := ( b 1 , …, b i , v i + 1 , …, v n ) 1 … m m + 1 … k n • Let p be last in with u p ( b p ) < u p ( b p – 1 )

Technical Lemma 2-GSP ⇒ no coalition K with ∀ i ∈ K : b i ∈ { – 1, b ∞ } is ever successful b i := ( b 1 , …, b i , v i + 1 , …, v n ) 1 … m m + 1 … k n • Let p be last in with u p ( b p ) < u p ( b p – 1 ) = 0

Technical Lemma 2-GSP ⇒ no coalition K with ∀ i ∈ K : b i ∈ { – 1, b ∞ } is ever successful b i := ( b 1 , …, b i , v i + 1 , …, v n ) 1 … m m + 1 … k n • Let p be last in with u p ( b p ) < u p ( b p – 1 ) = 0 • Then: p gains service, loses utility: u i ( b p ) < 0

Technical Lemma 2-GSP ⇒ no coalition K with ∀ i ∈ K : b i ∈ { – 1, b ∞ } is ever successful b i := ( b 1 , …, b i , v i + 1 , …, v n ) 1 … m m + 1 … k n • Let p be last in with u p ( b p ) < u p ( b p – 1 ) = 0 • Then: p gains service, loses utility: u i ( b p ) < 0 • u p ( b |b p ) ≤ u p ( b p |b p ) , i.e., x p ( b ) ≥ x p ( b p ) > v p

Technical Lemma 2-GSP ⇒ no coalition K with ∀ i ∈ K : b i ∈ { – 1, b ∞ } is ever successful b i := ( b 1 , …, b i , v i + 1 , …, v n ) 1 … m m + 1 … k n • Let p be last in with u p ( b p ) < u p ( b p – 1 ) = 0 • Then: p gains service, loses utility: u i ( b p ) < 0 • u p ( b |b p ) ≤ u p ( b p |b p ) , i.e., x p ( b ) ≥ x p ( b p ) > v p

2-GSP + Separability ⇒ GSP • Assume ( k – 1)-GSP & ∃ successful K of size k • By Lemma: k < n and, w.l.o.g., u n ( b ) < u n ( v ) 1 … … k n u i ( b ) > u i ( v ) or q i ( b ) > q i ( v ) M i ( b ) = M i ( v ) and x i ( b ) = v i • Utilities are stationary in phase 2 • Not everything can happen in phase 1

Other results WGSP GSP + This work Implications 2-WGSP 2-GSP + by definition Mutuswami, 2005 SP + weakly utility separable + non-bossy upper- (outcome) + continuous non-bossy

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