The Power of Two Prices Beyond Cross-Monotonicity Yvonne - - PowerPoint PPT Presentation

Introduction Significance The Power of Two Prices Conclusion

The Power of Two Prices

Beyond Cross-Monotonicity Yvonne Bleischwitz Burkhard Monien Florian Schoppmann Karsten Tiemann

Department of Computer Science International Graduate School Dynamic Intelligent Systems University of Paderborn

August 30, 2007

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 1 / 23

Introduction Significance The Power of Two Prices Conclusion

Sharing the Outcome-Dependent Cost of a Binary Service

Protocol

• 1. Customers submit binding bids for service
• 2. Service provider determines set of served customers S and how

to distribute the incurred cost C(S) among them Fundamental in economics, e.g.:

◮ Sharing cost of public infrastructure projects ◮ Distributing volume discounts ◮ Allocating development costs of built-to-order products

Mechanism Design Problem

◮ Strategic bidding: Cheating possibilities ◮ Incentives for truthful bidding needed

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 2 / 23

Introduction Significance The Power of Two Prices Conclusion

Sharing the Outcome-Dependent Cost of a Binary Service

Protocol

• 1. Customers submit binding bids for service
• 2. Service provider determines set of served customers S and how

to distribute the incurred cost C(S) among them Fundamental in economics, e.g.:

◮ Sharing cost of public infrastructure projects ◮ Distributing volume discounts ◮ Allocating development costs of built-to-order products

Mechanism Design Problem

◮ Strategic bidding: Cheating possibilities ◮ Incentives for truthful bidding needed

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 2 / 23

Introduction Significance The Power of Two Prices Conclusion

Sharing the Outcome-Dependent Cost of a Binary Service

Protocol

• 1. Customers submit binding bids for service
• 2. Service provider determines set of served customers S and how

to distribute the incurred cost C(S) among them Fundamental in economics, e.g.:

◮ Sharing cost of public infrastructure projects ◮ Distributing volume discounts ◮ Allocating development costs of built-to-order products

Mechanism Design Problem

◮ Strategic bidding: Cheating possibilities ◮ Incentives for truthful bidding needed

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 2 / 23

Introduction Significance The Power of Two Prices Conclusion

A Cost-Sharing Scenario (1/2)

Computing center with large cluster of parallel machines

◮ Offering customers

(uninterrupted) processing times

◮ Cost proportional to

makespan

Customer 1 Customer 2 Customer 4 Customer 5 Makespan({1, 2, 3, 4, 5 }) = 7 Customer 3 Machine A Machine B Machine C

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 3 / 23

Introduction Significance The Power of Two Prices Conclusion

A Cost-Sharing Scenario (2/2)

Each customer “owns” a job Different valuations for having job processed

◮ Cost for processing job oneself ◮ Virtual cost for not processing ◮ Costs of competing offers

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 4 / 23

Introduction Significance The Power of Two Prices Conclusion

A Cost-Sharing Scenario (2/2)

Each customer “owns” a job

v1 = 2 v2 = 4 v3 = 3 v4 = 2 v5 = 1.5

Different valuations for having job processed

◮ Cost for processing job oneself ◮ Virtual cost for not processing ◮ Costs of competing offers

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 4 / 23

Introduction Significance The Power of Two Prices Conclusion

A Cost-Sharing Scenario (2/2)

Each customer “owns” a job

Makespan([5]) = 7 v1 = 2 v2 = 4 v3 = 3 v4 = 2 v5 = 1.5

Different valuations for having job processed

◮ Cost for processing job oneself ◮ Virtual cost for not processing ◮ Costs of competing offers

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 4 / 23

Introduction Significance The Power of Two Prices Conclusion

The Model

◮ n ∈ N players:

◮ Have private valuations vi ∈ R for service, v := (vi)i∈[n] ◮ Submit bids bi ∈ R to service provider, b := (bi)i∈[n]

◮ Service provider uses mechanism to determine outcome:

Definition (Cost-Sharing Mechanism)

Mechanism (“White Box”) Rn → 2[n] × Rn (Q × x) : b Q(b) x(b) player set [n ]

◮ Desirable that b = v but this cannot be a priori guaranteed

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 5 / 23

Introduction Significance The Power of Two Prices Conclusion

Common Assumptions for Cost-Sharing Mechanisms

Consider only mechanisms with the following properties ∀i ∈ [n]:

◮ NPT (No Positive Transfer) = no negative payments:

xi(b) ≥ 0

◮ VP (Voluntary Participation) = obey bids:

xi(b) ≤ bi and (i / ∈ Q = ⇒ xi(b) = 0)

◮ CS (Consumer Sovereignty):

∃b+

i ∈ R : ∀b ∈ Rn : (bi ≥ b+ i =

⇒ i ∈ Q(b)) Note that players may opt not to receive the service: ∀b ∈ Rn : (bi < 0 = ⇒ i / ∈ Q(b))

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 6 / 23

Introduction Significance The Power of Two Prices Conclusion

Modeling Assumptions

◮ Players are rational, only maximize

ui(b) :=

• vi − xi(b)

if i ∈ Q(b) if i / ∈ Q(b) (That is, utilities are quasi-linear)

◮ A coalition K ⊆ [n] forms if ∃v ∈ Rn for which it is successful.

Successful means: There are bids bK ∈ RK such that

◮ ui(v −K, bK) ≥ ui(v) for all i ∈ K and ◮ ui(v −K, bK) > ui(v) for at least one i ∈ K.

Players do not sacrifice their own utility to help others!

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 7 / 23

Introduction Significance The Power of Two Prices Conclusion

Desirable Properties of Mechanisms

◮ GSP (Group-Strategyproofness): No coalition is ever successful

Definition (n-Player Cost Function)

Function C : 2[n] → R≥0 with C(A) = 0 ⇐ ⇒ A = ∅ Typically, costs stem from a combinatorial optimization problem Convention: C denotes cost of service provider if he uses an

• ptimal solution (tractable in this work)

◮ β-BB (β-Budget-Balance, with β ≥ 1):

C(Q(b)) ≤

• i∈[n]

xi(b) ≤ β · C(Q(b))

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 8 / 23

Introduction Significance The Power of Two Prices Conclusion

Implications of GSP

Theorem (Moulin, 1999)

Let (Q × x) be a GSP mechanism, b, b′ ∈ Rn, and Q(b) = Q(b′). Then x(b) = x(b′).

◮ Payments independent of bids ◮ Bids only determine set of served players

Definition (n-Player Cost-Sharing Method)

Function ξ : 2[n] → Rn. β-BB defined as before: ∀A ⊆ [n] : C(A) ≤

• i∈[n]

ξi(A) ≤ β · C(A)

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 9 / 23

Introduction Significance The Power of Two Prices Conclusion

Implications of GSP

Theorem (Moulin, 1999)

Let (Q × x) be a GSP mechanism, b, b′ ∈ Rn, and Q(b) = Q(b′). Then x(b) = x(b′).

◮ Payments independent of bids ◮ Bids only determine set of served players

Definition (n-Player Cost-Sharing Method)

Function ξ : 2[n] → Rn. β-BB defined as before: ∀A ⊆ [n] : C(A) ≤

• i∈[n]

ξi(A) ≤ β · C(A)

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 9 / 23

Introduction Significance The Power of Two Prices Conclusion

Moulin Mechanisms

◮ Basically only one general approach to design GSP mechanisms ◮ GSP relies on cross-monotonic cost-sharing method ξ, i.e.,

∀A, B ⊆ [n] and ∀i ∈ A : ξi(A) ≥ ξi(A ∪ B) Algorithm (for computing Mξ : Rn → 2[n] × Rn

≥0, Moulin, 1999)

Input: ξ : 2[n] → Rn

≥0, b ∈ Rn; Output: Q ∈ 2[n], x ∈ Rn ≥0

1: Q := [n] 2: while ∃i ∈ Q: bi < ξi(Q) do Q := {i ∈ Q | bi ≥ ξi(Q)} 3: x := ξ(Q)

Theorem (Moulin, 1999)

Mξ satisfies GSP and β-BB if ξ is cross-monotonic and β-BB.

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 10 / 23

Introduction Significance The Power of Two Prices Conclusion

Previous Research (1/2)

Characterizations.

◮ Submodular costs: For all A ⊆ B ⊆ [n] and i /

∈ B: C(A ∪ {i}) − C(A) ≥ C(B ∪ {i}) − C(B).

Theorem (Moulin, 1999, for submodular costs)

Any GSP, 1-BB mechanism has cross-monotonic cost-shares. Conversely, a 1-BB cross-monotonic ξ exists.

◮ Upper-Continuous mechanisms: If i ∈ Q(b−i, bi) for every bid

bi > b∗

i , then i ∈ Q(b−i, b∗ i ).

Theorem (Immorlica et. al., 2005)

Any upper-continuous, GSP, β-BB mechanism has cross-monotonic cost-shares. Trivially, Mξ is upper-continuous.

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 11 / 23

Introduction Significance The Power of Two Prices Conclusion

Previous Research (1/2)

Characterizations.

◮ Submodular costs: For all A ⊆ B ⊆ [n] and i /

∈ B: C(A ∪ {i}) − C(A) ≥ C(B ∪ {i}) − C(B).

Theorem (Moulin, 1999, for submodular costs)

Any GSP, 1-BB mechanism has cross-monotonic cost-shares. Conversely, a 1-BB cross-monotonic ξ exists.

◮ Upper-Continuous mechanisms: If i ∈ Q(b−i, bi) for every bid

bi > b∗

i , then i ∈ Q(b−i, b∗ i ).

Theorem (Immorlica et. al., 2005)

Any upper-continuous, GSP, β-BB mechanism has cross-monotonic cost-shares. Trivially, Mξ is upper-continuous.

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 11 / 23

Submodular seems natural (“marginal costs only decrease”), but:

◮ Example: makespan scheduling

C([1]) = 1, C([2]) = 1, C([3]) = 1, C([4]) = 2 Note: Equivalent to above definition is for all A, B ⊆ [n] C(A) + C(B) ≤ C(A ∪ B) + C(A ∩ B).

Introduction Significance The Power of Two Prices Conclusion

Previous Research (1/2)

Characterizations.

◮ Submodular costs: For all A ⊆ B ⊆ [n] and i /

∈ B: C(A ∪ {i}) − C(A) ≥ C(B ∪ {i}) − C(B).

Theorem (Moulin, 1999, for submodular costs)

Any GSP, 1-BB mechanism has cross-monotonic cost-shares. Conversely, a 1-BB cross-monotonic ξ exists.

◮ Upper-Continuous mechanisms: If i ∈ Q(b−i, bi) for every bid

bi > b∗

i , then i ∈ Q(b−i, b∗ i ).

Theorem (Immorlica et. al., 2005)

Any upper-continuous, GSP, β-BB mechanism has cross-monotonic cost-shares. Trivially, Mξ is upper-continuous.

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 11 / 23

Introduction Significance The Power of Two Prices Conclusion

Previous Research (2/2)

Good BB. Examples for cross-monotonic cost-sharing methods: Authors Problem β Moulin, Shenker (2001) submodular costs 1 Jain, Vazirani (2001) MST 1 Steiner tree, TSP 2 Pál, Tardos (2003) Facility location 3 Gupta et. al. (2003) Single-Source-Rent-or-Buy 4.6 Leonardi, Schäfer Connected Facility Location 30 Könemann et. al. (2005) Steiner forest 2 Bleischwitz, Monien (2006)

• Ident. Scheduling on m links

2m m+1

Computability.

◮ Tractable mechanisms if optimal costs NP-hard to compute

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 12 / 23

Introduction Significance The Power of Two Prices Conclusion

Our Contribution

It is known that cross-monotonicity bears limitations:

◮ Example: 2m m+1-BB exact lower bound for id. makespan costs

Systematic first step: Concentrate on BB and symmetric costs, i.e., for A, B ⊆ [n] : |A| = |B| = ⇒ C(A) = C(B)

◮ Surprisingly: Symmetry not sufficient for GSP and 1-BB

For c : [4] → R≥0 defined by i 1 2 3 7 c(i) 1 3 6 7 no GSP, 1-BB mechanism exists

◮ For 3 or less players, a GSP and 1-BB mechanism always exists ◮ Two-Price Mechanisms: Not upper-continuous, better BB

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 13 / 23

Introduction Significance The Power of Two Prices Conclusion

Our Contribution

It is known that cross-monotonicity bears limitations:

◮ Example: 2m m+1-BB exact lower bound for id. makespan costs

Systematic first step: Concentrate on BB and symmetric costs, i.e., for A, B ⊆ [n] : |A| = |B| = ⇒ C(A) = C(B)

◮ Surprisingly: Symmetry not sufficient for GSP and 1-BB

For c : [4] → R≥0 defined by i 1 2 3 7 c(i) 1 3 6 7 no GSP, 1-BB mechanism exists

◮ For 3 or less players, a GSP and 1-BB mechanism always exists ◮ Two-Price Mechanisms: Not upper-continuous, better BB

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 13 / 23

Introduction Significance The Power of Two Prices Conclusion

Two-Price Cost-Sharing Forms (1/2)

◮ Preference order. Cost-sharing method

defined by cost vectors ξj ∈ Rj

≥0:

ξi(A) :=

• ξ|A|

Rank(i,A)

if i ∈ A

• therwise.

2) 2, (4, 5 4 3 2 1 5 4 3 2 1 ξ|A| =

◮ At most 2 different cost-shares for any set of players A ⊆ [n]

Definition (Two-Price Cost-Sharing Form, 2P-CSF)

Tuple F = (n, h, l, d) where for each cardinality i ∈ [n]

◮ hi ∈ R≥0 is higher and li ∈ [0, hi] lower cost share ◮ di ∈ [i]0 is number of disadvantaged agents

For i ∈ [n] : ξi = (hi, . . . , hi

• di elements

, li, . . . , li) ∈ Ri

≥0

The Power Of One Price

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 14 / 23

Introduction Significance The Power of Two Prices Conclusion

Two-Price Cost-Sharing Forms (2/2)

Recall: A 2P-CSF form defines cost vectors: For i ∈ [n] : ξi = (hi, . . . , hi

• di elements

, li, . . . , li) ∈ Ri

≥0

A segment is a contiguous range {s, s + 1, . . . , t} ⊆ [n]0 with ds = dt+1 = 0, and dk > 0 for k ∈ {s + 1, . . . , t}. Valid cost-sharing form:

◮ li+1 ≤ li ◮ li+1 < li =

⇒ di+1 = 0

◮ hi+1 > hi =

⇒ di+1 = 0

◮ di+1 ≤ di + 1 ◮ hi+1 < hi =

⇒ di+1 ≤ 1 Example: i hi li di ξi 1 4 2 0 (2) 2 4 2 0 (2, 2) 3 4 2 1 (4, 2, 2) 4 3 2 1 (3, 2, 2, 2) 5 5 1 0 (1, 1, 1, 1, 1) 6 5 1 1 (5, 1, 1, 1, 1, 1)

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 15 / 23

Introduction Significance The Power of Two Prices Conclusion

The New Two-Price Mechanisms: Example

Algorithm (for Two-Price Mechanisms)

Jump to formal version

1: Find max. k ∈ [n]0 such that k players bid ≥ lk =: l 2: Reject all players with bid < l 3: loop (Decrease k when rejecting players) 4:

If possible: Include remaining players with bid > l for price l, by rejecting a suitable subset of indifferent players; stop

5:

Else: Least preferred player is included for hk or is rejected Example for b = (5

2, 3, 3, 2, 0, 0): ◮ k := 4, reject agents 5, 6 ◮ only agent 4 is indifferent ◮ Can’t include 1,2,3 even w/o 4 ◮ Reject agent 1 because 5 2 = bi < hk = 3, Set k := 3 ◮ Include 2,3 by rejecting 4

i hi li di ξi 1 4 2 0 (2) 2 4 2 0 (2, 2) 3 4 2 1 (4, 2, 2) 4 3 2 1 (3, 2, 2, 2) 5 5 1 0 (1, 1, 1, 1, 1) 6 5 1 1 (5, 1, 1, 1, 1, 1)

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 16 / 23

Introduction Significance The Power of Two Prices Conclusion

The New Two-Price Mechanisms: Example

Algorithm (for Two-Price Mechanisms)

Jump to formal version

1: Find max. k ∈ [n]0 such that k players bid ≥ lk =: l 2: Reject all players with bid < l 3: loop (Decrease k when rejecting players) 4:

If possible: Include remaining players with bid > l for price l, by rejecting a suitable subset of indifferent players; stop

5:

Else: Least preferred player is included for hk or is rejected Example for b = (5

2, 3, 3, 2, 0, 0): ◮ k := 4, reject agents 5, 6 ◮ only agent 4 is indifferent ◮ Can’t include 1,2,3 even w/o 4 ◮ Reject agent 1 because 5 2 = bi < hk = 3, Set k := 3 ◮ Include 2,3 by rejecting 4

i hi li di ξi 1 4 2 0 (2) 2 4 2 0 (2, 2) 3 4 2 1 (4, 2, 2) 4 3 2 1 (3, 2, 2, 2) 5 5 1 0 (1, 1, 1, 1, 1) 6 5 1 1 (5, 1, 1, 1, 1, 1)

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 16 / 23

Introduction Significance The Power of Two Prices Conclusion

The New Two-Price Mechanisms: Example

Algorithm (for Two-Price Mechanisms)

Jump to formal version

1: Find max. k ∈ [n]0 such that k players bid ≥ lk =: l 2: Reject all players with bid < l 3: loop (Decrease k when rejecting players) 4:

If possible: Include remaining players with bid > l for price l, by rejecting a suitable subset of indifferent players; stop

5:

Else: Least preferred player is included for hk or is rejected Example for b = (5

2, 3, 3, 2, 0, 0): ◮ k := 4, reject agents 5, 6 ◮ only agent 4 is indifferent ◮ Can’t include 1,2,3 even w/o 4 ◮ Reject agent 1 because 5 2 = bi < hk = 3, Set k := 3 ◮ Include 2,3 by rejecting 4

i hi li di ξi 1 4 2 0 (2) 2 4 2 0 (2, 2) 3 4 2 1 (4, 2, 2) 4 3 2 1 (3, 2, 2, 2) 5 5 1 0 (1, 1, 1, 1, 1) 6 5 1 1 (5, 1, 1, 1, 1, 1)

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 16 / 23

Introduction Significance The Power of Two Prices Conclusion

The New Two-Price Mechanisms: Example

Algorithm (for Two-Price Mechanisms)

Jump to formal version

1: Find max. k ∈ [n]0 such that k players bid ≥ lk =: l 2: Reject all players with bid < l 3: loop (Decrease k when rejecting players) 4:

If possible: Include remaining players with bid > l for price l, by rejecting a suitable subset of indifferent players; stop

5:

Else: Least preferred player is included for hk or is rejected Example for b = (5

2, 3, 3, 2, 0, 0): ◮ k := 4, reject agents 5, 6 ◮ only agent 4 is indifferent ◮ Can’t include 1,2,3 even w/o 4 ◮ Reject agent 1 because 5 2 = bi < hk = 3, Set k := 3 ◮ Include 2,3 by rejecting 4

i hi li di ξi 1 4 2 0 (2) 2 4 2 0 (2, 2) 3 4 2 1 (4, 2, 2) 4 3 2 1 (3, 2, 2, 2) 5 5 1 0 (1, 1, 1, 1, 1) 6 5 1 1 (5, 1, 1, 1, 1, 1)

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 16 / 23

Introduction Significance The Power of Two Prices Conclusion

The New Two-Price Mechanisms: Example

Algorithm (for Two-Price Mechanisms)

Jump to formal version

1: Find max. k ∈ [n]0 such that k players bid ≥ lk =: l 2: Reject all players with bid < l 3: loop (Decrease k when rejecting players) 4:

If possible: Include remaining players with bid > l for price l, by rejecting a suitable subset of indifferent players; stop

5:

Else: Least preferred player is included for hk or is rejected Example for b = (5

2, 3, 3, 2, 0, 0): ◮ k := 4, reject agents 5, 6 ◮ only agent 4 is indifferent ◮ Can’t include 1,2,3 even w/o 4 ◮ Reject agent 1 because 5 2 = bi < hk = 3, Set k := 3 ◮ Include 2,3 by rejecting 4

i hi li di ξi 1 4 2 0 (2) 2 4 2 0 (2, 2) 3 4 2 1 (4, 2, 2) 4 3 2 1 (3, 2, 2, 2) 5 5 1 0 (1, 1, 1, 1, 1) 6 5 1 1 (5, 1, 1, 1, 1, 1)

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 16 / 23

Introduction Significance The Power of Two Prices Conclusion

The New Two-Price Mechanisms: Example

Algorithm (for Two-Price Mechanisms)

Jump to formal version

1: Find max. k ∈ [n]0 such that k players bid ≥ lk =: l 2: Reject all players with bid < l 3: loop (Decrease k when rejecting players) 4:

If possible: Include remaining players with bid > l for price l, by rejecting a suitable subset of indifferent players; stop

5:

Else: Least preferred player is included for hk or is rejected Example for b = (5

2, 3, 3, 2, 0, 0): ◮ k := 4, reject agents 5, 6 ◮ only agent 4 is indifferent ◮ Can’t include 1,2,3 even w/o 4 ◮ Reject agent 1 because 5 2 = bi < hk = 3, Set k := 3 ◮ Include 2,3 by rejecting 4

i hi li di ξi 1 4 2 0 (2) 2 4 2 0 (2, 2) 3 4 2 1 (4, 2, 2) 4 3 2 1 (3, 2, 2, 2) 5 5 1 0 (1, 1, 1, 1, 1) 6 5 1 1 (5, 1, 1, 1, 1, 1)

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 16 / 23

Introduction Significance The Power of Two Prices Conclusion

The New Two-Price Mechanisms: Example

Algorithm (for Two-Price Mechanisms)

Jump to formal version

1: Find max. k ∈ [n]0 such that k players bid ≥ lk =: l 2: Reject all players with bid < l 3: loop (Decrease k when rejecting players) 4:

If possible: Include remaining players with bid > l for price l, by rejecting a suitable subset of indifferent players; stop

5:

Else: Least preferred player is included for hk or is rejected Example for b = (5

2, 3, 3, 2, 0, 0): ◮ k := 4, reject agents 5, 6 ◮ only agent 4 is indifferent ◮ Can’t include 1,2,3 even w/o 4 ◮ Reject agent 1 because 5 2 = bi < hk = 3, Set k := 3 ◮ Include 2,3 by rejecting 4

i hi li di ξi 1 4 2 0 (2) 2 4 2 0 (2, 2) 3 4 2 1 (4, 2, 2) 4 3 2 1 (3, 2, 2, 2) 5 5 1 0 (1, 1, 1, 1, 1) 6 5 1 1 (5, 1, 1, 1, 1, 1)

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 16 / 23

Introduction Significance The Power of Two Prices Conclusion

Two-Price Mechanisms are GSP

Theorem

Two-Price Mechanisms are GSP. Proof (Sketch). Let v ∈ Rn be true valuation vector, b ∈ Rn other bid vector and K ⊆ [n] such that b−K = v −K. We show: ∃i ∈ K : ui(v −K, bK) > ui(v) = ⇒ ∃j ∈ K : uj(v −K, bK) < uj(v) Outline of proof, let k := |Q(b)|:

◮ k and |Q(v)| in same segment and k ≤ |Q(v)| ◮ Assumptions imply: i /

∈ Q(v) or xi(v) > lk, but xi(b) = lk

◮ Only two options:

◮ ∃j ∈ [i] : bj ≥ hk > vj or ◮ ∃j ∈ {i + 1, . . . , n} : bj ≤ lk < vj

It follows that j ∈ K and uj(b) < uj(v)

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 17 / 23

Introduction Significance The Power of Two Prices Conclusion

Basic Idea for Computing 2P-CSFs: Scheduling Example

Algorithm: Cost Vectors: i c(i) li hi di ξi 1 1 c(1) = 1 ∞ (1) 2 1

c(2) 2

= 1

2

∞ (1

2, 1 2)

3 1

c(3) 3

= 1

3

∞ (1

3, 1 3, 1 3)

4 2

1 3

c(4) − 3 · l4 = 1 1 (1, 1

3, 1 3, 1 3)

Optimal Makespan: Note: Maximum recoverable for cardinality 5 would be 2 · h4 + 3 · l3 = 3.

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 18 / 23

Introduction Significance The Power of Two Prices Conclusion

Basic Idea for Computing 2P-CSFs: Scheduling Example

Algorithm: Cost Vectors: i c(i) li hi di ξi 1 1 c(1) = 1 ∞ (1) 2 1

c(2) 2

= 1

2

∞ (1

2, 1 2)

3 1

c(3) 3

= 1

3

∞ (1

3, 1 3, 1 3)

4 2

1 3

c(4) − 3 · l4 = 1 1 (1, 1

3, 1 3, 1 3)

Optimal Makespan: Note: Maximum recoverable for cardinality 5 would be 2 · h4 + 3 · l3 = 3.

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 18 / 23

Introduction Significance The Power of Two Prices Conclusion

Basic Idea for Computing 2P-CSFs: Scheduling Example

Algorithm: Cost Vectors: i c(i) li hi di ξi 1 1 c(1) = 1 ∞ (1) 2 1

c(2) 2

= 1

2

∞ (1

2, 1 2)

3 1

c(3) 3

= 1

3

∞ (1

3, 1 3, 1 3)

4 2

1 3

c(4) − 3 · l4 = 1 1 (1, 1

3, 1 3, 1 3)

Optimal Makespan: Note: Maximum recoverable for cardinality 5 would be 2 · h4 + 3 · l3 = 3.

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 18 / 23

Introduction Significance The Power of Two Prices Conclusion

Basic Idea for Computing 2P-CSFs: Scheduling Example

Algorithm: Cost Vectors: i c(i) li hi di ξi 1 1 c(1) = 1 ∞ (1) 2 1

c(2) 2

= 1

2

∞ (1

2, 1 2)

3 1

c(3) 3

= 1

3

∞ (1

3, 1 3, 1 3)

4 2

1 3

c(4) − 3 · l4 = 1 1 (1, 1

3, 1 3, 1 3)

Optimal Makespan: Note: Maximum recoverable for cardinality 5 would be 2 · h4 + 3 · l3 = 3.

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 18 / 23

Introduction Significance The Power of Two Prices Conclusion

Good Budget-Balance

Theorem

Jump to formal algorithm

For any subadditive symmetric cost function c : [n] → R≥0, there is a 2P-CSF with BB β :=

√ 17+1 4

≈ 1.281. Proof (Sketch). Denote γ(i) := di · hi + (i − di) · li. For all cardinalities i ∈ [n], in increasing order:

◮ If c(i) i

= minj∈[i]

c(j) j , set di := 0 and li := β · c(i) i

and f := i

◮ Else try hi := min{β · c(i) − (i − 1) · li, hi−1} and di := 1 ◮ If γ(i) < 2 · c(f ) then ensure:

◮ hi ≥ (β2 − β) · c(f ), by setting di := 0 and hi = ∞ if necessary ◮ hi = (β2 − β) · c(f ) if possible and di = 0, by reducing hi ◮ γ(i) ≥ c(i), by setting di := 2 if necessary

Use β · c(f ) + 2 · (β2 − β) · c(f ) = 2 · c(f ) and c(2f )

2f

≤ c(f )

f .

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 19 / 23

For all A, B ⊆ [n] : C(A) + C(B) ≥ C(A ∪ B).

Introduction Significance The Power of Two Prices Conclusion

Good Budget-Balance

Theorem

Jump to formal algorithm

For any subadditive symmetric cost function c : [n] → R≥0, there is a 2P-CSF with BB β :=

√ 17+1 4

≈ 1.281. Proof (Sketch). Denote γ(i) := di · hi + (i − di) · li. For all cardinalities i ∈ [n], in increasing order:

◮ If c(i) i

= minj∈[i]

c(j) j , set di := 0 and li := β · c(i) i

and f := i

◮ Else try hi := min{β · c(i) − (i − 1) · li, hi−1} and di := 1 ◮ If γ(i) < 2 · c(f ) then ensure:

◮ hi ≥ (β2 − β) · c(f ), by setting di := 0 and hi = ∞ if necessary ◮ hi = (β2 − β) · c(f ) if possible and di = 0, by reducing hi ◮ γ(i) ≥ c(i), by setting di := 2 if necessary

Use β · c(f ) + 2 · (β2 − β) · c(f ) = 2 · c(f ) and c(2f )

2f

≤ c(f )

f .

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 19 / 23

Introduction Significance The Power of Two Prices Conclusion

β =

√ 17+1 4

• BB is Lower Bound for all 2P-CSFs (1/2)

Theorem

∀α ∈ (1, β), there are scheduling instances (identical jobs) for which no α-BB 2P-CSF form exists.

• Proof. Let α ∈ (1, β). Scheduling with m := k + l + 1 machines:

i 1 · · · k k + 1 · · · k + l m m + 1 c(i) 1 · · · 1 β − β−1

β1

· · · β − β−1

βl

β 2 Assume: (n, h, l, d) is α-BB. Let γ(i) := di · hi + (i − di) · li.

◮ dm ≥ 1. Otherwise

γ(m) = m · lm ≤ m · lk ≤ m · α k = α ·

• 1 + l + 1

k

• < β = c(m)

◮ dm+1 = dm + 1. Otherwise

γ(m + 1) ≤ α · β + lm+1 < β2 + α k < 2 = c(m + 1)

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 20 / 23

Introduction Significance The Power of Two Prices Conclusion

β =

√ 17+1 4

• BB is Lower Bound for all 2P-CSFs (2/2)

i 1 · · · k k + 1 · · · k + l m m + 1 c(i) 1 · · · 1 β − β−1

β1

· · · β − β−1

βl

β 2 Let g := max{j ∈ [m + 1] | dj = 0}. Note: g < m, hm+1 ≤ hg+1. In any case: γ(m + 1) ≤ α · β + hg+1 < β2 + (β2 − β) = 2. E

◮ Case g ∈ {k, k + 1, . . . , k + l − 1}. Let j := g + 1 − k. Then

hg+1 ≤ α · c(g + 1) − c(g) = α · c(k + j) − c(k + j − 1) = α ·

• β − β − 1

βj

• −
• β − β − 1

βj−1

• < β2 − β − 1

βj−1 −

• β − β − 1

βj−1

• = β2 − β .

◮ Similarly: g ∈ [k − 1] and g = k + l = m − 1

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 21 / 23

Introduction Significance The Power of Two Prices Conclusion

Efficiency is Poor

Social Cost. Further desirable property for mechanisms:

◮ α-EFF (α-efficient, with α ≥ 1):

Let SC : 2[n] → R, SC(A) := C(A) +

i / ∈A vi. Then:

SC(Q(v)) ≤ α · min

A⊆[n]{C(A) + i / ∈Avi}

(where · : R → R≥0, x := max{0, x}) i 1 · · · m m + 1 · · · 2m c(i) 1 · · · 1 2 · · · 2 ξi (1) . . . ( 1

m, . . . )

(1, 1

m, . . . )

. . . ( 1

m, 1 m, . . . )

Let v = ( 1

m − ε, 2 m − ε, . . . , 1 − ε, 1 m, . . . , 1 m).

SC({m + 1, . . . , 2m}) ≈ m + 1 2 + 1 but SC([2m]) = 2

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 22 / 23

Introduction Significance The Power of Two Prices Conclusion

Efficiency is Poor

Social Cost. Further desirable property for mechanisms:

◮ α-EFF (α-efficient, with α ≥ 1):

Let SC : 2[n] → R, SC(A) := C(A) +

i / ∈A vi. Then:

SC(Q(v)) ≤ α · min

A⊆[n]{C(A) + i / ∈Avi}

(where · : R → R≥0, x := max{0, x}) i 1 · · · m m + 1 · · · 2m c(i) 1 · · · 1 2 · · · 2 ξi (1) . . . ( 1

m, . . . )

(1, 1

m, . . . )

. . . ( 1

m, 1 m, . . . )

Let v = ( 1

m − ε, 2 m − ε, . . . , 1 − ε, 1 m, . . . , 1 m).

SC({m + 1, . . . , 2m}) ≈ m + 1 2 + 1 but SC([2m]) = 2

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 22 / 23

Introduction Significance The Power of Two Prices Conclusion

Conclusion

Motivation:

◮ Mechanism Design: Align players’ incentives to global objective

Results presented in this talk:

◮ Symmetric costs

◮ New GSP Two-Price cost-sharing mechanisms ◮

√ 17+1 4

• BB for subadditive costs

◮ Best our new technique can yield

Further Research:

◮ Better BB possible? Further characterizations? ◮ Generalizations of two-price mechanisms? ◮ Can Moulin be improved both w.r.t. BB and EFF?

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 23 / 23

Appendix

The Power Of One Price

More general and intuitive than submodular costs:

◮ Subadditive Costs: For all A, B ⊆ [n]:

C(A) + C(B) ≥ C(A ∪ B). Consider symmetric and subadditive costs c : [n] → R≥0 Retrospective motivation to look at two prices:

◮ One price essentially means cross-monotonicity ◮ Define ξ : 2[n] → Rn ≥0 by ξi(A) := 2 · mini∈[|A|] c(i) i ◮ ξ is clearly cross-monotonic ◮ ξ is also 2-BB since 2c(i) ≥ c(2i) ≥ c(j) for all j ∈ [2i] ◮ In general, 2-BB is the best cross-monotonicity can yield

Back to Two Prices

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 24 / 23

Appendix

The New Two-Price Mechanisms: Formal Algorithm

Algorithm (for Two-Price Mechanisms)

Back to intuitive version

Input: b ∈ Rn; Output: Q ∈ 2[n], x ∈ Rn

1: k := max

• i ∈ [n]0
• |{j ∈ [n] | bj ≥ li}| ≥ i
• ; l := lk

2: Q := {i ∈ [n] | bi ≥ l} 3: I := {i ∈ [n] | bi = l} 4: D := ∅ 5: loop 6:

q := max{i ∈ [|Q|]0 | di = |D|}

7:

if q ≥ |Q \ I| then

8:

Q := Q \ {|Q| − q smallest elements of I}

9:

break

10:

ℓ := min(Q \ D)

11:

if bℓ ≥ h|Q| then D := D ∪ {ℓ}

12:

else Q := Q \ {ℓ}; I := I \ {ℓ}

13: xi := h|Q| for i ∈ D; xi := l for i ∈ Q \D; xi := 0 for i ∈ [n]\Q

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 25 / 23

Appendix

Computing a 2P-CSF for Subadditive Costs

Algorithm (for computing a 2P-CSF)

Back to sketch of algorithm

Input: cost function c : [n] → R≥0; Output: 2P-CSF (n, h, l, d)

1: l0 := l1 := β · c(1); h0 := h1 := ∞; d0 := d1 := 0; f := 1 2: for i := 2, . . . , n do 3:

if β · c(i)

i

≤ lf then li := β · c(i)

i ; hi := ∞; di := 0 ; f := i

4:

else

5:

li := li−1; hi := min{β · c(i) − (i − 1) · li, hi−1}

6:

if hi + (i − 1) · li < c(i) then

7:

di := 2

8:

else if hi + (i − 1) · li ≥ 2 · c(f ) then

9:

di := 1

10:

else if hi ≥ (β2 − β) · c(f ) then

11:

di := 1

12:

if (β2−β)·c(f )+(i−1)·li ≥ c(i) then hi := (β2−β)·c(f )

13:

else

14:

di := 0; hi := ∞

• Intern. Grad. School, University of Paderborn

Florian Schoppmann · 26 / 23