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A Strategy-proof Pricing Scheme for Multiple Resource Type Allocations Marian Mihailescu and Yong Meng Teo Department of Computer Science National University of Singapore Overview Introduction and Related Work Our Approach Proposed

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A Strategy-proof Pricing Scheme for Multiple Resource Type Allocations

Marian Mihailescu and Yong Meng Teo Department of Computer Science National University of Singapore

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Overview

  • Introduction and Related Work
  • Our Approach
  • Proposed Mechanism
  • Example
  • Simulation Results
  • Conclusions and Future Work

2 38th International Conference on Parallel Processing, 22-25 September 2009, Vienna, Austria

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Introduction

  • Large scale resource sharing
  • Grid
  • Peer-to-peer
  • Cloud Computing
  • Fundamental problem: resource allocation
  • Difficulty: rational users
  • Maximize their own interest in sharing
  • Affect the performance of the system

3 38th International Conference on Parallel Processing, 22-25 September 2009, Vienna, Austria

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Mechanism Design

Problem

  • Mechanism design problem
  • Outcome specification
  • Set of user valuations for a

specific outcome

Solution

  • Mechanism
  • Social choice function f

determines the outcome

  • User payment

4

  • Provides a framework to design protocols that give rational agents

incentives to interact in particular ways, such that social welfare is “maximized” at equilibrium

M = ( f , p1,..., pn) f (t1…tn) = maxo ui

i

∑

pi vi(ti,o)

38th International Conference on Parallel Processing, 22-25 September 2009, Vienna, Austria

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  • Computational Efficiency
  • Optimal allocation requires

NP-complete algorithm

Desired Properties

Economic

  • Multiple Resource Types
  • A buyer request contains more than one

resource type

  • Strategy-proof
  • Users gain higher welfare from participating

and have no incentives to declare false information

  • Budget Balance
  • Sum of all user payments is 0, and allocations

do not result in deficit or surplus

  • Economic Efficiency
  • Resources are allocated to the user that values

them the most; total welfare is maximized

Computational

5 38th International Conference on Parallel Processing, 22-25 September 2009, Vienna, Austria

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  • Computational Efficiency
  • Optimal allocation requires

NP-complete algorithm

Myerson-Satterthwite Impossibility Theorem:

no mechanism achieves strategy-proof, budget balance and economic efficiency at the same time

Desired Properties

Economic

  • Multiple Resource Types
  • A buyer request contains more than one

resource type

  • Strategy-proof
  • Users gain higher welfare from participating

and have no incentives to declare false information

  • Budget Balance
  • Sum of all user payments is 0, and allocations

do not result in deficit or surplus

  • Economic Efficiency
  • Resources are allocated to the user that values

them the most; total welfare is maximized

Computational

6 38th International Conference on Parallel Processing, 22-25 September 2009, Vienna, Austria

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Related Work

Property Proportional Share Bargaining Auctions Combinatorial Auctions Economic Multiple Resource Types

✔ ✔ ✕ ✔

Strategy-proof

✕ ✕ ✔ ✔

Budget Balance

✔ ✔ ✔ ✕

Pareto Efficiency

✕ ✕ ✕ ✔

Computational Algorithm Complexity

low low low high

Tycoon (2004) [8] REXEC (2000) [4] Nimrod/G (2002) [2] Popcorn (1998) [14] Spawn (1992) [18] Mirage (2005) [3] Bellagio (2004) [1]

7 38th International Conference on Parallel Processing, 22-25 September 2009, Vienna, Austria

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Our Approach

8

Pareto Efficiency Computational Efficiency Budget Balance Strategy-proof Multiple Resource Types

Trade-off Trade-off

38th International Conference on Parallel Processing, 22-25 September 2009, Vienna, Austria

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Market-based Resource Allocation Problem

  • Buyers submit requests for multiple resource types
  • Buyer private information
  • Maximum price the buyer is willing to pay such that, for each

resource type, resources are allocated to satisfy its request

  • Sellers publish each resource type separately
  • Seller private information for each resource type
  • Underlying costs for the respective resource type, such as power

consumption, bandwidth costs, etc.

  • For a particular request, the goal is to allocate resources such that

the underlying costs are minimized

9 38th International Conference on Parallel Processing, 22-25 September 2009, Vienna, Austria

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Winner Determination

  • Centralized market-maker
  • Manage requests and resources
  • Determine winners and compute payments
  • Reverse Auction based Winner Determination
  • Select one request (buyer winner)
  • For each resource type in the request
  • Select resources with minimum cost (seller winner)

10 38th International Conference on Parallel Processing, 22-25 September 2009, Vienna, Austria

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Payment Functions

  • Seller payment function
  • Buyer payment function

ps = s does not contribute resources to allocate the request −cM |s=∞ + cM |s=0 s contributes with resources to allocate the request   

pb = − ps

s∈S

∑

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cM |s=∞ minimum cost to allocate the request without the resources of seller s cM |s=0 minimum cost to allocate the request when the resource cost of seller s is 0 VCG payment function: strategy-proof, Pareto-efficient, NOT budget-balanced

38th International Conference on Parallel Processing, 22-25 September 2009, Vienna, Austria

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Payment Functions

  • Seller payment function
  • Buyer payment function

ps = s does not contribute resources to allocate the request −cM |s=∞ + cM |s=0 s contributes with resources to allocate the request   

pb = − ps

s∈S

∑

12

cM |s=∞ minimum cost to allocate the request without the resources of seller s cM |s=0 minimum cost to allocate the request when the resource cost of seller s is 0

38th International Conference on Parallel Processing, 22-25 September 2009, Vienna, Austria

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Achieved Properties

  • Multiple Resource Type
  • Seller Payment Function:
  • Strategy-proof
  • Economic Efficiency
  • Buyer Payment Function:
  • Strategy-proof (FCFS buyer requests)
  • Budget Balance
  • Computational Efficiency

13 38th International Conference on Parallel Processing, 22-25 September 2009, Vienna, Austria

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Example

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S1 CPU $1 S2 DISK $2 S3 DISK $1 S2 CPU $2 B1 CPU+DISK $5 B2 CPU+DISK $6

38th International Conference on Parallel Processing, 22-25 September 2009, Vienna, Austria

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Proposed Mechanism

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Market Maker Resources Requests

CPU [S1] = $1 CPU [S2] = $2 DISK [S2] = $2 DISK [S3] = $1 CPU + DISK [B1] = $5 CPU + DISK [B2] = $6

Winner Determination

Buyers Sellers CPU DISK B1 ($5) S1 ($1) S2 ($2) B2 ($6) S2 ($2) S3 ($1)

38th International Conference on Parallel Processing, 22-25 September 2009, Vienna, Austria

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Payment Computation

Proposed Mechanism

16

Market Maker Resources Requests

CPU [S1] = $1 CPU [S2] = $2 DISK [S2] = $2 DISK [S3] = $1 CPU + DISK [B1] = $5 CPU + DISK [B2] = $6

Winner Determination

Agent Payment S1 2 + 1 = 3 0 + 1 = 1 -3 + 1 = -2 S3 1 + 2 = 3 1 + 0 = 1 -3 + 1 = -2 B1

  • 2 + 2 = 4

cM|s=∞ cM|s=0

38th International Conference on Parallel Processing, 22-25 September 2009, Vienna, Austria

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Optimal Allocation

17

Market Maker Resources Requests

CPU [S1] = $1 CPU [S2] = $2 DISK [S2] = $2 DISK [S3] = $1 CPU + DISK [B1] = $5 CPU + DISK [B2] = $6

Winner Determination

Total Welfare Exchange w/o S1 6 – 2 – 1 = 3 B2 buys from S2, S3 w/o S2 6 – 1 – 1 = 4 B2 buys from S1, S3 w/o S3 6 – 1 – 2 = 3 B2 buys from S1, S2 w/o B1 6 – 1 – 1 = 4 B2 buys from S1, S3 w/o B2 5 – 1 – 1 = 3 B1 buys from S1, S3 maximum 6 – 1 – 1 = 4 B2 buys from S1, S3

38th International Conference on Parallel Processing, 22-25 September 2009, Vienna, Austria

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Payment Computation

Optimal Allocation

18

Market Maker Resources Requests

CPU [S1] = $1 CPU [S2] = $2 DISK [S2] = $2 DISK [S3] = $1 CPU + DISK [B1] = $5 CPU + DISK [B2] = $6

Winner Determination

Agent Payment S1

  • 1 – (4 – 3) = -2

S3

  • 1 – (4 – 3) = -2

B2 6 – (4 – 3) = 5

38th International Conference on Parallel Processing, 22-25 September 2009, Vienna, Austria

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Implementation

  • Discrete event auctions simulator
  • jCase – open-source combinatorial

auctions simulator

  • FreePastry-based implementation
  • n PlanetLab

Impact of Untruthful Users

5.5 6 6.5 7 7.5 8 8.5 1000 2000 3000 4000 5000 6000 7000 8000 Number of Successful Requests (log) Number of Requests truthful 10% untruthful, 10% price change 10% untruthful, 20% price change 30% untruthful, 10% price change 30% untruthful, 20% price change

19 38th International Conference on Parallel Processing, 22-25 September 2009, Vienna, Austria

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Comparison with Traditional One-sided Auctions

2 3 4 5 6 7 8 9 10 24 48 72 96 120 144 168 Number of Successful Requests (log) Simulation Time (hours) traditional auctions, 1 rt traditional auctions, 4 rt traditional auctions, 8 rt traditional auctions, 16 rt proposed mechanism, 1 rt proposed mechanism, 4 rt proposed mechanism, 8 rt proposed mechanism, 16 rt

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Price Diversity (%) Successful Buyer Requests (%) Traditional Auctions Proposed Increase (%) Under-Demand 10 66.4 78.9 26.5 20 66.4 79 27.1 40 66.3 79 26.6 Balanced Market 10 54.7 69.2 18.9 20 54.6 69.3 19.0 40 54.5 69.0 19.2 Over-Demand 10 33.5 39.1 16.7 20 33.8 39.1 15.6 40 33.6 39.1 16.3

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Comparison with Combinatorial Auctions

Pricing Mechanism Number

  • f Users

Properties Performance IC BB EE Runtime

  • Succ. Buyer

Requests (%)

  • Alloc. Seller

Items (%) Combinatorial Auctions (VCG)

20 40 80 ✔ ✔ ✔

  • 1,402
  • 1,544
  • 1,557

2,470 6,321 14,384 9.6 min 2.5 hrs 67.4 hrs 44.5 52.5 54.2 44.8 57.2 64.0

Combinatorial Auctions (Threshold)

20 40 80 ✕ ✕ ✕ 5 9 6 2,491 6,223 14,567 9.8 min 2.5 hrs 49.5 hrs 44.4 49.6 58.8 48.3 59.8 65.1

Proposed

20 40 80 100 200 500 ✔ ✔ ✔ ✔ ✔ ✔ 1,871 5,483 11,561 14,369 28,564 65,948 1 sec 3 sec 5 sec 7 sec 20 sec 1.9 min 36.3 48.5 52.8 54.1 53.5 52.6 32.5 55.3 68.0 71.6 76.5 80.2

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  • Scalability
  • Centralized market-maker that processes requests sequentially
  • Vertical – increase the number of resource types
  • Horizontal – increase the number of users
  • Monopolistic Sellers [Pham, H.N. et.al., An Approach to Vickrey-based Resource

Allocation in the Presence of Monopolistic Sellers, In Proc. 7th Australasian Symposium on Grid Computing and e-Research (AusGrid 2009), pp. 77-83, Wellington, New Zealand]

Limitations

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S1

CPU $1

S3

DISK $1

B1

CPU+DISK $5

S2

CPU $2

S2

DISK $2

B2

CPU+DISK $6

38th International Conference on Parallel Processing, 22-25 September 2009, Vienna, Austria

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Conclusions

  • Resource pricing and allocation scheme that:
  • Allocates multiple resource types
  • Provide incentives for rational buyers and sellers
  • Achieves budget balance
  • Computational efficiency
  • Future Work
  • Distributed pricing scheme – improve horizontal and

vertical scalability

23 38th International Conference on Parallel Processing, 22-25 September 2009, Vienna, Austria

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Questions ?

marianmi@comp.nus.edu.sg

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Thank you !

marianmi@comp.nus.edu.sg

  • Y. M. Teo and M. Mihailescu, A Strategy-proof Pricing Scheme for Multiple Resource Type

Allocations, in Proceedings of 38th International Conference on Parallel Processing, pp. 172-179, IEEE Computer Society Press, Vienna, Austria, September 22-25, 2009