Introduction to algorithmic aspect of auction theory

Introduction to Algorithmic Aspect of Auction Theory

2010-01-08Abner Huang

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CSBB Lab, Dept. of CS, NTHU.

Outline

• Connections between Computer Science and Economics

• Algorithmic Game Theory• Algorithmic Mechanism Design

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Herbert Alexander Simon

• Herbert Alexander Simon (June 15, 1916– February 9, 2001) was an American political scientist, economist, and psychologist.

• Notable awards – Turing Award 1975– Nobel Prize in Economics

1978

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Thomas Crombie Schelling

Thomas Crombie Schelling (born 14 April 1921) was awarded the 2005 Nobel Memorial Prize in Economic Sciences (shared with Robert Aumann) for "having enhanced our understanding of conflict and cooperation through game-theory analysis."

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Micromotives and macrobehavior

• Schelling analyzed apartheid with cellular automata, which is invented by John von Neumann and is well-studied by computer scientists.

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Market Equilibrium--Arrow-Debreu model

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Market Equilibrium

• Starting in the 60’s, economists have attempted to use the Arrow-Debreu model and the general equilibrium theory to realistically model actual economies, with the goal of evaluating alternate policy options.– Scarf’s algorithm,– Homotopy methods,– Global Newton’s method

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Market Equilibrium

• There are several algorithmic results in the last 6 years, i.e., PTAS, FPTAS, and polynomial time algorithm based on diophantine approximation and ellipsoid algorithm.

• Fisher’s Model with Linear Utility Functions is in P.

• Arrow-Debreu’s Model with Linear Utility Functions is in P.

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Outline

• Connections between Computer Science and Economics

• Algorithmic Game Theory• Algorithmic Mechanism Design

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Traffic Flow Problem

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Braess’s Paradox

Initial Network:

s tx 1

½

x1½

½

½

Cost = 1.5

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Braess’s Paradox

Initial Network: Augmented Network:

s tx 1

½

x1½

½

½

Cost = 1.5 Cost = 2

s tx 1

x10

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Braess’s Paradox

Initial Network: Augmented Network:

All traffic incurs more cost! [Braess 68]

• also has physical analogs [Cohen/Horowitz 91]

s tx 1

½

x1½

½

½

Cost = 1.5 Cost = 2

s tx 1

x10

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Inefficiency of Nash Flows

Note: selfish routing does not minimize average delay (observed informally by [Pigou 1920])

• Cost of equilibrium flow = 1•1 + 0•1 = 1• Cost of optimal (min-cost) flow = ½•½ +½•1 = ¾

s t

x

10

1 ½

½

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Performance Guarantees

Good news: in theoretical CS, have lots of techniques for measuring inefficiency.

• motivated by NP-completeness, real-time algorithms, etc.

Definition: approximation ratio (w.r.t. some objective function):

optimal obj fn value

protagonists's obj fn value the closer to 1 the better

Price of anarchy := equilibrium/OPT ratio

Market Equilibrium

• Fisher’s model

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Market Equilibrium

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Arrow-Debreu model

An Auction-Based AlgorithmGarg et al. proposed an auction-based PTAS for the linearcase of the Arrow-Debreu model.

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Remarks

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Outline

• Connections between Computer Science and Economics

• Algorithmic Game Theory• Algorithmic Mechanism Design

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Algorithmic mechanism design

• Algorithmic mechanism design (AMD) lies at the intersection of economic game theory and computer science.

• Noam Nisan and Amir Ronen, from the Hebrew University of Jerusalem, first coined "Algorithmic mechanism design" Games and Economic Behavior (35): 166–196.

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Algorithmic mechanism design

• It typically employs the analytic tools of theoretical computer science, such as worst case analysis and approximation ratios, in contrast to classical mechanism design in economics which often makes distributional assumptions about the agents.

• It also considers computational constraints to be of central importance: mechanisms that cannot be efficiently implemented in polynomial time are not considered to be viable solutions to a mechanism design problem

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Worst-Case Time Complexity

n=1000 ， Time cost for different computer

time Time 11000 steps/s

Time 22000 steps/s

Time 34000 steps/s

Time 48000 steps/s

log2n 0.010 0.005 0.003 0.001

n 1 0.5 0.25 0.125

nlog2n 10 5 2.5 1.25

n1.5 32 16 8 4

n2 1,000 500 250 125

n3 1,000,000 500,000 250,000 125,000

1.1n 1039 1039 1038 1038

P vs. NP-Hard

• P = the set of problems which can be solved in polynomial time ,e.g.,– 50*n0.2,3*n+2000, 0.01*n+19861234, 0.001*n80

– Finding maximum of a set of integers is in P. (linear time in fact.)

• EXP = the set of problems which can be solved in exponential time.

• NP-Hard = the set of problems which might be able to be solved in polynomial time (at least).

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Distributed artificial intelligence

• Mainstreams in DAI research included the following:– Parallel problem solving: mainly deals with how classic AI

concepts can be modified, so that multiprocessor systems and clusters of computers can be used to speed up calculation.

– Distributed problem solving (DPS): the concept of agent, autonomous entities that can communicate with each other, was developed to serve as an abstraction for developing DPS systems.

– Multi-Agent Based Simulation (MABS): a branch of DAI that builds the foundation for simulations that need to analyze not only phenomena at macro level but also at micro level, as it is in many social simulation scenarios.

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Multi-Agent Based Simulation

• Too crowded or too desolate in PUB?

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Applications of Algorithmic mechanism design

• Load balancing: The aggregate power of all computers on the Internet is huge. In a “dream world” this aggregate power will be optimally allocated online among all connected processors. One could imagine CPU-intensive jobs automatically migrating to CPU-servers, caching automatically done by computers with free disk space, etc. Access to data, communication lines and even physical attachments (such as printers) could all be allocated across the Internet. tightly linked systems, and is addressed, in various forms and with varying

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Applications of Algorithmic mechanism design

• Routing: When one computer wishes to send information to another, the data usually gets routed through various intermediate routers. So far this has been done voluntarily, probably due to the low marginal cost of forwarding a packet. However, when communication of larger amounts of data becomes common (e.g. video), and bandwidth needs to be reserved under various quality of service (QoS) protocols, this altruistic behavior of the routers may no longer hold. If so, we will have to design protocols specifically taking the routers’ self-interest into account.

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Ticket Allocation Game

Project lead (Ticket Allocator) (rational and intelligent)

Maintenance Engineers (rational and intelligent)

effort, time

effort, time

effort, time

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Resource Allocation in Grid Computing

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?

Incentive Compatible Broadcast in Ad hoc Wireless Networks

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Tier 1

Tier 2

Tier 3

Tier 1: UU Net, Sprint, AT&T, GenuityTier 2: Regional/National ISPsTier 3: Residential/Company ISP

Internet Routing

VGC mechanism

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Example: Shortest PathGoal: route packets along the lowest-cost path from S to T.

Example: Shortest Path

• Each edge is an agent• People want to send messages to other

people

Winner Determination Problem

• Winner Determination Problem– Agents want to bid some goods with all-or-nothing

style, and will report their private values.

• Maximum Independent Set Problem– Given undirected graph G = (V,E), MIS is to find a

maximum independent set X V in G. A subset of ⊆vertices Y V is independent if u, v Y, (u, v) ⊆ ∀ ∈not in E

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Maximum Independent Set Problem

• Theorem Winner determination (WD) problem is NP-hard.

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In general, we can’t approximate VCG-based

mechanisms

• Inapproximability for VCG-Based Combinatorial Auctions– By Dave Buchfuhrer et al. To appear in the proceedings of SODA 10.

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Can we cope with it?

• In general, it is hardly possible.• Some ways:

– Solve some special cases.– Approximate some special cases.– Change problem formulations.

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The Task Allocation Problem

The Task Allocation Problem

• The MinWork MechanismThe idea is to minimize the total work done. This is no very good solution since our agents are able to work in parallel.– Each task is allocated to the agent who is capable of doing

it in a minimal amount of time.– Each agent is given payment equal to the time of the

second best agent for every task:

The Task Allocation Problem

• Theorem: MinWork is a truthful n-approximation mechanism.

• Proof:

The Task Allocation Problem

• Proof (continue): MinWork belongs to the VGC family ( is truthful)⇒

VGC mechanism

• An output function of a VGC mechanism is required to maximize the objective function. In many cases (e.g. combinatorial auctions (Harstad, Rothkopf and Pekec (1995))), this makes the mechanism computationally intractable. Replacing the optimal algorithm with a non-optimal approximation usually leads to untruthful mechanisms

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VGC mechanism

• Thus, a VGC mechanism essentially provides a solution for any utilitarian problem (except for the possible problem that there might be dominant strategies other than truth-telling). utilitarian problems (Green and Laffont (It is known that (under mild assumptions) VGC mechanisms are the only truthful implementations for utilitarian problems (Green and Laffont (1977)).for 1977)).

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Thank You

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