PPT

Domination Game: When Game Theory Meets Data

Mining

Zhenjie Zhang

with Laks Laksmannan and Anthny Tung

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Outline

Motivation

Domination Game

Nash Equilibrium and Its Complexity

Best Response Query

Experimental Results

Future Work and Conclusion

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Motivation

Information Explosion

Transaction Records

Customer Information

Database for Business Analysis

Market Analysis

Decision Support

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Motivation

Current Database Techniques

Data Warehouse OLAP Techniques

Data mining techniques Association Rule

Clustering

Database Queries Top-k Query

Skyline Query

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Motivation

Drawbacks

Static environment

Without taking competition into consideration

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MotivationExample

One product with static cost 2$

Given the customers and their acceptable prices, what is the price maximizing the profit?

The optimal price is 9$

What if there is another

provider in the market?

Cust. Acceptable Price

C1 3$

C2 4$

C3 6$

C4 9$

C5 10$

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Motivation

Example If another provider offers the product with 6$

To compete with this provider, is it better to have a lower price?

Cust. Acceptable Price

C1 3$

C2 4$

C3 6$

C4 9$

C5 10$

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Motivation

What if we know more about the customers, and have a more flexible product design

Cust. Price Quality Warranty

C1 3$ Fair 1 Year

C2 4$ Medium 1 Year

C3 6$ Good 2 Year

C4 9$ Medium 2 Year

C5 10$ Good 3 Year

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Motivation

Game Theory

A powerful tool for competition analysis

Nash Equilibrium Given k players in the market, a Nash Equilibrium is a

stable configuration (a set of positions) on the strategies of players.

No incentive for any player to change his strategy, if all of the others keep their strategies in the Nash Equilibrium

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Motivation

Algorithmic Game Theory

The complexity of finding Nash Equilibrium can be high Nash proved the existence in 1960, without any

method to find them

Many types of Nash Equilibrium can be found in polynomial time, like congestion game and exchange game

Not applicable in large database

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Outline

Motivation

Domination Game

Nash Equilibrium and Its Complexity

Best Response Query

Experimental Results

Future Work and Conclusion

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Domination Game

A market with customers and providers

The requirement of a customer is represented by a vector in d numerical dimensions

Smaller value indicates better quality on that dimension

A provider positions his service on some profit constraint hyper-plane

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Domination

A market with customers and providers

A provider dominates a customer if his service satisfies the requirement on every dimension

Given multiple services dominating the same customer, the customer will buy one of these services with equal probability

The utility of a provider is the expected number of buyers

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Domination Game

Standby time

Memory

Expected customers= 1+1/2+1/2

Expected customers= 2+1/2+1/2

Smart Phone Market

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Domination Game

Nash Equilibrium of Domination Game

A configuration (positions of) of the providers

None of them is willing to change

A basic theorem (best response assumption)

In Nash Equilibrium, the position of any provider derives the highest number of expected customers with respect to the positions of all other providers.

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Domination Game

Standby time

Memory

Nash Equilibrium now? NoYes Now

Other Interesting Questions

Given k-1 competitors and their current positions in the market, can we predict the evolution of the competition in the market?

Given a manufacturer with N products, how should he/she position these products to gain the most market share, without internal competition?

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Outline

Motivation

Domination Game

Nash Equilibrium and Its Complexity

Best Response Query

Experimental Results

Future Work and Conclusion

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Existence of NE

A constructive method to prove the existence of NE

Randomly choose the positions of the providers

In one iteration, every provider tries to find a better position to improve his utility (Best Response Query) in a round robin

Stops when no provider can improve any more

The final configuration must be an NE

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Existence of NE

If there are n customers and m providers in our analysis

Question 1: How many iterations before convergence?

Question 2: How much time for one iteration?

Existence of NE

How many iterations before convergence?

If one provider improves in one iteration, the utilities of other provides may drop

However, something keeps increasing

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Dominator Num 0 1 2

Beore 2 3 2

After 1 5 1

Change -1 +2 -1

Number of customers by the number of dominators

Existence of NE

A Simple Analysis

For the whole market, the weighted Harmonic number

Before: 3*H1+2*H2 = 6

After: 5*H1+1*H2 = 6.5

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Dominator Num 0 1 2

Beore 2 3 2

After 1 5 1

Change -1 +2 -1

Hi=1+1/2+…1/i

Existence of NE

Upper bound on weighted Harmonic number

With n customers, no larger than nlogn

Lower bound on the increase in each iteration

With m providers, no smaller than 1/m

The number of iterations

No more than mnlogn

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Existence of NE

How much time for one iteration?

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At most nd cells

Intersection test in O(d) time

Utility computation in O(dn) time

The best response query is O(nd+1(d+n))

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Existence of NE

In summary, the complexity of Nash Equilibrium is polynomial to the number of customers and the number of providers

However, the basic best response query is exponential to the number of dimensions

Social Utility

Social Utility

The total number of customers with at least one satisfying product

Every NE is 2-approximate solution to maximum social utility problem

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Outline

Motivation

Domination Game

Nash Equilibrium and Its Complexity

Best Response Query

Experimental Results

Future Work and Conclusion

Why not DADA

Cuiping Li, Beng Chin Ooi, Anthony K. H. Tung, Shan Wang. "DADA: A Data Cube for Dominant Relationship Analysis", on SIGMOD 2006.

Drawback of DADA

Limited Resolution on the dimensions

Our convergence proof is based on exact solution

Does approximate solution work?

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Best Response Query

R-Tree for efficient dominance counting

weight = 1/2

weight = 1

Range Query here

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Best Response Query

Find the best setting based on dominated customers

Dominance region

Effective Dominance Region

At least one customer on each face of effective dominance region

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Best Response Query

A customer set defines a good effective dominance region if

1) Every customer is the only one on at least one face

2) The left-bottom corner is above the hyper-plane

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Best Response Query

A customer set defines a good effective dominance region if

1) Every customer is the only one on at least one face

2) The left-bottom corner is above the hyper-plane

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Best Response Query

A customer set defines a good effective dominance region if

1) Every customer is the only one on at least one face

2) The left-bottom corner is above the hyper-plane

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Best Response Query

On the customer lattice, find the best customer combination with optimal dominance utility

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Best Response Query

Pruning Strategy

Is it possible to estimate the best result in the sub-tree?

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Best Response Query

Utility upper bound can be estimated by the point here

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Outline

Motivation

Domination Game

Nash Equilibrium and Its Complexity

Best Response Query

Experimental Results

Future Work and Conclusion

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Experimental Results

Data Sets

Synthetic data: Correlated, Independent, Anti-Correlated, Clustered

Trip Advisor: a hotel review set on the hotels in Sydney, on 4 attributes: value, cleanliness, service and room

Algorithms

Naïve, BFS, DFS

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Experimental Results

Efficiency Results on 3D Synthetic Data with 1000 customers and 2 providers

Anti. Corr. Ind. Clu.

Naïve 5326 472 2163 31

DFS 130 27 69 23

BFS 132 29 73 25

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Experimental Results

Efficiency Results on TripAdvisor with 997 customers and 2 providers

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Future Work and ConclusionFuture Work

Extending the current model Subspace dominance

Dominance between providers

The introduction of game theory to other database queries Top-k query, Nearest Neighbor Query

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Future Work and Conclusion

Conclusion

The proposal of Domination Game

Existence proof of Nash Equilibrium over Domination Game

Nash Equilibrium computation with database’s support

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Question & Answer