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Domination Game: When Game Theory Meets Data
Mining
Zhenjie Zhang
with Laks Laksmannan and Anthny Tung
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23/4/8 Domination Game Analysis for Microeconomic Data Mining 2
Outline
Motivation
Domination Game
Nash Equilibrium and Its Complexity
Best Response Query
Experimental Results
Future Work and Conclusion
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23/4/8 Domination Game Analysis for Microeconomic Data Mining 3
Motivation
Information Explosion
Transaction Records
Customer Information
Database for Business Analysis
Market Analysis
Decision Support
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23/4/8 Domination Game Analysis for Microeconomic Data Mining 4
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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23/4/8 Domination Game Analysis for Microeconomic Data Mining 5
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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23/4/8 Domination Game Analysis for Microeconomic Data Mining 9
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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23/4/8 Domination Game Analysis for Microeconomic Data Mining 12
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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23/4/8 Domination Game Analysis for Microeconomic Data Mining 13
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
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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?
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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
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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
23/4/8 Domination Game Analysis for Microeconomic Data Mining 22
Dominator Num 0 1 2
Beore 2 3 2
After 1 5 1
Change -1 +2 -1
Hi=1+1/2+…1/i
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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
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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
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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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23/4/8 Domination Game Analysis for Microeconomic Data Mining 32
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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23/4/8 Domination Game Analysis for Microeconomic Data Mining 33
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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23/4/8 Domination Game Analysis for Microeconomic Data Mining 37
Outline
Motivation
Domination Game
Nash Equilibrium and Its Complexity
Best Response Query
Experimental Results
Future Work and Conclusion
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23/4/8 Domination Game Analysis for Microeconomic Data Mining 38
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