Domination Game: When Game Theory Meets Data Mining

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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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Motivation

• Example
• 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?

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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?

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Motivation

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

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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
Smart Phone Market
Memory
Expected customers 21/21/2
Standby time
Expected customers 11/21/2
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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
Nash Equilibrium now?
No
Yes Now
Memory
Standby time
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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

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 3H12H2 6
• After 5H11H2 6.5

Hi11/21/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?

At most nd cells
Intersection test in O(d) time
Utility computation in O(dn) time
The best response query is O(nd1(dn))
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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

Range Query here
weight 1/2
weight 1
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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

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Experimental Results

• Efficiency Results on TripAdvisor with 997
  customers and 2 providers

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Future Work and Conclusion

• Future 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 databases
  support

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Question Answer
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