Multiple Criteria Decision Analysis with Game-theoretic Rough Sets

1. Multiple Criteria Decision Analysis with Game-theoretic Rough Sets Nouman Azam and JingTao Yao Department of Computer Science University of Regina CANADA S4S 0A2 azam200n@cs.uregina.ca jtyao@cs.uregina.ca http://www.cs.uregina.ca/~azam200nhttp://www.cs.uregina.ca/~jtyao

2. Probabilistic Rough Sets (PRS) • Defines the approximations in terms of conditional probabilities. • Introduces a pair of threshold denoted as (α, β) to determine the rough set approximations and regions • Lower approximation • Upper approximation • The three Regions are defined as

3. A Key Issue in Probabilistic Rough Sets • Two extreme cases. • Pawlak Model: (α, β) = (1,0) • Large boundary. Not suitable in practical applications. • Two-way Decision Model: α = β • No boundary: Forced to make decisions even in cases of insufficient information. • Determining Effective Probabilistic thresholds. • The GTRS model. • Finds effective values of thresholds with a game-theoretic process among multiple criteria.

4. Multiple Criteria and PRS Utilities for Criterion C1 0.6 (α1, β1) 0.7 (α2, β2) 0.9 (α3, β3) Rankings based on C1 1 2 3 3 4 5 (α4, β4) 0.6 0.3 (α5, β5) 0.2 (α6, β6)

5. Multiple Criteria and PRS Utilities for Criterion C2 • Dilemma: • Ranking of C1 vs C2 • Which pair to select Rankings based on C2 1 2 3 4 4 6 0.7 (α1, β1) 0.1 (α2, β2) 0.4 (α3, β3) (α4, β4) 0.6 0.8 (α5, β5) 0.4 (α6, β6)

6. Game Theory for Solving Dilemma • Game theory is a core subject in decision sciences. • The components in a game. • Players. • Strategies. • Payoffs.

7. Game Theory: Basic Idea • Prisoners Dilemma. • A classical example in Game Theory. • Players = prisoners. • Strategies = confess, Don’t confess. • Utility or Payoff functions = years in gail.

8. Game-theoretic Rough Set Approach • Utilizing a game-theoretic setting for analyzing multiple criteria decision making problems in rough sets. • Multiple criteria as players in a game. • Each criterion enters the game with the aim of increasing its benefits. • Collectively they are incorporated in an interactive enviroment for analyzing a given decision making problem.

9. Probabilistic Rough Sets and GTRS • Determining an (α, β) pair with game-theoretic analysis. C1

10. The Need for GTRS based Framework • The GTRS has focused on analyzing specific aspects of rough sets. • The classification ability. • Further multiple criteria decision making problems may be investigated with the model. • Multiple criteria rule mining or feature selection. • A GTRS based framework is introduced for such a purpose.

11. Components of the Framework • Multiple Criteria as Players in a Game. • Strategies for Multiple Criteria Analysis. • Payoff Functions for Analyzing Strategies. • Implementing Competition for Effective Solutions.

12. Multiple Criteria as Players in a Game • The players are multiple influential factors in a decision making problem. • Including measures, parameters and variables that affect the decision making process. • Different criteria may provide competitive or complimentary aspects. • Accuracy versus generality: Providing competitive aspects of rough sets classification.

13. Strategies for Multiple Criteria Analysis • Strategies are formulated as changes in variables that affects the considered criteria. • Changes in probabilistic thresholds may be realized as strategies for different criteria in analyzing PRS.

14. Payoff Functions • The utilities, benefits or performance gains obtained from a strategy. • When measures are considered as players. • A measure value in response to a strategy may be realized as payoff.

15. Implementing Competition • Expressing the game as a competition or corporation in a payoff table. • Payoff tables. • Listing of all possible actions and their respective utilities or payoff functions. • Obtaining effective solution with game-theoretic equilibrium analysis. • For instance, Nash equilibrium.

16. A Payoff Table • A two player game with n actions for each player.

17. Confidence vs Coverage Game Example • Considering positive rules for a concept C. • The measures may be defined as, • The Pawlak model can generate rules with confidence of 1 but may have low coverage. • By weakening the requirement of confidence being equal to 1, one expects to increase the coverage.

18. Probabilistic Information for a Concept • Information about a concept C with respect to 15 equivalence classes.

19. The Measures in Case of Pawlak Model • This means that Pawlak model can generate positive rules that are 100% accurate but are applicable to only 19.55% of the cases.

20. Different Thresholds versus Measures • Utilizing the GTRS based framework to find a suitable solution.

21. A GTRS based Solution • The players. • Confidence(PRSC) versus Coverage(PRSC) . • The strategies. • Possible decreases in threshold α. • N = no change or decrease in α. • M = moderate decrease in α. • A = aggressive decrease in α.

22. The Game in a Payoff Table • Payoff table with a starting value of (α = 1). • Cells in bold represents Nash equilibrium. • None of the players can achieve a higher payoff given their opponents chosen action

23. Repeating the game • The game may be repeated several times based on updated value of α. • The game may be stopped when the measures fall in some predefined acceptable range.

24. Conclusion • A key issue in probabilistic rough sets. • Determination of effective probabilistic thresholds. • The GTRS model. • Incorporating multiple criteria in a game-theoretic environment to configure the required thresholds. • The GTRS based Framework. • Introduced for investigating further multiple criteria decision making problems in rough sets. • The framework may enable further insights through simultaneous consideration of multiple aspects.