Multiple Criteria Decision Analysis with Game-theoretic Rough Sets

Nouman Azam and JingTao Yao

Department of Computer Science University of ReginaCANADA S4S 0A2

azam200n@cs.uregina.ca jtyao@cs.uregina.cahttp://www.cs.uregina.ca/~azam200n http://www.cs.uregina.ca/~jtyao

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

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.

Multiple Criteria and PRSUtilities for Criterion C1

0.6

0.7

0.9

0.6

0.3

0.2

(α1, β1)

(α2, β2)

(α6, β6)

(α3, β3)

(α4, β4)

(α5, β5)

Rankings based on C1

1 2 3 3 4 5

Multiple Criteria and PRSUtilities for Criterion C2

0.7

0.1

0.4

0.6

0.8

0.4

(α1, β1)

(α2, β2)

(α6, β6)

(α3, β3)

(α4, β4)

(α5, β5)

Rankings based on C2

1 2 3 4 4 6• Dilemma:

– Ranking of C1 vs C2– Which pair to select

Game Theory for Solving Dilemma

• Game theory is a core subject in decision sciences.

• The components in a game.– Players.– Strategies.– Payoffs.

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.

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.

Probabilistic Rough Sets and GTRS

• Determining an (α, β) pair with game-theoretic analysis.

C1

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.

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.

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.

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.

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.

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.

A Payoff Table

• A two player game with n actions for each player.

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.

Probabilistic Information for a Concept

• Information about a concept C with respect to 15 equivalence classes.

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.

Different Thresholds versus Measures

• Utilizing the GTRS based framework to find a suitable solution.

α Confidence(PRSC) Coverage(PRSC)

0.90.80.70.60.5

0.97240.95070.90550.85540.8212

0.54220.61820.72480.83700.9049

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.

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

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.

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.