FORMULATING THREE-WAY DECISION MAKINGWITH GAME-THEORETIC ROUGH SETS

Nouman Azam

Department of Computer Science, University of Regina, Regina, Saskatchewan, Canada S4S 0A2Email: azam200n@cs.uregina.ca

ABSTRACTThree-way decisions are important for effective decision mak-ing in many real life problems found in medical, business,management and scientific research. The game-theoretic roughset (GTRS) model is a recent proposal in rough sets for ef-fective decision making when multiple criteria are involved.This article illustrates the formulation and interpretation ofthree-way decisions with the GTRS model. In particular, atwo-player game is implemented between different criteria re-alized as players in a game in order to obtain three-way de-cisions. The presented interpretations may relate the GTRSmodel to existing studies on three-way decision-making, lead-ing to more exciting applications.

Index Terms— rough sets, probabilistic rough sets, game-theoretic rough sets, game theory, three-way decisions

1. INTRODUCTION

Three-way decisions are sometimes more useful than conven-tional two-way decisions and may represent a more accuratestate of a problem. Two-way decisions are generally basedon crisp judgments that leads to stringent outcomes such asyes or no, accept or reject and true or false. This sometimesmay not be feasible from practical perspective especially insituations where the evidence is insufficient or weak due toabsence or lack of reliable information. One therefore mayprefer an additional option such as delaying or deferring a de-cision. For instance, a doctor obtains some test results to lookfor the presence of cancer. A false decision in this case maytake a human life. If the evidence is not enough for reachinga certain decision, the doctor may want to conduct additionaltests thereby deferring a definite decision for a while. In suchcases, three-way decisions may be viewed as an extension oftwo-way decisions with an added option [1].

The concept of three-way decisions was recently incor-porated in rough set theory by providing an alternative inter-

This work is conducted under the guidance of my Ph.D supervisor Dr.JingTao Yao and is partially supported by a discovery grant from NSERCCanada awarded to him and the FGSR Dean’s Scholarship to the author. Theauthor would like to acknowledge the support from Department of ComputerScience, Faculty of Science and Faculty of Graduate Studies and Research atthe University of Regina for attending CCECE’13.

pretation of the three rough set regions [1, 2, 3, 4, 5, 6]. Inparticular, the positive, negative and boundary regions maybe used to induce rules for obtaining decisions of acceptance,rejection and deferment, respectively. The positive and neg-ative rules are associated with strict conditions which some-times lead to many deferment decisions in practical applica-tions [7, 8]. Probabilistic rough sets was introduced with theaim of relaxing these conditions thereby reducing the num-ber of deferment decisions. The notion of level of confidencewith a pair of thresholds was introduced for making a deci-sion. A decision of acceptance is made when the confidenceis at or above an upper threshold. The rejection decisionis made when the confidence is at or below a lower thresh-old. A deferment decision is made when the confidence isbetween the two thresholds. The determination and interpre-tation of these thresholds are among the fundamental issuesin the field [9, 10].

The game-theoretic rough set (GTRS) model is a recenttrend in determination of effective threshold values for prob-abilistic rough sets [11, 12, 13]. In this article, another ap-proach with the model is proposed to construct three-way de-cisions. A two-player game is considered between multiplecriteria realized as game players. These players evaluate andcooperate to reach an effective decision based on an equiva-lence class. Two possible strategies are formulated for eachplayer representing a positive or negative evaluation result foran equivalence class corresponding to a concept. Three-waydecisions are constructed based on different game outcomesand the associated strategies. It is suggested that the proposedapproach will help in the interpretation and understandabilityof three-way decisions with GTRS.

2. THREE-WAY DECISIONS WITH PROBABILISTICROUGH SETS

The problem of three-way decisions was recently outlined byYao in [1]. Considering U as a finite nonempty set of objectsand C as a set of criteria or objectives. The problem of three-way decisions is to divide U based on C into three disjointregions, POS, NEG and BND called as positive, negative andboundary regions, respectively. The POS region consists ofobjects that satisfy the criteria in C to a certain level. The

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NEG region contain objects that does not satisfy the criteria inC to another level. The BND region consist of objects whoseinclusion in both POS and NEG regions are inconclusive.

The three regions may be used to construct rules for ob-taining three-way decisions. The POS region generates rulesfor decisions of acceptance, the NEG region generates rulesfor rejection and the BND generates rules for deferment. Basedon the above description, probabilistic rough sets may be in-terpreted as one form of three-way decisions [1].

Suppose E ⊆ U×U is an equivalence relation on U . Theequivalence class of E containing object x ∈ U is given by[x]E (or for simplicity by [x]). The set or family of all equiv-alence classes that generates a partition of U is represented asU/E = {[x]|x ∈ U}. For a nonempty set C ⊆ U containinginstances of a concept, the probability P (C|[x]) denotes theconditional probability of an object in C given that the objectis in [x].

The (α, β)-probabilistic rough set positive, negative andboundary regions can be defined as follows [1, 4, 14]:

POS(α,β)(C) = {x ∈ U |P (C|[x]) ≥ α},NEG(α,β)(C) = {x ∈ U |P (C|[x]) ≤ β},BND(α,β)(C) = {x ∈ U |β < P (C|[x]) < α}. (1)

The three regions may be interpreted in terms of three-waydecisions. A decision for an object in C based on [x] is madeas follows,

Acceptance: if P (C|[x]) ≥ α,Rejection: if P (C|[x]) ≤ β, andDeferment: if β < P (C|[x]) < α. (2)

The conditional probability may be recognized as a levelof confidence in making a decision. Particularly, For an objecty having the same description as x, we accept it as belongingto C if the confidence is greater than or equal to level α, i.e.P (C|[x]) ≥ α. The same object y may be rejected to be in Cif the confidence is at or below the level β, i.e. P (C|[x]) ≤β. A deferment decision may be made if the confidence isbetween the two levels, i.e. β < P (C|[x]) < α.

The problem of three-way decisions for probabilistic roughsets can now be interpreted based on Equations 1 - 2. Con-sidering U as a finite set of objects, the problem of three-waydecisions is to obtain the disjoint regions POS, NEG and BNDbased on criterion P (C|[x]). This means that a single crite-rion is used to construct three-way decisions. It is suggestedthat three-way decisions defined with multiple criteria may beuseful for obtaining more effective and robust decisions. TheGTRS model may be used for such a purpose.

3. REVIEW OF GAME-THEORETIC ROUGH SETS

The game-theoretic rough sets model provides a frameworkfor analyzing decision making problems as a competitive or

cooperative game involving multiple entities. Majority of thecurrent studies, if not all, focused on the issue of effectivethresholds determination using different approaches and for-mulations such as classification configuration, effective rulemining and rule measures tradeoff [14, 15, 16]. The thresh-olds determined with the GTRS are afterwards used in theprobabilistic rough set framework to obtain three-way deci-sions as outlined in Equation 2. This suggest the GTRS tobe an implicit or indirect mechanism for obtaining three-waydecisions by setting the threshold values. This article outlinesan approach that can directly obtain the three-way decisionswith the model. This may provide another motivation for ap-plying and studying the model in future research. The GTRSmodel is now being reviewed for the sake of completeness.

The GTRS model formulates a game based on a set ofplayers, a set of strategies for each player and the respectivepayoff functions for each strategy. The players are selected inorder to analyze critical factors or criteria that are used to e-valuate different aspects of decision making. From three-waydecision making perspective, these criteria will be incorporat-ed in a game-theoretic framework to determine the partitionof the universe into three disjoint regions. Measures for eval-uating decisions may be set as game players in this scenario.

The strategies represent different situations that the play-ers may encounter during a game. Each player selects an ac-tion to improve its position in the game and any future situ-ations it may encounter. From decision making perspective,the strategies may incorporate the factors that affect decisionmaking, including costs, loss functions or performance levelsassociated with different decisions.

Table 1. The game for three-way decision making

p2

s1 s2 ...

p1s1 u1(s1, s1), u2(s1, s1) u1(s1, s2), u2(s1, s2) ...

s2 u1(s2, s1), u2(s2, s1) u1(s2, s2), u2(s2, s2) ...

... ... ... ...

The payoff functions involve the notions of gains, benefitsor advantages achieved in performing or taking an action. Inthe context of decision making, the evaluations of differen-t criteria which may be in the form of measure’s value mayrepresent the payoff functions. In order to find an effectivesolution within the game, the solution concept of Nash equi-librium and payoff table are generally utilized. A payoff tablelists all possible actions of different players with their respec-tive payoffs. Table 1 represents a payoff table for the twoplayers. Finally, Nash equilibruim is determined within thepayoff table. This consists of finding a game state or situationin which no player can benefit by changing its strategy giventhe other player’s chosen strategy.

The essential relationship between multiple entities in-volved in a game and three-way decisions is of interest in thisarticle. This is elaborated in the next section.

4. THREE-WAY DECISIONS WITHGAME-THEORETIC ROUGH SETS

A two-player game is formulated with GTRS to implementthree-way decisions.

From Section 2 it was noted that the three-way decision-s in probabilistic rough sets are based on the evaluation of aparticular equivalence class with respect to a given concep-t. A similar idea is utilized in this study to define a game butinvolving different evaluations based on multiple criteria. Par-ticularly, a two-player game is considered where the playersare in the form of evaluation criteria that determine the in-clusion of an equivalence class into one of the three possibleregions. A single game will be implemented for each equiv-alence class where the two involved criteria will compete todetermine one of the three possible decisions, i.e. acceptance,rejection or deferment.

Two strategies for each player are considered in this game,namely s1 = P and s2 = N . Strategy s1 = P is desired insituations when the evaluation results obtained with a partic-ular criterion are convincing enough to accept an equivalenceclass as positively belonging to a concept. On the other hand,strategy s2 = N will be of interest when the evaluation resultsare convincing to reject an equivalence class as negatively be-longing to a concept. The two players realized as evaluationcriteria for an equivalence class will determine their respec-tive utility for considering different strategies using suitablepayoff functions.

The conditional probability P (C|[x]) may be realized asone form of an evaluation criterion for an equivalence classthat may be used to determine the two possible strategies. Us-ing P (C|[x]) as a criterion, the utility or payoff functions fortaking strategies P andN denoted by u(P ) and u(N) respec-tively, may be given by,

u(P ) = P (C|[x]),u(N) = 1− P (C|[x]) = P (Cc|[x]). (3)

This means that the strategy P reflects the player utility inaccepting [x] to be in C while the strategy N determines theplayer utility in rejecting [x].

More interesting measures for evaluation of an equiva-lence class as belonging to a concept C may be incorporatedin the game e.g. the measure of Shannon entropy. Consid-ering a partition of the universe formed with respect to theconcept C as πC = {C,Cc}, the entropy or uncertainty in πCdue to [x] may be given by,

H(πC |[x]) = −P (C|[x])logP (C|[x])−P (Cc|[x])logP (Cc|[x]) (4)

The corresponding utility functions u(P ) and u(N) may bedefined for this criterion as discussed above.

Table 2. The game for three-way decision making

C2

P N

C1

P uC1(P, P ), uC2

(P, P ) uC1(P,N), uC2

(P,N)

N uC1(N,P ), uC2(N,P ) uC1(N,N), uC2(N,N)

Table 2 represents the suggested two-player game betweencriterion C1 and C2, in the form of a payoff table. The rowscorrespond to criterion (or player) C1 and the columns to C2.Each cell in the table corresponds to a strategy profile con-taining a payoff pair. There are four possible outcomes of thisgame based on the considered strategies. Each outcome is ofthe form (si, sj), where player C1 selected strategy si whileplayer C2 selected strategy sj .

The solution concept of Nash equilibrium may be utilizedto determine the possible outcome of the game which may becategorized into the following three groups.

Both players select (si = P, i ∈ {1, 2}): This case will bethe Nash equilibruim or game outcome if the following con-ditions hold.

uC1(P, P ) ≥ uC1

(N,P ) & uC2(P, P ) ≥ uC1

(P,N). (5)

This means that the players C1 and C2 can not increase theirrespective payoffs by switching their strategies from P to N .In this situation there is no disagreement or uncertainty re-garding the decision about an equivalence class. Due to con-sensus of players about an equivalence class being positivelyevaluated as belonging to a concept, we may interpret the setof all equivalence classes associated with such outcome asbelonging to the positive region. The decision of acceptancemay be defined for equivalence classes associated with gameconditions presented as Equation 5.

Both players reject (si = N, i ∈ {1, 2}): The game mayresult in this outcome if the following conditions hold.

uC1(N,N) ≥ uC1(P,N) & uC2(N,N) ≥ uC1(N,P ) (6)

This suggests that none of the players can attain additionalbenefits by switching their respective strategies from N to P .Again, there is no disagreement regarding the decision aboutan equivalence class. We may interpret the set of all equiv-alence classes associated with such outcome as belonging tothe negative region. The decision of rejection may be associ-ated with equivalence classes satisfying conditions in Equa-tion 6.

One player selects (si = P ) and the other rejects (sj =N, i, j ∈ {1, 2}): The game will result in this outcome ifany of the following pair of conditions are true.

uC1(P,N) ≥ uC1

(N,N) & uC2(P,N) ≥ uC1

(P, P ) (7)uC1

(N,P ) ≥ uC1(P, P ) & uC2

(N,P ) ≥ uC1(N,N) (8)

The conditions in Equation 7 refer to a case where player C1

selects strategy P while player C2 selects strategy N . Simi-larly, Equation 8 represents a game output when C1 selectsNwhile C2 selects P . In any of these two cases, there is a dis-agreement about an equivalence class. A deferment decisionmay be considered in this uncertain situation.

The three cases discussed as possible game outcomes maybe used to generate three-way decision rules as follow,

Acceptance: if uC1(P, P ) ≥ uC1

(P,N) &uC2

(P, P ) ≥ uC2(N,P ),

Rejection: if uC1(N,N) ≥ uC1

(N,P ) &uC2

(N,N) ≥ uC2(P,N),

Deferment: otherwise (9)

The three-way decisions obtained in this framework in-corporates multiple criteria in decision making. It is sug-gested that the framework may provide another perspectiveof GTRS in making and defining effective decisions.

5. CONCLUSION

The concept of three-way decisions provides a new perspec-tive and motivation for studying the rough set theory. Thepositive, negative and boundary regions are interpreted as re-gions where the decisions of acceptance, rejection, and defer-ment may be made. The probabilistic rough set model definesthe three regions and the implied three-way decisions usingthe criterion of conditional probability of a concept with an e-quivalence class. In this article, a GTRS based approach is de-fined which obtains the three-way decisions by incorporatingmultiple criteria in a game-theoretic framework. The gamestrategies and the solution concept are utilized to constructedthree-way decisions with GTRS. The incorporation of multi-ple aspects in making decisions may increase the overall con-fidence in obtaining accurate and effective decisions.

6. REFERENCES

[1] Y. Y. Yao, “An outline of a theory of three-way decisions,” inProceedings of Rough Sets and Current Trends in Computing(RSCTC’12), Lecture Notes in Computer Science 7413, 2012,pp. 1–17.

[2] Y. Y. Yao, “Three-way decision: An interpretation of rules inrough set theory,” in Proceedings of 4th International Confer-ence on Rough Sets and knowledge Techonology (RSKT’09),Lecture Notes in Computer Science 5589, 2009, pp. 642–649.

[3] Y. Y. Yao, “Three-way decisions with probabilistic rough sets,”Information Science, vol. 180, no. 3, pp. 341–353, 2010.

[4] Y. Y. Yao, “The superiority of three-way decisions in proba-bilistic rough set models,” Information Sciences, vol. 181, no.6, pp. 1080–1096, 2011.

[5] Y. Y. Yao and X. F. Deng, “Sequential three-way decision-s with probabilistic rough sets,” in Proceedings of the 10thIEEE International Conference on Cognitive Informatics andCognitive Computing, (ICCI*CC’11), 2011, pp. 120–125.

[6] X. F. Deng, “Three-way decisions with artificial neural net-works,” in Proceedings of Canadian Conference on Electricaland Computer Engineering (CCECE’13), 2013.

[7] Y. Y. Yao and Yan Zhao, “Attribute reduction in decision-theoretic rough set models,” Information Sciences, vol. 178,no. 17, pp. 3356–3373, 2008.

[8] X. P. Yang and J. T. Yao, “A multi-agent decision-theoreticrough set model,” in Proceedings of 5th International Confer-ence Rough Set and Knowledge Technology (RSKT’10), Lec-ture Notes in Computer Science 6401, 2010, pp. 711–718.

[9] X. F. Deng and Y. Y. Yao, “An information-theoretic interpre-tation of thresholds in probabilistic rough sets,” in Proceedingsof Rough Sets and Current Trends in Computing (RSCTC’12),Lecture Notes in Computer Science 7413, 2012, pp. 232–241.

[10] Y. Y. Yao, “Two semantic issues in a probabilistic rough setmodel,” Fundamenta Informaticae, vol. 108, no. 3-4, pp. 249–265, 2011.

[11] N. Azam and J. T. Yao, “Classifying attributes with game-theoretic rough sets,” in Proceedings of 4th International Con-ference on Intelligent Decision Technologies (IDT’12), SmartInnovation, Systems and Technologies, Vol. 15, 2012, pp. 175–184.

[12] Yan Zhang, “Optimizing gini coefficient of probabilistic roughset regions using game-theoretic rough sets,” in Proceedings ofCanadian Conference on Electrical and Computer Engineer-ing (CCECE’13), 2013.

[13] N. Azam and J. T. Yao, “Game-theoretic rough sets for featureselection,” in Rough Sets and Intelligent Systems - ProfessorZdzislaw Pawlak in Memoriam, Intelligent Systems ReferenceLibrary, vol. 44, pp. 61–78. 2012.

[14] N. Azam and J. T. Yao, “Multiple criteria decision analysiswith game-theoretic rough sets,” in Proceedings of 7th Inter-national Conference on Rough Sets and knowledge Techonolo-gy (RSKT’12), Lecture Notes in Computer Science 7414, 2012,pp. 399–408.

[15] J. P. Herbert and J. T. Yao, “Game-theoretic rough sets,” Fun-damenta Informaticae, vol. 108, no. 3-4, pp. 267–286, 2011.

[16] Yan Zhang and J. T. Yao, “Rule measures tradeoff using game-theoretic rough sets,” in Proceedings of International Confer-ence on Brian Informatics (BI’12), Lecture Notes in ComputerScience 7670, 2012, pp. 348–359.