An equilibrium in group decision and its association with the Nash equilibrium in game theory

Journal Pre-proofs An equilibrium in group decision and its association with the Nash equilibrium in game theory Fujun Hou, Yubing Zhai, Xinli You PII: DOI: Reference:

S0360-8352(19)30607-2 https://doi.org/10.1016/j.cie.2019.106138 CAIE 106138

To appear in:

Computers & Industrial Engineering

Received Date: Revised Date: Accepted Date:

8 November 2018 23 October 2019 25 October 2019

Please cite this article as: Hou, F., Zhai, Y., You, X., An equilibrium in group decision and its association with the Nash equilibrium in game theory, Computers & Industrial Engineering (2019), doi: https://doi.org/10.1016/ j.cie.2019.106138

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier Ltd.

An Equilibrium in group decision and its association with the Nash equilibrium in game theory Fujun Hou ∗ , Yubing Zhai, Xinli You School of Management and Economics Beijing Institute of Technology Beijing, China, 100081 October 23, 2019

Abstract An equilibrium for group decision is introduced, and its association with the Nash equilibrium in game theory is examined. The experts’ preferences in the group decision situation are assumed to be ties-permitted ordinal rankings and represented by preference sequence vectors (PSVs). In particular, this paper is concerned with: (1) the introduction of a group decision equilibrium which is described by using PSVs; (2) the description of the Nash equilibrium by using PSVs; (3) the approach for calculating the Nash equilibrium by means of PSVs; (4) the comparison of the introduced PSV approach and an existing method for calculating the Nash equilibria; and (5) the relationship of the group decision equilibrium in group decision making and the Nash equilibrium in game theory. The observations are helpful for perceiving the relationship between the group decision making and the game theory. Keywords: group decision making (GDM), consensus, fixed point, Nash equilibrium

1

Introduction

Acknowledgment This work was supported by the National Natural Science Foundation of China (No. 71571019). ∗

Tel.: +86 10 6891 8960; Fax: +86 10 6891 2483; email: [email protected].

1

An equilibrium in group decision and its association with the Nash equilibrium in game theory Author’s name ∗ Author’s Institution October 23, 2019

Abstract An equilibrium for group decision is introduced, and its association with the Nash equilibrium in game theory is examined. The experts’ preferences in the group decision situation are assumed to be ties-permitted ordinal rankings and represented by preference sequence vectors (PSVs). In particular, this paper is concerned with: (1) the introduction of a group decision equilibrium which is described by using PSVs; (2) the description of the Nash equilibrium by using PSVs; (3) the approach for calculating the Nash equilibrium by means of PSVs; (4) the comparison of the introduced PSV approach and an existing method for calculating the Nash equilibria; and (5) the relationship of the group decision equilibrium in group decision making and the Nash equilibrium in game theory. The observations are helpful for perceiving the relationship between the group decision making and the game theory. Keywords: group decision making (GDM), consensus, fixed point, Nash equilibrium

1

Introduction

In group decision making (GDM) and/or social choice (SC), consensus is an everlasting research topic which can be traced back at least to 18th century Condorcet’s work. This is true because reaching consensus so as to achieve a consensus collective choice is one of the important objectives of a GDM/SC problem (Fishburn,1977; Dong et al., 2018; Liu & Chen, 2018; Liu et al., 2018; Sun et al., 2019; Wu et al., 2018; Yu et al., 2017; Yu et al., 2018; Zhang & Guo, 2017; Zhang et al., 2017c). And in the consensus reaching process, consensus measure is the first and essential step. In most cases, the consensus level is defined by distance function (Zhang et al., 2017b; Zhang et al., 2018b). Beyond that, Zhang et al. (2018a) proposed the additive consistency of an HFPR to reflect the consistency level. Hou and Triantaphyllou (2019) introduced a concept of premetric to define consensus gap that may reach consensus even if their preference vectors are not identical. Then, the feedback mechanism is required to support experts in increasing the group consensus. There are two kinds of consensus rules: (a) Identification rule and direction rule (Li et al., 2018), i.e., identify the experts who contribute less to consensus and modify their preferences based on the recommendations generated by the direction rules; (b) Optimization-based consensus rule (Zhang & Guo, 2016; Zhang et al., 2017a; Wu et al., 2018; Wu et al., 2019), which minimize preference-loss through linear programming model and generate the optimal adjusted preferences to modify experts preferences. To evaluate the efficiency of different consensus reaching processes in GDM models, Zhang et al. (2019) designed a general framework and reported several comparison results for this regard. These concrete results, among many others, have deepened people’s understanding of consensus in GDM, meanwhile, promoted the development of research regarding consensus-related topics. In game theory (GT), a game concerns the optimal strategic interactions between individuals. The Nash equilibrium is a fundamental concept which is closely related to the solution of a game problem, and it can be formulated by a certain kind of fixed-point equation in some cases (Nash 1950a, 1950b). ∗

Tel.: telephone-number; Fax: fax-number; Author’s e-mail.

1

Recently, a mathematical definition of consensus for GDM problems where individuals’ preferences are provided by ties-permitted ordinal rankings was given, and the definition was proved to be equivalent to a particular kind of fixed-point equation (Hou 2015b). In the GDM context considered by Hou (2015a, 2015b), the experts’ preferences were assumed to be ties-permitted ordinal rankings represented by preference sequence vectors (PSVs). A PSV stands for a ties-permitted ordinal ranking of which the alternatives in a tie are ranked at identical positions, and these positions are represented by continuous integer numbers. Hou and Triantaphyllou (2019) presented an iterative approach for achieving consensus in GDM problems where the individuals’ preferences are represented by PSVs. The PSV-based GDM method has some advantages over some classical GDM methods in that it can reflect consensus properly in GDM: (1) Two individuals may reach a certain consensus even if their preferences are not identical; and (2) Consensus does not necessarily satisfy the transitivity property. These advantages were highlighted by Hou and Triantaphyllou (2019) from an axiomatic view. Now that a GDM problem and a game problem both involve multiple individuals with the objective of seeking optimal choices or actions which can be formulated by some fixed-point equations in particular cases, a question thus arises: Is there any possible inter-relationship between the GDM and the GT from the perspective of fixed-point equations? This paper attempts to answer this question. Specifically, this paper concentrates on a group decision equilibrium in GDM and its association with the Nash equilibrium in GT. When we use PSVs to introduce an equilibrium in GDM and describe the Nash equilibrium in GT by means of PSV, as will be shown in sections 3, 4 and 5, we will observe that the equilibrium in GDM and the Nash equilibrium in GT take similar form of fixed-point equations. In Section 2 some notation and definitions to be used are introduced. Of key importance is the notion of the PSV and the definition of consensus in GDM. In Section 3 a definition of group decision equilibrium is introduced by using PSVs. Its existence conditions are also investigated, of which one is a sufficient condition, and the other is a necessary and sufficient condition taking the form as a particular fixed-point equation. In Section 4 the Nash equilibrium is defined by using the PSVs and a related fixed-point equation is also presented. How to calculate the Nash equilibrium by means of the PSV approach is discussed in Section 5. A Prisoner’s Dilemma game is examined for an illustrative purpose. The comparison of the PSV approach and an existing method for computing Nash equilibria is also made. The relationship of the group decision equilibrium and the Nash equilibrium is discussed in Section 6 and Section 7 concludes the paper.

2

Basic notation and definitions

Some notation and definitions are needed for our discussion which can be found in literatures of Hou (2015a), Hou (2015b) and Hou (2016). Denote by {A1 , A2 , . . . , Am } the alternative set with 1 < m < +∞ and by {E1 , E2 , . . . , En } the expert group with 1 < n < +∞. The experts’ preferences are provided as ties-permitted ordinal rankings. We assume that the alternatives in a tie are to be ranked at identical positions, and these positions are consecutive positive integers. Under this assumption, a ties-permitted ordinal ranking can be represented by a preference sequence vector (PSV), whose entries are sets indicating the alternatives’ possible ranking positions. Definition 1 A vector P S = [Vi ]m×1 is called a PSV over I = {1, 2, . . . , m} if the following conditions are satisfied: (1) ∀i(Vi ⊆ I and Vi 6= ∅). (2)

m S

Vi = I.

i=1

(3) ∀i, j ∈ I, either Vi = Vj or Vi ∩ Vj = ∅.

2

(4) The elements of Vi are consecutive natural numbers from I. (5) An entry Vi appears |Vi | times as the entries of [Vi ]m×1 , where |Vi | is the cardinal number of Vi . A ties-permitted ordinal ranking implies a weak order relation hence an ordered partition of the alternative set. In mathematics, a partition corresponds to an equivalence relation and vice versa. The items in Def.1 indicate that, different entries of a PSV constitute a partition of the set {1, 2, . . . , m}, and one entry corresponds to one partition block. Therefore, a PSV (an ordered partition of position set {1, 2, . . . , m} in nature) can be used to represent the alternatives’ possible ranking positions derived from a ties-permitted ordinal ranking (an ordered partition of alternative set {A1 , A2 , . . . , Am } in nature). Example 1 Consider a GDM problem where 4 experts’ ordinal preferences over 5 alternatives are provided as follows: E1 : A1 ∼ A2  A3 ∼ A4  A5 , E2 : A1 ∼ A2  A3  A4 ∼ A5 , E3 : A1  A2  A3  A4  A5 , E4 : A1  A2  A3  A5  A4 . According to Def.1, the experts’ preferences can be represented by PSVs: A1 A2 A3 A4 A5

     

E1 : {1, 2} {1, 2} {3, 4} {3, 4} {5}

      ,    

E2 : {1, 2} {1, 2} {3} {4, 5} {4, 5}

      ,    

E3 : {1} {2} {3} {4} {5}

      ,    

E4 :  {1} {2}   {3}  . {5}  {4}

As illustrated by Example 1 where experts provide their preferences by using weak order relations, an expert’s PSV indicates an ordered partition of the alternative set. For instance, expert E1’s PSV implies the ordered partition {A1 , A2 }  {A3 , A4 }  {A5 }, which is derived from expert E1’s preference, that is, the weak order relation A1 ∼ A2  A3 ∼ A4  A5 . The elements in an entry of a certain PSV indicate an alternative’s possible ranking positions according to a certain expert’s preference. For instance, according to expert E1’s preference, alternatives A1 and A2 are tied together occupying positions 1 and 2, alternatives A3 and A4 are tied together occupying positions 3 and 4, and alternative A5 is ranked at position 5. Because a PSV is a vector whose entries are sets, such relation or operations as the inclusion ⊆, the intersection ∩ and the union ∪ of two PSVs can be defined or manipulated inP an entry-by-entry manner. Moreover, the cardinal number of a PSV P S = [Vi ]m×1 is defined by |P S| = m i=1 |Vi |. Whether the experts are in consensus or not is determined by examining whether their PSVs have common positions or not. (1) (2) Definition 2 Assume that the experts’ PSVs are P S (1) = [Vi ]m×1 , P S (2) = [Vi ]m×1 , . . . , P S (n) = (n) (1) (2) (n) [Vi ]m×1 . The experts are in consensus if, and only if ∀i(Vi ∩ Vi ∩ . . . ∩ Vi 6= ∅); or, equivalently, (j) (k) ∀i, j, k(Vi ∩ Vi 6= ∅). To illustrate, we again consider Example 1 but just take into account the first three PSVs: A1 A2 A3 A4 A5

     

E1 : {1, 2} {1, 2} {3, 4} {3, 4} {5}

      ,    

E2 : {1, 2} {1, 2} {3} {4, 5} {4, 5}

      ,    

E3 :  {1} {2}   {3}  . {4}  {5}

If a group is composed only of experts E1, E2 and E3, then, the experts in this group are said to be in consensus, since the above three PSVs satisfy the consensus requirements defined in Def.2, namely, all the intersection sets of the PSVs’ corresponding entries are not empty sets. 3

We have the following results: — The experts’ PSVs can be written into a matrix called the preference sequence matrix, denoted by (j) [Vij ]m×n where Vij = Vi , i = 1, . . . , m and j = 1, . . . , n. Moreover, the intersection of a preference sequence matrix and a PSV is defined by [Vij ]m×n ∩ [Wi ]m×1 = [Vi1 ∩ Vi2 ∩ . . . ∩ Vin ∩ Wi ]m×1 . For example, the PSVs in Example 1 can be written into a preference sequence matrix as follows:      

{1, 2} {1, 2} {3, 4} {3, 4} {5}

{1, 2} {1, 2} {3} {4, 5} {4, 5}

{1} {2} {3} {4} {5}

{1} {2} {3} {5} {4}

   .  

In addition, the intersection between the above matrix and a PSV, say ({1}, {2}, {3}, {4}, {5})T , is calculated as follows:         {1, 2} {1, 2} {1} {1} {1} {1, 2} ∩ {1, 2} ∩ {1} ∩ {1} ∩ {1} {1}  {1, 2} {1, 2} {2} {2}   {2}   {1, 2} ∩ {1, 2} ∩ {2} ∩ {2} ∩ {2}   {2}           {3, 4} {3} {3} {3}  ∩  {3}  =  {3, 4} ∩ {3} ∩ {3} ∩ {3} ∩ {3}  =  {3}  .          {3, 4} {4, 5} {4} {5}   {4}   {3, 4} ∩ {4, 5} ∩ {4} ∩ {5} ∩ {4}   ∅  {5} {4, 5} {5} {4} {5} {5} ∩ {4, 5} ∩ {5} ∩ {4} ∩ {5} ∅

— If the experts are in consensus, then the intersection vector of their PSVs is still a PSV. As illustrated before, the first three PSVs in Example 1 are in consensus. Their intersection vector is ({1}, {2}, {3}, {4}, {5})T , which is still a PSV according to Def.1. — The experts are in consensus if, and only if, there exists at least one PSV over I, denoted [Wi ]m×1 , satisfying the following fixed-point equation: [Vij ]m×n ∩ [Wi ]m×1 = [Wi ]m×1 .

(1)

According to Def.2, the first two experts in example 1 are in consensus. Their PSVs constitute a preference sequence matrix in the following:   {1, 2} {1, 2}  {1, 2} {1, 2}     {3, 4} {3}  .    {3, 4} {4, 5}  {5} {4, 5} One can observe that the above matrix satisfies the following three fixed-point equations:       {1, 2} {1, 2} {1} {1}  {1, 2} {1, 2}   {2}   {2}         {3, 4} {3}  ∩  {3}  =  {3}  ,        {3, 4} {4, 5}   {4}   {4}  {5} {4, 5} {5} {5}      

{1, 2} {1, 2} {3, 4} {3, 4} {5}

{1, 2} {1, 2} {3} {4, 5} {4, 5}





    ∩    

4

{2} {1} {3} {4} {5}





    =    

{2} {1} {3} {4} {5}

   ,  

and      

{1, 2} {1, 2} {3, 4} {3, 4} {5}

{1, 2} {1, 2} {3} {4, 5} {4, 5}





    ∩    

{1, 2} {1, 2} {3} {4} {5}





    =    

{1, 2} {1, 2} {3} {4} {5}

   .  

— When the experts are in consensus, their consensus ranking, say [Wi ]m×1 , is the unique solution of the fixed-point equation of (1) that has the maximal cardinality, and it is calculated by:     

W1 W2 .. . Wm





    =   

(j)

∩nj=1 V1 (j) ∩nj=1 V2 .. . (j)

∩nj=1 Vm

    .  

(2)

In Example 1, experts E1 and E2 are in consensus. Their consensus ranking is A1 ∼ A2  A3  A4  A5 , since the intersection of their PSVs is ({1, 2}, {1, 2}, {3}, {4}, {5})T which is calculated by       {1, 2} {1, 2} {1, 2}  {1, 2}   {1, 2}   {1, 2}         {3, 4}  ∩  {3}  =  {3}  .        {3, 4}   {4, 5}   {4}  {5} {4, 5} {5} In addition, from the illustration for formula (1), we know that the intersection vector has the maximal cardinal number among all the fixed point vectors, since their cardinal numbers are       {1} {2} {1, 2}  {2}   {1}   {1, 2}         {3}  = 5,  {3}  = 5 and  {3}  = 7,        {4}   {4}   {4}  {5} {5} {5} respectively.

3

An equilibrium in group decision

A PSV stands for a ties-permitted ordinal ranking. According to Def.1, the i-th entry of a PSV indicates the alternative Ai ’s ranking position or positions. If some alternatives are ranked by the group at the first position, we say that the group reaches an equilibrium in the group decision making. To get a clear description of the group equilibrium, we introduce some notations in the following. (j)

— Denote by P SVi the i-th entry of expert Ej ’s PSV. For instance, those first entries of the four (1) (2) (3) (4) PSVs in Example 1 are P SV1 = {1, 2}, P SV1 = {1, 2}, P SV1 = {1}, and P SV1 = {1}, respectively. (j)

— Denote by P AS (j) the preferred alternative set of expert Ej , where P AS (j) = {Ai |P SVi ⊇ {1}, i = 1, 2, . . . , m}. For instance, the preferred alternative sets of the four experts in Example 1 are P AS (1) = {A1 , A2 }, P AS (2) = {A1 , A2 }, P AS (3) = {A1 } and P AS (4) = {A1 }, respectively.

5

Definition 3 The alternative set P AS ∗ in a group decision making problem is a group decision equilibrium if, for every expert Ej and every alternative Ai ∈ P AS ∗ , Ai is ranked at the first position by expert Ej . Mathematically, for every alternative Ai ∈ P AS ∗ , (j)

P SVi

⊇ {1} for every expert Ej .

(3)

Or equivalently, for every alternative Ai ∈ P AS ∗ , Ai ∈ P AS (j) for every expert Ej .

(4)

To illustrate, we consider Example 1. According to Def.3, the alternative set {A1 } is a group decision (1) (2) (3) equilibrium of Example 1. This is true because we have P SV1 = {1, 2}, P SV1 = {1, 2}, P SV1 = {1}, (4) (j) and P SV1 = {1}, hence P SV1 ⊇ {1}, j = 1, . . . , 4. We also have A1 ∈ P AS (j) , j = 1, . . . , 4 as a result of the fact that P AS (1) = {A1 , A2 }, P AS (2) = {A1 , A2 }, P AS (3) = {A1 } and P AS (4) = {A1 }. For a particular GDM problem, the group decision equilibrium may exist (i.e., P AS ∗ 6= ∅) or may not exist (i.e., P AS ∗ = ∅). However, from Def. 2 we have the following result. Proposition 1 In a GDM problem, if the experts are in consensus, then the group decision equilibrium exists. Proof. Assume that a GDM problem involves m alternatives and n experts, and that [P Sij ]m×n is the (j)

GDM problem’s preference sequence matrix where P Sij = P SVi . Let {A1 , A2 , . . . , Am } be the alternative set. According to formula (1), if the experts are in consensus, then there exists at least one PSV over {1, 2, . . . , m}, say [F Si ]m×1 , satisfying: [P Sij ]m×n ∩ [F Si ]m×1 = [F Si ]m×1 , which is equivalent to the following formula:  
(j) ∩nj=1 P Sij ∩ F Si = ∩nj=1 P SVi ∩ F Si = F Si , ∀i.

(∗)

Because [F Si ]m×1 is a PSV, according to Items (1) and (2) of Def.1, we know that there exists at least one index i∗ corresponding alternative Ai∗ and satisfying {1} ⊆ F Si∗ .

(∗∗)

From formulae (*) and (**), we have (j)

{1} ⊆ P SVi∗ , ∀j. Therefore, formula (3) holds regarding alternative Ai∗ , and thus our considered GDM has an equilibrium {Ai∗ }. The existence of the group decision equilibrium means that the group reaches a consensus with regard to some alternatives. ’Group in consensus’ means that the group reaches consensus with regard to every alternative. Proposition 1 presents a sufficient rather than a necessary condition for the existence of the group decision equilibrium, since ’consensus regarding some alternatives’ does not necessarily mean ’consensus regarding all the alternatives’. In example 1, for instance, the four experts are not in consensus. However, the GDM problem in Example 1 has a group decision equilibrium of {A1 }, since all the experts are willing to rank the alternative A1 at the first position. From formula (1), the following result can be deduced directly. Proposition 2 An alternative set P AS ∗ is the group decision equilibrium if, and only if, for every alternative Ai ∈ P AS ∗ , the following fixed-point equations are verified: (j)

P SVi

∩ {1} = {1} for every expert Ej . 6

(5)

Proof. On the one hand, formula (3) is equivalent to formula (5) as a result of the fact that (j)

{1} ⊆ P SVi

(j)

⇔ P SVi

∩ {1} = {1}.

(j)

On the other hand, formula P SVi ∩ {1} = {1} can be seen as a fixed-point equation which takes the form of f (x) = x. Therefore, Proposition 2 holds.

4

The description of Nash equilibrium by using of PSVs

This section is concerned with a possible description of the Nash equilibrium by means of PSVs. The Nash equilibrium of a strategic game with ordinal preferences is described as follows (Osborne 2004): ”Definition 23.1 The action profile a∗ in a strategic game with ordinal preferences is a Nash equilibrium if, for every player i and every action ai of player i, a∗ is at least as good according to player i’s preferences as the action profile (ai , a∗−i ) in which player i chooses ai which every other player j choose a∗j . Equivalently, for every player i, ui (a∗ ) ≥ ui (ai , a∗−i ) for every action ai of player i, where ui is a payoff function that represents player i’s preferences.” Let Bi (a−i ) be the best response of player i to a−i . The Nash equilibrium can also be defined by using best response functions (Osborne 2004): ”Definition 36.1 The action profile a∗ is a Nash equilibrium of a strategic game with ordinal preferences if and only if every player’s action is a best response to the other player’s actions: a∗i is in Bi (a∗−i ) for every player i.”

(6)

The above definitions which was originally presented by Nash (1950a, 1950b) remind us of: (1) in a finite strategic game with ordinal preferences, a player’s preference over her own various actions can be represented by a PSV, since the PSV can describe the decision maker’s ordinal preference effectively; (2) the Nash equilibrium of a finite strategic game with ordinal preferences might be described by using the PSV; and (3) some condition for the Nash equilibrium’s existence might be obtained by examining the players’ PSVs. We do this below. Let G = {S1 , . . . , Sn ; u1 , . . . , un } be a finite strategic game with ordinal preferences. Let N = {1, 2, . . . , n} be the set of players, Si = {si1 , . . . , simi } the set of player i’s actions, and ui the payoff function of player i’s ordinal preferences. In a static case, the order of the players’ actions does not matter, and thus the action profile (s1 , s2 , . . . , sn ) is equivalent to (si , s1 , s2 , . . . , si−1 , si+1 , . . . , sn ) = (si , s−i ). 0 Let s be an action profile, in which player i’s action is si . Let si be any action of player i. We denote 0 by (si , s−i ) the action profile in which every player except i select their actions as specified by s−i , while 0 player i selects her action as si . When players except i choose their actions as specified by s−i , the player i’s payoffs regarding her different actions can be ranked based on ui (si1 , s−i ), ui (si2 , s−i ), . . . , ui (simi , s−i ). Let the corre(i)

0

0

sponding PSV be P SV (i) (si , s−i ). Similar to Section 3, we denote by P SVk (si , s−i ) the k-th entry of 0 0 0 P SV (i) (si , s−i ), and we denote the preferred action set (PAS) from P SV (i) (si , s−i ) by P AS (i) (si , s−i ) = (i) 0 {sik |P SVk (si , s−i ) ⊇ {1}, k = 1, 2, . . . , mi }. 7

We note that here the ’preferred action set’ is also shorted as ’PAS’ just like that in Section 3 for the ’preferred alternative set’. This is a tolerable coincidence because the actions play the same role in game theory as the alternatives do in the GDM. We also remark that the ’preferred action set’ is just the ’best response set’ but described by the PSV. Similar to the definition of Nash equilibrium described by Osborne (2004) as reproduced above, the Nash equilibrium defined by formula (6) in a finite strategic game with ordinal preferences can be described by using PSVs as follows. Definition 4 Let G = {S1 , . . . , Sn ; u1 , . . . , un } be a finite strategic game with ordinal preferences, where the set of players is assumed as N = {1, 2, . . . , n}, Si = {si1 , . . . , simi } represents the player i’s action set, and ui represents the payoff function of player i’s ordinal preferences. The action profile s∗ = (s∗1 , . . . , s∗i , . . . , s∗n ) = (s1k1 , . . . , siki , . . . , snkn ) is a Nash equilibrium if, for every player i and every 0 0 action si of player i, s∗i is at least as good according to player i’s preferences as the action profile (si , s∗−i ) 0 in which player i chooses si while every other player j choose s∗j . Mathematically, (i)

0

P SVki (si , s∗−i ) ⊇ {1} for every player i, (i)

0

(7)

0

where P SVki (si , s∗−i ) is the ki -th entry of P SV (i) (si , s∗−i ). Or equivalently, 0

siki ∈ P AS (i) (si , s∗−i ) for every player i, 0

(8)

0

where P AS (i) (si , s∗−i ) is the preferred action set of P SV (i) (si , s∗−i ). Def.4 indicates an approach of seeking for the Nash equilibrium of a finite strategic game. We will give an illustration for this approach in next section. Evidently, from Def.4 we have the following results. Proposition 3 If s∗ is a Nash equilibrium, then 0

0

0

s∗ ∈ P AS (1) (s1 , s∗−1 ) × . . . × P AS (i) (si , s∗−i ) × . . . × P AS (n) (sn , s∗−n ).

(9)

Proof. Assume that s∗ is a Nash equilibrium and s∗ = (s∗1 , . . . , s∗i , . . . , s∗n ) = (s1k1 , . . . , siki , . . . , snkn ). From formula (8) we have 0 siki ∈ P AS (i) (si , s∗−i ), ∀i, hence 0

0

0

(s1k1 , . . . , siki , . . . , snkn ) ∈ P AS (1) (s1 , s∗−1 ) × . . . × P AS (i) (si , s∗−i ) × . . . × P AS (n) (sn , s∗−n ). Therefore, we have formula (9) and thus Proposition 3 holds. From formula (1) we have: Proposition 4 An action profile s∗ = (s∗1 , . . . , s∗i , . . . , s∗n ) = (s1k1 , . . . , siki , . . . , snkn ) is a Nash equilibrium if, and only if the following fixed-point equation holds: (1)

0

0

(i)

(n)

0

(P SVk1 (s1 , s∗−1 ) ∩ . . . ∩ P SVki (si , s∗−i ) ∩ . . . ∩ P SVkn (sn , s∗−n )) ∩ {1} = {1}.

(10)

Or equivalently, (i)

0

P SVki (si , s∗−i ) ∩ {1} = {1} for every player i.

(11)

Proof. Assume that s∗ is a Nash equilibrium and s∗ = (s∗1 , . . . , s∗i , . . . , s∗n ) = (s1k1 , . . . , siki , . . . , snkn ). From formula (7) we have (i) 0 {1} ⊆ P SVki (si , s∗−i ), ∀i, which is equivalent to (i)

0

P SVki (si , s∗−i ) ∩ {1} = {1}, ∀i. We thus have formula (11). In addition, formula (11) is equivalent to formula (10). Therefore, Proposition 4 holds. 8

We remark that our considered game is a finite static strategic game, and the discussion is concerned only with the pure strategy equilibrium rather than a mixed strategy equilibrium.

5

Computing the Nash equilibrium by using PSVs and comparison

According to Definition 4, we get an approach for finding the Nash equilibrium(s) (if exist) for a finite strategic game: • For every action profile (s1 , . . . , si , . . . , sn ) ∈ (S1 × . . . × Si × . . . × Sn ), 0

– obtain the P SV (i) (si , s−i ) for every player i according to player i’s preferences; 0

– get the P AS (i) (si , s−i ) for every player i; 0

– Check whether the formula (8), namely si ∈ P AS (i) (si , s−i ) for every player i, holds or not. If it holds, then the action profile under consideration is a Nash equilibrium. • Write out all the Nash equilibrium(s). To illustrate, we consider the ’Prisoner’s Dilemma’ which is known as a standard example in game theory. Two suspects of a crime are held in separate cells. The prosecutors have enough evidence to convict each of them of a minor offense, but lack sufficient evidence to convict either of them of the major charge unless one of them betrays the other. Each suspect can either fink the other or remain silent. The payoff is: • If they both fink, they will get both sentenced to 3 year in prison. • If they both remain silent, they will get both sentenced to 1 years in prison. • If one and only one finks, he will be set free, while the other will serve 5 years in prison.

Table 1: Payoffs of players suspect 2 silent fink silent (-1,-1) (-5,0) suspect 1 fink (0,-5) (-3,-3) The payoffs can be summarized in Table 1 in which the suspect 1’s payoff is listed first. We know that this example is a finite strategic game which can be represented by G = {1, 2; {silent, f ink}, {silent, f ink}}. Next we use the PSVs for obtaining the Nash equilibrium. To make it intuitive, we show the computation process by Table 2, from which we know that the action profile (f ink, f ink) is the only Nash equilibrium of this game. Notably, when it is for a two-person finite strategic game, the above approach can be implemented in a more compact table, in which the rows made up of the second parts in cells of the table represent a player’s PSVs, while the other player’s PSVs are represented by the columns made up of the first parts in the table cells. The criterion for testing a Nash equilibrium is to check whether the PSV entries in a cell have the common subset of {1}. We illustrate this compact approach by again investigating the above ’Prisoner’s Dilemma’ game. The illustration is included in Table 3. The Nash equilibrium is the action profile corresponding to the cell with an underline which indicates that the PSV entries in this cell have the common subset of {1}. The above procedure of the PSV approach for finding Nash equilibria in finite strategic games is quite similar to the ’best response’ approach as described in most game theory textbooks for instance in Osborne (2004). The ’best response’ method is a popular approach for computing Nash equilibria. In 9

Table 2: Computation process Action profile (s1 , s2 ) (silent, silent)

(silent, f ink)

(f ink, silent)

(f ink, f ink)

0

0

P SV (i) (si , s−i ) 0

P SV (1) (s1 , s−1 )   silent {2}   f ink {1}   silent {2}   f ink {1}   silent {2}   f ink {1}   silent {2}   f ink {1}

0

P AS (i) (si , s−i ) 0

P SV (2) (s2 , s−2 )   silent {2}   f ink {1}   silent {2}   f ink {1}   silent {2}   f ink {1}   silent {2}   f ink {1}

0

si ∈ P AS (i) (si , s−i )? 0

P AS (1) (s1 , s−1 )

P AS (2) (s2 , s−2 )

s1

s2

{f ink}

{f ink}

NO

NO

{f ink}

{f ink}

NO

YES

{f ink}

{f ink}

YES

NO

{f ink}

{f ink}

YES

YES

Table 3: Illustration for a two-person game suspect 2

suspect 1

silent

fink

silent

{2},{2}

{2},{1}

fink

{1},{2}

{1},{1}

a two-person finite strategic game, this method attaches a star for each player’s best response, and if the payoffs in a cell of the payoff table are both marked, then the corresponding action profile is a Nash equilibrium. On the one hand, the description of the Nash equilibrium by using the PSV (i.e., formula (8)) is quite similar to that described by the ’best response’ (i.e., formula (6)). The ’preferred action set (PAS)’ in the PSV approach acts just the same role as the ’best response set’ does in the ’best response’ approach. In other words, the PSV provides another mathematical way for describing the ’best response set’ in finite strategic games. On the other hand, constructing a PSV in the proposed approach takes more time than finding a best response in the ’best response’ approach, since ranking all the actions is more complicated than just finding a best one. Assume that we have n numbers. To find the maximal one we need to do comparisons of n − 1 times. To rank the numbers from large to small, we have to do more comparisons than n − 1 times. For instance, when the well-known Bubble Sort Method is used, the time complexity will be of O(n2 ). Therefore, when the numbers of decision makers (players) and alternatives (strategies) are large, the computation of the PSV method for computing Nash equilibrium will become exhausted. This is a disadvantage of the PSV approach. As mentioned in the introduction, our main objective in this paper is to investigate some possible relationships of the GDM and game theory from the prospective of fixedpoint equations. For computing Nash equilibria, we do not recommend the use of the PSV approach, especially when the game involves large number of players and/or strategies.

10

6

Relationship of the ’group decision equilibrium’ and the Nash equilibrium and possible extensions

As shown by formulae (3) and (7), the introduced ’group decision equilibrium’ is formally similar to the Nash equilibrium described by using of the PSV. This similarity coincides with a well-known knowledge that the group decision making has some inner relationship with the game theory. One can even say that a group decision making problem is a simplified version of a game problem. Therefore, the PSV is a helpful tool for perceiving the relationship between the group decision making and the game theory. Moreover, it is known that the Nash equilibrium is a kind of fixed-point (Nash 1950a,b). Interestingly, as shown by Proposition 2, the group decision equilibrium is also a particular kind of fixed-point that satisfies the fixed-point equation of (5). We remark that this paper made an attempt associating the group decision with game theory from the prospective of fixed-point equation. We should admit that the observed relationship between the group decision equilibrium and Nash equilibrium cannot be seen as a theoretical contribution to the theory of decision making or the theory of game. Moreover, we do not recommend using the PSV method when seeking Nash equilibrium in finite strategic games. As afore mentioned, the computation process of the PSV method is complex and of low efficiency for problems of large scale. By using the PSV this paper unveils a connection between the group decision and the game theory. In the last 70 years, the GDM/SC problems have been well investigated with the help of game theory, and vice versa (see, e.g., Kaneko & Nakamura, 1979; Peleg, 1978; Shapley, 1962; Wilson, 1972). In 1944, von Neumann and Morgenstern distinguished a certain kind of games termed as simple games (von Neumann & Morgenstern, 1944). Each coalition in a simple game is ”either all-powerful or completely ineffectual” (Shapley, 1962). A certain class of simple games that may be of interest here is the ’majority game’. We adopt the following notations inspired by Shapley (1962) for a majority game, Mn,k , where ”there are n players in all and it takes k or more to win”. Shapley (1962) distinguished the following cases: • Mn,(n+1)/2 , with an odd number n; • Mn,(n+2)/2 , with an even number n; • Mn,n . Evidently, we can associate the consensus in GDM with the winning coalitions in majority games, since they both involve such concepts as ’agree’ or ’be in cooperation’ with the experts or players in all or in a part. The previous discussion in this paper may be helpful to further exploration into the relationships of the GDM and the game theory by means of the PSV. However, the consensus definition (Def.2) used in this paper is too restrictive for further application, since it only touched on the full consensus case. In practice, a majority voting or a majority game may involve such kinds of decision rules as ’more than half’ or ’at least 2/3’, etc. Sometimes, only some rather than all of the alternatives (actions) are concerned with. Therefore, the consensus definition (Def.2) needs to be generalized to more practical situations. We do this below. Definition 5 Let {A1 , A2 , . . . , Am } be an alternative set with 1 < m < +∞ and {E1 , E2 , . . . , En } (1) an expert group with 1 < n < +∞. Assume that the experts’ PSVs are P S (1) = [Vi ]m×1 , P S (2) = (2) (n) [Vi ]m×1 , . . . , P S (n) = [Vi ]m×1 . Suppose that S is a nonempty subset of the alternatives’ indices set, i.e., S ⊆ I where I = {1, 2, . . . , m} and S 6= ∅, and T a nonempty subset of the experts’ indices set, i.e., T ⊆ N where N = {1, 2, . . . , n} and T 6= ∅. The experts indicated by T are in consensus on the (j) ranking positions of the alternatives indicated by S if, and only if ∀i ∈ S(∩j∈T Vi 6= ∅); or, equivalently, (j) (k) ∀i ∈ S, Vi ∩ Vi 6= ∅ hold for any j, k ∈ T . For simplicity, we denote by RankingConsensus(I, N ; S, T ) the consensus case, as defined by Def.5, that the experts indicated by T are in consensus on the ranking positions of the alternatives 11

indicated by S. One can see that, Definition 2 is a special case of Definition 5, i.e., the case of RankingConsensus(I, N ; I, N ). The notation of RankingConsensus(I, N ; S, T ) provides a flexibility for describing some special consensus cases under some special majority rules by means of PSVs. For instance, a particular ranking consensus under a ’more than 1/2’ majority rule can be formulated as • RankingConsensus(I, N ; S, T ), where |N | is an odd number, T ⊆ N , and |T | ≥ • RankingConsensus(I, N ; S, T ), where |N | is an even number, T ⊆ N , and |T | ≥

|N |+1 2 ; |N |+2 2 .

As a demonstration of this idea, let us focus on the alternative A5 in Example 1. A closer examination of the PSVs in Example 1 reveals that more than half (exactly, 3) of the experts have reached a consensus regarding the alternative A5 ’s ranking position(s). This observation can be formulated as RankingConsensus({1, 2, 3, 4, 5}, {1, 2, 3, 4}; {5}, {1, 2, 3}). We note that in Definition 5 S and/or T can be single element sets. This is by no means a trivial case in that it may bring some convenience to theoretical discussion. In the future, we shall use the previous developments to investigate more relationships between the GDM and the game theory, for instance, the relationship between a certain case of consensus and the core of a cooperative game.

7

Conclusion

The aim of the paper was to introduce an equilibrium definition for the group decision and to examine its relationship with the Nash equilibrium in game theory. The following results were obtained. (1) The group decision equilibrium is defined as a set containing those alternatives that are ranked by all the experts at the first positions (alternatives in a tie that are ranked at first position are allowed). After representing the experts’ preferences by PSVs, a mathematical description is given. A PSV stands for a ties-permitted ordinal ranking of which an entry indicates an alternative’s ranking position or positions. A fixed-point equation is shown to be equivalent to the definition of the group decision equilibrium. (2) The Nash equilibrium is described by using PSVs. The description is shown to be equivalent to a particular fixed-point equation. (3) An approach for computing the Nash equilibrium by using the PSVs is proposed, and its comparison with an existing approach is discussed. It is pointed out that, the proposed approach has a greater time complexity than the classical ’best response’ method. (4) The similarities of the group decision equilibrium and the Nash equilibrium are examined. When the Nash equilibrium is described by the PSVs, the group decision equilibrium resembles the Nash equilibrium in at least two aspects: (a) the similar form of their representations; and (b) the similar fixed-point equation of their equivalence conditions. The introduced GDM equilibrium does not always exist in real situations, since it requires all the individuals agree about the alternative or alternatives that is to be ranked at the first position, or are tied together to be ranked at first positions. In future research, we will discuss about how to help individuals achieve an equilibrium in a GDM context. This indicates a particular consensus reaching process, namely, approaching consensus regarding the best alternative(s). Some literature will be a great help regarding this topic. For example, Zhang et al. (2019) proposed a feedback adjustment process for the consensus reaching process along with a framework and some criteria for evaluating the efficiency of different consensus reaching processes in GDM problems. The literature of Zhang et al. (2019) can help us to select/design a more efficient consensus reaching process when trying to achieve a group decision equilibrium. 12

In addition to the process of approaching consensus regarding the best alternative(s), how to deduce the set of best alternative(s) is another topic on which deserves our future concerns. This is a topic related to one of the four fundamental problems in MCDM (Roy, 1996): choice (choosing the best alternative), ranking (ranking alternatives from best to worst), sorting (sorting the alternatives into categories), and description (describing the alternatives into details). The GDM equilibrium is the set of best alternative(s) agreed by all the individuals. As afore mentioned, when the PSV is used as the describing tool, the GDM equilibrium has the same representation as that of the Nash equilibrium. Therefore, making optimal choice in GDM and MCDM may be realized by some means of seeking Nash equilibrium in game theory. This fascinating problem will be our additional work in the future. Acknowledgments

References [1] Dong, Y., Zha, Q., Zhang, H., Kou, G., Fujita, H., Chiclana, F., & Herrera-Viedma, E. (2018). Consensus reaching in social network group decision making: Research paradigms and challenges. Knowledge-Based Systems, 162: 3-13. [2] Fishburn, P.C. (1977). Condorcet social choice functions. SIAM Journal on Applied Mathematics, 33(3), 469489. [3] Hou, F. (2015a). A consensus gap indicator and its application to group decision making. Group Decision and Negotiation, 24(3), 415-428. [4] Hou, F. (2015b). The prametric-based GDM selection procedure under linguistic assessments. Fuzzy Systems (FUZZ-IEEE), 2015 IEEE International Conference on. IEEE, 2015: 1-8. [5] Hou, F. (2016). The prametric-based GDM procedure under fuzzy environment. Group Decision and Negotiation, 25(5), 1071-1084. [6] Hou, F., & Triantaphyllou, E. (2019). An iterative approach for achieving consensus when ranking a finite set of alternatives by a group of experts. European Journal of Operational Research, 275(2), 570-579. [7] Kaneko, M., & Nakamura, K. (1979). The Nash social welfare function. Econometrica: Journal of the Econometric Society, 423-435. [8] Li, C., Dong, Y., & Herrera, F. (2018). A consensus model for large-scale linguistic group decision making with a feedback recommendation based on clustered personalized individual semantics and opposing consensus groups. IEEE Transactions on Fuzzy Systems, 27(2), 221-233. [9] Liu, P., & Chen, S.M. (2018). Multiattribute group decision making based on intuitionistic 2-tuple linguistic information. Information Sciences, 430, 599-619. [10] Liu, Y., Dong, Y., Liang, H., Chiclana, F., & Herrera-Viedma, E. (2018). Multiple attribute strategic weight manipulation with minimum cost in a group decision making context with interval attribute weights information. IEEE Transactions on Systems, Man, and Cybernetics: Systems, https://doi.org/10.1109/TSMC.2018.2874942. [11] Nash, J. (1950a). The bargaining problem. Econometrica, 18, 155-162. [12] Nash, J. (1950b). Equilibrium points in n-person games. Proceedings of the National Academy of Sciences U.S.A., 36(1), 48-49. [13] Osborne, M.J. (2004). An introduction to game theory. Oxford University Press, New York, NY, USA. [14] Peleg, B. (1978). Representations of simple games by social choice functions. International Journal of Game Theory, 7(2), 81-94. [15] Roy, B. (1996). Multicriteria Methodology for Decision Aiding. Kluwer, London. [16] Shapley, L.S. (1962). Simple games: an outline of the descriptive theory. Behavioral Science, 7(1), 59-66.

13

[17] Sun, B., Ma, W. & Chen, X. (2019). Variable precision multigranulation rough fuzzy set approach to multiple attribute group decision-making based on -similarity relation. Computers & Industrial Engineering, 127: 326343. [18] von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behavior. Princeton: Princeton University Press. [19] Wilson, R. (1972). The game-theoretic structure of Arrow’s general possibility theorem. Journal of Economic Theory, 5(1), 14-20. [20] Wu, J., Dai, L., Chiclana, F., Fujita, H., & Herrera-Viedma, E. (2018). A minimum adjustment cost feedback mechanism based consensus model for group decision making under social network with distributed linguistic trust. Information Fusion, 41, 232-242. [21] Wu, Y., Dong, Y., Qin, J., & Pedrycz, W. (2019). Flexible Linguistic Expressions and Consensus Reaching With Accurate Constraints in Group Decision-Making. IEEE transactions on cybernetics, DOI: 10.1109/TCYB.2019.2906318. [22] Yu, G.F., Li, D.F., Qiu, J.M., & Zheng, X.X. (2018). Some operators of intuitionistic uncertain 2-tuple linguistic variables and application to multi-attribute group decision making with heterogeneous relationship among attributes. Journal of Intelligent & Fuzzy Systems, 34, 599-611. [23] Yu, W., Zhang, Z., Zhong, Q., & Sun, L. (2017). Extended TODIM for multi-criteria group decision making based on unbalanced hesitant fuzzy linguistic term sets. Computers & Industrial Engineering, 114, 316-328. [24] Zhang, H., Dong, Y., & Chen, X. (2017a). The 2-rank consensus reaching model in the multigranular linguistic multiple-attribute group decision-making. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 48(12), 2080-2094. [25] Zhang, H., Dong, Y., Chiclana, F., & Yu, S. (2019). Consensus efficiency in group decision making: A comprehensive comparative study and its optimal design. European Journal of Operational Research, 275(2), 580-598. [26] Zhang, H., Dong, Y., & Herrera-Viedma, E. (2017b). Consensus building for the heterogeneous large-scale GDM with the individual concerns and satisfactions. IEEE Transactions on Fuzzy Systems, 26(2), 884-898. [27] Zhang, Z., & Guo, C. (2016, July). Minimum adjustment-based consistency and consensus models for group decision making with interval pairwise comparison matrices. In 2016 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE) (pp. 1701-1708). IEEE. [28] Zhang, Z., & Guo, C. (2017). Deriving priority weights from intuitionistic multiplicative preference relations under group decision-making settings. Journal of the Operational Research Society, 68(12), 1582-1599. [29] Zhang, Z., Guo, C., & Martnez, L. (2017c). Managing multigranular linguistic distribution assessments in large-scale multiattribute group decision making. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47(11), 3063-3076. [30] Zhang, Z., Kou, X., & Dong, Q. (2018a). Additive consistency analysis and improvement for hesitant fuzzy preference relations. Expert Systems with Applications, 98, 118-128. [31] Zhang, Z., Kou, X., Yu, W., & Guo, C. (2018b). On priority weights and consistency for incomplete hesitant fuzzy preference relations. Knowledge-Based Systems, 143, 115-126.

14

 A group decision equilibrium (GDE) for group decision making (GDM) is introduced.  A fixed-point equation is shown to be equivalent to the definition of the GDE.  The similarity of the GDE and the Nash equilibrium in game theory (GT) is examined.  The findings are helpful for perceiving the relationship of the GDM and the GT.