A Bargaining Game Model for Multi-person Conflict Decision with MUltiple Criteria

Copyright © IFAC Large Scale Systems, Beijing, PRC, 1992

A BARGAINING GAME MODEL FOR MULTI-PERSON CONFLICT DECISION WITH MULTIPLE CRITERIA Zhu Shijing and Chen Ting Systems Engineering Insitute, Huazhong University of Science and Technology, Wuhan, 430074, PRC

Abstract, In this paper, the multi-person confLict probLem in which various objective s are concerned is studied and a bargaining game modeL for this kind of problem is presented. The model can achieve a Pareto optimal solution to aLL the decision makers in conflict on the base of individual preferences and individual optimization. A new optimization method for individual decision making is proposed and a compromise solution function which i s the onLy one that satisfies a set of axioms is designed , The properties of the solution function is analyzed and the procedure to use the modeL is demonstrated . Key words, multi-person conflict, individual preferences, bargaining game, solution function, compromise solution.

the OMs groups, which means that no one can increase his utiLity without decrease that of the other DMs. The two stages of optimization should be proceeded simuLtaneously in order to keep the two LeveLs of Pareto optimaLity of the compromise soLution.

t. Introduction Considering the multi-person multi-criteria decision problem (MMDP) as foLlows, max [FI ( x), .. . , F' ( x)] xE X

(la) (lb)

FormerLy, the MMDP was usuaLLy studied by virtue of utility functions . Assuming that the utiLity functions of every OM and the DMs group were known, many methods were proposed to achieve a optimaL soLution by handLing the utiLity functions. In fact, the utiLity functions can not be formuLated accurateLy, so those methods in which the utiLity function is invoLed can not be used efficientLy and app ropri ate Ly.

where F' ( x)= [ fl (x), .. . , f.: (x) 1E R" is the objective function vector of the i-th decis ion maker(DMi) , i E N is the index of decision makers (DMs) . In problem (l), if not all the DMs can achieve their optimal solution simultaneously, there exists conflict among the DMs and the compromise solution shOUld be reached . Two stages of optimization mus t be proceeded t o reach the compromise solution, first, every one of the OMs optimize their own decision problems and the solution must be individual optimal to eveTJ'one, that is to say, no one can improve one of his objectives without infr ing ing the 0 ther obj ec t i ves; second, the DMs optimaze the MMDP on the base of the individual optimazation and jointLy reach a soLution. The solution must be Pareto optimal to

In this paper, the objective functions are directLy concerned in the bargaining game modeL in stead of utiLity functions, which wouLd be appLicable in reaLity . A set of axioms abuot the optimaLity and equity is proposed and a solution function which is proved satisfying the axioms uniqueLy is designed.

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Z. Optimization Metnod for Individual Decision

So, we can design the following optillization problem,

Decomposing the MMDP (1), we can get the following decision problem (DPi), iE N, max Fi (x):: ( f; (x), . . . , f.: (x) J xE X wne r e f ~ (x) is t ne j - t nob j e c t i ve

0

max A s. t. F'(X)::( l-A )F'.t A F'. xE X

( 2a) (2b)

Tneorem 1. Let Xi' denote tne optimal solution to problem (i), tnen x:E PO i(XL

(2) (1)

Proof. Suppose x:E POi(X), tnen tnere is x' E X sucn tnat Fi(x) > F;eX:L Froll proposi tion 1. tnere exits A' and Fi(x')::(l-A')F i,tA'F' •. SO C1-A)F',tAF'",wnicn mean tnat A'>A and x: is not tne optimal solution to problem (1), wnicn contradicts tne assumption of tne tneroll, so tnis tneorell nOlds. Q.E.D.

(3)

Usuall;,'. X'::<\l,wnich means that there exits conflict among the DMs and tne compromise solution snould be reacned. Let Vi::{ vE R·i I xE X. Fi(x)::v} be the objective space of DMi. Definition 1. The following PO i

Factually, tnis tneorem is very obvious, the optimal solution x: to problem (I') maxillizes F'(X) on tne line that connects F'.E V, and F' ... so the po i.n.t F'(.)(.) IIIUS t be on. the boundary c ~ V" so it must be noninferior.

(V,)

PO, (V,)::{vE V, I Vv' E R oi, if v' >v then v' E V,} is called the noninferior values and the following PO ,(X)

(4a)

Tneorem 2. For every xE pO'eX), tnere exi ts F;,E V; sucn that the optimal solution to problem (7) is x.

set to DMi

pO ieX)::{xE XI x' E Ro'. if F'(x' »F'(x) (4b) thenx' EX}

Proof. Straigntforwardly froll proposition 1.

is called the noninferior solution set of DMi.

Tneorell 2 implies tnat every noninferior solution of DMi can be reacned by adjusting the value of F;,. So DMi can control tne optimizat ion outcolle by F;. to acnieve nis pre ference outcome.

Let F'.:: ( f; "•...• f.:" 1 denote the maxillum of the objective F'(x). wnile

Let x:::opt(F;(x), X) denote tnat x: optimizes F'(x) in tne feasible domain X by DMi and F;(x:)::opt(F;(x), V,) denote tnat F;(x:) optiIlizes F;( x) in tne space Vi.

and F',:: [ f\', .. .• f.:' 1 denote minillUll value of F'(x) that DMi wishes to achieve, where f\' is tne minimUll value tnat DMi wishes to ar.nieve on objective f i(X). Obviously. F',E Vi. Proposistion 1. For every xE X, A ,0< A < 1 and FioE V, such that F'(x)::(l-A )F ',tAF '•.

(ic)

For simplicity, denote tne optimal solution to problem (0;) by x,', tnen tnere nold the following tneorells .

f OM i.

Suppose tne optimal solution set to (DPi) is X:, t ne n the op t i ma Iso l uti on to MMDP is

Cia)

(o;b)

Theroem 3. Suppose x: is the optillaL soLution to prob Lell (7) and F;( x:) is the op t ima L objective value, then x: and F;(x:) satisfy the condition of [nvariance under Positive Affine Transforllation O and hi' j::l, ...• 11 1, let G;(x)::(a\ f; (x)tb" ... , a..;t,(x)tb.;) and Vi' ::hE R"; Ixf X, v::G ; (.~~J,

there exits

This proposition is very obvious and need not to be proved. Furtherllore, for every x/E X/,

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then

has been defined in the former part of this section, c is called the conflict point and represents the outcome that no joint decision is reached.

and x.'=opt(G'( X), X) G'( x.')=opt(G'(x), V,'). Proof. Multipl1ing a, and adding bj on botlt hands of formula (7b), then the optimization problem of Gi(x) can be getted. It is equivalent to problem (7) . So the optimal solutions are the same, denoted as x,', so x, '=opt(G'(x), X), G'(x ,')=opt(G,(x), V , /) Q. E. D.

Denote the set of bargaining decision problem by L, and the set of bargaining domain by L I • Definition 3. Denote the ideal point of bargaining game problem (B , c,s) by z(B), whiLe

The condition of IPAT is very important to the axioms of bargaining game theory, so it is also required in the bargaining deci s ion problems. Not all the optimization methods and corresponding solutions satisfy this condit ion.

z(B)= (FI(X}'), Fl(x2' ) ,

• • • ,

F"(x,')) T

(Sa)

where F' ( x, ') is the optimaL objective value to problem (7), i . e., F'( x,')=max{ v, E V, Iv,=C 1- i\ )Fo+ i\ F.. 0<; i\<;1}

3.Decision model based on bargaining

(Sb)

Ohviousll , z(B)E E, otherwise there i; no conf Li et among the DMs.

3.1. Formulation of the model

Furthermore,we adopt the fOLLowing notations,

For simplicity, we consider the conflict prob lell in the obj ec t i ve space \,=V}x V2 X • •• x V, and assume that V}' V2, •• • , V, are of the same number of demens ions, i. e. , V}' V2, • • • , Vn E R·. So F( x)=v=(v}, VI' . .. , v,) TE R" ·, v.=(v }' , VI" ... ' v.' ) E V,=R', v, '=f j' ( x). Let B={VE R· ·n I there exi ts xE X such that v=F( x)} denote the joint objective space of the DMs . Let s=F,={F,I, F,2, . .. , F:} lienote the minimum value that the DMs wish to achieve on their objectives, where F, is defined in las t sec t ion. In this paper, only the case sE B is considered and the case sE B will he studied in the other paper.

B(u)={vlvE B, v>u}

S~'mmetric

Transformation n ;;Cv), For every

V=(V

.. . , Vi' . . . , V j ,

• •• ,

V,,)T,

"

n ' j(v)=Cv}, . .. , vj, . .. , v" . .. , vn)T, n ,jCB)={vE R"'I there exi ts v' E B,

v= n

, j(

v' )}

Positive Affine Transformation A(v), For v=CVj ;)E Rn ••, ACv)=(a ,jv/+b ,)E Rn••, a'j>O;a' j' b' jE RI. Let A' denote the set of alL the positive affine transformations. Definition 4. A solution function for the bargaining game problem (B,c,s)E L is any function g. L __ Rnx . such that gCE, c,S)E B. The point gCB , c, s) is the solution point of the bargaining situation represented by (B,c,s).

Definition 2. A triplet (B, c, s) is called a multi-person bargaining game with multiple criteria problem if the following conditions are satisfied. (D. B=R'" is convex and compact; (2), sE B, cE B, c<;s and there exits vE B such that V>S; (3), B is comprehensive, Le . , for vE B, ifc<;u<;v, then u<;B .

3. 2. Axiomatic solution About bargaining game modeL, many researchers, such as Nash 151, KaLai-Smorodinsky 161, Thomson 171 and Gupta l81 , have presented various axiom set:; and corresponding solution functions. Basing on these work,we propose the following axioms.

In this definition,B is called the bargaining donain, s is called the reference point and

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Axiom 1. Relevant Domain (RD). For every (B, c, s)E L, g(B, c, s)=g(B [c], c, s).

Before proving this theorem, lemma

Axiom 2. Weak Pareto Optimality (WPO), For every (B, c, s)E L, g(B, c, s)E WPO(B)={vE BluE R" ·, u>v-uE B}

Lemma 1. For every (B,c,s)E L, if z(B[cj),c, sE f:::., then there is (B', c, s)E L such that B' =B; n:;(B')=B(i,j=1.2, ... ,n), g*(B,c,s)=g*(B',

we give out two

r..• s).

Axiom 3. Invariance under Positive Affine Transformation (IPAT), For every (B, c,s)E Land AE A' , g(A
Proof. Because c,s,z(B[c])E f:::., g*(B,c,s)E 6.. So n ,;(g*(B, c, s»=g*(B, c, s)( i, j=1. 2, .. . , n), and g*( n ;;(B), c, s)=g*(B, c, s)E n ij(B)n 6. (i, j=1.2, ... ,nL Let B'=n ;.;n,;CB)nB,obviously n ,;(B' )=B and B' is convex and g*(B, c, s)E B' . So g*(B', c, s)=g*( n ;. ;n ;;(B), c, s)=g*(B, c, s). Q.E.D.

Axiom 4. SYlllletry(SYM),For every (B,c,s)E L, if V\=V 1=... =\,", sE f:::., cE 6., and n ,;(B)=B
Lemma 2.For every (B,c,s)E L,if z(B[c ) ,c,s are not on the same line, then there is (B', c' , s) E L such that z( B' )=z( B), B' =B, WPO( B' )= WPO(B) and s is on the Line that connects c' and z(B').

Axiom 6.Limited Sensitivity to Changes in the Conflict Point (LSCCP),For every pair (B,c,s) (B,c',s)E L if z(B [cj)=z(ll [c']), then g(B,c, s )=g(B, c'. s) .

Proof. Let B"=BUD', where

0' ={v ER" ·1 v= i\. s +( 1- A. ) z (B [c] ),

Axiom RD guarantees the individual rationalit y, i. e. , for everyone of the OMs, the solut ion outcome is preferable to the conflict outcome. Axiom WPO, IPAT, SYM are usually used in the bargaining game model and have been discussed extensively. Axiom RMO is an extension of the Individual Monotonicity requirement used by Nash. Axiom LSCCP defines the connection belween solution outcome and conflict outcome, i . e. , in the same bargaining domain, if the conflicl si tualion changes and only causes the cnange of the conflict point, then the solution outcome does not change.

So, for arbitrary vE 0', v
The solution function we propose is denoted by g*. For every (B, c, s)E L, it is defined by g*(B, c, s)=max{vE Blv=(1-A. )s+ A.z(B [c] ),O<),
i\. ;;. 1 },

(9)

The solution is a generalization of the solution proposed by Raffa and axiomatically charaterized by KaLai-Smorodinsky and by Gupta for tne reference solution case.

c, s, z(B) are not on the same Line. By lelllla 2, there is (B', c', s)E L such that c', s, z( B') are on the same line and z( B' [c'] )= z(B' )=z(B)=z(B [c). From the proof above, we

(2).

Theorem 4. The unique function satisfying the axiom 1--axiom 6 is g*, which is defined by forllula (9) .

know

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9(B',",S)=9*
c',s)=g(B',c',s)=g*(B',c',s). By LSCCP, g(B', c, S)=9(B', c!, S)=9*(B', c', s). By RMO, 9(B, c, S)= 9(B', c, s)=!I*(B', c!, s). Furthermore, g*(B', c!, s)=g*(n, c, s), So gCn, c, s)=g*CB, c, s).

their preferabLe soLutions by adjusting the value of F•. In this stage, severaL interactions are needed to get the optimal solution; Stage 2. Formulating the bargaining decision problem (B,c,s) and solve the compromise solution g*(B, c, s) . This is a one-shot process on the base of perfect information. Of cause, the condition sE B must be satisfied, otherwise the model can not be used.

From the two cases proved, this theorem hOlds. Q. E. D. 3.3.

Some properties of the solution and the solution procedure

The solution function g* is very convergent, which wiLL be advantageous in handling the prob lell in real i ty. I t is easy to show that g* has the folLowing properties.

The compromise solution is reached basing on the preferences of the DMs, i.e., sand z(B). So every OM can control the reference point s and ideal point z(B) so that they can reach a solution that is preferable to all of them.

Property l.For every CB,c,s)E I:,g*CB,c,s);>s. This property guarantees that the soLution outcome will never be worse than the reference point s,which supplies incentive for the DMs to make joint decision.

4. ConClusion Multi-person conflict decision with multicriteria probLems exist extensiveLy in reality. More and more researchers begin to pay attentions to this kind of problems.Bargining game model proposed by Nash can anaLyze multi person conflict decision problem with one objective , but can not directly dealing with multi-criteria decision problems . Moreover, the utility functions of the DMs must be concerned in the bargaining game model, which hinders the application of bargaining game model.This study integrates bargaining game model with multi-triteria decision method and presents a bargaining decision model. This modeL is concerned with the objective functions not utility functions of the DMs, so it is more applicable . By this model, everyone of the DMs can adjust their own minimum values that he wishes to achieve on his objectives and reach a preferable compromise solution outcome .

Property 2. For eveq' (B, c, s)E I:, if there is a sequence {s.(B)} in B satisf~;ing s.(B)-s(B), then g*(B, c, s.CB))-g*(B, c, 5). Property 3. For every (B, c, s)E I:, if there is a se'luence {Bn} in I:' satisfying Bn-B, then g*(Bn, c, s)-g*(B, c, s). Properties 2 and 3 mean that if the reference point and bargaining domain converge, the convergence of the solution in application is guaranteed . Property 4 (" . For every pair (B,c,s),(B',c,s) cL , if B' =B, B',B' =B [cJ, z(B' [cJ )=z(B [cJ ), g",(B', c,s»c and gl(B',c,S)E WPO(B), then g*(B,c,s)= gl( B, c, (B ', c, s» . Property 4 states that the solution of function g* tan guarantee the invariance under multiple stages bargaining.

References, [1l. Chen, T., An approach for arbitrating a dispute about large reservoir, Decision Making Mode 1s for Managellen t and Manufacturing. R.Kulikowski and I.Stefanski eds. Ollni tch Press. Warsaw. 1990 . (2J. Krus, L. and Bronisz. P.• Decis i on suppor t in negotiation on a joint development Decision Making ModelS for Management and Manufacturing, R. Kulikowski and I. Stefanski eds. Ollni tch Press, Warsaw,

From the formulation of the model in the forlIer sections, we know that the model consists of two parts, individual decision making problem (,) and multi-person bargaining decision problem (9).So the solution procedure for the model consists of two stages, Stage 1. Everyone of the DMs optimizing their own decision making problem (7) and achieve

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