Szidarovszky
					, 
						Ferenc

English

Università del Salento: ESE - Salento University Publishing

- 18 -3. The classical oligopoll-BameIn thia section we will discuss a special economic gamewith seta of strategies- (~ > 0. k=1.2 •••••n)~ = Lo. ~J (19 )and pay-off functionan'fk (xl····.~ ) = ~ f LXi - ~(~). (20)i=lwhel'e the functions f and ~ muat have the properties:nS lf) = [O. LJ. whel' eclassical oli~oll-game.Before discUBsing the equilibrium problem of this game weShbW how the game appears in seme applicationa.!2Plication l. Assume that n factories manufacture thesame product and they sell i t on the same market. let f be theunit price of the product being a function of the total productionlevel. and let Rk be the cost function of the manufacturer k.Then ~ is the production bound for manufacturer k andf k (xl''' •• X n ) is i ts netto income assumìng that xi is theproduction level of the manufacturer i for i=1.2 •••••n.AE,Plication 2. Assume that a multipur'pose water supplysystem has to be designed. let the water users denoted by k(k=1.2 •••••n) and let the water quantity given to user k be- 19 -be the investment cost being a functionI~tk=1.2 •••••n.nof L ~. 1etk=lforcapacity bounde of the usere are denotedby xk • If thethen obvious1ybydenotedVk(~) and wk(~) be the production costo income and the economieloss of the water shortage /penalty e.t.c./ of user' k.respective1y. ~t us assume. that the total investment cost i~devided by the users in the rate of the water quantity used bythe water users. Thus the tota1 incorne of user k can be determinedby the functionn~ I L Xi- nL Xi i=li=l(21)By introduceing the notationsn nf L l I LXix. - -~ ni=l LXi i=1i=lfunction <21~ has immediate1y form (20).~lication J, ~t us now assume that n factories areon the bank of a r'iver and they send a certain quanti ty ofwaste-water to the river. It is also assumed that the totalpenalty paid by the factories i8 a function of the total waste--water quanti ty sent to the river and i t is devided among thefactories proportional1y to the waste-water quanti tysent to theriver' by the different factories. I.et ~ be the total waste-- 20 --water quantity produced by factory k, let xk be the waste-waterquantity sent to the river' by factory k. Then the total "income"of factory k can be given by the formula( 22)where Pis the penalty function, ek is the cleaning cost offactory k. letnp L Xi 'i=ll= - nL Xii=lthen the function (22) immidiately has the form of (20).Firat we shaw that the equilibrium problem of the classicaloligopoly game ia equivalent to a fixed point problem of a onedimension point-to-set mapping. It will be much more convenientthan the application of the fixed point problem of lemma 1, sincethe latter ia an n-dimensionaI problem.let'+'k(s, ~, t k ) = t k r(s - ~ + t k ) - Y"k(tk )•for k=l,2, •• ,n J 8 ~ [O, q, ~ ~ CO, ~] and t k ~ [o, ~] ,where t'k = min {~, .b - 8 + X k\. Since f k :; 0, tne intervalfor t k can not be empty. Far k=l,2, •• ,n; 8 ~ [o, 1J; Xk ~ [O, ~1let- 21 -and for k=1.2 •• •• n ; s ~ [O. 1] letLemma h A vector !,H = e H ii).Xl' •••• Xh ~s an equilibrium pointof the classical oligopoly game if andn(k=1.2 ••••• n). where SH = 2: x: .k=lonlv if-Proof. The definition of the equilibrium point impliesthat a strategy vector !l:.Jf. = (x~.....~) is an equilibriulI' pointif and only iffor k=1.2 •••• ,n and t k E [o, ~J. /It is easy to observe thatnfor sI( ~~ i'k = ~/ Inequali ty (23) is equivalent= ,i=lto the fact that x!l e: Tk (sJl. ~) , that is xH i ~ (sft). •k kLet us finally introduce the following one dimensionalpoint-to-set mapping:nXls) = {u lu = LXi' Xi ~Xils)} (s € [O, 1J). (24)i=lLemma 1. and definition (24) imply the following important result.Theorem 3. A vector equilibriumpoint of thenli L òlI:s - ~-i=lclassical oliaopoly game if and only if for,• •pOJ..n"G- 22 -Remark. The solution of the ~ame has two steps::3tep l: the solution of the one dimensionaI i'ixeciproblem sft E X(SH);Step 2: the cietermination o~' sets Xl( (SH) and the computationof the vectors xH - (x~, .•• ,x~ ) such that--ilH, :s: (s*) (k= l, 2, ••• ,n) and il: L x HX. S - •K 1{k=lIn the followin6 parts oi tllis section we will assume thatthe conditions given below are satisfied.l. There exists a constant ~ >° such thatal f (s) -°for 8 ::. ~ .- - ,-bi f l8 continuous, concave and strictly decreasingln the interval [o, n·2. For k=l, 2, ••• , n function Y1: is continuous, convexand strictly increasing in the interval [O, ~J._o'Iheorem ~ Under the above conditions the game has atleast one equilibrium point.Proof. The proof consists of several steps.-al First we proveequilibrium point, thenthat if x*nLxt:~tk=l•lS an• Let us suppose thatand L ~ + X k >( • This impliesi'knL H >çxk •Ic=l°< X\c <HXk-Then there are positive x*k and such that- 23 -(X~, ••. ,~, ..• ,X~ )=( ili li ii)xl' •• , t Xk , ••• ,~ , which is a contraàiction to inequalitybi I.etx = {~ I~ = (xl'··· ,Xn ),nL X k .:::--" l.. lo.=le=1,2, ••• ,n } •Next we prDve that any equilibrium point ~1i of the generalizeugame r = (n; Xl' ••• '):n' X; fl' ... ,f n ) gives an equilibriumpoint for the classical oligopoly game. Let xk ~ [O, ~].If ( ii j( ii li)r!'xl' •• "xk _ l , x k ' xlc+l'.'.'~ 'II!: ,C, then the equilibriumproperty for same r •<:;~ves'f k (~, ••• ,X~ , ••• ,~ ) ~ \f' k (x~, ..• ,xk ' ....X~ ) ,and if (x~, ••• ,xk "'. ,X~ ) f.:::, then= O.f L X~ - 11((0) = f k (x~, ... ,O' •• "~)~ fk(x~, ... ,x~"",~}ilks inc e (x~, ••• , 0, ••• ,~ ) ~ x.cl Next we prove tl~t if function h is continuous, concaveand strictly decreasing in a nonnegative interval CA, B], tnenthe function xh(x) is concave in the same interval.Let U8 first assume that h is twice continuouslydifferentiab le.Then- 24 -{Xh(Xl}' " xh'(x) T n(x),{xhlx)) " " 2h'(x) + xh"( X) < \.,which implies the assertion.If h :lS continuous, tl,en let \n ([;j"1,2, •• ) ue twicecontinuously àifferenciable, concave, strictly decreasingfunctiom such that lirJ , ,DU; " Il.Let A ~ x <: y !f. Lj 0(, (>J ~ G; -x _,o f" " l, then l'or :J],,1,2 •••Jy the limi t relation m _00> we obtain(<XX + i?y)b ( <Xx T ,r>, y):- o',xb(xì + hyh(y),thus xh(x) is concave.di The parts al ana bi imply that the classica~ oligopolygame and the generaIized game r = (n; ;;l""'Xn ' X; -f I ,··., f r,)have the same equilibrium points. Under the assumptions of tnetheorem X is a convex, closed,continuous and part cl impliesbounded subset of Rn , 'f kthat \fk is concave in,:lSThus the conditions of the Nikaido-Isoda theorem are satisfied,consequently the game has at Ieast one equilibrium point. ..no\;Remark. 'Ine uniqueness of the equilibrium point iSfassureGin general as the followin[ example sbows.Example ~ Let n=2; ~"12 = 1,2 ;- 25 -1,75 - 0,5s, if O~ s ~ 1,5f (s) - 2,5 - s, ii' 1,5f;s&2,5°, i.f s > 2,5(x;;- O).•,de will prove that ano arbi trary point of the setg~ves an equilibrium point of the game.Lét ? E [0,5 ; lJ oe fixed, ana let\f!(x) :: xf(1,5 - x + x*) - I\(x)It is easy to verify that( k::l, 2. ).and'+" (x* - O):: x~(-0.5) - l - 0,5:: 0,5(1 - x~)= O,*' (xil "' O) :: Xii ( -l) + l - 0,5 - O C"- , .J ~- x .c O.-Part cl implies that function Y is concave ln v"" ,consequently from the inequalities q-" (x~ - O) ~ O and11'" (x~ + O) we conclude that !li maximumpoint ofcan x ~s athe function 4J • Thus arbitrary x!li € XIi lS an equilibrium point.Next we discuss a numerical algorithm for findig theequilibrium points of the classical oligopoly game. Under theassumptinns of Theorem 4. the following statements are true.Lemma 8.al For s e- [O, L] , ::\c(sì is not empty and is a closeiinterval ["\(s), 1\(s)J ' (k:::i.,2, ••• ,n) ;- 26 -b/ for ° ~ s < s'!:. L the inequality I\;(s') ;: ~(s)holds for k=1,2, ••• ,n ;c/ if f is differentiable at the point s, then~ls) = 1\ (S) •,d/ if f is differentiable in the inte:rval [o, I,J , then-~(s) is a continuous function of s.Proof. Parts a/ and b/ can be proven by simple modifica-tions of parts e/al and G/bl of the proof of Theorem l. in•paper (lOJ. The statements cl and d/ are proven :in the C/a,b,cpart of the proof of Theorem l. in paper [lOJ.Lemma 9. If !i:,H = (x~, •• " ~) and !J8f. = (x~,. •• ,~- )are equilibrium pointsof the classical oligopoly game havingthe properties given in Theorem 4., then•n nL~=L~H3t·k=l k=l n nS H _ \ 1 x..'" < siiii = '" x..1fili •Preoi. Assume that L- K L Kk=l k=lThennsH = L x: :;k=ln n~ L I\; (sJi!!) ~ L x~ =k=l k=lwhich is a contradiction. •Corollary. The point-to-set mapping X(s) has exactly onefixed point, which can be computed by the usual bisection method(see F; Szidarovszky, S.Yakowitz [12J).- 27 -Theorem 2' Asaume that the conditions of Theoreo 4. aresatisfied. Let s* be the 1J~jque fixed point of the mappincX(s). Then alI equilibrium points of the c1assica1 oligopo1:>'bame can be obtained by the solution of the system of 1inearequations and inequa1ities:,\(sft) ~ ~ : l\:(s*)n'\' x = SliL_ kk=l(k=l,2, • •• ,n)preof. The statefuent is a consequence of Lemma u. arilllemma 9. •Corollary. If in addition to the conditions of Theorem 4.function f is differentiab1e on the interval [o, LJ, then tllliequilibrium point is unique.Remark l. It is interesting to observe that the game lSnot linear but the set of equi1ibJ:"ium points is a simplex.Remark 2. The uniqueness of the equi1ibrium point dependson the differentiabi1ity of a function and not on strictconcavity as i t is usual in the theory of nonlinear programming •.§'pecial cases.l. In case of t and ~ (l ~ k ~ n) being twice differentiab1ethe uniqueness was proved by O. Opi tz [7] without giving anyalgoJ:"i thm for finding i t.2. Under the assumptions of O. Opi tz, F. Szidarovszky [JJproved the existence and uniqueness of the equi1ibrium point- 28 -and also gave an iterative algoritbm for computing it.3. If the cost functions l~ are idenoical anù thed t O Op . t h E B Lrl] provedcon i iona of • ~'Z are satisfied, t en • urgerthe existence and uniqueness of the equilibrium point and alsogave an algoritbm to compute it. ',7e remark that the algorithmof Szidarovszky is a generalization of Burger's method.4. If the functions f and Kk (k=1,2, ••• ,n) are linear,then the existence and uniqueness was proved by li. Matias,[4] •who gave an algoritbm which is independent of th~ method ofSzidarovszky. ~e remark that using the result of Theorem 5.the equilibrium point in this special case can be given inclosed form (see pp. 37-39 of [lOJ).iJ The group eguilibrium problemIn this paragraph we will discuss the general~zed versionof the classical oligopoly game r having the srategy setsand pay-off functions••• (l f k ~ n) (25)•~k~ x"L K~i=l,n ~-tf L LX{j.(. =1 j=l(26)where for k=1,2, ••• ,n'!k = (Xkl"",xkik) E~ • This gamecan oeeur when the players of the classical oligopoly gamefom disjoint groups and they tend to the optimal income ofthe group. If the number of members in group k is equal to

The classical oligopoly game

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The classical oligopoly game

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