UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN BOOKSTACKS Digitized by the Internet Archive in 2011 with funding from University of Illinois Urbana-Champaign http://www.archive.org/details/onsymmetriccourn1327khan FACULTY WORKING PAPER NO. 1327 FE? ; - On Symmetric Cournot-Nash Equilibrium Distributions in a Finite- Action, Atomless Game M. Alt Khan Ye Neng Sun College of Commerce and Business Administration Bureau of Economic and Business Research University of Illinois, Urbana-Champaign BEBR FACULTY WORKING PAPER NO. 132 7 College of Commerce and Business Administration University of Illinois at Urbana-Champaign February 1987 On Symmetric Cournot-Nash Equilibrium Distributions in a Finite-Action, Atomless Game M. Ali Khan, Professor Department of Economics Ye Neng Sun Department of Mathematics On Symmetric Cournot-Nash Equilibrium Distributions in a Finite-Action, Atoraless Garnet by M. Ali Khan* and Ye Neng Sun** January 1987 Abstract . We show that in a finite-action, atomless game, every Cournot-Nash equilibrium distribution can be "symmetrized." This yields an elementary proof of a result of Mas-Colell. tThis research was supported by a N.S.F. grant to the University of Illinois. *Department of Economics, University of Illinois, 1206 South Sixth Street, Champaign, IL 61820. **Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801. 1. Introduction In [A] , Mas-Colell showed the existence of a Cournot-Nash equi- librium distribution (CNED) as a consequence of the Fan-Glicksberg theorem. Mas-Colell also showed the existence of a symmetric CNED in finite-action, atomless games as a consequence of the Kakutani fixed point theorem and results in the theory of integration of correspon- dences. These results consist, in particular, of Lyapunov's theorem on the range of a vector measure, Aumann's measurable selection theorem, as well as his theorem on the upper heraicontinuity of the integral of a correspondence with upper-heraicontinuous values; on all of this [1] is a standard reference. In this note, we show that in a finite-action, atomless game every CNED can be "symmetrized" to yield a symmetric CNED. This allows us to deduce Mas-Colell' s result on the existence of a symmetric CNED from his first result on the existence of a CNED. The proof of our result is elementary in the sense that it uses only Lyapunov's theorem on the convexity of the range of a scalar measure. Section 2 recalls the model and presents the results. Section 3 gives the basic idea of the proof and Section 4 is devoted to the formalities of the proof. Section 5 concludes with a remark. 2. The Model and Results We recall for the reader's convenience the basic definitions from [4]. Let A be a compact, metric space of actions , /tL the set of Borel probability measures on A endowed with the weak * topology and £/ is the space of continuous from AxHt into R and endowed with the supremum-norm topology. A game is a Borel probability measure on vC A* -2- A Borel probability measure t is said to be a Cournot-Nash equilibrium distribution (CNED) of the game y if the marginal of t on Ci . , T , is y and x(B ) = 1 where B = {(a,u) e Ax^f : u(a,T.) ^> u(a,x ) for all a e A} and t. denotes the marginal of t on A. t is said to be a A symmetric Cournot-Nash equilibrium distribution if t is a CNED and there exists a measurable function h: £o. ■*■ A such that t (graph A h) = 1. We shall say that every CNED t can be symmetrized if there s exists a symmetric CNED t such that B = B . ST T We can now state Theorem . Every Cournot-Nash equilibrium distribution of a game y with action set A can be symmetrized if y is atomless and A is finite . This yields as a corollary Corollary (Mas-Colell) : A symmetric Cournot-Nash equilibrium distri- bution exists for a game y with action set A whenever y is atomless and A is finite . The Corollary is an easy consequence of our theorem and Theorem 1 of [4]. 3. Heuristics of the Proof We illustrate the basic idea of the proof of our theorem by con- sidering an action set with two elements. The reader may wish to keep Figure 1 in mind as we go through the argument. Let t be the CNED of a game y with action set {a ,a }. Let the set B of all pay-offs and corresponding pay-off maximizing actions be > Z 7 i «- V 2^ /^- u. -> ? — 1" v i2- u in\ A\V 4 #, — _> Figure 1 -3- denoted by the set (a.xU.) U( a o x U ? ). Unlike Figure 1, U. and U need not necessarily be connected sets. Suppose, again unlike Figure 1, that U. C\ U = $. Since U \J U = £/ , t can be shown to be symmetric CNED simply by letting h(u) = a for all u e U. , for all i = 1, 2. Certainly h is measurable and t (graph h) = 1. Thus, in the case U, C\ U- = 4>, there is nothing to prove. Suppose U. HU» * $• The basic idea in this case is to "dis- * jointify" U, and U~ , i.e., to construct measurable subsets U. (Z. U. * * for all i = 1, 2, such that U H U = $• Since y is atomless, this can be done in a number of ways but the important consideration is to do this in such a way that the marginal of x on A, x , does not change. Pi. Since B depends only on x., this ensures that B does not change. We T A T now briefly spell out the mechanics of such a procedure. Let V = U - U. , i = 1, 2, j * i, and V = U HU . Find mea- 12 12 12 surable subsets V^, V of V^ such that V^HV^ = , V U V = V 12 and u(V^) = T(a *V ), 1 - 1, 2. Since Z x(a.xV ) - 2 i=l t(_U (a i xV l2 )} = T({ a 1 » a 2 > xV 1 2 ) = t a (V 12 ) = "tS^ll*' L y a P unov ' s i = l theorem on the range of an atomless scalar measure guarantees that 12 * i V and V can be found. Now let U = V. (J V , i = 1, 2. These are the sets that work by letting h: LC.. + A be a function such that * s h(u) = a for all u e U. , for all i = 1, 2. Now let t (B) = y{u e \Ji .'• (h(u),u) e B} for any measurable subset B of Ax (JL . . t is the symmetric CNED. The only point which needs to be checked is that x^ » x. But T.({a. }) = x(a .x£t.) - x(a xU.) = x(a xV ) + x(a.xV 10 ) = AA Ai iA ii ii il2 u(V.) + u(V^) = M (V i VJvJ 2 ) = M (U*) = y{u e Ct^ (h(u),u) e (a.x^)} = x^((a i} ). -4- 4. Proof of the Theorem We begin with an elementary lemma. Lemma 1 . Let A ( i = 1 , . . . , k) and B be arbitrary sets. Then ~~ k k U (A xB) = (( A )xB). i=l i=l Proof : Straightforward. II Our next lemma is a simple consequence of Lyapunov's theorem on the range of a scalar measure. L emma 2 . Let (S,/^ , u) be an atomless measure space. If V e A , n u(V) = E X with X >_ for all i , there exist for all i = 1, ... t n, V 1 £^ such that V 1 A V J - (i * j ) , \J V = V and y(V ) = X. . i=l X Proof: We shall prove the lemma by induction. The lemma is trivially true for n = 1. Assume it to be true for n = k and let V £ /Q with k+1 y(V) = E X., X. _> for all i = 1, ..., k+1. If X. = for any i, i=l X X X we are reduced to the case of n = k and the proof is completed by letting V. = <(> for that i. Thus, suppose X. > for all i. Let k 1 k+1 X(l) = E X./ E X. and X(2) = 1 - X(l). By Lyapunov's theorem 1=1 X i=1 X k+1 / k+1 [1, p. 45], we can find V e A such that y(V ) = X, . . Since k+1 (/ k+1 k (V-V ) e ^ , and jj(V-V ) = E X. , we use the induction hypothesis i=l 1 to complete the proof. II Before we present the proof of Theorem, we develop some notation. Let I denote the set {1, 2, ..., n} and P(l) the set of subsets of I, including the empty set. For any tt z P(I), let tt denote the comple- ment of tt in I. Let P (I) = {it e P(I): m e tt}. We shall use the -5- convention that a union over the empty set is the empty set. We also use the same notation for a point and a set consisting solely of that point. Proof of Theorem Let t be the Cournot-Nash equilibrium distribution of the game \i. Let U. = proj /,. (B n(a.xCO) for all iel 1 fX\ T i a en U », ■ ^ iel rtainly U. C ££. f or Iel. On the other hand, let u e <-c A • Cer- J i A A iel Ce tainly there exists k e I such that u(a, ,t) _> u(a x). Then (a, ,u) e B and hence u e U, . k t k (2) B t = \J (a.xU.) iel Certainly (a xUj CL B for all iel. Now any element x of B can be lit t written as (a.,u) for some iel and some u e £c« Hence u e U. and 1 A 1 x e ( a . xU . ) . l i (3) tU.xU.) = i(a.x# ) 11 l A Since (a x(J. ) (TT,acP(I),Tr^a), (c) ^ V = U, n/TN ir A tt a n i /TN iri TreP(I) ttcP (I) For (a), pick u e CC.. Let or - {I e I: u e U. }. By (1), a # . Then u e V . On the other hand, u e v ,J V implies that there irePd) exists a e P(I), a # such that u e V . Hence u e U for all i e a a i and hence, by (1), u e "£v.« For (h) , suppose there exists tt, o in P(I) such that tt t a and V f\ V # 4. Since V and V are nonempty, tt and tt a tt a a are nonempty. Then there exists i e tt, i k o. Now u e V f|V tt a implies u e U.. Since i e a , u e V which is a contradiction. For (c), pick u e * — i — ' V . Then there exists tt e P (I) such that ireP 1 (I) * u e V . Since i e tt, u e U.. On the other hand, for any u e U. , let TT 1 1 O - {j el. ueU.} and tt = {i } U a. Certainly u e V and tt e P (I). J TT (5) For any tt e P(I), -(measurable V 1 (iel), V 1 f\ V^ = d>(i*J), I J V 1 = V ~~^ TT tt TT , > *' TT tt 1 ETT and yCV 1 ) = x(a.xV ) TT 1 TT Observe that y(V ) - ?., (V ) - t(AxV ) = t( ( LJ a.)xV ) which, by TT ££ TT TT . T 1 TT A 1 el Lemma 1, equals t(( I) a . xV ) ) = E x(a.xV ). We can now apply . t i tt . T 1 TT iel lei Lemma 2 to complete the proof of (5). Now let U. = W-> V 1 . 1 D l/ T s TT tteP (I) (6) For all i e I, (a) U* Gl^, (b) U* C\ U* - cf> (i*j), (c) [J U* = li^ i el * i To see (a), pick u e U. . Then there exists tt e P (I) such that u e V . This implies u e V . Since i e tt, u e U.. (b) follows from TT Tt 1 the fact that for i * j , V H V - 4 on the one hand, and from TT TT -7- V f\ V = <}> for it * a on the other. For (c) , note that tt o uvuu V U U O v 1 = U U v 1 - U v = iel iel ireP (I) iel TreP(I) v tteP(I) iel w ireP(I) " Cc , the last step from (4a). (7) p(U.) = x(a xU ) for all iel. The left hand side equals p( I — ,— > (V 1 )). Since V 1 d V by (5), and 1, _ N TT TT TT •rreP (I) V f\ V ■ (Ji for tt * o by (Ac), this equals Z y(V ). By (5), this TT eP ( I ) equals E. x(a. xV ) which equals t( ' — ; — ' (a.xV )). By Lemma 1, this can be written as x(a.x » — : — ' V ) and hence by (4b) as x(a.xij.). i D i,.v tt 3 l i ireP (I) We are now ready to construct our symmetric Cournot-Nash equilib- rium distribution. Let h: vC, + A be such that h(u) = a J for all A i * i * u e U, , for all iel. Since V are measurable, U. are measurable, i tt i s Moreover, from (vi), h is a well-defined function. Now let x be a measure on Ax {£. such that for any measurable B, x (B) = y {u e Cc .: s (h(u),u) e B}. Given measurability of h and the identity map, x is well-defined. Also x S (graph h) = p{u e (L k : (h(u),u) e (graph h) } = u {u e ££ A > = 1. g All that remains to be shown is that x is a Cournot-Nash equilibrium s distribution. Towards this end, we first show that x = y. Pick ^A any measurable subset W of {A k • Then x, (W) = x (AxW) = A CL u {u e £t k : (h(u),u) e AxW} = y {u e (fi^W)} = u(W). s Next, we show x = x . Pick any measurable subset of A. If this set is empty, there is nothing to be shown. Hence, let this set be -8- (J a. for some * e P(I). Now t®( (J a ) = t S ( (J a x^ ) = u {u e^: (h(u),u) e (( (J a £ ) x tlj } = y {u ef^: h(u) = a lf ieir i e tt} = n((J h _1 (a.)) = I u(U*). Now E (U*) = £ T(a i xU 1 ) (by (7)) ieir ieir - Z T(a iX ^ A ) (by (2)) i eir t((J (a^^)) ieir = t(( U a i )x ^ A ) (by Lemma 1} ieir - T A (U . ± ). ieir We are done. s Since t. = T. and since B depends only on t. , B = B . Thus A A t Ast T to show t (B ) = 1. But by the definition of h, graph h OB . Since s T T s s s t (graph h) = 1, t (B ) = t (B ) = 1. The proof of the theorem is o T T complete. II 5. Concluding Remark In [2, 3], the authors present an alternative formulation of Mas-Colell's result in games where pay-offs are represented by preference relations or by functions which are upper-semicontinuous in actions. We remark that the theorem proved here applies to that generalized set-up. -9- Ref erences [1] Hildenbrand, W. , Core and Equilibria of a Large Economy , Princeton University Press, Princeton, New Jersey, 1974. [2] Khan, M. Ali, On a Variant of a Theorem of Schmeidler, B.E.B.R. Faculty Working Paper No. 1306. Revised in December 1986.. [3] Khan, M. Ali and Y. Sun, On a Reformulation of Cournot-Nash Equilibria, University of Illinois , mimeo , January 1987. [4] Mas-Colell, A., On a Variant >ef~a» Theorem of Schmeidler, Journal of Mathematical Economics, 13 (1984), 201-206. D/92 HECKMAN BINDERY INC. DEC 95 t 01 £ N. MANCHESTER, |Bound-To.Vl«.s^ , ND1AN A 46962