Mathematical Social Sciences 7 (l?H4) 83-102

North-Holland

I.

DOMINANCE SOLVABILITY AND COURNOT STABILPTY

Hervc MOlJLIN CEPREMA P. 142, rue du Chevaierec, 1’,‘013 Paris, France

Communicated by K.H. Kim

Received 21 December 1983

In normal form games with single-valued best reply functions it is shown that dominanctr- solvability (resulting from successive elimination of dominated strategies) implies the global stability of the Cournot tatonnement process. When only two players are present, and the jtrateg\ spaces are one dimensional, these two notions actually coincide. A computational characterkarion of the two properties is given in a local sense as well as a sufficient condition for global dominance-solvability: an application to the Cournot-oligopoly model is proposed

KAY words: Dominance solvability; Cournot stability oligopoly model.

Two non-cooperative stories: Cournot tatonnement and swcessive elimination

of dominated strategies

The familiar postulate of non-cooperative (normal form) game theory is that only Nash equilibria (NE) emerge. These outcomes are in effect the only self-fulfilling propheties (alternatively, the only self-enforcing agreements) consistent with the normal form of the game (where each agent’s final decision is private). However, the justification of the NE concept is severely limited since it fails to propose an explicit behavioural scenario for the concerned individual player. When several NE outcomes are present some coordination among the players is unavoidable to achieve any one of them, which threatens the plausibility of the non-cooperative context. Therefore in this paper we are interested in results that (i) guarantee existence and uniqueness of the NE outcome and/or (ii) provide an explicit dynamical story for reaching a particular NE outcome.

Following Cournot’s inspiration (Cournot (1839, ch. 7)), sufficient conditions are known under which the two above requirements are simultaneously met. We assume that the game is played repeatedly, while each player adjusts his strategy by using tomorrow a best reply strategy to his opponents’ strategies today. This adjustment process is called the Cournot-tatonnement: the game is Cournot-stable (in short C- stable) if the Cournot-tatonnement always converges toward the same outcome. whatever the initial position. Although it relies on a fairly naive behaviour (each agent myopically assumes that his opponents’ strategies will not change tomorrow from today, a prediction constantly violated along the past trajectory) this concept

0165-4896/84/$3.00 2 1984, Elsevier Science Publishers B.V. (North-Holland)

a4 H. Moulin / Dominance solvability and Cournot stability

is quite appealing when the agents have virtually no information about their op- ponents’ utility, so that coordination can only be achieved via mutual observation of moves. In a C-stable game the common limit of the Cournot-tatonnement

uences is also the unique NE outcome. Sufficient conditions for C-stability have been ex#ored mainly for oligopoly games (see McManus and Quandt (1960), Theocharis (1059). Okuguchi (1976)); some general results can be found in Rosen (l%5), Luenberger (1978) and Gabay and Moulin (1980).

In this paper we explore an alternative strategical behaviour of a dynamical in- spiration which in turn yields uniqueness of the NE outcome and enhances its non- cooperative rat ionale. This is the sophisticated behaviour (Farqharson (1969)) where each player independently performs the successive elimination of dominated wategies, i.e. he deletes as irrational his own dominated strategies as well as those of his fellow players. Then he repeats his operation again and again. If this process (on the limit) shrinks all strategy sets to a singleton, then the game is said to be dominance-solvable (Moulin ( 1979)), in short D-solvable, and the resulting singleton is its dominance-solution. It is also the unique NE outcome of the game. Sufficient clrnditions for D-solvability already exist for games with finite strategy sets (see Moulin (1982)); here we give new such conditions for continuous strategy sets (‘Theorem 4 below).

The interpretations of C-stability and D-solvability are diametrically opposed: the fo;rmer is a myopic adjustment that goes on for ever; the latter relies upon an induc- tion argument that agents mutually aware of their preferences perform instan- taneously to pick a strategy in a one-shot game. However, these two concepts show deep mathematical connections to which this paper is devoted. Under the assump- tion that each player has a unique best reply to any feasible choice of his opponents, we prove first that dominance-solvability implies Cournot-stability (Theorem 1, Sec- tion 2). The sequential versions of these concepts (where adjustment or elimination of strategies are performed sequentially instead of simultaneously) display the same logical connection: henceforth (simultaneous) D-solvability implies sequential D- solvability as well as simultaneous or sequential C-stability (Corollary to Theorem 1, Section 2).

In the remaining sections v/e focus on the class of ‘nice’ games where each strategy set is one dimensional and each player’s utility ii strictly quasi-concave with respect to his own strategy. There we show that the suticessive elimination of dominated strategies relates quite simply to the best re@y mappings (Lemma 2, Section 3), implying that Cournot-stability and dominance-solv&ility coincide for two player’s games only. Next we define a local version o?” our two concepts and compare their first-order characterizations (Theorems 2 and 3, S&on 4). Finally we propose a sufficient condition for (global) dominance-,iolvability of nice games (Theorem 4, Section 5) that proves useful when applied to the quantity setting Cournot oligopoly model.

H. Moulin .* Dorninartce :ohahility and Courttot stability H5

2. Cournot stable and dominance-solvable games

We denote by G = (X,, ujr ic N) a normal form game where: (a) N is the finite set of agents, and (b) Xi is agent i’s strategy set with current element _Q and ui his payoff or utilitv .

function. Throughout the paper we assume that agent i’s strategy is a compact metric space,

his utility is continuous and his best reply defines a single-valued function denoted

Bi: for all x_; E X_ i, x, = L$(x_,) i\: the unique strategy in X, such that

Ui(Xi, X_ i) = mas Ui(yl, S-;) y, E s,

(under the notation X_, = X,, , ,ot Xj with current element s_,). Thus, Bi is a continuous mapping from X_, into X,. The (discrete) Cournot-tatonnement of G (in short CT) is the follo~~ing

dynamic8al 3 ;:ster? .

.q ’ =Bj(XL;), all HEN,

summarized AS:

x ‘+‘=B(x’), where x’,x’%X. (1)

Given an outcome x in X we denote by A’(S) the set of rectangular neighbourhoods of x, namely those neighbourhcods R of A- that can be twit ten as:

R = X Ri, where Ri is a neighbourhood of s, in X,. IE .\

Definition 1. The game G is Cournot-stable (in short C-stable) if there exists an out- come x* (necessarily unique) such that:

(i) for all x0 E X, lim,, + oD x’ =x*, where (x’) is the CT starting at x0;

(ii) for all R E A’(x*) there exists R’E 2(x*) such that for all S’E R’, the CT starting at x0 remains within R :x’ E R for all t.

In that case x* is the unique Nash equilibrium outcome of G, and is said tc be a Cournot-stable NE outcome.

Condition (i) has an obvious interpretation in terms of the myopis CourIic?i behaviour (see the introduction). Condition (ii) is a 10~ * st,lbility condition. Its

failure means that for some starting points arbitrarily close to A-*, the corresponding CT wanders outsiide of some fixed neighbourhood R of A-*. Thus, if (8 holcl~~ true whereas (ii) fails (this situation can occur only if till Xl are at least two dlmen- sional, or nz 3) the CT starting from these points can wander around for an ar- bitrary large number of periods before it is trapped within R forever. Thus. both conditions (i) and (ii) are necessary to sustain the plausibility of A-* as the (ap)Prosi- mate) outcome of a myopic Cournot-like behaviour.

8th H. Moulin / Donrinance solvabilit_y and Cow-not stabi/it_,v

We denote by .# the set of rectangular subsets of X. Given an element R of .$ and a player i, wc denote by i/‘i(R) the set of player i’s undominated strategies in the restriction of G to R:

xie s/i(R)WXiERi and for no YiERi tii(xir x_i)S Mi(_Yi, X-i), all X-i E R-i,

u~(x,, X-i)< Ui(Yi, X-i), some X-i E R-i.

BcnnMon 2. The successive elimination of dominated strategies (in short SEDS) of G is the following decreasing sequence .R’ of 9:

RO=X; R” ’ = X r/i(R’). IE .V

We say that G is dominance-solvable (in short D-solvable) if the sequence R’ converges to a singleton as t goes to infinity:

(! R’=(x*}, for some s*EX. (2) t .

In that case we say that s* is the D-solution of G. From the assumption that the best reply Bi are single valued for all i E N, it

follows that the D-solution x* of a D-solvable game G is the unique NE outl;ome of G: check first that for any NE outcome x of G (x E R’) implies (X E R’+ ’ ) hence n, R’ contains all NE outcomes. Next observe that for any outcome SE X (xe R’, B(x) E R’) implies (B(x) E R*+ I)), hence from (2) we get B(.u*) = s*.

. I’heonm 1. If G is D-solvable, then it is (Y-statmle as weI/.

Proof. For any R E .#, we prove first the fotlowing inclusion:

B,(R_,)nRic 2i(R), all iw’V (3)

P.ick X, E Ri such that for some X_i E R-i, Xi = Bi(X_i). For any yi E R;, uniqueness of the best reply implies:

Hence, Xi cannot be a dominated strategy. Let (R’) be the SEDS of G. From (3) we get:

B-(RO_-)=B.(R”.)T’rR?cR! 1 I I --I I I’

FBy induction, suppose Bil(RI-; ‘)C Rf. Then .Bi(R!i))C R’ (since Ri is decreasing with respect to t) so that

Bi(R!.,)= Bi(R!_i)rI RfC R:’ ‘a

Hence, (4) holds for all t r 0 and all i E N. Define now the mapping d from 2 into itself:

(4)

for all R E 2: .A!R)= X Ri(R_i).. iE.V (5)

Relations (4) are then rewritten as:

8’

Since d is

Observe

d(R’)C R’ + ‘, all t 10.

monotonic (RChf’=d((A)Cil(R’)) and R”=X, this implies:

d’(X)CR’, all IrO.

now that for all R E .d:

B(R)CkI(R).

(That this inclusion is not in general an equality will be commented upon: belo: - see Section 3.) Combining these two in& ision we get:

B’(X)C3’(X)CR’, a!1 t>rs.

In particular:

?, R’= {s*)=, n B’(X) = (s*). r@z.V

Our theorem now follows immediately from a topological lemma.

Lemma 1. Let X be a compact nletric space and B a cominuous nlappirg fiorr~ B into itself. The two following properties are thw equivalent:

(a) n,,,. B’(X) is a singleton (x*1, and (8) the dynamical system x’ A ’ = B&V’) satisfies (i) and (ii) _ftom Dqfinitimr 1 ,for

outcome s*.

The proof of Lemma 1 is postponed to the Appendix.

The sequential Cournot-tatonnernent and/or elimination of dorninated strategies

Definition 1 (resp. Definition 2) supposes a simultaneous adjustment (resp. elimination) process: at each step each player uses a best reply strategy (resp. eliminates all his dominated strategies). For any fixed ordering I, 2, . . . , II of rhe

agent both definitions can be given a sequential form; namely the sequential CT is the dynamical system:

x! + ’ = E&!_,), 1

whenever t - i is a multipiz of n,

=A-;, otherwise.

Whereas the sequent ial SEDS is given by:

Rf+‘= I’ 1 LJi(R’), whenevr;tr t - i is a multiple @ n,

=Rf, otherwise.

Thecxoof of Theorem 1 is then adapted step by step so as to .;how that if G is sequcutially D-solvable (for a given ordering) then it is sequentially C-stable (for the

88 H. Moulin / Dominance solvability and Cournot stability

same ordering). Next we invoke the familiar robustness result for dominance solvability (due to Rochet (1980) and Gretlein (1980); for a short statement see Lemma 3 Chapter 2 in Moulin (1982)) which implies here that whenever a game G is D-solvable it is also sequentially D-solvable for any ordering of the agents. Gathering these remarks yields

ComUmy to Tbeorem 1. If G is D-solvable, then it is sequentially D-solvable and sequentially C-stable as well for any orderirtg of the agents.

Remah 1. Notice that (simultaneous) C-stability does not imply sequential C- stability (the converse implication does not hold either). Counterexamples are easily constructed: see, for example, Exercise 6, Chapter 3 in Moulin (1982).

Emmpk 1. To illustrate Theorem 1, consider the two-person game described on Fig. 1. The strategy sets are real compact intervals. To visualize whether the proposed game actually is C-stable, we note that for a two-player game, stability of the (simultaneous) 43T sysem,

is equivalent to r:hat of the sequential Cournot-tatonnement:

6; * ‘, ~4’ r ) = (Br (xi), xi), all t odd,

(Xi+’ , Xi+ ’ ) = (xf, B&Vi)), all t even.

The latter corresponds to the familiar ‘cobweb’ process depicted on the figure. Since the proposed game is not C-stable (check that any cobweb not starting from

the NE cutcome is non-convergent) we deduce from Theorem 1 that it is not D- solvable either. This can be proved directly by exhibiting a rectangular subset such as R, satisfying

BRi(R,i)= Ri, i= 1,2; j#i.

Namely, such a rectangular subset will always survive along the SEDS: R C R’, for alll @N. (By induction, assume RC R’ and pick XE R. Then B(x) E B(R) = R; herwe, (XE Rf and B(x) E R’) implying B(x) E R”““. Therefore B(R) = R is a subset of R’+‘, completing the induct ion argumenti _J

That the converse implication of Theorem 1 does not hold but for a very restric- tive subclass of games is the subject-matter af the next section. (Example 2 in par- ticular is a C-stable, not D-solvable game.)

3. The class of nice games

The game G is said to be nice if for all i(eM (i) the strategy set Xi is a one-dimensional compact interval, and

H. Mot&n 1 Dominance solvahili~_v and (iwnot siabilitr SC)

Fig. 1. A non-C-stable, non-D-solvable game; strategy sets are Xi = [a,, 6,], i = 1,2. The two tick lines are

the graphs of each player’s best reply. The game has a unique NE outcome at the origin.

(ii) the utility function Ui is continuous over X and strictly quasi-concave with *

respect t0 Xi:

Ui(AXi + (1 - A)X;, X-i) > inf{ Ui(Xi, X-i), U;(X;, -U-j)} 9

all Xi, Xi)EXi, X_i EX_jp A E]O, I[.

In particular, Bi is a single-valued function.

Lemma 2. If G is nice its associated SEDS is entirely determined by its best rep& mappings Bi, kNz

Rj=Bi(R’_;:‘), for all ieN, all ttzN, t> 1. (6)

90 H. Moulin / Lkminance solvability and Cowrnot stability

Proof. We choose a nice game G and we fix R = XlriV Rj E 2, where each Rj is a compact subinterval of -Xi.

WC claim that for all i,~ IV, player i’s undominated strategies in the restriction of G to ,R are precisely those that are achieved as the best reply to some strategies of the other players:

pi(R) = proj~, Bj(R-i), (7)

where projA is the projection over the subinterval A. Fix ,%ny X_iE R-i. Then B,(X_i) is player i’s best reply to X_i in the unrestricted

game G. In the restriction of G to Rig by the strict quasi-concavity of Ui, player i’s best reply to x_, is y; = projR,B;(X_;) (remember that Rj is a one-dimensional inter- val). Since yi is the unique best reply to X-i, it cannot be a dominated strategy in the restriction of G to R:

II= projR, B;(R_i)C Y;(R).

To prove the converse inclusion, we pick a strategy xi E Ri\ Ii and derive that it is dominated. By the continuity of Bi and the compactness of the Rj, 1; is a com- pact subinterval of Ri, say I, = [ai, bi]; hence, we have Xi C aj or b,C...i. Assuming for instance xi <ai, we get for any fixed x_; E R-j:

Xi < a; S _Yi = projR, Bi(X-i). (8)

Since U;( l ,x_;) reaches its maximum over Ri at yj and is strictly quasi-concave, system (8) implies:

Uj(xjp x-j) C Uj(aj, X-j),

which holds for all X_iE R_;, thereby proving that xi is dominated (by a;) in the restriction of G to R. This concludes the proof of (7).

We describe now the SEDS of G with the help of the mapping d from g into itself, introduced in the proof of Theorem I (relation (5)).

We prove by induction on t L 1 that:

h?’ is a product of compact intervals and R’=d(R’- I). (9)

Property (9) for t= 1 follows immediately frcm (7). Assuming that (9) holds for 1, we apply (7) to get:

R’!’ r = proj, Bi(R!_i), I for all i 6~ N.

As we observed above, continuity of Bi implies that R:+ ’ is a compact interval. Since R: is a dbweasing sequence, we get:

Bi(.W’i)CBi(R~~‘)=R~~Rf”=iBi(Rli),

which completes the proof of (9) for t+ I, and that of Lemma 2. El

Assuming ornly that Bi was single valued, we proved in Section 2 (proof of

H. Moulin i Dominance solvability and Cortrnot J-tahi’litr,

Theorem 1) that

A'(X)C R’, all te N.

In the (much) smaller subclass of nice games we have in fact:

A’(X)=R’, all t&V. w

There all R’ are products of compact intervals, and so is n, R’. In the rest of this paper we concentrate on nice games (or locally nice games -

Section 4) and derive several corollaries of formulas (6) and (10). First observe that for n =2, the mapping A takes the form:

A(R) = B,(R2) x Bz(R,) = B(R), all R = R, x R2 E R_

Thus by (10) G is D-solvable iff n,,, B’(X) is a singleton, which by Lemma 1

amounts to saying that G is C-stable.

Corollary of Lemmas 1 and 2. L2t G be Q two-player nice game. Then G is D-

solvable if and only if it is C-stable.

Not withstanding the narrowness of its applicability (two-person games with one- dimensional strategy sets and one’s payoff strictly quasi-concave w.r.t. one’s strategy) this result has a striking interpretation: two dramatically distinct behavioural scenarios (myopic Cournot-tatonnement on one hand and sophisticated mutual anticipation of strategical choices on the other) pinpoint exactly rhe same well-behaved games.

Fig. 2 proposes a D-solvable two-person nice game where the SEDS is visualized with the help of Lemma 2.

When at least three players are in G, the mappings d and B’ no longer coin<&: only the inclusion B(R)C A(R), all R E 2, survives. This should be clear from c he definition itself:

. ..EB(R)%?~ER. Vi E N, -Yj = Bi(y-, ), dei

XEA(R)= t’id% ZYE R, -Vi = Bi(y-,)*

A nice game G being D-solvable (resp. C-stable) iff the sequence 1’(X) (rcl;p. B’(X)) shrinks to a singleton, we can design the inclusion B’(A’K -f’(S) in such a way that B’(X) does shrink to a singleton whereas A’(X) does not.

Example 2: A C-stable nice game which is not D-solvable. Let S= i 1.2.3 j ;and

Xj = [- 1, + l] all i. Let

4 = - +xf + “‘Xf - a_upJ

u2= -a_tj A-2 + $5 + ax?.t-3

u,=ax$-3 -aY$-3-$t~ 1

for some a>0 to be specified later.

H. Moulin / Dominance solvability and Cownot stabilit_v

Fig. 2. A D-solvable (C-stable) game; successive rectangles represent the SEDS.

Calmputing au, /k, = -xl +cw2-cu3 we get that

h(x2s x3) = wd[- I. + I] (ax2 - m3) I

Altogether this gives:

B(x) = p 0 i&x),

where # is the linear operator

0 a -a 8= -a 0 a

[ 1 a -a 0

H. Moulin /I DorGtance solrabiiit? and Cournot stability 93

and p is the coordinate-wise projection over [- 1, + I]. The spectral radius of 8 (i.e. the maximal modulus of its eigenvalues) is worth f3 l a, hence for CK i 313, the dynamical system x’ + ’ = Bx’ converges geometrically to zero. We leave the careful reader check that the same is true for the system A-” ’ = B-t-‘, whence our game ic C- stable. On the other hand we have:

B,([-1, +l] x I-1, +l])= I-20, +(r].

Therefore

Bj([- 1, + l] x [- 1, + 1)) = [- 1, + 1)

as soon as cr2+. In that case we have d(X) = A’ so that the SEDS of G is c~wz- tant: R’=Xall teN. We conclude that for (Y, +%z<f3:3, the game G is C-stahie

while not D-solvable.

4. Local C-stability and local D-solvability

Throughout this section and the following one we consider a game G where for all i= 1, . . . . n, player i’s strategy set Xi is a one-dimensional compact-interA. We no longer assume that G is nice, yet we still suppose that its best reply is single valued.

Definition 3. Le: G be an n-person game (X,, . . . , A&, ZQ , . . . , u,]) and iet s* be r\ NE outcome of G. We say that x* is a regular NE outcome if for all i = 1,. . . . II:

(i) xi* is an interior point of Xi, (ii) Uj is twice continuously differentiable in a neighbourhood of s*, and

ihlj -jj (.P)=O; 3 (x*)<o.

i i

At a regular NE outcome, the game G is locally nice. Thus, the assumptions made in Definition 3 are the (local) counterparts for smooth games of properties (ill and (ii) for nice games (Section 3).

Definition 4. Let x* be a NE outcome of G. We ;sy that G is locally Cownor-st Me (in short LC-stable) at x* (or equivalentlv that .Y* is a LC-stable NE outcome) if: _

where xt is the CT starting at x0. (Recall that ,k (x) is made up of the rrstanguliar neighbourhoods of x.)

Definition 4 exactly parallels Definition 1: in *particular C-stability of G Aplies

H. Monlin / &mtinance s0lvabilit.v and Cwmot stubihly

LX-stability at the corresponding NE outcome. A locally stable NE outcome is an isolated NE; however, G might possess several distinct LC-stable NE outcomes.

At a rcgrulw NE outcome x9 player i’s best; reply function is locally given by the implicit equation:

It is a single-valued continuously differentiable function with derivatives at s*:

Hence, the CT dynamical system x’+ ’ = &.a!) ic (first-order) approximated at x4 at:

xt* I -x+ = B’(x*) l (x’ -x’),

B’(X*)=[b~l~j= ~,...,n I bii=O b aBi ij = z (Xri)

i

The standard results on the local stability of discrete dynamical systems give us:

Theoma 2. Let x* be a regular NE outcome of G. Let p(A) be the spectml tadius

of any mwix A (the maximal modulus of its egenvalues). JJ p(B’(x*))< 1, then G is L&table at x*. If G is LC-stable at x*, then p(B’(x*))r 1.

Roof. Apply Ostrowski’s theorem, i.e. Theorem 10.1.13 in Ortega and Rheinboldt (1970).

&#I&Iort 5. Let x* be a NE outcome of G. We say that G is locally dominance- s&able (in short LD-solvable) at x* if there is a Cartesian product of compact inter- vals R% 2(x*) such that the restriction of G to R” is R-solvable:

n R’ is a singleton, where R’ is the SEDS starting at R”. IEN

Since x* is a NE outcome in the restriction of G to R” ;as well, and since the D- solution of a D-solvable game is also its uni.que NE outcome, it folIows that x* is the D-solution of G restricted to R” : n&itf = (x+}.

Observe that the diameter of R’ goes to zero as t goes to infinity and that the restriction of G to R’ is dominance-solvabret as well. If Ir!’ is a neighbourhood of x* for alI t (this holds true in particular if x* is a regular NE - we let the reader check this point), then an equivalent formula%ion of LD-solvability obtains: for any

R E 3(x*) there is a product of compact intervals R’ E ~‘(A-*) such that R’c PC and the restriction of G to R ’ is D-solvable.

We note next that if the restriction of G to R* is D-solvable, it is also C-stable (Theorem 1). If .Y* is a regular NE, the CT of the restriction of G to R” and nhat of G coincide on any sufficiently small neighbourhood of s*. Therefore A-* actually is LC-stable in the original game G:

(G is LD-solvable at a regular NE outcome s*} * (G is L&table at s*) .

We now give a computational characterization of LD-solvability. We first raise

intuition of the result by approximating in a neighbourhood of A-* the mapping B by its derivative at A-*, i.e. B’(P). The key equation for the SEDS. namely (6). is

t htts approximated as:

Rf” = (B’(x*));(R; x 9.. x R;,). (;12)

Suppose that R: is a symmetrical neighbourhood of ..$:

R~=[$-r:,xF+r,?], all i= l,...,n,

then (12) amounts to the induction equation:

Therefore r’ = (r{, . . . , ri) varies with t as:

r ‘+ ’ = :B’i (_y*)#, where :f?‘: (A-*)= [ b, IL/= ,, _. .n.

This suggests that the spectral radius of I B’; (_P) should be related to LD-solubilit> of G at A?. Specifically we have:

Theorem 3. Let x* be a regdar NE outcome of G and let 1 B’ (P) be de rr~rix wirh entry ibiit i, j= I,..., n.

If p(! B' f (P)) c 1, then G is LDsolvable at s*. If G is LD-solvable at x*, then p(t B’ j (PC)) I 1.

Proof. The Perron-Frobenius theorem for non-negative matrices implies the fdl~~\-- ing:

Lemma 3. Let A = [ati] be an n x n non-negative math-:

atiM, all i,_k 1, . . . . n.

For any real number a, such that p(A)<cr, there esists a strictly positive v=tor K

ri>O, all i= l,..., n,

such that

c CIijQICYri, dl i= 1, . . . . Ia. J= I,....n

96 H. Riot&n / Dominance solvaMit_v and Counrsr stability

For any real srsmber ,!?, such that 0 I /R&4), there exists a strictly positive vector F such that:

C piiqZ/3~, all i= l,..., n. (14) j= I.....n

The proof of this lemma is omitted for the sake of brevity: it is easily deduced from theorems in Gantmacher (1966) vol. 2. Now to the proof of Theorem 3.

Suppose first that fill?‘1 (x*)) c 1. Then we pick CL: ,Iz( 1 B'l (x*)) c a < 1, and by Lemma 3 a strictly positive r = (r,, . . . , rJ such that (13) holds with iE$ in place of

%* Without loss of generality we take x* to be the origin of the n-dimensional

euclidian space containing our strategy space X. We work with the following norm on 2”:

ixil r = sup - 9

Is&n ri

of which the balls centred at x+= 0 take the form:

We also denote, for all i= 1, . . . ,n:

R-&t) = X Rj(a), J=I,....n

J*i

I I for all X-i E R-i(a): [X-i!,= ,:ylrn $- . 11

j#i J

Next we choose a’,a<a’< 1 and some ~0 such that:

for u>O: R(a) = Z? Ri(a), where Ri(a) = [-ria, +ria). i=l

E +a%$, all ieN. ri

Since B is differentiable at x * = 0 (x* is regular] we can pxk some q >O such that:

for all i=l,...,n

for all X-i E R-i(q) 1 IBi(X_i)-Bi(Xri)--B~(Xfi).(X_i-X:fi)((e* IlX-i-X!ill,.

Taking into account that x = 0, and B(X) = Y this yields:

I Bi(X-i)- C buXj s&* bx_illrm j. j+i

Hence, by (13) and (15):

(16)

H. Moulin / llominancv solvaldity and Cournot stabilit\ 9’

These inequalities can be formulated as:

for all i= l,...,n

for all q’, OC~‘l~ 1 B;(R-i(q’))C Ri(CY’q’). (17)

We prove now that the restriction of G to R(q) actually is dominance-solvable. From (17) we get Bi(RJq))C Ri(q)- Hence, by (7) (proof of Lemma 2):

r/;(R(rl))=~rOjR,(rl)Bi(R-i(rl))=Bi(R-i(rl)).

Thus, the SE’DS of G restricted to R(q) starts with:

R’ = R(q); R’ =A(R*)c R(a’q).

We prove by induction that the rth term R’ of that SEDS is such that:

R’=A(R’-‘)cR(a”q). (IS)

Suppose (18) holds for t and compute:

.;+I= prOjR:Bi(R’i).

We have

Bi(R!_i)C Bi(Rf_y ’ ) = Rf

(since R’ is, by construction, a decreasing sequence). Therefor,)

[R;+‘= Bi(R’,) all i], i.e. I?‘+ ’ =A(R’).

Moreover by (17):

Bi(R~i)CBi(R_i(a”~))CRi(~““~),

which concludes the proof of (18). Since a’< 1, it implies the desired dominance- solvability.

It remains to prove the converse statement: Supose p(’ B” (P)) > 1, then G is not LD-solvable. Pick j& 1 < @$(I B’I (x*)) and choose (by Lemma 3) a strictly positive vector r such that (14) holds with 1 b, 1 instead of ati. Next fis E >O such that:

/?>l+:, all i=l,..., n. ri

Associated with E is an q >0 such that (16) holds for all i = 1, *. . , II 2nd al! X-i E R-i(q).

Choosing 2, = 5 l sign(b,) l q, all j# i, and taking (14) into account, \w get:

H. Moulin / Dominance solvabiilily and Coumot stability

Henczforah:

BilR-ifrt)) DRitV) t all i= 1, . . ..n. all q small enough.

(19)

Consider now the restriction of G to some R E 2(x*), the Cartesian product of compact intervals around xi’. Next take q small enough so that (19) holds and R(q)C R. We claim that in the SEDS of G restricted to R, the fth term R’ contains R(q) for all 1, thus precluding dominance-solvability. Namely, we have:

Rf = Pi(R)= proj, Bi(R_i)

BiiR- i) 2 BifR-ittlll DRilVl *Rj >ProjRiRi(~)= R,(V),

therefore proving the claim for f = 1. Next, if for some integer t, the inclusion R:D R,(q) holds true, then:

Rt+b = projR: Bi(R!.i)

Bi(R’,) 3) Bi(R-i(q))DRi(q) I * Rf ” >projR; R is) =Ri(q)g

’

which concludes the proof of the claim, and that of Theorem 3. iI?

From the inequality

(another consequence of the Perron-Frobenius theorem) combined with Theorems 2 and 3, the comparison of the LC-stability and LD-solvability is quite complete. In particular, if the entries bti of B’(P) all have the same sign, then p(F) =p( 1 B’ 1). Therefore as long as p(B’) # 1, LC-stability and LD-solvability are equivalent. See Example 3 below.

Remark 2. Theorem 3 points out a mistake in the original paper on Cournot stability and dominance-solvability by Gabay and Moulin (1980). Their Theorem 7.1 asserts wrongly that local D-solvability is essentially equivalent to local S-stability.

5. A suffiiient condition for global D-solv8!bility

We consider a nice and twrce differentiabk game G. More precisely, we assume:

a2Ui 7-7 (x)<O, all l EX, all i= 1, . . . . n. .

i (20)

Thus, agent i’s utility ui i:, strictly concave with respect to Xi. We generalize the global C-stability resullt under diagonally-dominance assump-

tioru (Gabay and Moulin (1980, Theorem 4.1)) by deriving under the same premises the global D-solvability of G.

H. Moulin / Dorninanc*e solvability and Cournot stabiiitr w

Theorem 4. Let G be a nice game satisfying (20). Suppose, moreover, thrrr rw BWW:

Then G is D-solvable (hence C-stable).

Proof. From Lemma 2 (Section 2) we know that the SEDS of G is given by I?[‘= S, R’+‘=A(R’), all WV. We set:

k = sup c j bu(x) 1, a%4,/a.4-t a.~,

where b,(x) = - p_ (At I- i.....W j,J*r a$a_t-f

By assumption (21) we have k< 1. We claim next that for all i = 1, . . . , N, agent i’s

best reply mapping is k-lipschitzian on X_j with respect to the supremuhn norm:

1Bj~~~_j)-Bj(y_j)jIkjl_~_,-y_,j;, ail S,,_!-,EX,. (23

Namely, B,(x_j) either is a boundary point of Xi or is given by the implicit

equation:

2 (Bj(X_ j), X-j) = 0. - 1

Therefore the function (p(A) = Bi(kY_j + (1 - A&_,) is right-hand differentiahlc on [O, 11, its derivative being either zero or

C bG(Z-i(A)) 9 (-Yj -Yj), where z_j(E,)=jLS_; + (1 -i)J-:. J._ifl

In any case we have:

Thus, property (22) follows from Rolle’s theorem. Given a rectangular subset R of A+, denote S(R) its diameter for the stipro:mum

norm: 6(R)=sup{ ~,u--c\l~,/.u,~~ R}.

Property (22) is reformulated as:

&(Bj(R_i)) I k&R-j) 5 k&R)*

Since this holds for all i we get:

&NR)) s k&R).

In view of the construction of the SEDS of G this yields:

6(Rt+‘)rk6(Rt), all t,

and the conclusion that G is dominance-solvable. 1 C

tab H. Moulin / Dominance solvabiiit_v and Cournot stability

We apply now our computational res’ults (Theorems 2, 3 and 4) to a standard example.

$kan@e 3t The quantity-setting Cournot oligopoly model. Player i produces a quantity xi, OSXiSai, of a given commodity at a cost Ci(Xi). Let p be the inverse demand function relating the market price to the total quantity R= Ci=,,....n_~i sup- plied by the n players. We face the game:

and we assume:

p’<O: p”s0,

d>O; c” >O (decreasing returns to scale).

These conditions imply that G is nice, satisfies (20) and possesses a NE outcome A-*. By taking the a, large enough we can make .Y* an interior point of X.

Next we compute the entry bii of the matrix B’(x*):

Since b,, ilr! negative, B’(.u*) and 1 B’ 1 (.x*) have the same spectral radius so that LC- stability and LPsolvability are together characterized by

(23)

If in particular we take j3, c I +‘(n - I), all i, the above condition is always satisfied. Therefore the following system of inequalities implies LC-stability and I-D-solvability:

(LI-~)\P’~+(I~-~)x,% ip”I<cy, all i= 1,...,n (24)

(all derivatives being taken at $). When the cost functions are ident ical for all players, there is a symmetrical NE

outcome s+(x~ - -x7 all i, j) at which fii does not depend on i, so that condition (23) amounts to j?, c 1 l(n - 1).

Turning now to Theorem 4, we compute inequality (21) It comes out as:

(n- 3). p’(X)/ + (n - 2)Si l f p”(X) <c,y(_y). (25)

Clearly, as n grows, condition (25) (as wehl .as (24)) gets more difficult to satisfy: D-solvability (as well as C-stability) is more problematic among more players.

Aelrnowkdgements

This work was supported partly by National Science Foundation Grant

SES80-06654 at the Institute for Mathematical Studies in the Social Sciewm. ,ud partly by the contract ‘Planification Decentralisk‘ from the Commissarislr Gtirkra~ du Plan, at the Laboratoire d’EconomCtrie de 1’Ecole Polytechnique. 1 Gsh to acknowledge very stimulating discussions with Daniel Gabay and Jean-Charles

Rochet. The latter in particular suggested a proof of Lemma 2.

Appendix: Proof of Lemma 1

satisfies (i) and (ii) from DsJinition 1 (whew $(A-) stmrds ,fi,r tk w c?.s

neighbourhoods of s) if and on& if the sequewe B’(X) corwergt-s to c: sin:,a/twra.-

,?\ B’(X) = (P), for some s* E X. (Z611 . .

Proof. Suppose first that (26) holds. It follows that the diameter d’ of B ‘$ Y I:

d’ = sup{d(_\; _v)/x, yE B’(X)), where d is the distance of A:,

goes to zero as f goes to infinity, which implies that for any .a- in S, tht: W+W~O‘C x’=B’(x) converges to A? as t goes to infinity. We haye also B(s*b -=A * w~ct

B(.Y*) E n.._, B’(X) = {A?]. To prove property (ii) we choose G mctiuluo of wn- . _. a tinuity p for 8, namely a real-valued function defined on ]@ + =[ such ah&ll\r:

for all 00 O<&)l&

d(x, y)sp(&)*d(B(s~ B(_r))se, for all A-. JQE A’.

Next we pick any neighbourhood R of A -*. For some e >O, the ball Gth center .C * and radius 8 is contained in R. Nest, for T large enough d ’ is bounded atm c k! e. Then we set tl =pr@) se, ami we claim that the ball R *, with cmt:t;“r x* awl

radius q, meets our requirements. Indeed, we have:

d(x*. s) s tl = p(p ‘-‘(&))=4(x*, B(s))cyT- l(c)

Moreover dT S& implies BT(X)c R (because Br(5) contains A-*). Hen;c

R3 B’(X)> Br+ ‘(X)3 l -- I B’(X). all1 t 2 7:

From (27) and (28) we conclude that the s~uence B’(x) stays u it bin R. \\ hiA

was to be proved. Conversely, we assume that the system s’ * 1 = l&y’) saiisfies (i) and (rib and \‘: c

have to prove (26). We set

a compact subset of X, and observe that B(x”) =A? inclusion B(X”)CX” is ckar. Conversely. fis any s in X7 For all t there is an s, in B’(X) such that et = &(.v,) (invoke x E #I’ + l(X)). Let _t” be any limit point of the sequence x8. Siwz B)‘(X) is clo& and decreases when t increases. _v txlongs to B’(X). for all I. i.e. ,VE X*. By continuity of B we have also AT= &,I), and this concludes the proof of scX”,=X”.

We must prove finally that X O” is a singleton. Suppose not: there is an s E X** xtx*. Sin&v the restriction of B to X” is onto X” we construct inductively a wquence . ..‘.u... within X” by:

@.v = .\7 ’ _ ‘NE B-‘(x,,nx,.

Choose next a limit wint _v of that sequence. Since B is continuous and @‘_+z’ ’ x, it follows that f4i.v) is a limit point of the sequence ‘.x= as well. Simi- larly. B’(p), . . . , B’(p), . . . are alk limit points of that sequence. Recall that limit points of a given sequence form a closed set; moreover, by assumption (i). lirc, B’(y) =x*. Therefore .Y* is a limit point of ‘x. Choose now a neighbourhood R cpf .r* nof containing x. and by property (ii) a subneighbourhood R' of A-*, For

me f@* % is in R’(P is a limit point). By (ii) B'(%) stays within R for all I. WoWVet:

&rOv) = 4I - Ix; B’( ‘dx) = ‘0 - txP _. _ , Bt@(fQ_Q = _y,

which implies XE R, a contradiction.

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