OF THE EQUILIBRIUM IN CHOICE
MARIA VIORICA S¸TEF ˘ANESCU and ANTON S¸TEF ˘ANESCU
Generalizing the concept of Nash equilibrium, the equilibrium in choice can be defined when the players options are represented as a choice profile. The main results of the paper establish sufficient conditions for the existence of equilibria in choice. As particular cases, known results on the existence of classical solutions are found. Thus, our approach can be also seen as a general method for proving the existence of different solutions for noncooperative games.
AMS 2000 Subject Classification: 91A06, 91B50.
Key words: Nash equilibrium, equilibrium in choice, fixed point.
1. INTRODUCTION
The Nash equilibrium (equilibrium point) is the most important solution concept of the noncooperative game theory. It is defined in terms of thenormal form of a game as a strategy combination with the property that no player can gain by unilaterally deviating from it. In the original definition of Nash [4], [5], the players options were expressed by utility functions defined on the product of the individual strategy spaces, and the most significant existence results refer to this formalization.
Later, the original definition was extended to cover more general situa- tions met in the noncooperative competitions. Two such extensions may be mentioned: one of them concerns the case when some exogenous factors re- strain the individual choices ([2], [7]), the second one considers more general representation of the players options [8].
Motivated by the problem of the implementation in noncooperative so- lutions of the voting operators, in [9] has been introduced a new concept of equilibrium called Nash equilibrium in choice form. Rephrased in terms of game strategies and renamed as equilibrium in choice, this concept is dis- cussed in the present paper. The formal framework for the definition of the equilibrium in choice is the game in choice form,represented as the family of the sets of individual strategies and a choice profile. Intuitively, a choice pro- file specifies the desirable outputs for each player, and since each output of the
MATH. REPORTS11(61),3 (2009), 249–258
game is associated with a game strategy, it can be represented as a collection of subsets of the set of all game strategies. In particular, when the players options are represented by utility functions or by preference relations, a choice profile may be the family of the graphs of players best reply mappings, and then the set of equilibria in choice coincides with the set of Nash equilibria. In fact, the definition of the equilibrium in choice captures the main idea of the
“best reply” from the definition of the Nash equilibrium, but the new concept is more general, responding to various representations of the players options.
The main results of the paper are three existence theorems of the equi- librium in choice proved in Section 3. In Section 4 it is pointed out the fact that the Nash equilibrium is a special case of the equilibrium in choice. Hence, some representative existence results for Nash equilibria are easily restated as corollaries of our main results.
2. EQUILIBRIUM IN CHOICE
In the following, as well as in the classical model of a game in normal form, N = {1,2, . . . , n} is the set of players and (Xi)i∈N is the family of the individual sets of strategies. The Cartesian product X of all Xi is the set of game strategies. Set X−i =Q
j6=iXj ifi ∈N, and represent the game strategy x = (x1, . . . , xn) as (x−i, xi), where x−i ∈ X−i and xi ∈ Xi. Also, write (x−i, Xi) for the set{(x−i, xi)|xi ∈Xi}.
Definition 1. A choice profile is any collection C = (Ci)i∈N of subsets of X.
Definition 2. A game in the choice form is a double family ((Xi)i∈N, (Ci)i∈N),where C= (Ci)i∈N is a choice profile.
In the present formalism, a choice profile is a primary element and, in order to sustain the definition below, we think of Ci as the set of all game strategies that determines outputs which are desirable for the player i. How- ever, we can imagine several constructive definitions, starting from classical representations of a noncooperative game, which lead to particular but more precise meanings for the present model. We exemplify below such possibilities.
Example1. Let ((Xi)i∈N,(ri)i∈N) be a game in normal form, whereri is the preference relation (reflexive, complete and transitive binary relation on X) of player i. For each i∈N, ther’s best reply mapping Bi :X−i 7→2Xi of player iis defined by
Bi(x−i) ={xi ∈Xi|(x−i, xi)ri(x−i, yi) for allyi ∈Xi}.
Setting Ci = GrBi for each i ∈ N, where GrBi = {(x−i, yi) | yi ∈ Bi(x−i), x−i ∈ X−i} is the graph of Bi, the game in choice form ((Xi)i∈N,
(Ci)i∈N) is well defined and can be explained according to the above description.
Example 2. Let ((Xi)i∈N,(ri)i∈N),where eachri is a bi-level preference relation, i.e. for each i there are two disjoint sets Ci and Di such that X = Ci∪Diandxpiyiffx∈Ci,y∈Di andxeiyiffx, y∈Ciorx, y∈Di.(Here,ei and pi stand for the symmetric, respectively, asymmetric parts of ri.) Then, ((Xi)i∈N,(Ci)i∈N) satisfies both the formal and the intuitive definition of a game in choice form.
Remark 1. In the example above we haveCi ⊆GrBi,the strict inclusion being possible.
Example 3. Let ((Xi)i∈N,(ui)i∈N) be a game in normal form, where the preferences are represented by the utility functions ui. Set Ci = {x ∈ X | ui(x)≥µi}for given µi ∈R.In particular,µi = supxi∈Xiinfx−i∈X−iui(x).
Definition 3. The game strategyx∗ is an equilibrium in choice (denoted EC) if
(1) ∀i∈N, (x∗−i, Xi)∩Ci6=∅ ⇒x∗∈Ci.
Denoting by Ci(x−i) the ith section through x−i of the set Ci, i.e., Ci(x−i) ={yi ∈Xi |(x−i, yi)∈Ci},relation (1) becomes
(2) x∗i ∈Ci(x∗−i) for everyi∈N for which Ci(x∗−i)6=∅ Let us define the correspondencesγi:X−i7→2Xi by
γi(x−i) =
( Ci(x−i) ifCi(x−i)6=∅ Xi ifCi(x−i) =∅, and define γ on X by γ(x) =Q
i∈Nγi(x−i) Then, the result below is obvious.
Proposition1. x∗ is an equilibrium in choice if and only if it is a fixed point of γ.
Obviously, a stronger form of (1) is
(3) x∗∈\
i∈NCi
and we will refer tox∗ in this case as to astrong equilibrium in choice (SEC).
3. MAIN EXISTENCE RESULTS
In this section we provide existence results for equilibria in choice. In an increasing order of generality we will consider the underlying spaces of strate- gies as Euclidean, locally convex and general topological vector spaces, but
the other necessary assumptions make that none of our three results overlaps another.
Since, as in the case of the Nash equilibrium the proofs of existence of the equilibrium in choice lean on fixed point theory for correspondences, let us first recall some basic definitions and results in this field.
Let beXandY two topological spaces andϕ:X7→2Y a correspondence (multifunction, point-to-set mapping).
Definition 4. ϕ is said to be upper semi-continuous at the point x∈X if for each open set O ⊆Y such that ϕ(x) ⊆ O there exists a neighborhood Vx of x for which ϕ(x0)⊆O for everyx0 ∈Vx.
Definition 5. ϕis said to be lower semi-continuous at the pointx∈X if for each open set O ⊆Y such that ϕ(x)∩O6=∅ there exists a neighborhood Vx of x for which ϕ(x0)∩O6=∅for everyx0 ∈Vx.
Proposition2. ϕ is upper semi-continuous on X if and only if the set {x∈X |ϕ(x)⊆O} is open in X for every open subset O of Y.
Proposition3. ϕ is lower semi-continuous on X if and only if the set {x∈X |ϕ(x)⊆C} is closed in X for every closed subset C of Y.
Proposition4.If Y is a compact in a Hausdorff topological space, then ϕ is upper semi-continuous on X and compact valued if and only if it has closed graph.
Definition 6. ϕ is said to have open lower sections if the set ϕ−1(y) = {x∈X|y ∈ϕ(x)} is open inX, for everyy∈Y.
To prove our main results, we will invoke the well-known fixed point theorems of Kakutani [3] and Browder [1], and a relatively recent result of Wu [11].
TheoremA ([3]). Let X be a nonvoid convex compact in the Euclidean space Rn. If the correspondence ϕ:X 7→2X is nonempty and convex valued and has closed graph, then there exists x∗ ∈ϕ(x∗).
Theorem B ([1], Theorem 1). Let X be a nonvoid convex compact in a Hausdorff topological vector space. If the correspondence ϕ : X 7→ 2X is nonempty and convex valued and has open lower sections, then there exists x∗∈ϕ(x∗).
Theorem C ([11], Theorem 1). Let I be an index set. For each i ∈ I let Xi be a nonempty convex compact subset of a locally convex space, and ϕi, ψi :X=Q
i∈IXi 7→2Xi two correspondences such that (i)for each x∈X, ψi(x)6=∅ and cl(coψi(x))⊆ϕi(x);
(ii)ψi is lower semi-continuous.
Then there exists a point x∗ ∈X such that x∗i ∈ϕi(x∗) for each i∈I.
Now, we establish an existence results for games over finite-dimensional spaces of strategies.
Theorem1. Let ((Xi)i∈N,(Ci)i∈N) be a game in choice form. Assume for each i∈N that
a)Xi is a nonvoid convex compact in the Euclidean space Rmi; b) Ci is nonempty and closed in X;
c)the ith sections of Ci are convex.
Then, the game admits equilibria in choice.
Proof. If m = Pn
i=1mi and d(·,·) denotes the distance in Rm, let us define the correspondences ϕi :X−i 7→2Xi,i= 1, . . . , n, by
ϕi(x−i) = n
xi∈Xi |d(x, Ci) = min
yi∈Xid((x−i, yi), Ci) o
.
Obviously, ϕi has nonempty values. We can easily verify that it has closed graph. Define now the correspondences ψi:X−i7→2Xi by
ψi(x−i) = coϕi(x−i).
(Here, coA stands for the convex hull of the set A.) We show that each ψi has closed graph, too. For that, let (xk−i)k ⊆X−i be a convergent sequence with limit x0−i and suppose that zik∈ψi(xk−i) for each kand zik −→
k→∞zi0.This means that, for eachk,
(4) zki =
mi+1
X
j=1
λkjyijk,
for some λkj ≥ 0, j = 1, . . . , mi + 1 with Pmi+1
j=1 λkj = 1 and for some ykij ∈ ϕi(xk−i), j = 1, . . . , mi+ 1.Since all sequences (λkj)k, j= 1, . . . , mi+ 1,(ykij)k, j = 1, . . . , mi+1, lay in some compacts ([0,1],respectively,Xi), a subsequence (kt) of subscripts can be chosen such that
λkjt −→
kt→∞λ0j, j = 1, . . . , mi+ 1; ykijt −→
kt→∞y0ij, j = 1, . . . , mi+ 1.
Obviously, λ0j ≥0, j = 1, . . . , mi+ 1,Pmi+1
j=1 λ0j = 1, and, since ϕi is closed, yij0 ∈ϕi(x0−i) for allj= 1, . . . , mi+1.It follows from (4) thatzi0=Pmi+1
j=1 λ0jyij0, so that zi0 ∈ψi(x0−i).Since the fact thatψi is nonempty and convex valued is trivial, the correspondence ψ:X7→2X defined by
ψ(x) =
n
Y
i=1
ψi(x−i)
satisfies all assumptions of Kakutani’s fixed point theorem (Theorem A).
To end the proof, we will prove that every fixed point of ψ is an equili- brium in choice. Let x∗ be a game strategy such that x∗ ∈ ψ(x∗), i.e., x∗i ∈ ψi(x∗−i) for i = 1, . . . , n. Suppose that (x∗−i, Xi)∩Ci 6= ∅ for some i ∈ N.
Then d((x∗−i, yi), Ci) = 0 for every yi ∈ ϕi(x∗−i). Since Ci is closed, we have (x∗−i, yi) ∈ Ci if and only if yi ∈ ϕi(x∗−i), so that ϕi(x∗−i) = Ci(x∗−i). But x∗i ∈ ψi(x∗−i) = coϕi(x∗−i) and since ϕi(x∗−i) is the ith section trough x∗−i of Ci which is convex, we havex∗i ∈ϕi(x∗−i), hence x∗ = (x∗−i, x∗i)∈Ci.
In a more general topological framework, we obtain two existence results.
For the first one we will invoke Wu’s fixed point theorem (Theorem C).
Theorem2. Let ((Xi)i∈N,(Ci)i∈N) be a game in choice form. Assume for each i∈N that
a)Xi is a nonvoid convex compact in a metrizable locally convex space;
b) Ci is nonempty and closed in X;
c) (intCi)(x−i)6=∅ whenever Ci(x−i)6=∅ for some x−i∈X−i; d) the ith sections of intCi are convex.
Then the game admits equilibria in choice.
Proof. Define the correspondences ϕi : X−i 7→ 2Xi, ψi : X−i 7→ 2Xi, i= 1, . . . , n,by
ϕi(x−i) =
( Ci(x−i) ifCi(x−i)6=∅, Xi ifCi(x−i) =∅, and
ψi(x−i) =
( (intCi)(x−i) ifCi(x−i)6=∅, Xi ifCi(x−i) =∅.
Eachψihas nonempty convex values, and sinceCiis closed, we have clψi(x−i)
⊆ϕi(x−i).
Let us show thatψi is lower semi-continuous onX−i.It suffices for that to prove that the set
{x−i∈X−i |ψi(x−i)⊆D}
is closed inX−i wheneverDis closed inXi.The non trivial case is whenDis a proper subset of X−i. Suppose that xk−i →x0−i ask→ ∞and ψi(xk−i) ⊆D for all k. Because D ⊂ Xi, we should have Ci(xk−i) 6= ∅ for all k. Then for each k there exists yki ∈Dsuch that (xk−i, yik)∈Ci.Since the sequence (yki)k lay in the compact D,there exists a subsequence convergent to some yi0∈D.
The closedness of Ci implies that (x0−i, yi0)∈Ci,so that Ci(x0−i)6=∅ and this means that
ψi(x0−i) = (intCi)(x0−i).
To prove that ψi(x0−i) ⊆ D, assume for a contradiction that there exists some yi ∈ ψi(x0−i) such that yi ∈ Xi\D. Then (x0−i, yi) ∈ intCi while some neighborhoods Vx0
−i and Vyi of x0−i, respectively yi, can be found such that Vx0
−i×Vyi ⊆intCi.Obviously, xk−i ∈Vx0
−i for all but finitely many k.There- fore, (xk−i, yi) ∈ intCi or, equivalently, yi ∈ ψi(xk−i). Since yi ∈/ D, we have reached a contradiction.
Now, by Theorem C there exist x∗ ∈ X with x∗i ∈ ϕi(x∗−i) for each i∈N.To verify thatx∗ is an equilibrium in choice, suppose thatCi(x∗−i)6=∅ for some i. Then x∗i ∈ϕi(x∗−i) =Ci(x∗−i),i.e.,x∗∈Ci.
For the third result we will invoke Browder’s fixed point theorem (The- orem B).
Theorem3. Let ((Xi)i∈N,(Ci)i∈N) be a game in choice form. Assume for each i∈N that
a)Xiis a nonvoid convex compact in a Hausdorff topological vector space;
b) Ci is nonempty and closed in X;
c) there exists a sequence (ϕik)k of nonempty convex valued correspon- dences, ϕik :X−i 7→2Xi, k= 1,2, . . . ., with the properties
c1) ϕik has open lower sections, k= 1,2, . . .;
c2) ϕik(x−i)⊇ϕik+1(x−i) for everyx−i ∈X−i, k= 1,2, . . .;
c3)for every open setGwith G⊃Cithere exists ksuch that Grϕik ⊆ G.
Then, the game admits strong equilibria in choice.
Proof. For each k set ϕk(x) =Qn
i=1ϕik(x−i). The correspondence ϕk : X 7→2X has nonempty convex values. Since the set
ϕ−1k (y) =
n
\
i=1
{x∈X|yi ∈ϕik(x−i)}=
n
\
i=1
ϕ−1ik (yi)
is open for eachy∈X,Browder’s fixed point theorem implies the existence of a fixed point xk of ϕk. Hence xki ∈ ϕik(xk−i) or, equivalently, xk ∈Grϕik, for every i∈N and k= 1,2, . . ..
Since the sequence (xk)klays in the compactX,it contains a subsequence that converges to somex∗∈X.Relabelling the terms of this subsequence, we have xk −→
k→∞x∗ ask→ ∞ and let us show thatx∗ is aSEC.
Let us assume for a contradiction thatx∗∈/ Cifor somei∈N.SinceCiis a nonempty compact, a neighborhood Vx∗ ofx∗ and an open setGcontaining Ci can be found such that Vx∗∩G=∅.Then, by c3) and c2), there isk1 such that Grϕik ⊆G fork≥k1.On the other hand, Vx∗ contains all terms of the convergent subsequence beginning with some rank k2.Hence xk ∈/ Grϕik for k≥max{k1, k2}, thus contradicting the above assertion.
4. COROLLARIES AND DISCUSSIONS
Turning back to the choice profiles considered in Section 2, we can argue that the equilibrium in choice is a general concept of equilibrium, including as special cases usual solution concepts of noncooperative game theory.
Recall first that aNash equilibrium of a game ((Xi)i∈N,(ri)i∈N) in nor- mal form is any game strategy x∗ such that
(5) ∀i∈N, x∗ri(x∗−i, xi) for all xi ∈Xi.
If the player preferences are represented by utility functions ui, i ∈ N, then (5) becomes
∀i∈N, ui(x∗)≥ui(x∗−i, xi) for all xi ∈Xi
As is known,x∗ is a Nash equilibrium if and only if it is a fixed point of the game best reply mapping B:X 7→2X,defined asB(x) =Q
i∈NBi(x−i).
Remark 2. x∗ is a Nash equilibrium if and only if∀i∈N,x∗∈GrBi. Proposition5.Let ((Xi)i∈N,(ri)i∈N) be a game in normal form. Then x∗ is a Nash equilibrium of ((Xi)i∈N,(ri)i∈N) if and only if it is a strong equilibrium in choice of the game in the choice form ((Xi)i∈N,(Ci)i∈N), where Ci = GrBi for every i∈N.
Remark 3. The case Ci = GrBi can be also considered when Bi are allowed to have empty values. For such a situation, the definition of the Nash equilibrium can be relaxed to
(6) x∗ri(x∗−i, xi), ∀xi ∈Xi for everyi∈N for whichBi(x∗−i)6=∅.
In fact, we can consider (6) as the definition of a weaker concept of equi- librium and we will call it weak Nash equilibrium (W N E). Then an obvious version of Proposition 5 is as follows.
Proposition6. Let ((Xi)i∈N,(ri)i∈N)be a game in normal form. Then x∗ is a W N E of ((Xi)i∈N,(ri)i∈N) if and only if it is a EC of the game in the choice form ((Xi)i∈N,(Ci)i∈N), where Ci = GrBi for every i∈N.
Finally, let us consider the case in Example 3. As an alternative to the best reply principle, sometime the maximin principle can be considered. We recall that for a game ((Xi)i∈N,(ui)i∈N) amaximin strategy of playeriis any x∗i ∈Xi such that
x−iinf∈X−i
ui(x−i, x∗i)≥ inf
x−i∈X−i
ui(x−i, xi), ∀xi ∈Xi, i.e., inf
x−i∈X−i
ui(x−i, x∗i) = max
xi∈Xi inf
x−i∈X−i
ui(x−i, xi).
It is known that a combination of maximin strategy of the players may not be a Nash equilibrium. However, it is always an equilibrium in choice for a properly defined game in choice form.
Proposition7.Let ((Xi)i∈N,(ui)i∈N) be a game in normal form. If x∗ is a game strategy formed by maximin strategies then it is a SRC of the game in choice form ((Xi)i∈N,(Ci)i∈N), where Ci = {x ∈ X | ui(x) ≥ µi} and µi = max
xi∈Xi inf
x−i∈X−iui(x−i, xi), i∈N.
As a consequence of Proposition 6 and Theorem 1, we easily obtain a slight generalization of the well-known Nikaido-Isoda theorem (see [6]).
Theorem4.Let ((Xi)i∈N,(ui)i∈N) be a game in normal form. Assume for each i∈N that
a)Xi is a nonvoid convex compact in the Euclidean space Rmi; b) ui is upper semi-continuous on X;
c)ui(·, xi) is lower semi-continuous on X−i, for each xi ∈Xi; d) Bi(x−i) is convex, for each x−i∈X−i.
Then the game has Nash equilibria.
Proof. It is straightforward to verify that b) and c) imply that the best reply mappingsBi have closed graph. ForCi = GrBithe game in choice form ((Xi)i∈N,(Ci)i∈N) satisfies all requirements of Theorem 1. Hence it admits equilibria in choice, and since all Bi are nonempty valued (that means that the ith sections of each Ci are nonempty), these are Nash equilibria of the game ((Xi)i∈N,(ui)i∈N).
As a corollary of Theorem 3 we can obtain the following result from [10].
Theorem5. Let ((Xi)i∈N,(ui)i∈N) be a game in normal form. Assume for each i∈N that
a)Xiis a nonvoid convex compact of a Hausdorff topological vector space; b) ui is upper semi-continuous on X;
c)ui(·, xi) is lower semi-continuous on X−i, for each xi ∈Xi; d) ui(x−i,·) is quasi-concave on Xi, for each x−i ∈X−i. Then, the game has Nash equilibria.
Proof. IfCi = GrBi,then b) and c) imply b) of Theorem 3.
Setϕik(x−i) ={yi ∈Xi |ui(x−i, yi)>maxzi∈Xiui(x−i, zi)−1/k}.It is obvious thatϕik(x−i) is nonempty while its convexity follows from d). It is also easy to verify that the correspondences ϕik satisfy c1) and c2) of Theorem 3.
To verify c3), let us suppose for a contradiction that for somei∈N and for some open subset G of X we have Ci ⊂ X and for each k there exists xk∈Grϕik butxk∈/ G. The sequence (xk)k lays in the compactGC =X\G, therefore it contains a convergent subsequence (xkr)kr with a limit x0 ∈GC.
Since
ui(xk−ir, xkir)> max
zi∈Xiui(xk−ir, zi)−1/kr it follows from b) and c) that ui(x0−i, x0i) ≥ max
zi∈Xiui(x0−i, zi), and this means that x0 ∈Ci,and we have reached a contradiction.
Thus, all assumptions of Theorem 3 are satisfied by the game in choice form ((Xi)i∈N,(Ci)i∈N).By Proposition 6, any strong equilibrium in choice of this game is a Nash equilibrium of ((Xi)i∈N,(ui)i∈N).
Acknowledgements.This research has been supported by CNCSIS Grant #2126.
REFERENCES
[1] F.E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces.Math. Ann.177(1968), 283–301.
[2] G. Debreu,A social equilibrium existence theorem.Proc. Nat. Acad. Sci. USA38(1952), 386–393.
[3] S. Kakutani,A generalization of Brouwer’s fixed point theorem.Duke Math. J.8(1941), 457–459.
[4] J.F. Nash,Equilibrium points in n-person games.Proc. Nat. Acad. Sci. USA36(1950), 48–49.
[5] J.F. Nash,Non-cooperative games. Ann. of Math. (2)54(1951), 286–295.
[6] H. Nikaido and K. Isoda, Note on noncooperative convex games. Pacific J. Math. 5 (1955), 807–815.
[7] J.B. Rosen,Existence and uniqueness of equilibrium points for concave n-person games.
Econometrica33(1965), 520–533.
[8] W. Shafer and H. Sonnenschein, Equilibrium in abstract economies without ordered preferences.J. Math. Econom.2 (1975), 345–348.
[9] A. Stefanescu and M. Ferrara, Implementation of voting operators.J. Math. Econom.
42(2006), 315–324.
[10] K-K. Tan, J. Yu and X-Z. Yuan, Existence theorems of Nash equilibria for non- cooperative N-person games.Internat. J. Game Theory24(1995), 217–222.
[11] X. Wu, A new fixed point theorem and its applications. Proc. Amer. Math. Soc.125 (1997), 1779–1783.
Received 10 April 2009 Academy of Economic Studies Department of Mathematics stefanescu.viorica@csie.ase.ro
University of Bucharest
Faculty of Mathematics and Computer Science 050711 Bucharest 5, Romania
astefanescu@fmi.unibuc.ro
