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Continuity properties of the Nash equilibrium correspondence in a discontinuous setting
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Continuity properties of the Nash equilibrium correspondence in a discontinuous setting Vincenzo Scalzo Department of Economics and Statistics (DISES), University of Naples Federico II, via Cinthia 21, 80126 Napoli, Italy
a r t i c l e
i n f o
Article history: Received 14 June 2018 Available online xxxx Submitted by J.A. Filar Keywords: Discontinuous non-cooperative games Continuity of correspondences Existence of Nash equilibria Essential stability of Nash equilibria
a b s t r a c t We introduce a new complete metric space of discontinuous normal form games and prove that the Nash equilibrium correspondence is upper semicontinuous with non-empty and compact values. So, using the Theorem of Fort (1949), we obtain that the correspondence is also lower semicontinuous in a dense subset. We introduce new topological assumptions on the payoff functions and a strengthening of standard quasi-concavity properties. Examples show that our results cannot be obtained from the previous ones. © 2019 Elsevier Inc. All rights reserved.
1. Introduction The existence of Nash equilibria and the continuity properties of the Nash equilibrium correspondence in normal form games, where the usual continuity assumptions on the payoffs are weakened (henceforth discontinuous games), are issues that have captured the attention of many authors. For the existence of Nash equilibria, let us refer to the seminal papers [11], [5] and [17], and the more recent ones [3,9,15,4,21,10, 14,16,19]. The upper semicontinuity of the Nash equilibrium correspondence in the setting of discontinuous games have been investigated in [27] and [20], while the continuity of the correspondence (that means upper and lower semicontinuity) has been obtained in several classes of discontinuous games: see [26,7,21,8]. More precisely, we focus on the upper (and lower) semicontinuity when the Nash equilibrium correspondence is defined on metric spaces of discontinuous games where the existence of Nash equilibria is guaranteed, and the metric is given by the sup-norm. The search for more and more general classes of discontinuous games where Nash equilibria exist has led to a condition introduced in [18] and called Single Deviation Property. Even if such a property refers to a reasonable behavior of players on the strategy profiles which are not Nash equilibria, a counterexample shows that it is not sufficient to guarantee the existence of equilibria in games satisfying standard quasi-concavity E-mail address: [email protected]. https://doi.org/10.1016/j.jmaa.2019.01.021 0022-247X/© 2019 Elsevier Inc. All rights reserved.
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conditions (see [18, Section 3.1]). However, the Single Deviation Property plays a role in identifying the class of games where the Nash equilibria can be characterized by means of some kind of weak Nash equilibria: see [23]. Moreover, nothing is known about the continuity properties of the Nash equilibrium correspondence in classes of games where Single Deviation Property-like assumptions are given. The aim of the present paper is to provide new results on both the existence of Nash equilibria and the continuity (upper and lower semicontinuity) of the Nash equilibrium correspondence in games satisfying Single Deviation Property-like assumptions.1 We identify a new class of discontinuous games where the Nash equilibrium correspondence is continuous and non-empty valued. More precisely, we introduce a generalization of the Single Deviation Property, called Generalized Single Deviation Property, and define an independent quasi-concavity condition, called C-condition, which guarantee the existence of Nash equilibria when the sets of strategies are compact and convex subsets of locally convex Hausdorff topological vector spaces. For every non-equilibrium z, the Generalized Single Deviation Property allows to find a wellbehaved map ξ defined in some open neighborhood of z so that, given any z in the open neighborhood and x ∈ ξ(z ), at least one player profitable deviates using her/his strategy in the profile x . Note that the player who deviates depends on both z and x ; while, in the Single Deviation Property, the deviating player is not depending on x because ξ is constant-valued.2 Furthermore, we obtain the upper semicontinuity of the Nash equilibrium correspondence by using a strengthening of the Generalized Single Deviation Property, called Positive Generalized Single Deviation Property. Finally, we have that the Nash equilibrium correspondence is continuous and non-empty and compact valued in a dense subset of the class of games satisfying the Positive Generalized Single Deviation Property and the C-condition. Let us highlight that our interest in new results on the existence of Nash equilibria and continuity of the Nash equilibrium correspondence is focused on those topological and quasi-concavity conditions which are independent of each other. In fact, our C-condition holds in games where no topological assumptions are required. The paper is organized as follows. The setting and the definitions are given in Section 2, where we also present a new Nash equilibrium existence result. Finally, Section 3 deals with the continuity properties of the Nash equilibrium correspondence. 2. Setting, definitions and equilibrium existence Assume that X and Y are topological spaces and let T : Y ⇒ X be a correspondence (a map) from Y to X. We recall that T is said to be: upper semicontinuous if, for any y ∈ Y and any open set U which includes T (y), there exists an open neighborhood Oy of y such that T (y ) ⊂ U for all y ∈ Oy ; closed if the graph of T (that is: {(y, x) ∈ Y × X : x ∈ T (y)}) is a closed set in Y × X; lower semicontinuous if, for any y ∈ Y and any open set U so that T (y) ∩ U = ∅, there exists an open neighborhood Oy of y such that T (y ) ∩ U = ∅ for all y ∈ Oy ; continuous if it is both upper and lower semicontinuous. Let N be a finite set of players (N denotes also the number of players), and, for any i ∈ N , assume that Xi is a non-empty, convex and compact subset of a locally convex Hausdorff topological vector space. Set X = i∈N Xi , that is the set of strategy profiles (any element x of X is also denoted by (xi , x−i ) where x−i ∈ j=i Xj ), ui : X −→ R is the payoff function of player i. The list of data G = Xi , ui i∈N is a normal form game (game in short) and a strategy profile x∗ is a Nash equilibrium of G if ui (x∗ ) ≥ ui (xi , x∗−i ) for all xi ∈ Xi and all i ∈ N . EG denotes the set of Nash equilibria of G. A game G is called quasi-concave if 1 The upper semicontinuity of the Nash equilibrium correspondence refers to the notion of Hadamard well-posedness introduced in Optimization Theory: see [12], while the lower semicontinuity is nothing but the essential stability of equilibria introduced in [25]. Hadamard well-posedness and essential stability are stability properties of equilibria with respect to perturbations on the characteristics of players. 2 The idea to consider a locally defined well-behaved map for every non-equilibrium strategy profile has been often used in Discontinuous Games: see, among the others, [4,21,19].
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ui (·, x−i ) is a quasi-concave function for any x−i ∈ X−i and any i ∈ N . The following definition is given in [18]: Definition 1. G satisfies the Single Deviation Property (SDP in short) if, whenever z ∈ X is not a Nash equilibrium of G, there exists an open neighborhood Oz of z and a strategy profile x such that, for each z ∈ Oz , at least one player i gets ui (xi , z−i ) > ui (z ). As it is shown by a counterexample in [18] (see also [19]), the SDP does not guarantee the existence of Nash equilibria in quasi-concave games: Example 1. [18, Section 3.1] Let G be the 3-player game where the set of strategies is [0, 1], x, y and z are the choices of players 1, 2 and 3, respectively, and the outcomes are given as below:
x ∈ [0, 1/2] x ∈ ]1/2, 1]
y ∈ [0, 1/3] (u(x), v(y), u(z)) (u(x), v(y), u(z))
y ∈ ]1/3, 2/3[ (v(x), v(y), u(z)) (v(x), v(y), v(z))
y ∈ [2/3, 1] (v(x), v(y), v(z)) (v(x), v(y), v(z))
if z ∈ [0, 1/2]
x ∈ [0, 1/2] x ∈ ]1/2, 1]
y ∈ [0, 1/3] (u(x), u(y), u(z)) (u(x), u(y), u(z))
y ∈ ]1/3, 2/3[ (u(x), u(y), u(z)) (u(x), u(y), v(z))
y ∈ [2/3, 1] (v(x), u(y), v(z)) (v(x), u(y), v(z))
if z ∈ ]1/2, 1]
where u(t) = 0 if t ∈ ]0, 1] and u(0) = 1, while v(t) = 0 if t ∈ [0, 1[ and v(1) = 1. G is a quasi-concave game which satisfies the SDP. It is easy to see that EG = ∅. So, if one aims to obtain a Nash equilibrium existence result for games satisfying the SDP, one has to strengthen the assumption of quasi-concavity, and it is interesting to look for a new condition which have no connections with both the topological structure of the game and the SDP. So, we introduce the following condition: Definition 2. G is said to satisfy the C-condition if, for every finite set F of strategy profiles and for every z ∈ scoF , there exists x ∈ F such that ui (xi , z−i ) ≤ ui (z) for all i ∈ N .3 It is clear that any game which satisfies the C-condition is quasi-concave, but the converse does not hold (in fact, we will have that the C-condition allows the existence of Nash equilibria in games that have the SDP). Basically, the C-condition requires some uniformity among players in having the same strategy profile x where, with respect to z, no profitable unilateral deviations are possible. Nevertheless, the C-condition does not imply concavity or continuity on the payoff functions, as Example 2 shows. First, we give a characterization of the C-condition in terms of set-valued mappings. Given a game G = Xi , ui i∈N , define Pi : X ⇒ Xi by Pi (x) = {zi ∈ Xi : ui (zi , x−i ) > ui (x)} for any x ∈ X and any i ∈ N . We have the following proposition. Proposition 1. Given a game G and the corresponding mappings Pi , let P : X ⇒ X be defined as below: P (x) = {z ∈ X : there exists i ∈ N such that zi ∈ Pi (x)} ∀ x ∈ X .
(1)
Then, G satisfies the C-condition if and only if x ∈ / coP (x) for all x ∈ X. 3 Given a set A, we denote by coA the convex hull of A and by scoA the subset of strict convex combinations of all elements of A.
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Proof. Assume that G satisfies the C-condition and x ∈ coP (x) for some x ∈ X. So, there exists a finite subset F ⊂ P (x) so that x ∈ scoF . The C-condition guarantees that, for at least one x ¯ ∈ F, x ¯i ∈ / Pi (x) for all i ∈ N , that is: x ¯∈ / P (x), and we get a contradiction. On the other hand, suppose that z ∈ scoF , where F is a finite set of strategy profiles. Since z ∈ / coP (z), we have x ¯∈ / P (z) for at least one x ¯ ∈ F , that is x ¯i ∈ / Pi (z) for all i ∈ N , which proves that the C-condition is satisfied. 2 Example 2. Let G = Xi , ui i=1,2 be such that X1 = X2 = [0, 1] and ⎧ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎨ u1 (x) = 1 ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎩1 u2 (x) =
if x1 > x2 > 0 or x1 = x2 > 0 if 0 < x1 < x2 or x1 = x2 = 0 if x1 = 0 and x2 ∈ ]0, 1] or x2 = 0 and x1 ∈ ]0, 1] if x1 ∈ [0, 1] and x2 = 1 if x1 = 1 and x2 ∈ [0, 1] , u1 (x) if x ∈ {z : z1 = z2 } ∪ ({0} × [0, 1]) ∪ ([0, 1] × {0})
1 − u1 (x) otherwise .
It is easy to see that u1 and u2 are neither concave nor continuous. On the other hand, the C-condition holds true. Indeed, one can use Proposition 1, since we have: P1 (x) = ∅ and P2 (x) = [x1 , 1] if x1 > x2 > 0 P1 (x) = [x2 , 1] and P2 (x) = ∅ if 0 < x1 < x2 P1 (x) = P2 (x) = {1} if x1 = x2 = 0 P1 (x) = P2 (x) = ∅ otherwise . Note that EG = {x : x1 = x2 > 0} ∪ ({0}×]0, 1]) ∪ (]0, 1] × {0}). In order to obtain the existence of Nash equilibria in games where the SDP is weakened, the C-condition is the right strengthening of quasi-concavity. Below, we introduce a generalization of the SDP. Definition 3. G is said to satisfy the Generalized Single Deviation Property (GSDP in short) if, whenever z ∈ X is not a Nash equilibrium of G, there exists an open neighborhood Oz of z and a well-behaved4 map ξz : Oz ⇒ X such that, for every z ∈ Oz \EG and every x ∈ ξz (z ), there is a player i for whom ui (xi , z−i ) > ui (z ). The following example shows that the GSDP is a generalization of the SDP even in games which satisfy the C-condition. Example 3. Consider the game G = Xi , ui i=1,2 where X1 = X2 = [0, 1], u1 (x1 , x2 ) = 0 whenever x1 = x2 and x2 > 0, u1 (x) = 1 otherwise and u2 (x) = 1 for all x ∈ X. So, EG = {x : x1 = x2 } ∪ [0, 1] × {0}. If z ∈ / EG , let Oz be an open neighborhood of z such that z1 = z2 for each z ∈ Oz and ξz : Oz ⇒ X be defined by ξz (z ) = {z2 } × [0, 1]. The map ξz is upper semicontinuous with non-empty, convex and compact values, and we have u1 (x1 , z2 ) > u1 (z ) for every x ∈ ξz (z ) and every z ∈ Oz . So, G satisfies the GSDP. 4
We call well-behaved any upper semicontinuous map with non-empty, convex and compact values.
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Moreover, the SDP fails on G because, for any open neighborhood Oz and any strategy profile x , there are elements z ∈ Oz with x1 = z2 , which implies that u1 (x1 , z2 ) ≤ u1 (z ). Finally, one can easily prove that G satisfies the C-condition. Theorem 1. If G satisfies the GSDP and the C-condition, then EG is non-empty. Proof. By contradiction, assume that EG = ∅. So, for every x ∈ X, let Ox and ξx be, respectively, the open neighborhood and the map associated to x by the GSDP. Since X is compact, there exists a partition of the unity {β1 , ..., βk } subordinated to a finite covering {Ox1 , ..., Oxk } of X such that, for h = 1, ..., k, βh is k continuous on X, βh (x) > 0 implies x ∈ Oxh and h=1 βh (x) = 1 for all x ∈ X. Define the map ξ from X to itself as follows5 : ξ(x) =
βh (x)ξxh (x) ∀ x ∈ X .
(2)
βh (x)>0
Obviously, ξ is upper semicontinuous with non-empty, convex and compact values. So, Kakutani–Fan– Glisberg fixed point theorem guarantees the existence of a strategy profile x∗ such that x∗ ∈ ξ(x∗ ), that is:
x∗ =
βh (x∗ )z h
(3)
βh (x∗ )>0
where z h ∈ ξxh (x∗ ). Since x∗ ∈ Oxh whenever βh (x∗ ) > 0, from the GSDP we have that, for any h ∈ {1, ..., k} such that βh (x∗ ) > 0, there exists ih ∈ N so that uih (zihh , x∗−ih ) > uih (x∗ ). On the other hand, since ∗ ∗ h ∗ βh (x∗ )>0 βh (x ) = 1, from (3) one gets x ∈ sco{z : βh (x ) > 0}. So, the C-condition implies that, for at least one z ho with βho (x∗ ) > 0, we have ui (ziho , x∗−i ) ≤ ui (x∗ ) for all i ∈ N , and we get a contradiction. 2
Remark 1. The topological and quasi-concavity assumptions considered in Theorem 1, that are the GSDP and C-condition, are independent of each other, in the sense that the C-condition supposes neither the GSDP nor any other topological property. For instance, consider the very simple situation of 1-player game G where the payoff function u is defined on [0, 1] by u(x) = x if x ∈ [0, 1[ and u(1) = 0. Obviously, G satisfies the C-condition because it is quasi-concave and 1-player game; moreover, EG = ∅. On the other hand, if u is such that u(0) = 1, u(x) = x if x ∈ ]0, 1[ and u(1) = 0, the GSDP holds but the C-condition fails. Remark 2. The game G defined in Example 2 satisfies the GSDP (we have already seen that this game verifies the C-condition). More precisely, G has the SDP: in fact, it is sufficient to set x = (1, 1) for every z ∈ / EG . Note that for z = (0, 0) ∈ / EG and z in an open neighborhood of z, if z1 > z2 , the player who deviates is 2, while only player 1 deviates when z1 < z2 . Remark 3. Consider the game in Example 2. Assume that O is an open neighborhood of z = (0, 0) and ξ : O ⇒ X is a well-behaved map. Let z ∈ O. As observed in the Remark above, if z1 > z2 , the only player that could have profitable deviation strategies from the elements of ξ(z ) is player 2, while only player 1 could have good deviations if z1 < z2 . So, no one of the two players has profitable deviation strategies for all z ∈ O\EG , but the player who deviates depends on z . This circumstance is not covered by the Nash equilibria existence results given in the recent literature: for instance, see [3,15,4,16]. Hence, Theorem 1 cannot be deduced from the previous results. 5
Given Ai ⊂ X and a real number αi , for any i ∈ N ,
i∈N
αi Ai is the set of all
i∈N
αi yi with yi ∈ Ai .
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Furthermore, the Nash equilibrium existence result [16, Theorem 6] is given for games which satisfy the SDP and a weak quasi-concavity like property. In Theorem 1, we weaken the SDP assumption and strengthen the quasi-concavity property; however, Example 3 shows that the two theorems are independent of each other. Finally, the assumptions of [16, Theorem 4] fail to be verified on Example 2. In fact, following the bottom at page 525 in [16], with respect to the non-equilibrium z = (0, 0) and for any non-equilibrium z in every open neighborhood Oz of z, set ξz a well-behaved map defined on Oz , for each non-equilibrium z ∈ Oz , we have that z ∈ {t ∈ X : ui (ti , z−i ) − ui (z ) ≥ ui (yi , z−i ) − ui (z )} for some player i = 1, 2, some z ∈ Oz \EG and y ∈ ξz (z ), and this is in contrast with the assumption of [16, Theorem 4]. Remark 4. Suppose a unique player and a compact and convex subset X of a locally convex Hausdorff topological vector space. Theorem 1 ensures the existence of maximum points for any quasi-concave function u such that: (∗) x ∈ / {maximum points of u} implies that there exists a well-behaved map ξx : Ox ⇒ X, where Ox is an open neighborhood of x, so that u(x ) < u(z ) for all z ∈ ξx (x ) and all x ∈ Ox \{maximum points of u}. So, Theorem 1 is a generalization of the well known Weierstrass Theorem. An other generalization of the Weierstrass Theorem is provided in [24], where it has been proved that a function u has maximum points on a compact set if and only if u is transfer weakly upper continuous (t.w.u.c.), that is: x ∈ / {maximum points of u} implies that there exists an open neighborhood Ox of x and z so that u(x ) ≤ u(z ) for all x ∈ Ox .6 Now, assume that u is t.w.u.c. and x ∈ / {maximum points of u}. If z ∈ {maximum points of u}, it is sufficient to set ξx (x ) = z for any x ∈ Ox and we get that (∗) is satisfied. If z ∈ / {maximum points of u}, for some z , we have u(x ) ≤ u(z ) < u(z ) for all x ∈ Ox , and we deduce that (∗) holds with the map define by ξx (x ) = z for any x ∈ Ox . So, (∗) is a generalization of the transfer weak upper continuity. On the other hand, in light of Theorem 1, we have that any quasi-concave function (in the setting of the theorem) which satisfies (∗) is t.w.u.c. It remains still as open question to provide an example of a function, defined on a compact set, which satisfies (∗) but not the transfer weak upper continuity. Remark 5. Several papers deal with ordinal games, where any player i ∈ N is characterized by a map Pi : X ⇒ Xi , and a Nash equilibrium is a strategy profile x∗ so that Pi (x∗ ) = ∅ for all i ∈ N (the map Pi could be derived from the asymmetric part of a binary relation i on the set on strategy profiles, that is: Pi (x) = {yi ∈ Xi : (yi , x−i ) i x}): among the more recent ones, see [22,10,14,19]. The definition of the GSDP for ordinal games can be easily derived from Definition 3: now we require that for every z ∈ Oz \EG and every x ∈ ξz (z ), there is a player i for whom xi ∈ Pi (z ). Similarly for the C-condition. Now, Theorem 1 can be given for ordinal games accordingly (the proof works in the same way): below, we provide a formal statement. However, even in the framework of ordinal games, our result cannot be obtained from the previous literature, as the following example shows. Theorem 2. Let G = Xi , Pi i∈N be an ordinal game which satisfies the GSDP and the C-condition. Then, EG is non-empty.
1
2 1 Example 4. Let A = : n is a natural number , K1 = , n : n is odd and n > 4 and K2 = n n
1 3 n , n : n is even and n ≥ 4 . Consider the ordinal game G = Xi , Pi i=1,2 with X1 = X2 = [0, 1] and P1 and P2 are given as below: P1 (x) = ∅ and P2 (x) = {1} if x1 > x2 > 0 and x1 ∈ /A P1 (x) = {1} and P2 (x) = ∅ if 0 < x1 < x2 and x2 ∈ /A 6
A generalization of the Weierstrass Theorem in non-necessarily compact sets is given in [2].
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P1 (v1 , x2 ) = ∅ and P2 (x) = {1} if x ∈ K1 and v1 ∈ [0, 1] P1 (x) = {1} and P2 (x1 , v2 ) = ∅ if x ∈ K2 and v2 ∈ [0, 1] P1 (x) = P2 (x) = {1} if x1 = x2 = 0 P1 (x) = P2 (x) = ∅ otherwise . It is easy to see that G satisfies the GSDP: in fact, for every non-equilibrium z, it is sufficient to define ξz (z ) = {(1, 1)} for each z in some open neighborhood of z. Moreover, G satisfies the C-condition. So, all the assumptions of Theorem 2 hold true. Since none of the players has profitable unilateral deviation for every z in every open neighborhoods of the non-equilibrium z = (0, 0), the results given in [22,14,19] do not apply on G. Concerning [10], we build up an ordinal game where it is known the binary relation i of each player on the set of strategy profiles. For every x and y belonging to [0, 1]2 , define i by: x i y ⇐⇒
x−i = y−i and xi ∈ Pi (y) ,
where i = 1, 2. So, given any strategy profiles x and v, the set {wi ∈ [0, 1] : (wi , x−i ) i v} is non-empty if and only if x−i = v−i and Pi (v) = ∅. Obviously, the game satisfies the assumptions of Theorem 2. Consider the non-equilibrium strategy profile x = (0, 0) and the compact set K = K1 ∪ K2 ∪ {(0, 0)}, which is such that K ∩ EG = ∅. The condition introduced in [10, Definition 3] requires that, for at least one player i,
the set Fi (x ) = v∈πi (x ) wi ∈ Xi : (wi , x−i ) i v be non-empty for some π i : [0, 1]2 ⇒ [0, 1]2 and for all x ∈ Ox ∩ K, where Ox is a suitable open neighborhood of x = (0, 0). But here we have that F1 (x ) = ∅ for all x ∈ K1 ∩ Ox and F2 (y ) = ∅ for all y ∈ K2 ∩ Ox . In fact, we get P1 (v1 , x2 ) = ∅ for all v1 ∈ [0, 1] and P2 (y1 , v2 ) = ∅ for all v2 ∈ [0, 1]. Hence, the equilibrium existence results from [10] cannot be applied on the game Xi , i i=1,2 . 3. Continuity properties of the Nash equilibrium correspondence Assume that the set of players N and the sets of strategies X1 , ..., XN are fixed, any Xi is a non-empty, convex and compact subset of a metrizable locally convex topological vector space (d denotes the metric induced on X) and the payoffs are bounded functions: let us denote by g0 the space of such games. We endow g0 with the classical sup-norm metric: ρ(G, G ) =
i∈N
sup |ui (x) − ui (x)| .
x∈X
With N we denote the Nash equilibrium correspondence: N (G) = EG . The aim of this section is to study the continuity properties of N in a setting of games derived from that of Theorem 1. More precisely, we introduce a complete metric spaces of discontinuous games, denoted by g, where, with respect to the metric ρ, N is upper semicontinuous, with non-empty and compact values. Let us point out that, in our setting, a map is upper semicontinuous with compact values if and only if it is closed (see, for example, [6]). We note that the GSDP guarantees neither that N is closed nor that it has compact values: in fact, the set of Nash equilibria of the game given in Example 2 is not compact. So, in order to obtain the upper semicontinuity of N , we need to strengthen the GSDP. We introduce the following definition.
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Definition 4. G is said to have the Positive Generalized Single Deviation Property (PGSDP in short) if, whenever z ∈ X is not an equilibrium of G and uj (xj , z−j ) − uj (z) > t for some x ∈ X, j ∈ N and t > 0, there exists an open neighborhood Oz of z and a well-behaved map ξz : Oz ⇒ X such that, for every z ∈ Oz and every x ∈ ξz (z ), there is a player i for whom ui (xi , z−i ) − ui (z ) > t. The PGSDP allows the map N to be upper semicontinuous and compact valued (N is a closed map). So, since the game given in Example 2 has the GSDP and a non-closed set of Nash equilibria, we have that the GSDP does not imply the PGSDP. We denote by g the subspace of go of all games which satisfy both the PGSDP and the C-condition. Theorem 3. N is non-empty and compact valued and upper semicontinuous on g. Proof. First, let us prove that N is a closed map. Suppose that ρ (Gn , G) −→ 0, z n ∈ N (Gn ) for all n, z n −→ z and z ∈ / N (G). So, there exists an open neighborhood Oz of z, a well-behaved map ξz : Oz ⇒ X and t > 0 such that, for every z ∈ Oz and every x ∈ ξz (z ), there is a player i for whom ui (xi , z−i )− n n ui (z ) > t. For any n greater than a suitable no and for any x ∈ ξz (z ), there exists a player in so that n uin (xnin , z−i ) −uin (z n ) > t. This implies that there exists a subsequence (z n )n of (z n )n , a sequence (xn )n , n n where xn ∈ ξz (z n ) for all n , and a player i such that ui (xni , z−i ) − ui (z n ) > t for every n . On the other hand, since ρ (Gn , G) −→ 0, for n sufficiently large, we obtain: n n n ui (xni , z−i ) − ui (z n ) > t > ui (xni , z−i ) − ui (z n ) − uni (xni , z−i ) − uni (z n ) ,
that is: z n ∈ / N (Gn ), which is a contradiction. So, N is closed and it is obvious that N (G) is compact. Finally, N (G) is non-empty in light of Theorem 1. 2 The PGSDP and the C-condition have the following convergence properties with respect to the metric ρ. Proposition 2. Let (Gn )n be a sequence of games which have the PGSDP and G be such that ρ (Gn , G) −→ 0. Then, G has the PGSDP. / EG and uj (xj , z−j ) − uj (z) > t for some x ∈ X, j ∈ N and t > 0. Let t1 be such Proof. Assume that z ∈ that uj (xj , z−j ) − uj (z) > t1 > t. So, z ∈ / EGn for n sufficiently large and unj (xj , z−j ) − unj (z) > t1 . Since every game Gn has the PGSDP, we have: for n sufficiently large, there exists a well-behaved map ξzn : Ozn ⇒ X such that, for any z ∈ Ozn and any x ∈ ξzn (z ), some player in gets unin (xin , z−i ) − unin (z ) > t1 . n On the other hand, we obtain: sup (y,y )∈X×X
n ui (yi , y−i ) − uni (y ) − ui (yi , y−i ) − ui (y ) ≤ 2ρ(Gn , G) ∀ i ∈ N .
So, since ρ (Gn , G) −→ 0, for a suitable n, we get: sup (y,y )∈X×X
n ui (yi , y−i ) − uni (y ) − ui (yi , y−i ) − ui (y ) < t1 − t ∀ i ∈ N and
∀ z ∈ Ozn ∀ x ∈ ξzn (z ) ∃ in ∈ N such that unin (xin , z−i ) − unin (z ) − t > t1 − t , n
which implies:
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unin (xin , z−i ) − unin (z ) − t > t1 − t > unin (xin , z−i ) − unin (z ) − uin (xin , z−i ) − uin (z ) , n n n that is: uin (xin , z−i ) − uin (z ) > t. Finally, it is sufficient to set Oz = Ozn and ξx = ξzn , and the proof is n concluded. 2
Remark 6. Proposition 2 fails to be true in the class of games having the GSDP. Indeed, consider the sequence of 1-player games Gn = [0, 1], un , where
un (x) =
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x if x ∈ 0, 1 − n1
2 1 − n1 − x if x ∈ 1 − n1 , 1 0 if x = 1 .
(Gn )n converges to G = [0, 1], u , where u is the function defined in Remark 1, and u does not satisfy the GSDP. Proposition 3. Let (Gn )n be a sequence of games which satisfy the C-condition and G be such that ρ (Gn , G) −→ 0. Then, G satisfies the C-condition. Proof. Let F be a finite subset of strategy profiles and z = scoF . Since any Gn satisfies the C-condition, for any n there exists xn ∈ F such that uni (xni , z−i ) − uni (z) ≤ 0 for each i ∈ N . So, for at least a subsequence (xn )n of (xn )n , there exists x ∈ F so that uni (xi , z−i ) − uni (z) ≤ 0 for each i ∈ N and each n . Then, ui (xi , z−i ) − ui (z) ≤ 0 for each i ∈ N , which concludes the proof. 2 From Propositions 2 and 3 we have that g is a complete metric space with respect to the sup-norm ρ. This property allows to deduce that the map N is continuous (upper and lower semicontinuous) in a dense subset of g. In fact, since N is upper semicontinuous with non-empty and compact values, Theorem of Fort guarantees that N is lower semicontinuous on a dense subset of g.7 When the Nash equilibrium correspondence is lower semicontinuous on a set q, the Nash equilibria of any game belonging to q have a stability property called essential stability: Definition 5. [25] Let g ⊆ g0 and G ∈ g. A strategy profile x is said to be an essential Nash equilibrium of G relative to g if it is a Nash equilibrium of G and for any ε > 0 there exists δ > 0 such that any game G ∈ g with ρ(G, G ) < δ has at least one Nash equilibrium x such that d(x, x ) < ε. Previous results on the upper semicontinuity of the Nash equilibrium correspondence and the essential stability of equilibria have been provided in [26,20,7,21,8]. The assumptions considered in [26,20,7,8] are strengthening of those given in [17,3,9,15,4]; so, in light of Remark 3, g is not connected with the spaces of games introduced in the mentioned literature. In [21], classes of discontinuous games, where the essential stability of Nash equilibria holds in dense subsets, have been defined through properties on the aggregator of payoff functions ΦG (x, z) = i∈N [ui (xi , z−i ) − ui (z)]. It is easy to see that these properties imply those introduced in the present paper (Definitions 3 and 4). Nevertheless, because of the quasi-concavity like conditions, [21] and the present paper refer to different classes of discontinuous games. 7 Theorem of Fort. [13] Let Y and X be metric spaces and Y be a Baire’s space. Assume that T : Y ⇒ X is an upper semicontinuous map with non-empty and compact values. Then, there exists a dense Gδ subset Q of Y such that T is lower semicontinuous at any point belonging to Q. We recall that a subset of a topological space is a Gδ -set (in short Gδ ) if it is a countable intersection of open subsets. A Baire’s space is a topological space where any countable intersection of open dense subsets is a dense subset. Examples of Baire spaces are the complete metric spaces (see, for example, [1]).
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References [1] C.D. Aliprantis, K.C. Border, Infinite Dimensional Analysis, Springer, Berlin, Heidelberg, 1999. [2] A. Amini-Harandi, M. Fakhar, H.R. Hajisharifi, Some generalizations of the Weierstrass theorem, SIAM J. Optim. 26 (4) (2016) 2847–2862. [3] A. Bagh, A. Jofre, Reciprocal upper semicontinuity and better reply secure games: a comment, Econometrica 74 (2006) 1715–1721. [4] P. Barelli, I. Meneghel, A note on the equilibrium existence problem in discontinuous games, Econometrica 81 (2013) 813–824. [5] M.R. Baye, G. Tian, J. Zhou, Characterizations of the existence of equilibria in games with discontinuous and nonquasiconcave payoffs, Rev. Econ. Stud. 60 (1993) 935–948. [6] C. Berge, Espaces Topologiques. Fonctions Multivoques, Dunod, Paris, 1959. [7] O. Carbonell-Nicolau, Essential equilibria in nornal-form games, J. Econom. Theory 145 (2010) 421–431. [8] O. Carbonell-Nicolau, Further results on essential Nash equilibria in normal-form games, Econom. Theory 59 (2) (2015) 277–300. [9] G. Carmona, Understanding some recent existence results for discontinuous games, Econom. Theory 48 (2011) 31–45. [10] G. Carmona, K. Podczeck, Existence of Nash equilibrium in ordinal games with discontinuous preferences, Econom. Theory 61 (3) (2016) 457–478. [11] P. Dasgupta, E. Maskin, The existence of equilibrium in discontinuous economic games, I: theory, Rev. Econ. Stud. 53 (1986) 1–26. [12] A.L. Dontchev, T. Zolezzi, Well-Posed Optimization Problems, Springer-Verlag, Berlin, 1993. [13] M.K. Fort, A unified theory of semi-continuity, Duke Math. J. 16 (1949) 237–246. [14] W. He, N.C. Yannelis, Existence of Walrasian equilibria with discontinuous, non-ordered, interdependent and pricedependent preferences, Econom. Theory 61 (3) (2016) 497–513. [15] R. Nessah, Generalized weak transfer continuity and the Nash equilibrium, J. Math. Econom. 47 (2011) 659–662. [16] R. Nassah, G. Tian, On the existence of Nash equilibrium in discontinuous games, Econom. Theory 61 (3) (2016) 515–540. [17] P.J. Reny, On the existence of pure and mixed strategy Nash equilibria in discontinuous games, Econometrica 67 (1999) 1029–1056. [18] P.J. Reny, Further results on the existence of Nash equilibria in discontinuous games, Mimeo, University of Chicago, 2009. [19] P.J. Reny, Nash equilibrium in discontinuous games, Econom. Theory 61 (3) (2016) 553–569. [20] V. Scalzo, Hadamard well-posedness in discontinuous non-cooperative games, J. Math. Anal. Appl. 360 (2009) 697–703. [21] V. Scalzo, Essential equilibria of discontinuous games, Econom. Theory 54 (1) (2013) 27–44. [22] V. Scalzo, On the existence of maximal elements, fixed points and equilibria of generalized games in a fuzzy environment, Fuzzy Sets and Systems 272 (2015) 126–133. [23] V. Scalzo, Weak maximal elements and weak equilibria in ordinal games with applications to exchange economies, Econ. Theory Bull. 6 (1) (2018) 29–39. [24] G. Tian, J. Zhou, Transfer continuities, generalizations of the Weierstrass theorem and maximum theorem – a full characterization, J. Math. Econom. 24 (1995) 281–303. [25] W.T. Wu, J.H. Jiang, Essential equilibrium points of n-person noncooperative games, Sci. Sin. 11 (1962) 1307–1322. [26] J. Yu, Essential equilibria of n-person noncooperative games, J. Math. Econom. 31 (1999) 361–372. [27] J. Yu, H. Yang, C. Yu, Well-posed Ky Fan’s point, quasi-variational inequality and Nash equilibrium problems, Nonlinear Anal. 66 (2007) 777–790.
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Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Continuity properties of the Nash equilibrium correspondence in a discontinuous setting Vincenzo Scalzo Department of Economics and Statistics (DISES), University of Naples Federico II, via Cinthia 21, 80126 Napoli, Italy
a r t i c l e
i n f o
Article history: Received 14 June 2018 Available online xxxx Submitted by J.A. Filar Keywords: Discontinuous non-cooperative games Continuity of correspondences Existence of Nash equilibria Essential stability of Nash equilibria
a b s t r a c t We introduce a new complete metric space of discontinuous normal form games and prove that the Nash equilibrium correspondence is upper semicontinuous with non-empty and compact values. So, using the Theorem of Fort (1949), we obtain that the correspondence is also lower semicontinuous in a dense subset. We introduce new topological assumptions on the payoff functions and a strengthening of standard quasi-concavity properties. Examples show that our results cannot be obtained from the previous ones. © 2019 Elsevier Inc. All rights reserved.
1. Introduction The existence of Nash equilibria and the continuity properties of the Nash equilibrium correspondence in normal form games, where the usual continuity assumptions on the payoffs are weakened (henceforth discontinuous games), are issues that have captured the attention of many authors. For the existence of Nash equilibria, let us refer to the seminal papers [11], [5] and [17], and the more recent ones [3,9,15,4,21,10, 14,16,19]. The upper semicontinuity of the Nash equilibrium correspondence in the setting of discontinuous games have been investigated in [27] and [20], while the continuity of the correspondence (that means upper and lower semicontinuity) has been obtained in several classes of discontinuous games: see [26,7,21,8]. More precisely, we focus on the upper (and lower) semicontinuity when the Nash equilibrium correspondence is defined on metric spaces of discontinuous games where the existence of Nash equilibria is guaranteed, and the metric is given by the sup-norm. The search for more and more general classes of discontinuous games where Nash equilibria exist has led to a condition introduced in [18] and called Single Deviation Property. Even if such a property refers to a reasonable behavior of players on the strategy profiles which are not Nash equilibria, a counterexample shows that it is not sufficient to guarantee the existence of equilibria in games satisfying standard quasi-concavity E-mail address: [email protected]. https://doi.org/10.1016/j.jmaa.2019.01.021 0022-247X/© 2019 Elsevier Inc. All rights reserved.
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conditions (see [18, Section 3.1]). However, the Single Deviation Property plays a role in identifying the class of games where the Nash equilibria can be characterized by means of some kind of weak Nash equilibria: see [23]. Moreover, nothing is known about the continuity properties of the Nash equilibrium correspondence in classes of games where Single Deviation Property-like assumptions are given. The aim of the present paper is to provide new results on both the existence of Nash equilibria and the continuity (upper and lower semicontinuity) of the Nash equilibrium correspondence in games satisfying Single Deviation Property-like assumptions.1 We identify a new class of discontinuous games where the Nash equilibrium correspondence is continuous and non-empty valued. More precisely, we introduce a generalization of the Single Deviation Property, called Generalized Single Deviation Property, and define an independent quasi-concavity condition, called C-condition, which guarantee the existence of Nash equilibria when the sets of strategies are compact and convex subsets of locally convex Hausdorff topological vector spaces. For every non-equilibrium z, the Generalized Single Deviation Property allows to find a wellbehaved map ξ defined in some open neighborhood of z so that, given any z in the open neighborhood and x ∈ ξ(z ), at least one player profitable deviates using her/his strategy in the profile x . Note that the player who deviates depends on both z and x ; while, in the Single Deviation Property, the deviating player is not depending on x because ξ is constant-valued.2 Furthermore, we obtain the upper semicontinuity of the Nash equilibrium correspondence by using a strengthening of the Generalized Single Deviation Property, called Positive Generalized Single Deviation Property. Finally, we have that the Nash equilibrium correspondence is continuous and non-empty and compact valued in a dense subset of the class of games satisfying the Positive Generalized Single Deviation Property and the C-condition. Let us highlight that our interest in new results on the existence of Nash equilibria and continuity of the Nash equilibrium correspondence is focused on those topological and quasi-concavity conditions which are independent of each other. In fact, our C-condition holds in games where no topological assumptions are required. The paper is organized as follows. The setting and the definitions are given in Section 2, where we also present a new Nash equilibrium existence result. Finally, Section 3 deals with the continuity properties of the Nash equilibrium correspondence. 2. Setting, definitions and equilibrium existence Assume that X and Y are topological spaces and let T : Y ⇒ X be a correspondence (a map) from Y to X. We recall that T is said to be: upper semicontinuous if, for any y ∈ Y and any open set U which includes T (y), there exists an open neighborhood Oy of y such that T (y ) ⊂ U for all y ∈ Oy ; closed if the graph of T (that is: {(y, x) ∈ Y × X : x ∈ T (y)}) is a closed set in Y × X; lower semicontinuous if, for any y ∈ Y and any open set U so that T (y) ∩ U = ∅, there exists an open neighborhood Oy of y such that T (y ) ∩ U = ∅ for all y ∈ Oy ; continuous if it is both upper and lower semicontinuous. Let N be a finite set of players (N denotes also the number of players), and, for any i ∈ N , assume that Xi is a non-empty, convex and compact subset of a locally convex Hausdorff topological vector space. Set X = i∈N Xi , that is the set of strategy profiles (any element x of X is also denoted by (xi , x−i ) where x−i ∈ j=i Xj ), ui : X −→ R is the payoff function of player i. The list of data G = Xi , ui i∈N is a normal form game (game in short) and a strategy profile x∗ is a Nash equilibrium of G if ui (x∗ ) ≥ ui (xi , x∗−i ) for all xi ∈ Xi and all i ∈ N . EG denotes the set of Nash equilibria of G. A game G is called quasi-concave if 1 The upper semicontinuity of the Nash equilibrium correspondence refers to the notion of Hadamard well-posedness introduced in Optimization Theory: see [12], while the lower semicontinuity is nothing but the essential stability of equilibria introduced in [25]. Hadamard well-posedness and essential stability are stability properties of equilibria with respect to perturbations on the characteristics of players. 2 The idea to consider a locally defined well-behaved map for every non-equilibrium strategy profile has been often used in Discontinuous Games: see, among the others, [4,21,19].
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ui (·, x−i ) is a quasi-concave function for any x−i ∈ X−i and any i ∈ N . The following definition is given in [18]: Definition 1. G satisfies the Single Deviation Property (SDP in short) if, whenever z ∈ X is not a Nash equilibrium of G, there exists an open neighborhood Oz of z and a strategy profile x such that, for each z ∈ Oz , at least one player i gets ui (xi , z−i ) > ui (z ). As it is shown by a counterexample in [18] (see also [19]), the SDP does not guarantee the existence of Nash equilibria in quasi-concave games: Example 1. [18, Section 3.1] Let G be the 3-player game where the set of strategies is [0, 1], x, y and z are the choices of players 1, 2 and 3, respectively, and the outcomes are given as below:
x ∈ [0, 1/2] x ∈ ]1/2, 1]
y ∈ [0, 1/3] (u(x), v(y), u(z)) (u(x), v(y), u(z))
y ∈ ]1/3, 2/3[ (v(x), v(y), u(z)) (v(x), v(y), v(z))
y ∈ [2/3, 1] (v(x), v(y), v(z)) (v(x), v(y), v(z))
if z ∈ [0, 1/2]
x ∈ [0, 1/2] x ∈ ]1/2, 1]
y ∈ [0, 1/3] (u(x), u(y), u(z)) (u(x), u(y), u(z))
y ∈ ]1/3, 2/3[ (u(x), u(y), u(z)) (u(x), u(y), v(z))
y ∈ [2/3, 1] (v(x), u(y), v(z)) (v(x), u(y), v(z))
if z ∈ ]1/2, 1]
where u(t) = 0 if t ∈ ]0, 1] and u(0) = 1, while v(t) = 0 if t ∈ [0, 1[ and v(1) = 1. G is a quasi-concave game which satisfies the SDP. It is easy to see that EG = ∅. So, if one aims to obtain a Nash equilibrium existence result for games satisfying the SDP, one has to strengthen the assumption of quasi-concavity, and it is interesting to look for a new condition which have no connections with both the topological structure of the game and the SDP. So, we introduce the following condition: Definition 2. G is said to satisfy the C-condition if, for every finite set F of strategy profiles and for every z ∈ scoF , there exists x ∈ F such that ui (xi , z−i ) ≤ ui (z) for all i ∈ N .3 It is clear that any game which satisfies the C-condition is quasi-concave, but the converse does not hold (in fact, we will have that the C-condition allows the existence of Nash equilibria in games that have the SDP). Basically, the C-condition requires some uniformity among players in having the same strategy profile x where, with respect to z, no profitable unilateral deviations are possible. Nevertheless, the C-condition does not imply concavity or continuity on the payoff functions, as Example 2 shows. First, we give a characterization of the C-condition in terms of set-valued mappings. Given a game G = Xi , ui i∈N , define Pi : X ⇒ Xi by Pi (x) = {zi ∈ Xi : ui (zi , x−i ) > ui (x)} for any x ∈ X and any i ∈ N . We have the following proposition. Proposition 1. Given a game G and the corresponding mappings Pi , let P : X ⇒ X be defined as below: P (x) = {z ∈ X : there exists i ∈ N such that zi ∈ Pi (x)} ∀ x ∈ X .
(1)
Then, G satisfies the C-condition if and only if x ∈ / coP (x) for all x ∈ X. 3 Given a set A, we denote by coA the convex hull of A and by scoA the subset of strict convex combinations of all elements of A.
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Proof. Assume that G satisfies the C-condition and x ∈ coP (x) for some x ∈ X. So, there exists a finite subset F ⊂ P (x) so that x ∈ scoF . The C-condition guarantees that, for at least one x ¯ ∈ F, x ¯i ∈ / Pi (x) for all i ∈ N , that is: x ¯∈ / P (x), and we get a contradiction. On the other hand, suppose that z ∈ scoF , where F is a finite set of strategy profiles. Since z ∈ / coP (z), we have x ¯∈ / P (z) for at least one x ¯ ∈ F , that is x ¯i ∈ / Pi (z) for all i ∈ N , which proves that the C-condition is satisfied. 2 Example 2. Let G = Xi , ui i=1,2 be such that X1 = X2 = [0, 1] and ⎧ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎨ u1 (x) = 1 ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎩1 u2 (x) =
if x1 > x2 > 0 or x1 = x2 > 0 if 0 < x1 < x2 or x1 = x2 = 0 if x1 = 0 and x2 ∈ ]0, 1] or x2 = 0 and x1 ∈ ]0, 1] if x1 ∈ [0, 1] and x2 = 1 if x1 = 1 and x2 ∈ [0, 1] , u1 (x) if x ∈ {z : z1 = z2 } ∪ ({0} × [0, 1]) ∪ ([0, 1] × {0})
1 − u1 (x) otherwise .
It is easy to see that u1 and u2 are neither concave nor continuous. On the other hand, the C-condition holds true. Indeed, one can use Proposition 1, since we have: P1 (x) = ∅ and P2 (x) = [x1 , 1] if x1 > x2 > 0 P1 (x) = [x2 , 1] and P2 (x) = ∅ if 0 < x1 < x2 P1 (x) = P2 (x) = {1} if x1 = x2 = 0 P1 (x) = P2 (x) = ∅ otherwise . Note that EG = {x : x1 = x2 > 0} ∪ ({0}×]0, 1]) ∪ (]0, 1] × {0}). In order to obtain the existence of Nash equilibria in games where the SDP is weakened, the C-condition is the right strengthening of quasi-concavity. Below, we introduce a generalization of the SDP. Definition 3. G is said to satisfy the Generalized Single Deviation Property (GSDP in short) if, whenever z ∈ X is not a Nash equilibrium of G, there exists an open neighborhood Oz of z and a well-behaved4 map ξz : Oz ⇒ X such that, for every z ∈ Oz \EG and every x ∈ ξz (z ), there is a player i for whom ui (xi , z−i ) > ui (z ). The following example shows that the GSDP is a generalization of the SDP even in games which satisfy the C-condition. Example 3. Consider the game G = Xi , ui i=1,2 where X1 = X2 = [0, 1], u1 (x1 , x2 ) = 0 whenever x1 = x2 and x2 > 0, u1 (x) = 1 otherwise and u2 (x) = 1 for all x ∈ X. So, EG = {x : x1 = x2 } ∪ [0, 1] × {0}. If z ∈ / EG , let Oz be an open neighborhood of z such that z1 = z2 for each z ∈ Oz and ξz : Oz ⇒ X be defined by ξz (z ) = {z2 } × [0, 1]. The map ξz is upper semicontinuous with non-empty, convex and compact values, and we have u1 (x1 , z2 ) > u1 (z ) for every x ∈ ξz (z ) and every z ∈ Oz . So, G satisfies the GSDP. 4
We call well-behaved any upper semicontinuous map with non-empty, convex and compact values.
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Moreover, the SDP fails on G because, for any open neighborhood Oz and any strategy profile x , there are elements z ∈ Oz with x1 = z2 , which implies that u1 (x1 , z2 ) ≤ u1 (z ). Finally, one can easily prove that G satisfies the C-condition. Theorem 1. If G satisfies the GSDP and the C-condition, then EG is non-empty. Proof. By contradiction, assume that EG = ∅. So, for every x ∈ X, let Ox and ξx be, respectively, the open neighborhood and the map associated to x by the GSDP. Since X is compact, there exists a partition of the unity {β1 , ..., βk } subordinated to a finite covering {Ox1 , ..., Oxk } of X such that, for h = 1, ..., k, βh is k continuous on X, βh (x) > 0 implies x ∈ Oxh and h=1 βh (x) = 1 for all x ∈ X. Define the map ξ from X to itself as follows5 : ξ(x) =
βh (x)ξxh (x) ∀ x ∈ X .
(2)
βh (x)>0
Obviously, ξ is upper semicontinuous with non-empty, convex and compact values. So, Kakutani–Fan– Glisberg fixed point theorem guarantees the existence of a strategy profile x∗ such that x∗ ∈ ξ(x∗ ), that is:
x∗ =
βh (x∗ )z h
(3)
βh (x∗ )>0
where z h ∈ ξxh (x∗ ). Since x∗ ∈ Oxh whenever βh (x∗ ) > 0, from the GSDP we have that, for any h ∈ {1, ..., k} such that βh (x∗ ) > 0, there exists ih ∈ N so that uih (zihh , x∗−ih ) > uih (x∗ ). On the other hand, since ∗ ∗ h ∗ βh (x∗ )>0 βh (x ) = 1, from (3) one gets x ∈ sco{z : βh (x ) > 0}. So, the C-condition implies that, for at least one z ho with βho (x∗ ) > 0, we have ui (ziho , x∗−i ) ≤ ui (x∗ ) for all i ∈ N , and we get a contradiction. 2
Remark 1. The topological and quasi-concavity assumptions considered in Theorem 1, that are the GSDP and C-condition, are independent of each other, in the sense that the C-condition supposes neither the GSDP nor any other topological property. For instance, consider the very simple situation of 1-player game G where the payoff function u is defined on [0, 1] by u(x) = x if x ∈ [0, 1[ and u(1) = 0. Obviously, G satisfies the C-condition because it is quasi-concave and 1-player game; moreover, EG = ∅. On the other hand, if u is such that u(0) = 1, u(x) = x if x ∈ ]0, 1[ and u(1) = 0, the GSDP holds but the C-condition fails. Remark 2. The game G defined in Example 2 satisfies the GSDP (we have already seen that this game verifies the C-condition). More precisely, G has the SDP: in fact, it is sufficient to set x = (1, 1) for every z ∈ / EG . Note that for z = (0, 0) ∈ / EG and z in an open neighborhood of z, if z1 > z2 , the player who deviates is 2, while only player 1 deviates when z1 < z2 . Remark 3. Consider the game in Example 2. Assume that O is an open neighborhood of z = (0, 0) and ξ : O ⇒ X is a well-behaved map. Let z ∈ O. As observed in the Remark above, if z1 > z2 , the only player that could have profitable deviation strategies from the elements of ξ(z ) is player 2, while only player 1 could have good deviations if z1 < z2 . So, no one of the two players has profitable deviation strategies for all z ∈ O\EG , but the player who deviates depends on z . This circumstance is not covered by the Nash equilibria existence results given in the recent literature: for instance, see [3,15,4,16]. Hence, Theorem 1 cannot be deduced from the previous results. 5
Given Ai ⊂ X and a real number αi , for any i ∈ N ,
i∈N
αi Ai is the set of all
i∈N
αi yi with yi ∈ Ai .
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Furthermore, the Nash equilibrium existence result [16, Theorem 6] is given for games which satisfy the SDP and a weak quasi-concavity like property. In Theorem 1, we weaken the SDP assumption and strengthen the quasi-concavity property; however, Example 3 shows that the two theorems are independent of each other. Finally, the assumptions of [16, Theorem 4] fail to be verified on Example 2. In fact, following the bottom at page 525 in [16], with respect to the non-equilibrium z = (0, 0) and for any non-equilibrium z in every open neighborhood Oz of z, set ξz a well-behaved map defined on Oz , for each non-equilibrium z ∈ Oz , we have that z ∈ {t ∈ X : ui (ti , z−i ) − ui (z ) ≥ ui (yi , z−i ) − ui (z )} for some player i = 1, 2, some z ∈ Oz \EG and y ∈ ξz (z ), and this is in contrast with the assumption of [16, Theorem 4]. Remark 4. Suppose a unique player and a compact and convex subset X of a locally convex Hausdorff topological vector space. Theorem 1 ensures the existence of maximum points for any quasi-concave function u such that: (∗) x ∈ / {maximum points of u} implies that there exists a well-behaved map ξx : Ox ⇒ X, where Ox is an open neighborhood of x, so that u(x ) < u(z ) for all z ∈ ξx (x ) and all x ∈ Ox \{maximum points of u}. So, Theorem 1 is a generalization of the well known Weierstrass Theorem. An other generalization of the Weierstrass Theorem is provided in [24], where it has been proved that a function u has maximum points on a compact set if and only if u is transfer weakly upper continuous (t.w.u.c.), that is: x ∈ / {maximum points of u} implies that there exists an open neighborhood Ox of x and z so that u(x ) ≤ u(z ) for all x ∈ Ox .6 Now, assume that u is t.w.u.c. and x ∈ / {maximum points of u}. If z ∈ {maximum points of u}, it is sufficient to set ξx (x ) = z for any x ∈ Ox and we get that (∗) is satisfied. If z ∈ / {maximum points of u}, for some z , we have u(x ) ≤ u(z ) < u(z ) for all x ∈ Ox , and we deduce that (∗) holds with the map define by ξx (x ) = z for any x ∈ Ox . So, (∗) is a generalization of the transfer weak upper continuity. On the other hand, in light of Theorem 1, we have that any quasi-concave function (in the setting of the theorem) which satisfies (∗) is t.w.u.c. It remains still as open question to provide an example of a function, defined on a compact set, which satisfies (∗) but not the transfer weak upper continuity. Remark 5. Several papers deal with ordinal games, where any player i ∈ N is characterized by a map Pi : X ⇒ Xi , and a Nash equilibrium is a strategy profile x∗ so that Pi (x∗ ) = ∅ for all i ∈ N (the map Pi could be derived from the asymmetric part of a binary relation i on the set on strategy profiles, that is: Pi (x) = {yi ∈ Xi : (yi , x−i ) i x}): among the more recent ones, see [22,10,14,19]. The definition of the GSDP for ordinal games can be easily derived from Definition 3: now we require that for every z ∈ Oz \EG and every x ∈ ξz (z ), there is a player i for whom xi ∈ Pi (z ). Similarly for the C-condition. Now, Theorem 1 can be given for ordinal games accordingly (the proof works in the same way): below, we provide a formal statement. However, even in the framework of ordinal games, our result cannot be obtained from the previous literature, as the following example shows. Theorem 2. Let G = Xi , Pi i∈N be an ordinal game which satisfies the GSDP and the C-condition. Then, EG is non-empty.
1
2 1 Example 4. Let A = : n is a natural number , K1 = , n : n is odd and n > 4 and K2 = n n
1 3 n , n : n is even and n ≥ 4 . Consider the ordinal game G = Xi , Pi i=1,2 with X1 = X2 = [0, 1] and P1 and P2 are given as below: P1 (x) = ∅ and P2 (x) = {1} if x1 > x2 > 0 and x1 ∈ /A P1 (x) = {1} and P2 (x) = ∅ if 0 < x1 < x2 and x2 ∈ /A 6
A generalization of the Weierstrass Theorem in non-necessarily compact sets is given in [2].
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P1 (v1 , x2 ) = ∅ and P2 (x) = {1} if x ∈ K1 and v1 ∈ [0, 1] P1 (x) = {1} and P2 (x1 , v2 ) = ∅ if x ∈ K2 and v2 ∈ [0, 1] P1 (x) = P2 (x) = {1} if x1 = x2 = 0 P1 (x) = P2 (x) = ∅ otherwise . It is easy to see that G satisfies the GSDP: in fact, for every non-equilibrium z, it is sufficient to define ξz (z ) = {(1, 1)} for each z in some open neighborhood of z. Moreover, G satisfies the C-condition. So, all the assumptions of Theorem 2 hold true. Since none of the players has profitable unilateral deviation for every z in every open neighborhoods of the non-equilibrium z = (0, 0), the results given in [22,14,19] do not apply on G. Concerning [10], we build up an ordinal game where it is known the binary relation i of each player on the set of strategy profiles. For every x and y belonging to [0, 1]2 , define i by: x i y ⇐⇒
x−i = y−i and xi ∈ Pi (y) ,
where i = 1, 2. So, given any strategy profiles x and v, the set {wi ∈ [0, 1] : (wi , x−i ) i v} is non-empty if and only if x−i = v−i and Pi (v) = ∅. Obviously, the game satisfies the assumptions of Theorem 2. Consider the non-equilibrium strategy profile x = (0, 0) and the compact set K = K1 ∪ K2 ∪ {(0, 0)}, which is such that K ∩ EG = ∅. The condition introduced in [10, Definition 3] requires that, for at least one player i,
the set Fi (x ) = v∈πi (x ) wi ∈ Xi : (wi , x−i ) i v be non-empty for some π i : [0, 1]2 ⇒ [0, 1]2 and for all x ∈ Ox ∩ K, where Ox is a suitable open neighborhood of x = (0, 0). But here we have that F1 (x ) = ∅ for all x ∈ K1 ∩ Ox and F2 (y ) = ∅ for all y ∈ K2 ∩ Ox . In fact, we get P1 (v1 , x2 ) = ∅ for all v1 ∈ [0, 1] and P2 (y1 , v2 ) = ∅ for all v2 ∈ [0, 1]. Hence, the equilibrium existence results from [10] cannot be applied on the game Xi , i i=1,2 . 3. Continuity properties of the Nash equilibrium correspondence Assume that the set of players N and the sets of strategies X1 , ..., XN are fixed, any Xi is a non-empty, convex and compact subset of a metrizable locally convex topological vector space (d denotes the metric induced on X) and the payoffs are bounded functions: let us denote by g0 the space of such games. We endow g0 with the classical sup-norm metric: ρ(G, G ) =
i∈N
sup |ui (x) − ui (x)| .
x∈X
With N we denote the Nash equilibrium correspondence: N (G) = EG . The aim of this section is to study the continuity properties of N in a setting of games derived from that of Theorem 1. More precisely, we introduce a complete metric spaces of discontinuous games, denoted by g, where, with respect to the metric ρ, N is upper semicontinuous, with non-empty and compact values. Let us point out that, in our setting, a map is upper semicontinuous with compact values if and only if it is closed (see, for example, [6]). We note that the GSDP guarantees neither that N is closed nor that it has compact values: in fact, the set of Nash equilibria of the game given in Example 2 is not compact. So, in order to obtain the upper semicontinuity of N , we need to strengthen the GSDP. We introduce the following definition.
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Definition 4. G is said to have the Positive Generalized Single Deviation Property (PGSDP in short) if, whenever z ∈ X is not an equilibrium of G and uj (xj , z−j ) − uj (z) > t for some x ∈ X, j ∈ N and t > 0, there exists an open neighborhood Oz of z and a well-behaved map ξz : Oz ⇒ X such that, for every z ∈ Oz and every x ∈ ξz (z ), there is a player i for whom ui (xi , z−i ) − ui (z ) > t. The PGSDP allows the map N to be upper semicontinuous and compact valued (N is a closed map). So, since the game given in Example 2 has the GSDP and a non-closed set of Nash equilibria, we have that the GSDP does not imply the PGSDP. We denote by g the subspace of go of all games which satisfy both the PGSDP and the C-condition. Theorem 3. N is non-empty and compact valued and upper semicontinuous on g. Proof. First, let us prove that N is a closed map. Suppose that ρ (Gn , G) −→ 0, z n ∈ N (Gn ) for all n, z n −→ z and z ∈ / N (G). So, there exists an open neighborhood Oz of z, a well-behaved map ξz : Oz ⇒ X and t > 0 such that, for every z ∈ Oz and every x ∈ ξz (z ), there is a player i for whom ui (xi , z−i )− n n ui (z ) > t. For any n greater than a suitable no and for any x ∈ ξz (z ), there exists a player in so that n uin (xnin , z−i ) −uin (z n ) > t. This implies that there exists a subsequence (z n )n of (z n )n , a sequence (xn )n , n n where xn ∈ ξz (z n ) for all n , and a player i such that ui (xni , z−i ) − ui (z n ) > t for every n . On the other hand, since ρ (Gn , G) −→ 0, for n sufficiently large, we obtain: n n n ui (xni , z−i ) − ui (z n ) > t > ui (xni , z−i ) − ui (z n ) − uni (xni , z−i ) − uni (z n ) ,
that is: z n ∈ / N (Gn ), which is a contradiction. So, N is closed and it is obvious that N (G) is compact. Finally, N (G) is non-empty in light of Theorem 1. 2 The PGSDP and the C-condition have the following convergence properties with respect to the metric ρ. Proposition 2. Let (Gn )n be a sequence of games which have the PGSDP and G be such that ρ (Gn , G) −→ 0. Then, G has the PGSDP. / EG and uj (xj , z−j ) − uj (z) > t for some x ∈ X, j ∈ N and t > 0. Let t1 be such Proof. Assume that z ∈ that uj (xj , z−j ) − uj (z) > t1 > t. So, z ∈ / EGn for n sufficiently large and unj (xj , z−j ) − unj (z) > t1 . Since every game Gn has the PGSDP, we have: for n sufficiently large, there exists a well-behaved map ξzn : Ozn ⇒ X such that, for any z ∈ Ozn and any x ∈ ξzn (z ), some player in gets unin (xin , z−i ) − unin (z ) > t1 . n On the other hand, we obtain: sup (y,y )∈X×X
n ui (yi , y−i ) − uni (y ) − ui (yi , y−i ) − ui (y ) ≤ 2ρ(Gn , G) ∀ i ∈ N .
So, since ρ (Gn , G) −→ 0, for a suitable n, we get: sup (y,y )∈X×X
n ui (yi , y−i ) − uni (y ) − ui (yi , y−i ) − ui (y ) < t1 − t ∀ i ∈ N and
∀ z ∈ Ozn ∀ x ∈ ξzn (z ) ∃ in ∈ N such that unin (xin , z−i ) − unin (z ) − t > t1 − t , n
which implies:
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unin (xin , z−i ) − unin (z ) − t > t1 − t > unin (xin , z−i ) − unin (z ) − uin (xin , z−i ) − uin (z ) , n n n that is: uin (xin , z−i ) − uin (z ) > t. Finally, it is sufficient to set Oz = Ozn and ξx = ξzn , and the proof is n concluded. 2
Remark 6. Proposition 2 fails to be true in the class of games having the GSDP. Indeed, consider the sequence of 1-player games Gn = [0, 1], un , where
un (x) =
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x if x ∈ 0, 1 − n1
2 1 − n1 − x if x ∈ 1 − n1 , 1 0 if x = 1 .
(Gn )n converges to G = [0, 1], u , where u is the function defined in Remark 1, and u does not satisfy the GSDP. Proposition 3. Let (Gn )n be a sequence of games which satisfy the C-condition and G be such that ρ (Gn , G) −→ 0. Then, G satisfies the C-condition. Proof. Let F be a finite subset of strategy profiles and z = scoF . Since any Gn satisfies the C-condition, for any n there exists xn ∈ F such that uni (xni , z−i ) − uni (z) ≤ 0 for each i ∈ N . So, for at least a subsequence (xn )n of (xn )n , there exists x ∈ F so that uni (xi , z−i ) − uni (z) ≤ 0 for each i ∈ N and each n . Then, ui (xi , z−i ) − ui (z) ≤ 0 for each i ∈ N , which concludes the proof. 2 From Propositions 2 and 3 we have that g is a complete metric space with respect to the sup-norm ρ. This property allows to deduce that the map N is continuous (upper and lower semicontinuous) in a dense subset of g. In fact, since N is upper semicontinuous with non-empty and compact values, Theorem of Fort guarantees that N is lower semicontinuous on a dense subset of g.7 When the Nash equilibrium correspondence is lower semicontinuous on a set q, the Nash equilibria of any game belonging to q have a stability property called essential stability: Definition 5. [25] Let g ⊆ g0 and G ∈ g. A strategy profile x is said to be an essential Nash equilibrium of G relative to g if it is a Nash equilibrium of G and for any ε > 0 there exists δ > 0 such that any game G ∈ g with ρ(G, G ) < δ has at least one Nash equilibrium x such that d(x, x ) < ε. Previous results on the upper semicontinuity of the Nash equilibrium correspondence and the essential stability of equilibria have been provided in [26,20,7,21,8]. The assumptions considered in [26,20,7,8] are strengthening of those given in [17,3,9,15,4]; so, in light of Remark 3, g is not connected with the spaces of games introduced in the mentioned literature. In [21], classes of discontinuous games, where the essential stability of Nash equilibria holds in dense subsets, have been defined through properties on the aggregator of payoff functions ΦG (x, z) = i∈N [ui (xi , z−i ) − ui (z)]. It is easy to see that these properties imply those introduced in the present paper (Definitions 3 and 4). Nevertheless, because of the quasi-concavity like conditions, [21] and the present paper refer to different classes of discontinuous games. 7 Theorem of Fort. [13] Let Y and X be metric spaces and Y be a Baire’s space. Assume that T : Y ⇒ X is an upper semicontinuous map with non-empty and compact values. Then, there exists a dense Gδ subset Q of Y such that T is lower semicontinuous at any point belonging to Q. We recall that a subset of a topological space is a Gδ -set (in short Gδ ) if it is a countable intersection of open subsets. A Baire’s space is a topological space where any countable intersection of open dense subsets is a dense subset. Examples of Baire spaces are the complete metric spaces (see, for example, [1]).
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