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Hadamard well-posedness of the α-core
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Hadamard well-posedness of the α-core ✩ Zhe Yang a,b,∗ , Dawen Meng a,b a b
School of Economics, Shanghai University of Finance and Economics, Shanghai 200433, China Key Laboratory of Mathematical Economics (SUFE), Ministry of Education, Shanghai 200433, China
a r t i c l e
i n f o
Article history: Received 10 September 2016 Available online xxxx Submitted by M. Quincampoix Keywords: Hadamard well-posedness α-Core Abstract economy Pseudocontinuity Coalitionally C-secure game Essential stability
a b s t r a c t In this paper, we discuss the continuity property of the α-core with respect to data perturbations in different environments. We show that some collections of abstract economies (or normal games) with the nonempty α-core have the Hadamard well-posedness property. We also show that the α-core points of every game in a (dense) residual subset are essential. © 2017 Elsevier Inc. All rights reserved.
1. Introduction Two central solution concepts for normal-form games are Nash equilibrium (NE) and the core. Nash equilibrium is a noncooperative solution in which only the individual behaviors are considered, whereas the core is a cooperative solution involving the collective behaviors of players. The existence and continuity of solutions are two important topics in the game theory. The existence of Nash equilibria has been extensively studied in the literature (see Nash [15], Debreu [8], Dasgupta and Maskin [7], Simon and Zame [21], Tian and Zhou [22,23], Reny [16], McLennan et al. [11], Barelli and Meneghel [3]). Comparing with NE, the definition of cores is less unified since it varies with the “blocking” ways of a coalition. Among various blocking concepts defined in the literature, the α-core due to Aumann [2] attracts a significant attention. We adopt this concept throughout this paper. A coalition is said to α-block a given social state if it has a feasible strategy with which the coalition can ensure a social ✩ We are grateful to an associate editor and anonymous referees for their valuable comments and suggestions. Yang wishes to thank Jinfeng Liu for her assistance. Yang acknowledges the financial supports of National Natural Science Foundation of China (No. 11501349), Chen Guang Project sponsored by the Shanghai Municipal Education Commission and Shanghai Education Development Foundation (No. 13CG35), the open project of Key Laboratory of Mathematical Economics (SUFE), Ministry of Education (No. 201309KF02). Meng acknowledges the financial supports of National Natural Science Foundation of China (No. 71301094) and the open project of Key Laboratory of Mathematical Economics (SUFE), Ministry of Education (No. 201309KF02). * Corresponding author at: School of Economics, Shanghai University of Finance and Economics, Shanghai 200433, China. E-mail address: [email protected] (Z. Yang).
http://dx.doi.org/10.1016/j.jmaa.2017.03.038 0022-247X/© 2017 Elsevier Inc. All rights reserved.
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state preferred by all the agents in it regardless of the complementary coalition’s actions. The α-core is the set of social states that cannot be α-blocked by any coalition. There are a number of papers in the literature dealing with the nonemptiness of the α-core. Scarf [20] proved the nonemptiness of the α-core of normal games with nontransferable utilities. Zhao [31] extended the result to transferable utilities. Kajii [10] showed the nonemptiness of the α-core with nonordered preferences. Martins-da-Rocha and Yannelis [12] gave the nonemptiness result for games with nonordered and discontinuous preferences. More recently, Uyanik [24] provided sufficient conditions for the nonemptiness of α-cores in games with possibly discontinuous payoff functions for cases with and without transferable utilities, respectively. When it comes to the continuity property, we need to introduce two concepts, i.e., Hadamard wellposedness (H-wp) and essentiality. H-wp refers to the continuous dependence of a solution on small change of the game. A solution of a game is essential if any perturbed game has a solution approximated to it. Yu et al. [27] studied the well-posedness of Nash equilibrium points in some classes of noncooperative games. Scalzo [17] proved that some classes of games with weaker conditions (i.e., better-reply security and pseudocontinuity) also have the H-wp property. A classical reference for the essential stability of Nash equilibria is Wu and Jiang [25]. Later, a great deal of work has been done (cf. Yu [28], Carbonell-Nicolau [5,6], Scalzo [18, 19], Yang [26], Zhou [29,30]). It is still an open question whether the α-core correspondence is continuous. To the best of our knowledge, there are few existing literature investigating this issue. We will fill this gap in the present paper by discussing both H-wp and essential stability of the α-core in different collections of games. The rest of the paper is organized as follows. Section 2 recalls some preparatory definitions and results. In Section 3, we study the Hadamard well-posedness for three different models. In Section 4, we analyze the essential stability of the α-core. Section 5 concludes with some remarks. 2. Settings and preliminaries We consider an abstract economy with a finite set I := {1, · · · , N } of agents. A subset S ⊆ I represents a coalition and I = 2I \{∅} is the family of all admissible coalitions. Each agent i ∈ I chooses an individual strategy xi in his strategy set Xi . Xi is assumed to be a nonempty subset of a Hausdorff topological space. A strategy profile x ∈ X = i∈I Xi is called a social state, fi : X −→ R is the payoff function of player i. We represent by XS = i∈S Xi the strategy space of the coalition S and X−S = i∈I\S Xi its complement. GS : X ⇒ XS denotes the feasible strategy correspondence of the coalition S. Our abstract economy can therefore be represented by the family Γ = {I, (Xi )i∈I , (fi )i∈I , (GS )S∈I }. If GS (x) = XS for each x ∈ X and each S ∈ I, an abstract economy can be simplified to a normal game Γ = {I, (Xi )i∈I , (fi )i∈I }. A core allocation is a feasible social state that is robust to all possible coalitional deviations. Since actions of the agents outside the coalition affect the welfare of the members inside the coalition, it is necessary to consider the reacting way of outsiders when defining a core solution. Aumann [2] suggests a possible “blocking” way as follows: a coalition is said to α-block a given social state if it has a feasible strategy with which the coalition can ensure a social state preferred by all the agents in it regardless of strategies the other agents may choose. Precisely, a coalition S is said to α-block a social state x ∈ X if there exists zS ∈ GS (x) such that fi (zS , v−S ) > fi (x), ∀v−S ∈ X−S , ∀i ∈ S. A strategy profile x ∈ X is in the α-core of Γ if x ∈ GI (x) and no coalition S ∈ I can α-block x. We denote by C (Γ) the α-core of the abstract economy Γ. In this paper, the payoff functions of a game are allowed to be discontinuous. We next introduce some conditions about payoffs: pseudocontinuity, coalitional C-security and strongly coalitional C-security, which are all strictly weaker than continuity.
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Definition 2.1 ([13]). A real-valued function f , defined on a topological space Z, is said to be: upper pseudocontinuous at x0 ∈ Z if for all x ∈ Z such that f (x0 ) < f (x), we have lim supy→x0 f (y) < f (x); lower pseudocontinuous at x0 ∈ Z if −f is upper pseudocontinuous at x0 ; pseudocontinuous if it is both upper and lower pseudocontinuous. Definition 2.2 ([24]). A normal game Γ = {I, (Xi )i∈I , (fi )i∈I } is coalitionally C-secure (CCS) if for each x∈ / C (Γ), there exist vSx ∈ R|S| and ySx ∈ XS for each S ∈ I and an open neighborhood U x of x such that (i) fi (ySx , z−S ) − (vSx )i ≥ 0, ∀z−S ∈ X−S , ∀i ∈ S, ∀S ∈ I; (ii) for each x ∈ U x , there exists S ∈ I such that (vSx )i − fi (x ) > 0, ∀i ∈ S. Definition 2.3. A normal game Γ = {I, (Xi )i∈I , (fi )i∈I } is strongly coalitionally C-secure (SCCS) if for each x∈ / C (Γ), there exist {(vSx , ySx ) ∈ R|S| × XS : S ∈ I}, ε > 0 and an open neighborhood U x of x such that (i) fi (ySx , z−S ) − (vSx )i ≥ 0, ∀z−S ∈ X−S , ∀i ∈ S, ∀S ∈ I; (ii) for each x ∈ U x , there exists S ∈ I such that (vSx )i − fi (x ) > ε, ∀i ∈ S. Note that SCCS implies CCS, but the converse is not true. We will show in the next section that CCS cannot guarantee the Hadamard well-posed property of the α-core. So, a strengthened version, SCCS, is introduced here (see Theorem 3.3 and Example 3.2 for detailed discussions). We next give some preparatory definitions and lemmas on set-valued mappings (due to [1] and [17]) for the proofs of our main results in the subsequent sections. A set-valued mapping F : X ⇒ Y , where X and Y are two Hausdorff topological spaces, is said to be: upper semicontinuous at x ∈ X if, for any open subset O of Y with O ⊃ F (x), there exists an open neighborhood U (x) of x such that O ⊃ F (x ) for any x ∈ U (x); upper semicontinuous on X, if it is upper semicontinuous at each x ∈ X; lower semicontinuous at x ∈ X if, for any open subset O of Y with O ∩ F (x) = ∅, there exists an open neighborhood U (x) of x such that O ∩ F (x ) = ∅ for any x ∈ U (x); lower semicontinuous on X, if it is lower semicontinuous at each x ∈ X; closed if Graph(F ) = {(x, y) ∈ X × Y : y ∈ F (x)} is a closed subset of X × Y . Lemma 2.1 ([1]). A correspondence with compact Hausdorff range space is closed if and only if it is upper semicontinuous and closed-valued. Lemma 2.2 ([17]). A real-valued function f , defined on a Hausdorff topological space, is pseudocontinuous if and only if whenever f (x) < f (z), there exist an open neighborhood Nx of x and an open neighborhood Nz of z such that f (x ) < f (z ) for all x ∈ Nx and all z ∈ Nz . Lemma 2.3 ([17]). If a real-valued function f , defined on a connected topological space X, is pseudocontinuous on X, then the following property holds: f (x1 ) < f (x2 ) =⇒ ]f (x1 ), f (x2 )[∩f (X) = ∅. 3. Main results Let Y be a set of games endowed with a topology, and C : Y ⇒ X be the α-core correspondence. We say that a game y ∈ Y is generalized Hadamard well-posed if C (y) = ∅, and for any net (xβ , y β )β in X × Y with y β → y and xβ ∈ C (y β ), there exists a cluster point of (xβ )β belonging to C (y). If y is generalized Hadamard well-posed and C (y) is a singleton set, then we say that y is Hadamard well-posed. Besides, we
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say that Y has the Hadamard well-posedness property if every y ∈ Y is (generalized) Hadamard well-posed. Yu et al. [27] gave the following sufficient conditions for Hadamard well-posedness. Theorem 3.1 ([27]). Let X and Y be two Hausdorff topological spaces and F : Y ⇒ X be a set-valued mapping. (a) If F is upper semicontinuous at y ∈ Y , F (y) is nonempty and compact, then y is generalized Hadamard well-posed. (b) If F is upper semicontinuous at y ∈ Y and F (y) = {x∗ }, then y is Hadamard well-posed. Next, we study the Hadamard well-posedness for three different models. Model (A) concerns the abstract economies with pseudocontinuous payoffs and continuous feasible strategy correspondences. Model (B) deals with the strongly coalitionally C-secure games. Model (C) is about the abstract economies with nonordered preferences. Moreover, we show that the pseudocontinuous normal game is also strongly coalitionally C-secure if the space of action profiles is a nonempty and compact subset of a Hausdorff topological space, and the class of coalitionally C-secure games fails to satisfy the Hadamard well-posedness property. 3.1. Model (A) For each i ∈ I, let Xi be a nonempty connected compact subset of a metric space, HS be the Hausdorff metric on XS and d be the metric on X. We denote by Mp the set of abstract economies Γ = {I, (Xi )i∈I , (fi )i∈I , (GS )S∈I } having a common set of strategy profiles X, where (1) for each i ∈ I, fi : X −→ R is pseudocontinuous on X, (2) for each S ∈ I, GS : X ⇒ XS is a continuous set-valued mapping with nonempty compact values, (3) the α-core is nonempty, i.e., C (Γ) = ∅, ∀Γ ∈ Mp . Moreover, we define a metric ρ over Mp as follows ρ(Γ, Γ ) :=
i∈I
sup |fi (x) − fi (x)| + max sup HS (GS (x), GS (x)).
x∈X
S∈I x∈X
Theorem 3.2. Mp has the Hadamard well-posedness property. Proof. First, we need to show that the correspondence C : Mp ⇒ X is upper semicontinuous with nonempty compact values. It is obvious that C (Γ) = ∅ for each Γ ∈ Mp . By Lemma 2.1, it suffices to show that Graph(C ) is closed. Suppose that (Γβ , xβ )β is a net in Mp × X with xβ ∈ C (Γβ ) and (Γβ , xβ ) −→ (Γ, x) ∈ Mp × X, then xβ ∈ GβI (xβ ). Since GI is continuous and Γβ −→ Γ, we have d(x, GI (x)) ≤ d(x, xβ ) + d(xβ , GβI (xβ )) + HI (GβI (xβ ), GI (xβ )) + HI (GI (xβ ), GI (x)) ≤ d(x, xβ ) + HI (GI (xβ ), GI (x)) + ρ(Γβ , Γ) −→ 0. Therefore, x ∈ GI (x). We next show that x cannot be α-blocked by any coalition S. Suppose, by way of contradiction, that there exist S ∈ I and zS ∈ GS (x) such that fi (zS , v−S ) > fi (x), ∀v−S ∈ X−S , ∀i ∈ S. For any i, since fi is pseudocontinuous on X and X−S is compact, by Proposition 2.1 in [14], there exists 0 v−S ∈ X−S such that 0 fi (zS , v−S ) ≥ fi (zS , v−S ) > fi (x), ∀v−S ∈ X−S .
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By Lemma 2.3, there exist y1i , y2i ∈ X such that 0 fi (zS , v−S ) ≥ fi (zS , v−S ) > fi (y1i ) > fi (y2i ) > fi (x), ∀v−S ∈ X−S .
Given v−S ∈ X−S , in light of Lemma 2.2, there exist open neighborhoods Ovi −S (zS ), Oi (v−S ) and Oi (x) containing, respectively, zS , v−S and x such that fi (zS , v−S ) > fi (y1i ) > fi (y2i ) > fi (x ), ∀zS ∈ Ovi −S (zS ), ∀v−S ∈ Oi (v−S ), ∀x ∈ Oi (x). 1 L Since X−S is nonempty and compact, there exists a finite subset {v−S , · · · , v−S } of X−S such that L
X−S =
k Oi (v−S ).
k=1
Let Oi (zS ) =
L
Ovi k (zS ). −S
k=1
We get: fi (zS , v−S ) > fi (y1i ) > fi (y2i ) > fi (x ), ∀zS ∈ Oi (zS ), ∀v−S ∈ X−S , ∀x ∈ Oi (x). Set O(zS ) =
Oi (zS ),
i∈S
O(x) =
Oi (x),
i∈S
εS = min{fi (y1i ) − fi (y2i )} > 0. i∈S
Then, for any zS ∈ O(zS ), any v−S ∈ X−S , and any x ∈ O(x), fi (zS , v−S ) − fi (x ) > εS , ∀i ∈ S. Moreover, since HS (GβS (xβ ), GS (x)) ≤ HS (GβS (xβ ), GS (xβ )) + HS (GS (xβ ), GS (x)) ≤ ρ(Γβ , Γ) + HS (GS (xβ ), GS (x)) −→ 0, it follows that there exists a net (zSβ )β of XS such that zSβ ∈ GβS (xβ ) and zSβ −→ zS . So, there exists β1 such that zSβ ∈ O(zS ) and xβ ∈ O(x) whenever β β1 . Then, zSβ ∈ GβS (xβ ) and fi (zSβ , v−S ) − fi (xβ ) > εS , ∀v−S ∈ X−S , ∀i ∈ S. Additionally, since Γβ −→ Γ, then for
εS 2
> 0, there exists β2 such that for any β β2 ,
fiβ (z ) − fiβ (x ) > fi (z ) − fi (x ) −
εS , ∀(z , x ) ∈ X × X, ∀i ∈ S. 2
For any sufficiently large β, we have that zSβ ∈ GβS (xβ ) and
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fiβ (zSβ , v−S ) − fiβ (xβ ) > fi (zSβ , v−S ) − fi (xβ ) −
εS εS εS > εS − = > 0, ∀v−S ∈ X−S , ∀i ∈ S, 2 2 2
thereby contradicting the fact that xβ ∈ C (Γβ ). Finally, the Hadamard well-posedness property follows directly from Theorem 3.1.
2
3.2. Model (B) For each i ∈ I, let Xi be a nonempty compact subset of a Hausdorff topological space. We denote by Mc (resp. Msc ) the set of (resp. strongly) coalitionally C-secure games Γ = {I, (Xi )i∈I , (fi )i∈I } having bounded payoffs, a common set of strategy profiles X and a nonempty α-core. It follows directly from the definitions of SCCS and CCS that Msc ⊂ Mc . Moreover, we define the metric ρ on Mc as follows: ρ(Γ, Γ ) :=
i∈I
sup |fi (x) − fi (x)|.
x∈X
Theorem 3.3. Msc has the Hadamard well-posedness property. Proof. An argument analogous to that in Theorem 3.2 shows that the correspondence C : Msc ⇒ X is upper semicontinuous with nonempty compact values. The nonemptiness of the α-core is obvious. By Lemma 2.1, it suffices to show that Graph(C ) is closed. Suppose that (Γβ , xβ )β is a net in Msc × X with xβ ∈ C (Γβ ) and (Γβ , xβ ) −→ (Γ, x) ∈ Msc × X. The proof will be completed if we show that x ∈ C (Γ). Suppose that x ∈ / C (Γ). Then there exist {(vSx , ySx ) ∈ R|S| ×XS : S ∈ I}, ε > 0 and an open neighborhood x U of x such that (1) for each S ∈ I and each z−S ∈ X−S , fi (ySx , z−S ) − (vSx )i ≥ 0, ∀i ∈ S; (2) for each x ∈ U x , there exists S ∈ I such that (vSx )i − fi (x ) > ε, ∀i ∈ S. It implies that there exists β1 such that for any β β1 , there exists S ∈ I such that (vSx )i > fi (xβ ) + ε, ∀i ∈ S. As I is a finite set, it follows that there exist S0 ∈ I and a subnet {xβk } of {xβ } satisfying fi (ySx0 , z−S0 ) ≥ (vSx0 )i > fi (xβk ) + ε, ∀z−S0 ∈ X−S0 , ∀i ∈ S0 . Since Γβk −→ Γ, then for any sufficiently large βk , ε fiβk (ySx0 , z−S0 ) − fiβk (z ) > fi (ySx0 , z−S0 ) − fi (z ) − , ∀z−S0 ∈ X−S0 , ∀z ∈ X, ∀i ∈ S0 , 2 implying that fiβk (ySx0 , z−S0 ) − fiβk (xβk ) > fi (ySx0 , z−S0 ) − fi (xβk ) −
ε ε > ε − > 0, ∀z−S0 ∈ X−S0 , ∀i ∈ S0 . 2 2
It contradicts the fact that xβk ∈ C (Γβk ). Finally, the Hadamard well-posedness property of Msc follows directly from Theorem 3.1. 2 The next result shows that a pseudocontinuous normal game is also strongly coalitionally C-secure if the space of action profiles is a nonempty and compact subset of a Hausdorff topological space.
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Theorem 3.4. Assume that the normal game Γ = {I, (Xi )i∈I , (fi )i∈I } satisfies the following conditions: (1) for each i ∈ I, Xi is a nonempty compact subset of a Hausdorff topological space; (2) for each i ∈ I, fi is pseudocontinuous on X. Then the game Γ is strongly coalitionally C-secure. Moreover, if C (Γ) = ∅, then Γ is generalized Hadamard well-posed. / C (Γ), then there exist S0 ∈ I and ySx0 ∈ XS0 such that Proof. Suppose that x ∈ fi (ySx0 , z−S0 ) > fi (x), ∀z−S0 ∈ X−S0 , ∀i ∈ S0 . Since fi pseudocontinuous on X and Xi is compact for each i ∈ I, it follows from Proposition 2.1 of [14] that for each i ∈ S0 , there exists z i−S0 ∈ X−S0 such that fi (ySx0 , z−S0 ) ≥ fi (ySx0 , z i−S0 ) > fi (x), ∀z−S0 ∈ X−S0 . By Lemma 2.2, there exists an open neighborhood O(x) of x in X such that fi (ySx0 , z−S0 ) ≥ fi (ySx0 , z i−S0 ) > fi (x ), ∀z−S0 ∈ X−S0 , ∀i ∈ S0 , ∀x ∈ O(x). Since X is a nonempty compact subset of a Hausdorff topological space, it follows from Theorem 2.48 of [1] that X is normal. Then, there exists an open neighborhood U (x) of x in X such that U (x) ⊂ clU (x) ⊂ O(x) ⊂ X and for each i ∈ S0 , fi (ySx0 , z−S0 ) ≥ fi (ySx0 , z i−S0 ) > fi (x ), ∀z−S0 ∈ X−S0 , ∀x ∈ clU (x). Thus clU (x) is compact. Moreover, by Proposition 2.1 of [14], we have that for each i ∈ S0 , there exists x ∈ clU (x) such that i
fi (xi ) =
max
x ∈clU (x)
fi (x ),
fi (ySx0 , z−S0 ) ≥ fi (ySx0 , z i−S0 ) > fi (xi ) ≥ fi (x ), ∀z−S0 ∈ X−S0 , ∀x ∈ clU (x). Let ε=
1 min fi (ySx0 , z i−S0 ) − fi (xi ) > 0, 2 i∈S0
(vSx0 )i = fi (ySx0 , z i−S0 ), ∀i ∈ S0 , vSx0 = (vSx0 )i∈S0 ∈ R|S0 | , (vSx )i = min fi (z), ∀i ∈ S, ∀S = S0 , z∈X
vSx
=
(vSx )i∈S
∈ R|S| , ∀S = S0 .
Then, for each i ∈ S0 , fi (ySx0 , z−S0 ) ≥ (vSx0 )i > fi (x ) + ε, ∀z−S0 ∈ X−S0 , ∀x ∈ U (x). For any S = S0 , we can select a point ySx ∈ XS such that
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fi (ySx , z−S ) ≥ (vSx )i , ∀z−S ∈ X−S , ∀i ∈ S. Therefore, for each x ∈ X that is not in the α-core of Γ, there exist {(vSx , ySx ) ∈ R|S| × XS : S ∈ I}, ε > 0 and an open neighborhood U (x) of x such that (i) for each S ∈ I and each z−S ∈ X−S , fi (ySx , z−S ) − (vSx )i ≥ 0, ∀i ∈ S; (ii) for each x ∈ U (x), there exists S0 ∈ I such that (vSx0 )i − fi (x ) > ε, ∀i ∈ S0 . Therefore, Γ is strongly coalitionally C-secure. Finally, taking C (Γ) = ∅ into consideration, by Theorem 3.3, we obtain that Γ is generalized Hadamard well-posed. 2 Note that the strongly coalitional C-security does not imply the pseudocontinuity. See the following counterexample. Example 3.1. Consider the game Γ = (X1 , X2 , f1 , f2 ), where I = {1, 2}, X1 = X2 = [0, 1], X = X1 × X2 , f1 (x1 , x2 ) = x1 , x1 ∈ [0, 1], 0 x2 ∈ [0, 1) f2 (x1 , x2 ) = 1 x2 = 1. It is easy to verify that the normal game Γ is strongly coalitionally C-secure. Now 1 1 f2 (1, 1) = 1 > f2 ( , ), 2 2 1 1 lim inf f2 (y) = 0 = f2 ( , ). 2 2 y−→(1,1) Therefore, f2 is not pseudocontinuous on X. The following counterexample shows that Mc does not satisfy the Hadamard well-posedness property. Example 3.2. Let X = [0, 10], ⎧ ⎪ ⎨ 10 x = 0 f (x) = x x ∈ (0, 10) ⎪ ⎩0 x = 10. The one-person game Γ = (X, f ) is coalitionally C-secure and C (Γ) = {0}. We claim that Γ is not strongly coalitionally C-secure. Otherwise, from the fact that x = 10 is not in the α-core of Γ, it follows that there exist v x ∈ R, y x ∈ X, ε > 0 and an open neighborhood U (x) of x such that f (y x ) ≥ v x > f (x ) + ε, ∀x ∈ U (x). It follows that 10 = sup f (z) ≥ f (y x ) ≥ v x ≥ z∈X
We get a contradiction. Therefore, Γ ∈ Mc \Msc .
sup f (x ) + ε = 10 + ε. x ∈U (x)
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Let ⎧ ⎪ 10 − n3 ⎪ ⎪ ⎨x
f n (x) = ⎪ −x + 2 10 − n1 ⎪ ⎪ ⎩0
x=0
x ∈ 0, 10 − n1 x ∈ 10 − n1 , 10 x = 10.
It is easy to check that each Γn = (X, f n ) is coalitionally C-secure, Γn −→ Γ and C (Γn ) = {10 − Therefore, Γ is not Hadamard well-posed.
1 n }.
3.3. Model (C) In this subsection, we deal with an abstract economy with nonordered preferences. Each agent i has a preference correspondence Pi : X ⇒ X and each coalition S ∈ I has a feasible strategy corre = spondence GS : X ⇒ XS . This abstract economy can therefore be represented by the family Γ {I, (Xi )i∈I , (Pi )i∈I , (GS )S∈I }. A coalition S is said to α-block a social state x ∈ X if there exists zS ∈ GS (x) if x ∈ GI (x) and no such that {zS } × X−S ⊂ Pi (x), ∀i ∈ S. A social state x ∈ X is in the α-core of Γ coalition S ∈ I can α-block x. For each i ∈ I, let Xi be a nonempty compact subset of a metric space, d be the metric on X, HS be the Hausdorff metric on XS and H be the Hausdorff metric on X × X. We denote by Mg the set of abstract = {I, (Xi )i∈I , (Pi )i∈I , (GS )S∈I } having a common set of strategy profiles X, where (1) for economies Γ each i ∈ I, Pi : X ⇒ X has the open graph in X × X, (2) for each S ∈ I, GS : X ⇒ XS is a continuous set-valued mapping with nonempty compact values, (3) the α-core is nonempty. We consider the topology over Mg generated by the metric ρ defined in the following way Γ ) := max H(Graph(Pi )c , Graph(P )c ) + max sup HS (GS (x), G (x)). ρ(Γ, i S i∈I
S∈I x∈X
Theorem 3.5. Mg has the Hadamard well-posedness property. Proof. Applying Theorem 3.1, it suffices to show that the correspondence C : Mg ⇒ X is upper semicontinuous with nonempty compact values. It follows directly from the definition of Mg that C (Γ) = ∅, ∀Γ ∈ Mg . β , xβ )β is a net in Mg × X with From Lemma 2.1, we need to show that Graph(C ) is closed. Suppose that (Γ β β β β β β β β −→ Γ, x ∈ C (Γ ) and (Γ , x ) −→ (Γ, x) ∈ Mg × X, then x ∈ GI (x ). Since GI is continuous and Γ we have d(x, GI (x)) ≤ d(x, xβ ) + d(xβ , GβI (xβ )) + HI (GβI (xβ ), GI (xβ )) + HI (GI (xβ ), GI (x)) β , Γ) −→ 0. ≤ d(x, xβ ) + HI (GI (xβ ), GI (x)) + ρ(Γ Therefore, x ∈ GI (x). Suppose, by way of contradiction, that x can be α-blocked by some coalition S. Then {zS } × X−S ⊂ Pi (x), ∀i ∈ S, for some zS ∈ GS (x), that is, ({x} × {zS } × X−S ) Since
Graph(Pi )c = ∅, ∀i ∈ S.
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β , Γ) + HS (GS (xβ ), GS (x)) −→ 0, HS (GβS (xβ ), GS (x)) ≤ HS (GβS (xβ ), GS (xβ )) + HS (GS (xβ ), GS (x)) ≤ ρ(Γ it follows that there exists a net (zSβ )β of XS such that zSβ ∈ GβS (xβ ) and zSβ −→ zS . β , xβ ) −→ (Γ, x), we have From zSβ −→ zS and (Γ
{xβ } × {zSβ } × X−S
Graph(Piβ )c = ∅, ∀i ∈ S,
for any sufficiently large β. It implies that zSβ ∈ GβS (xβ ) and {zSβ } × X−S ⊂ Piβ (xβ ), ∀i ∈ S, β ). Finally, the Hadamard well-posedness property follows thereby contradicting the fact that xβ ∈ C (Γ directly from Theorem 3.1. 2 4. Essential stability In this section, we discuss the essential stability of the α-core. We denote by X a set of strategy profiles endowed with a metric d, M ∈ {Mp , Msc , Mg } represents a class of abstract economies (or normal games) endowed with certain metric ρM and Γ ∈ M . A point x ∈ C (Γ) is said to be essential if for any open neighborhood O of x, there exists δ > 0 such that C (Γ ) ∩ O = ∅ for any Γ ∈ M with ρM (Γ , Γ) < δ. We say that Γ is essential if every point of C (Γ) is essential. Note that Γ is essential if and only if the α-core correspondence C : M ⇒ X is lower semicontinuous at Γ. The standard approach to studying the essential stability consists in identifying a collection M with the following properties: (i) M is a complete metric space; (ii) every member of M has a nonempty set of solutions; (iii) the solution correspondence is compact-valued and upper semicontinuous. Items (i)–(iii), together with the following lemma, imply the essential stability. Lemma 4.1 ([9]). Suppose that X is a Hausdorff topological metric space and Y a metric space. Suppose further that F : X ⇒ Y is a compact-valued and upper semicontinuous correspondence with F (x) = ∅ for all x ∈ X. Then there exists a residual subset Q of X such that F is lower semicontinuous at every point in Q. The following lemma due to [5,6,18,26] strengthens Lemma 4.1. Lemma 4.2. Suppose that X is a complete metric space and Y a metric space. Suppose further that F : X ⇒ Y is a compact-valued and upper semicontinuous correspondence with F (x) = ∅ for all x ∈ X. Then there exists a dense residual subset Q of X such that F is lower semicontinuous at every point in Q. To apply Lemma 4.2, we need to prove the following result. Lemma 4.3. (Mg , ρ) is a complete metric space. ∈ Mg such that n }∞ in Mg , it suffices to show that there exists Γ Proof. For any Cauchy sequence {Γ n=1 n Γ −→ Γ under the distance ρ. For any ε > 0, there exists N0 > 0 such that for any n, m > N0 , max H(Graph(Pin )c , Graph(Pim )c ) + max sup HS (GnS (x), Gm S (x)) < ε. i∈I
S∈I x∈X
Referring to the proof of Theorem 2.2 in [26], we have: for each i ∈ I, there exists an open-graph correspondence Pi : X ⇒ X such that Graph(Pin )c −→ Graph(Pi )c . It is easy to show that for each S ∈ I,
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there exists a compact-valued and continuous correspondence GS : X ⇒ XS such that GS (x) = ∅ for all x ∈ X and supx∈X HS (GnS (x), GS (x)) −→ 0. Now, the only work left is to show that the α-core of = {I, (Xi )i∈I , (Pi )i∈I , (GS )S∈I } is nonempty. From the nonemptiness of α-core of Γ n , we have that there Γ n ). Since X is compact, without loss of generality, we assume that xn −→ x ∈ X. The proof exists xn ∈ C (Γ of x ∈ C (Γ) is omitted here since it is similar to the proof of Theorem 3.5. 2 Next, using Lemmas 4.1 and 4.2, we have the following results. Theorem 4.1. For any M ∈ {Mp , Msc }, there exists a residual subset Q of M such that every member of Q is essential. Theorem 4.2. There exists a dense residual subset Q of Mg such that every member of Q is essential. Note that Mp and Msc are not complete metric spaces, see the following counterexample. Example 4.1. Let X = [0, 1] and ⎧ ⎪ ⎨0 f (x) = 0 ⎪ ⎩1
2
⎧ ⎪ ⎪ ⎪ ⎨
x=0 x=1 x ∈ (0, 1),
0 0 f n (x) = 1 1 ⎪ + ⎪ 2 nx ⎪ ⎩−1x + 1 + n n
1 2
x=0 x=1 x ∈ [0, 12 ] x ∈ ( 12 , 1).
It is easy to check that ρ(f n , f ) −→ 0 as n → ∞, each f n is pseudocontinuous on [0, 1] and the one-person game (X, f ) is not strongly coalitionally C-secure. So, this example shows that the metric spaces Mp and Msc are not complete. The following examples show that C (·) is not necessarily lower semicontinuous for every member of M ∈ {Mp , Msc , Mg }. Example 4.2. Let I = {1, 2}, X1 = X2 = [0, 1], X = [0, 1] × [0, 1], f1 (x1 , x2 ) = 1, f2 (x1 , x2 ) = 2, GS (x) = XS , ∀x ∈ X, ∀S ∈ I. Then the game Γ = (X1 , X2 , f1 , f2 , (GS )S∈I ) ∈ Mp ∩ Msc and C (Γ) = [0, 1] × [0, 1]. For each n, define f1n , f2n : X −→ R by f1n (x1 , x2 ) = 1 − f2n (x1 , x2 ) = 2 +
1 x1 , n
1 x2 , ∀x ∈ X. n
Then the game Γn = (X1 , X2 , f1n , f2n , (GS )S∈I ) ∈ Mp ∩ Msc , C (Γn ) = {0} × {1} and ρ(Γn , Γ) −→ 0. Thus, C : Mp ⇒ X and C : Msc ⇒ X are not lower semicontinuous at Γ. Example 4.3. Let I = {1, 2}, X1 = X2 = [0, 1], X = [0, 1] × [0, 1], the preference correspondences P1 , P2 : X ⇒ X be defined as
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P1 (x1 , x2 ) = P2 (x1 , x2 ) = ∅, GS (x) = XS , ∀x ∈ X, ∀S ∈ I. = (X1 , X2 , P1 , P2 , (GS )S∈I ) ∈ Mg and C (Γ) = [0, 1] × [0, 1]. For each n, Then the abstract economy Γ n n define the correspondences P1 , P2 : X ⇒ X by P1n (x1 , x2 )
=
P2n (x1 , x2 )
=
1 ], x2 ∈ [0, 1] [0, x1 ) × [0, 1] x1 ∈ [0, 2n 1 1 [0, 2n ) × [0, 1] x1 ∈ ( 2n , 1], x2 ∈ [0, 1] 1 ] [0, 1] × [0, x2 ) x1 ∈ [0, 1], x2 ∈ [0, 2n 1 1 [0, 1] × [0, 2n ) x1 ∈ [0, 1], x2 ∈ ( 2n , 1].
n = (X1 , X2 , P n , P n , (GS )S∈I ) ∈ Mg , C (Γ −→ 0. n ) = {0} × {0} and ρ(Γ n , Γ) Then the abstract economy Γ 1 2 Thus, C : Mg ⇒ X is not lower semicontinuous at Γ. Remark 4.1. Yang [26] studied only the essential stability of the α-core for qualitative games. We make the following extensions in the present paper. (1) We obtain the essential stability of the α-core for abstract economies. (2) Using payoff perturbations, we obtain the essential stability of the α-core for the classes Mp and Msc . 5. Conclusions Nash equilibrium and α-core are two major solution concepts for normal form games. While both the existence and continuity of Nash equilibria have been studied extensively in the literature, the results on the α-core are only about its existence/nonemptiness (see [2,4,10,24]). Little attention has been devoted to the continuity of the α-core. In this paper, we fill this gap by analyzing the Hadmard well-posedness and essential stability of the α-core. We show that the Hadmard well-posedness property holds for the following three models: (i) games with pseudocontinuous payoffs and continuous feasible strategy correspondences; (ii) strongly coalitionally C-secure games; (iii) abstract economies with nonordered preferences. Also, we study the essential stability of the α-core by showing that every point of the α-core is essential for every game in a residual (dense) subset of the space of games. References [1] C. Aliprantis, K.C. Border, Infinite Dimensional Analysis, 3rd edn., Springer, Berlin, 2006. [2] R.J. Aumann, The core of a cooperative game without side payments, Trans. Amer. Math. Soc. 98 (1961) 539–552. [3] P. Barelli, I. Meneghel, A note on the equilibrium existence problem in discontinuous games, Econometrica 81 (2013) 813–824. [4] K.C. Border, A core existence theorem for games without ordered preferences, Econometrica 52 (1984) 1537–1542. [5] O. Carbonell-Nicolau, Essential equilibria in normal-form games, J. Econom. Theory 145 (2010) 421–431. [6] O. Carbonell-Nicolau, Further results on essential Nash equilibria in normal-form games, Econom. Theory 59 (2015) 277–300. [7] P. Dasgupta, E. Maskin, The existence of equilibrium in discontinuous economic games. Part I: theory, Rev. Econ. Stud. 53 (1986) 1–26. [8] G. Debreu, A social equilibrium existence theorem, Proc. Nat. Acad. Sci. 38 (1952) 886–893. [9] M.K. Fort, Points of continuity of semicontinuous functions, Publ. Math. Debrecen 2 (1951) 100–102. [10] A. Kajii, A generalization of Scarf’s theorem: an α-core existence theorem without transitivity or completeness, J. Econom. Theory 56 (1992) 194–205. [11] A. McLennan, P.K. Monteiro, R. Tourky, Games with discontinuous payoffs: a strengthening of Reny’s existence theorem, Econometrica 79 (2011) 1643–1664. [12] V. Martins-da-Rocha, N. Yannelis, Nonemptiness of the alpha core, working paper, Manchester School of Social Sciences, University of Manchester, 2011. [13] J. Morgan, V. Scalzo, Discontinuous but well-posed optimization problems, SIAM J. Optim. 17 (3) (2006) 861–870. [14] J. Morgan, V. Scalzo, Pseudocontinuous functions and existence of Nash equilibria, J. Math. Econom. 43 (2) (2007) 174–183.
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[15] J. Nash, Noncooperative games, Ann. of Math. 54 (1951) 286–295. [16] P.J. Reny, On the existence of pure and mixed strategy Nash equilibria in discontinuous games, Econometrica 67 (5) (1999) 1029–1056. [17] V. Scalzo, Hadamard well-posedness in discontinuous non-cooperative games, J. Math. Anal. Appl. 360 (2009) 697–703. [18] V. Scalzo, Essential equilibria of discontinuous games, Econom. Theory 54 (2013) 27–44. [19] V. Scalzo, Remarks on the existence and stability of some relaxed Nash equilibrium in strategic form games, Econom. Theory 61 (3) (2016) 571–586. [20] H.E. Scarf, On the existence of a cooperative solution for a general class of n-person games, J. Econom. Theory 3 (1971) 169–181. [21] L. Simon, W. Zame, Discontinuous games and endogenous sharing rules, Econometrica 58 (1990) 861–872. [22] G. Tian, J. Zhou, The maximum theorem and the existence of Nash equilibrium of (generalized) games without lower semicontinuities, J. Math. Anal. Appl. 166 (1992) 351–364. [23] G. Tian, J. Zhou, Transfer continuities, generalizations of the Weierstrass and maximum theorems: a full characterization, J. Math. Econom. 24 (1995) 281–303. [24] M. Uyanik, On the nonemptiness of the α-core of discontinuous games: transferable and nontransferable utilities, J. Econom. Theory 158 (2015) 213–231. [25] W.T. Wu, J.H. Jiang, Essential equilibrium points of n-person non-cooperative games, Sci. Sin. 11 (1962) 1307–1322. [26] Z. Yang, Essential stability of α-core, Internat. J. Game Theory 46 (2017) 13–28. [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. [28] J. Yu, Essential equilibrium points of n-person non-cooperative games, J. Math. Econom. 31 (1999) 361–372. [29] Y.H. Zhou, J. Yu, S.W. Xiang, Essential stability in games with infinitely many pure strategies, Internat. J. Game Theory 35 (2007) 493–503. [30] Y.H. Zhou, J. Yu, S.W. Xiang, L. Wang, Essential stability in games with endogenous sharing rules, J. Math. Econom. 45 (2009) 233–240. [31] J. Zhao, The existence of TU α-core in normal form games, Internat. J. Game Theory 28 (1999) 25–34.
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Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Hadamard well-posedness of the α-core ✩ Zhe Yang a,b,∗ , Dawen Meng a,b a b
School of Economics, Shanghai University of Finance and Economics, Shanghai 200433, China Key Laboratory of Mathematical Economics (SUFE), Ministry of Education, Shanghai 200433, China
a r t i c l e
i n f o
Article history: Received 10 September 2016 Available online xxxx Submitted by M. Quincampoix Keywords: Hadamard well-posedness α-Core Abstract economy Pseudocontinuity Coalitionally C-secure game Essential stability
a b s t r a c t In this paper, we discuss the continuity property of the α-core with respect to data perturbations in different environments. We show that some collections of abstract economies (or normal games) with the nonempty α-core have the Hadamard well-posedness property. We also show that the α-core points of every game in a (dense) residual subset are essential. © 2017 Elsevier Inc. All rights reserved.
1. Introduction Two central solution concepts for normal-form games are Nash equilibrium (NE) and the core. Nash equilibrium is a noncooperative solution in which only the individual behaviors are considered, whereas the core is a cooperative solution involving the collective behaviors of players. The existence and continuity of solutions are two important topics in the game theory. The existence of Nash equilibria has been extensively studied in the literature (see Nash [15], Debreu [8], Dasgupta and Maskin [7], Simon and Zame [21], Tian and Zhou [22,23], Reny [16], McLennan et al. [11], Barelli and Meneghel [3]). Comparing with NE, the definition of cores is less unified since it varies with the “blocking” ways of a coalition. Among various blocking concepts defined in the literature, the α-core due to Aumann [2] attracts a significant attention. We adopt this concept throughout this paper. A coalition is said to α-block a given social state if it has a feasible strategy with which the coalition can ensure a social ✩ We are grateful to an associate editor and anonymous referees for their valuable comments and suggestions. Yang wishes to thank Jinfeng Liu for her assistance. Yang acknowledges the financial supports of National Natural Science Foundation of China (No. 11501349), Chen Guang Project sponsored by the Shanghai Municipal Education Commission and Shanghai Education Development Foundation (No. 13CG35), the open project of Key Laboratory of Mathematical Economics (SUFE), Ministry of Education (No. 201309KF02). Meng acknowledges the financial supports of National Natural Science Foundation of China (No. 71301094) and the open project of Key Laboratory of Mathematical Economics (SUFE), Ministry of Education (No. 201309KF02). * Corresponding author at: School of Economics, Shanghai University of Finance and Economics, Shanghai 200433, China. E-mail address: [email protected] (Z. Yang).
http://dx.doi.org/10.1016/j.jmaa.2017.03.038 0022-247X/© 2017 Elsevier Inc. All rights reserved.
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state preferred by all the agents in it regardless of the complementary coalition’s actions. The α-core is the set of social states that cannot be α-blocked by any coalition. There are a number of papers in the literature dealing with the nonemptiness of the α-core. Scarf [20] proved the nonemptiness of the α-core of normal games with nontransferable utilities. Zhao [31] extended the result to transferable utilities. Kajii [10] showed the nonemptiness of the α-core with nonordered preferences. Martins-da-Rocha and Yannelis [12] gave the nonemptiness result for games with nonordered and discontinuous preferences. More recently, Uyanik [24] provided sufficient conditions for the nonemptiness of α-cores in games with possibly discontinuous payoff functions for cases with and without transferable utilities, respectively. When it comes to the continuity property, we need to introduce two concepts, i.e., Hadamard wellposedness (H-wp) and essentiality. H-wp refers to the continuous dependence of a solution on small change of the game. A solution of a game is essential if any perturbed game has a solution approximated to it. Yu et al. [27] studied the well-posedness of Nash equilibrium points in some classes of noncooperative games. Scalzo [17] proved that some classes of games with weaker conditions (i.e., better-reply security and pseudocontinuity) also have the H-wp property. A classical reference for the essential stability of Nash equilibria is Wu and Jiang [25]. Later, a great deal of work has been done (cf. Yu [28], Carbonell-Nicolau [5,6], Scalzo [18, 19], Yang [26], Zhou [29,30]). It is still an open question whether the α-core correspondence is continuous. To the best of our knowledge, there are few existing literature investigating this issue. We will fill this gap in the present paper by discussing both H-wp and essential stability of the α-core in different collections of games. The rest of the paper is organized as follows. Section 2 recalls some preparatory definitions and results. In Section 3, we study the Hadamard well-posedness for three different models. In Section 4, we analyze the essential stability of the α-core. Section 5 concludes with some remarks. 2. Settings and preliminaries We consider an abstract economy with a finite set I := {1, · · · , N } of agents. A subset S ⊆ I represents a coalition and I = 2I \{∅} is the family of all admissible coalitions. Each agent i ∈ I chooses an individual strategy xi in his strategy set Xi . Xi is assumed to be a nonempty subset of a Hausdorff topological space. A strategy profile x ∈ X = i∈I Xi is called a social state, fi : X −→ R is the payoff function of player i. We represent by XS = i∈S Xi the strategy space of the coalition S and X−S = i∈I\S Xi its complement. GS : X ⇒ XS denotes the feasible strategy correspondence of the coalition S. Our abstract economy can therefore be represented by the family Γ = {I, (Xi )i∈I , (fi )i∈I , (GS )S∈I }. If GS (x) = XS for each x ∈ X and each S ∈ I, an abstract economy can be simplified to a normal game Γ = {I, (Xi )i∈I , (fi )i∈I }. A core allocation is a feasible social state that is robust to all possible coalitional deviations. Since actions of the agents outside the coalition affect the welfare of the members inside the coalition, it is necessary to consider the reacting way of outsiders when defining a core solution. Aumann [2] suggests a possible “blocking” way as follows: a coalition is said to α-block a given social state if it has a feasible strategy with which the coalition can ensure a social state preferred by all the agents in it regardless of strategies the other agents may choose. Precisely, a coalition S is said to α-block a social state x ∈ X if there exists zS ∈ GS (x) such that fi (zS , v−S ) > fi (x), ∀v−S ∈ X−S , ∀i ∈ S. A strategy profile x ∈ X is in the α-core of Γ if x ∈ GI (x) and no coalition S ∈ I can α-block x. We denote by C (Γ) the α-core of the abstract economy Γ. In this paper, the payoff functions of a game are allowed to be discontinuous. We next introduce some conditions about payoffs: pseudocontinuity, coalitional C-security and strongly coalitional C-security, which are all strictly weaker than continuity.
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Definition 2.1 ([13]). A real-valued function f , defined on a topological space Z, is said to be: upper pseudocontinuous at x0 ∈ Z if for all x ∈ Z such that f (x0 ) < f (x), we have lim supy→x0 f (y) < f (x); lower pseudocontinuous at x0 ∈ Z if −f is upper pseudocontinuous at x0 ; pseudocontinuous if it is both upper and lower pseudocontinuous. Definition 2.2 ([24]). A normal game Γ = {I, (Xi )i∈I , (fi )i∈I } is coalitionally C-secure (CCS) if for each x∈ / C (Γ), there exist vSx ∈ R|S| and ySx ∈ XS for each S ∈ I and an open neighborhood U x of x such that (i) fi (ySx , z−S ) − (vSx )i ≥ 0, ∀z−S ∈ X−S , ∀i ∈ S, ∀S ∈ I; (ii) for each x ∈ U x , there exists S ∈ I such that (vSx )i − fi (x ) > 0, ∀i ∈ S. Definition 2.3. A normal game Γ = {I, (Xi )i∈I , (fi )i∈I } is strongly coalitionally C-secure (SCCS) if for each x∈ / C (Γ), there exist {(vSx , ySx ) ∈ R|S| × XS : S ∈ I}, ε > 0 and an open neighborhood U x of x such that (i) fi (ySx , z−S ) − (vSx )i ≥ 0, ∀z−S ∈ X−S , ∀i ∈ S, ∀S ∈ I; (ii) for each x ∈ U x , there exists S ∈ I such that (vSx )i − fi (x ) > ε, ∀i ∈ S. Note that SCCS implies CCS, but the converse is not true. We will show in the next section that CCS cannot guarantee the Hadamard well-posed property of the α-core. So, a strengthened version, SCCS, is introduced here (see Theorem 3.3 and Example 3.2 for detailed discussions). We next give some preparatory definitions and lemmas on set-valued mappings (due to [1] and [17]) for the proofs of our main results in the subsequent sections. A set-valued mapping F : X ⇒ Y , where X and Y are two Hausdorff topological spaces, is said to be: upper semicontinuous at x ∈ X if, for any open subset O of Y with O ⊃ F (x), there exists an open neighborhood U (x) of x such that O ⊃ F (x ) for any x ∈ U (x); upper semicontinuous on X, if it is upper semicontinuous at each x ∈ X; lower semicontinuous at x ∈ X if, for any open subset O of Y with O ∩ F (x) = ∅, there exists an open neighborhood U (x) of x such that O ∩ F (x ) = ∅ for any x ∈ U (x); lower semicontinuous on X, if it is lower semicontinuous at each x ∈ X; closed if Graph(F ) = {(x, y) ∈ X × Y : y ∈ F (x)} is a closed subset of X × Y . Lemma 2.1 ([1]). A correspondence with compact Hausdorff range space is closed if and only if it is upper semicontinuous and closed-valued. Lemma 2.2 ([17]). A real-valued function f , defined on a Hausdorff topological space, is pseudocontinuous if and only if whenever f (x) < f (z), there exist an open neighborhood Nx of x and an open neighborhood Nz of z such that f (x ) < f (z ) for all x ∈ Nx and all z ∈ Nz . Lemma 2.3 ([17]). If a real-valued function f , defined on a connected topological space X, is pseudocontinuous on X, then the following property holds: f (x1 ) < f (x2 ) =⇒ ]f (x1 ), f (x2 )[∩f (X) = ∅. 3. Main results Let Y be a set of games endowed with a topology, and C : Y ⇒ X be the α-core correspondence. We say that a game y ∈ Y is generalized Hadamard well-posed if C (y) = ∅, and for any net (xβ , y β )β in X × Y with y β → y and xβ ∈ C (y β ), there exists a cluster point of (xβ )β belonging to C (y). If y is generalized Hadamard well-posed and C (y) is a singleton set, then we say that y is Hadamard well-posed. Besides, we
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[m3L; v1.209; Prn:23/03/2017; 10:56] P.4 (1-13)
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say that Y has the Hadamard well-posedness property if every y ∈ Y is (generalized) Hadamard well-posed. Yu et al. [27] gave the following sufficient conditions for Hadamard well-posedness. Theorem 3.1 ([27]). Let X and Y be two Hausdorff topological spaces and F : Y ⇒ X be a set-valued mapping. (a) If F is upper semicontinuous at y ∈ Y , F (y) is nonempty and compact, then y is generalized Hadamard well-posed. (b) If F is upper semicontinuous at y ∈ Y and F (y) = {x∗ }, then y is Hadamard well-posed. Next, we study the Hadamard well-posedness for three different models. Model (A) concerns the abstract economies with pseudocontinuous payoffs and continuous feasible strategy correspondences. Model (B) deals with the strongly coalitionally C-secure games. Model (C) is about the abstract economies with nonordered preferences. Moreover, we show that the pseudocontinuous normal game is also strongly coalitionally C-secure if the space of action profiles is a nonempty and compact subset of a Hausdorff topological space, and the class of coalitionally C-secure games fails to satisfy the Hadamard well-posedness property. 3.1. Model (A) For each i ∈ I, let Xi be a nonempty connected compact subset of a metric space, HS be the Hausdorff metric on XS and d be the metric on X. We denote by Mp the set of abstract economies Γ = {I, (Xi )i∈I , (fi )i∈I , (GS )S∈I } having a common set of strategy profiles X, where (1) for each i ∈ I, fi : X −→ R is pseudocontinuous on X, (2) for each S ∈ I, GS : X ⇒ XS is a continuous set-valued mapping with nonempty compact values, (3) the α-core is nonempty, i.e., C (Γ) = ∅, ∀Γ ∈ Mp . Moreover, we define a metric ρ over Mp as follows ρ(Γ, Γ ) :=
i∈I
sup |fi (x) − fi (x)| + max sup HS (GS (x), GS (x)).
x∈X
S∈I x∈X
Theorem 3.2. Mp has the Hadamard well-posedness property. Proof. First, we need to show that the correspondence C : Mp ⇒ X is upper semicontinuous with nonempty compact values. It is obvious that C (Γ) = ∅ for each Γ ∈ Mp . By Lemma 2.1, it suffices to show that Graph(C ) is closed. Suppose that (Γβ , xβ )β is a net in Mp × X with xβ ∈ C (Γβ ) and (Γβ , xβ ) −→ (Γ, x) ∈ Mp × X, then xβ ∈ GβI (xβ ). Since GI is continuous and Γβ −→ Γ, we have d(x, GI (x)) ≤ d(x, xβ ) + d(xβ , GβI (xβ )) + HI (GβI (xβ ), GI (xβ )) + HI (GI (xβ ), GI (x)) ≤ d(x, xβ ) + HI (GI (xβ ), GI (x)) + ρ(Γβ , Γ) −→ 0. Therefore, x ∈ GI (x). We next show that x cannot be α-blocked by any coalition S. Suppose, by way of contradiction, that there exist S ∈ I and zS ∈ GS (x) such that fi (zS , v−S ) > fi (x), ∀v−S ∈ X−S , ∀i ∈ S. For any i, since fi is pseudocontinuous on X and X−S is compact, by Proposition 2.1 in [14], there exists 0 v−S ∈ X−S such that 0 fi (zS , v−S ) ≥ fi (zS , v−S ) > fi (x), ∀v−S ∈ X−S .
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By Lemma 2.3, there exist y1i , y2i ∈ X such that 0 fi (zS , v−S ) ≥ fi (zS , v−S ) > fi (y1i ) > fi (y2i ) > fi (x), ∀v−S ∈ X−S .
Given v−S ∈ X−S , in light of Lemma 2.2, there exist open neighborhoods Ovi −S (zS ), Oi (v−S ) and Oi (x) containing, respectively, zS , v−S and x such that fi (zS , v−S ) > fi (y1i ) > fi (y2i ) > fi (x ), ∀zS ∈ Ovi −S (zS ), ∀v−S ∈ Oi (v−S ), ∀x ∈ Oi (x). 1 L Since X−S is nonempty and compact, there exists a finite subset {v−S , · · · , v−S } of X−S such that L
X−S =
k Oi (v−S ).
k=1
Let Oi (zS ) =
L
Ovi k (zS ). −S
k=1
We get: fi (zS , v−S ) > fi (y1i ) > fi (y2i ) > fi (x ), ∀zS ∈ Oi (zS ), ∀v−S ∈ X−S , ∀x ∈ Oi (x). Set O(zS ) =
Oi (zS ),
i∈S
O(x) =
Oi (x),
i∈S
εS = min{fi (y1i ) − fi (y2i )} > 0. i∈S
Then, for any zS ∈ O(zS ), any v−S ∈ X−S , and any x ∈ O(x), fi (zS , v−S ) − fi (x ) > εS , ∀i ∈ S. Moreover, since HS (GβS (xβ ), GS (x)) ≤ HS (GβS (xβ ), GS (xβ )) + HS (GS (xβ ), GS (x)) ≤ ρ(Γβ , Γ) + HS (GS (xβ ), GS (x)) −→ 0, it follows that there exists a net (zSβ )β of XS such that zSβ ∈ GβS (xβ ) and zSβ −→ zS . So, there exists β1 such that zSβ ∈ O(zS ) and xβ ∈ O(x) whenever β β1 . Then, zSβ ∈ GβS (xβ ) and fi (zSβ , v−S ) − fi (xβ ) > εS , ∀v−S ∈ X−S , ∀i ∈ S. Additionally, since Γβ −→ Γ, then for
εS 2
> 0, there exists β2 such that for any β β2 ,
fiβ (z ) − fiβ (x ) > fi (z ) − fi (x ) −
εS , ∀(z , x ) ∈ X × X, ∀i ∈ S. 2
For any sufficiently large β, we have that zSβ ∈ GβS (xβ ) and
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[m3L; v1.209; Prn:23/03/2017; 10:56] P.6 (1-13)
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fiβ (zSβ , v−S ) − fiβ (xβ ) > fi (zSβ , v−S ) − fi (xβ ) −
εS εS εS > εS − = > 0, ∀v−S ∈ X−S , ∀i ∈ S, 2 2 2
thereby contradicting the fact that xβ ∈ C (Γβ ). Finally, the Hadamard well-posedness property follows directly from Theorem 3.1.
2
3.2. Model (B) For each i ∈ I, let Xi be a nonempty compact subset of a Hausdorff topological space. We denote by Mc (resp. Msc ) the set of (resp. strongly) coalitionally C-secure games Γ = {I, (Xi )i∈I , (fi )i∈I } having bounded payoffs, a common set of strategy profiles X and a nonempty α-core. It follows directly from the definitions of SCCS and CCS that Msc ⊂ Mc . Moreover, we define the metric ρ on Mc as follows: ρ(Γ, Γ ) :=
i∈I
sup |fi (x) − fi (x)|.
x∈X
Theorem 3.3. Msc has the Hadamard well-posedness property. Proof. An argument analogous to that in Theorem 3.2 shows that the correspondence C : Msc ⇒ X is upper semicontinuous with nonempty compact values. The nonemptiness of the α-core is obvious. By Lemma 2.1, it suffices to show that Graph(C ) is closed. Suppose that (Γβ , xβ )β is a net in Msc × X with xβ ∈ C (Γβ ) and (Γβ , xβ ) −→ (Γ, x) ∈ Msc × X. The proof will be completed if we show that x ∈ C (Γ). Suppose that x ∈ / C (Γ). Then there exist {(vSx , ySx ) ∈ R|S| ×XS : S ∈ I}, ε > 0 and an open neighborhood x U of x such that (1) for each S ∈ I and each z−S ∈ X−S , fi (ySx , z−S ) − (vSx )i ≥ 0, ∀i ∈ S; (2) for each x ∈ U x , there exists S ∈ I such that (vSx )i − fi (x ) > ε, ∀i ∈ S. It implies that there exists β1 such that for any β β1 , there exists S ∈ I such that (vSx )i > fi (xβ ) + ε, ∀i ∈ S. As I is a finite set, it follows that there exist S0 ∈ I and a subnet {xβk } of {xβ } satisfying fi (ySx0 , z−S0 ) ≥ (vSx0 )i > fi (xβk ) + ε, ∀z−S0 ∈ X−S0 , ∀i ∈ S0 . Since Γβk −→ Γ, then for any sufficiently large βk , ε fiβk (ySx0 , z−S0 ) − fiβk (z ) > fi (ySx0 , z−S0 ) − fi (z ) − , ∀z−S0 ∈ X−S0 , ∀z ∈ X, ∀i ∈ S0 , 2 implying that fiβk (ySx0 , z−S0 ) − fiβk (xβk ) > fi (ySx0 , z−S0 ) − fi (xβk ) −
ε ε > ε − > 0, ∀z−S0 ∈ X−S0 , ∀i ∈ S0 . 2 2
It contradicts the fact that xβk ∈ C (Γβk ). Finally, the Hadamard well-posedness property of Msc follows directly from Theorem 3.1. 2 The next result shows that a pseudocontinuous normal game is also strongly coalitionally C-secure if the space of action profiles is a nonempty and compact subset of a Hausdorff topological space.
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Theorem 3.4. Assume that the normal game Γ = {I, (Xi )i∈I , (fi )i∈I } satisfies the following conditions: (1) for each i ∈ I, Xi is a nonempty compact subset of a Hausdorff topological space; (2) for each i ∈ I, fi is pseudocontinuous on X. Then the game Γ is strongly coalitionally C-secure. Moreover, if C (Γ) = ∅, then Γ is generalized Hadamard well-posed. / C (Γ), then there exist S0 ∈ I and ySx0 ∈ XS0 such that Proof. Suppose that x ∈ fi (ySx0 , z−S0 ) > fi (x), ∀z−S0 ∈ X−S0 , ∀i ∈ S0 . Since fi pseudocontinuous on X and Xi is compact for each i ∈ I, it follows from Proposition 2.1 of [14] that for each i ∈ S0 , there exists z i−S0 ∈ X−S0 such that fi (ySx0 , z−S0 ) ≥ fi (ySx0 , z i−S0 ) > fi (x), ∀z−S0 ∈ X−S0 . By Lemma 2.2, there exists an open neighborhood O(x) of x in X such that fi (ySx0 , z−S0 ) ≥ fi (ySx0 , z i−S0 ) > fi (x ), ∀z−S0 ∈ X−S0 , ∀i ∈ S0 , ∀x ∈ O(x). Since X is a nonempty compact subset of a Hausdorff topological space, it follows from Theorem 2.48 of [1] that X is normal. Then, there exists an open neighborhood U (x) of x in X such that U (x) ⊂ clU (x) ⊂ O(x) ⊂ X and for each i ∈ S0 , fi (ySx0 , z−S0 ) ≥ fi (ySx0 , z i−S0 ) > fi (x ), ∀z−S0 ∈ X−S0 , ∀x ∈ clU (x). Thus clU (x) is compact. Moreover, by Proposition 2.1 of [14], we have that for each i ∈ S0 , there exists x ∈ clU (x) such that i
fi (xi ) =
max
x ∈clU (x)
fi (x ),
fi (ySx0 , z−S0 ) ≥ fi (ySx0 , z i−S0 ) > fi (xi ) ≥ fi (x ), ∀z−S0 ∈ X−S0 , ∀x ∈ clU (x). Let ε=
1 min fi (ySx0 , z i−S0 ) − fi (xi ) > 0, 2 i∈S0
(vSx0 )i = fi (ySx0 , z i−S0 ), ∀i ∈ S0 , vSx0 = (vSx0 )i∈S0 ∈ R|S0 | , (vSx )i = min fi (z), ∀i ∈ S, ∀S = S0 , z∈X
vSx
=
(vSx )i∈S
∈ R|S| , ∀S = S0 .
Then, for each i ∈ S0 , fi (ySx0 , z−S0 ) ≥ (vSx0 )i > fi (x ) + ε, ∀z−S0 ∈ X−S0 , ∀x ∈ U (x). For any S = S0 , we can select a point ySx ∈ XS such that
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[m3L; v1.209; Prn:23/03/2017; 10:56] P.8 (1-13)
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fi (ySx , z−S ) ≥ (vSx )i , ∀z−S ∈ X−S , ∀i ∈ S. Therefore, for each x ∈ X that is not in the α-core of Γ, there exist {(vSx , ySx ) ∈ R|S| × XS : S ∈ I}, ε > 0 and an open neighborhood U (x) of x such that (i) for each S ∈ I and each z−S ∈ X−S , fi (ySx , z−S ) − (vSx )i ≥ 0, ∀i ∈ S; (ii) for each x ∈ U (x), there exists S0 ∈ I such that (vSx0 )i − fi (x ) > ε, ∀i ∈ S0 . Therefore, Γ is strongly coalitionally C-secure. Finally, taking C (Γ) = ∅ into consideration, by Theorem 3.3, we obtain that Γ is generalized Hadamard well-posed. 2 Note that the strongly coalitional C-security does not imply the pseudocontinuity. See the following counterexample. Example 3.1. Consider the game Γ = (X1 , X2 , f1 , f2 ), where I = {1, 2}, X1 = X2 = [0, 1], X = X1 × X2 , f1 (x1 , x2 ) = x1 , x1 ∈ [0, 1], 0 x2 ∈ [0, 1) f2 (x1 , x2 ) = 1 x2 = 1. It is easy to verify that the normal game Γ is strongly coalitionally C-secure. Now 1 1 f2 (1, 1) = 1 > f2 ( , ), 2 2 1 1 lim inf f2 (y) = 0 = f2 ( , ). 2 2 y−→(1,1) Therefore, f2 is not pseudocontinuous on X. The following counterexample shows that Mc does not satisfy the Hadamard well-posedness property. Example 3.2. Let X = [0, 10], ⎧ ⎪ ⎨ 10 x = 0 f (x) = x x ∈ (0, 10) ⎪ ⎩0 x = 10. The one-person game Γ = (X, f ) is coalitionally C-secure and C (Γ) = {0}. We claim that Γ is not strongly coalitionally C-secure. Otherwise, from the fact that x = 10 is not in the α-core of Γ, it follows that there exist v x ∈ R, y x ∈ X, ε > 0 and an open neighborhood U (x) of x such that f (y x ) ≥ v x > f (x ) + ε, ∀x ∈ U (x). It follows that 10 = sup f (z) ≥ f (y x ) ≥ v x ≥ z∈X
We get a contradiction. Therefore, Γ ∈ Mc \Msc .
sup f (x ) + ε = 10 + ε. x ∈U (x)
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[m3L; v1.209; Prn:23/03/2017; 10:56] P.9 (1-13)
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Let ⎧ ⎪ 10 − n3 ⎪ ⎪ ⎨x
f n (x) = ⎪ −x + 2 10 − n1 ⎪ ⎪ ⎩0
x=0
x ∈ 0, 10 − n1 x ∈ 10 − n1 , 10 x = 10.
It is easy to check that each Γn = (X, f n ) is coalitionally C-secure, Γn −→ Γ and C (Γn ) = {10 − Therefore, Γ is not Hadamard well-posed.
1 n }.
3.3. Model (C) In this subsection, we deal with an abstract economy with nonordered preferences. Each agent i has a preference correspondence Pi : X ⇒ X and each coalition S ∈ I has a feasible strategy corre = spondence GS : X ⇒ XS . This abstract economy can therefore be represented by the family Γ {I, (Xi )i∈I , (Pi )i∈I , (GS )S∈I }. A coalition S is said to α-block a social state x ∈ X if there exists zS ∈ GS (x) if x ∈ GI (x) and no such that {zS } × X−S ⊂ Pi (x), ∀i ∈ S. A social state x ∈ X is in the α-core of Γ coalition S ∈ I can α-block x. For each i ∈ I, let Xi be a nonempty compact subset of a metric space, d be the metric on X, HS be the Hausdorff metric on XS and H be the Hausdorff metric on X × X. We denote by Mg the set of abstract = {I, (Xi )i∈I , (Pi )i∈I , (GS )S∈I } having a common set of strategy profiles X, where (1) for economies Γ each i ∈ I, Pi : X ⇒ X has the open graph in X × X, (2) for each S ∈ I, GS : X ⇒ XS is a continuous set-valued mapping with nonempty compact values, (3) the α-core is nonempty. We consider the topology over Mg generated by the metric ρ defined in the following way Γ ) := max H(Graph(Pi )c , Graph(P )c ) + max sup HS (GS (x), G (x)). ρ(Γ, i S i∈I
S∈I x∈X
Theorem 3.5. Mg has the Hadamard well-posedness property. Proof. Applying Theorem 3.1, it suffices to show that the correspondence C : Mg ⇒ X is upper semicontinuous with nonempty compact values. It follows directly from the definition of Mg that C (Γ) = ∅, ∀Γ ∈ Mg . β , xβ )β is a net in Mg × X with From Lemma 2.1, we need to show that Graph(C ) is closed. Suppose that (Γ β β β β β β β β −→ Γ, x ∈ C (Γ ) and (Γ , x ) −→ (Γ, x) ∈ Mg × X, then x ∈ GI (x ). Since GI is continuous and Γ we have d(x, GI (x)) ≤ d(x, xβ ) + d(xβ , GβI (xβ )) + HI (GβI (xβ ), GI (xβ )) + HI (GI (xβ ), GI (x)) β , Γ) −→ 0. ≤ d(x, xβ ) + HI (GI (xβ ), GI (x)) + ρ(Γ Therefore, x ∈ GI (x). Suppose, by way of contradiction, that x can be α-blocked by some coalition S. Then {zS } × X−S ⊂ Pi (x), ∀i ∈ S, for some zS ∈ GS (x), that is, ({x} × {zS } × X−S ) Since
Graph(Pi )c = ∅, ∀i ∈ S.
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β , Γ) + HS (GS (xβ ), GS (x)) −→ 0, HS (GβS (xβ ), GS (x)) ≤ HS (GβS (xβ ), GS (xβ )) + HS (GS (xβ ), GS (x)) ≤ ρ(Γ it follows that there exists a net (zSβ )β of XS such that zSβ ∈ GβS (xβ ) and zSβ −→ zS . β , xβ ) −→ (Γ, x), we have From zSβ −→ zS and (Γ
{xβ } × {zSβ } × X−S
Graph(Piβ )c = ∅, ∀i ∈ S,
for any sufficiently large β. It implies that zSβ ∈ GβS (xβ ) and {zSβ } × X−S ⊂ Piβ (xβ ), ∀i ∈ S, β ). Finally, the Hadamard well-posedness property follows thereby contradicting the fact that xβ ∈ C (Γ directly from Theorem 3.1. 2 4. Essential stability In this section, we discuss the essential stability of the α-core. We denote by X a set of strategy profiles endowed with a metric d, M ∈ {Mp , Msc , Mg } represents a class of abstract economies (or normal games) endowed with certain metric ρM and Γ ∈ M . A point x ∈ C (Γ) is said to be essential if for any open neighborhood O of x, there exists δ > 0 such that C (Γ ) ∩ O = ∅ for any Γ ∈ M with ρM (Γ , Γ) < δ. We say that Γ is essential if every point of C (Γ) is essential. Note that Γ is essential if and only if the α-core correspondence C : M ⇒ X is lower semicontinuous at Γ. The standard approach to studying the essential stability consists in identifying a collection M with the following properties: (i) M is a complete metric space; (ii) every member of M has a nonempty set of solutions; (iii) the solution correspondence is compact-valued and upper semicontinuous. Items (i)–(iii), together with the following lemma, imply the essential stability. Lemma 4.1 ([9]). Suppose that X is a Hausdorff topological metric space and Y a metric space. Suppose further that F : X ⇒ Y is a compact-valued and upper semicontinuous correspondence with F (x) = ∅ for all x ∈ X. Then there exists a residual subset Q of X such that F is lower semicontinuous at every point in Q. The following lemma due to [5,6,18,26] strengthens Lemma 4.1. Lemma 4.2. Suppose that X is a complete metric space and Y a metric space. Suppose further that F : X ⇒ Y is a compact-valued and upper semicontinuous correspondence with F (x) = ∅ for all x ∈ X. Then there exists a dense residual subset Q of X such that F is lower semicontinuous at every point in Q. To apply Lemma 4.2, we need to prove the following result. Lemma 4.3. (Mg , ρ) is a complete metric space. ∈ Mg such that n }∞ in Mg , it suffices to show that there exists Γ Proof. For any Cauchy sequence {Γ n=1 n Γ −→ Γ under the distance ρ. For any ε > 0, there exists N0 > 0 such that for any n, m > N0 , max H(Graph(Pin )c , Graph(Pim )c ) + max sup HS (GnS (x), Gm S (x)) < ε. i∈I
S∈I x∈X
Referring to the proof of Theorem 2.2 in [26], we have: for each i ∈ I, there exists an open-graph correspondence Pi : X ⇒ X such that Graph(Pin )c −→ Graph(Pi )c . It is easy to show that for each S ∈ I,
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there exists a compact-valued and continuous correspondence GS : X ⇒ XS such that GS (x) = ∅ for all x ∈ X and supx∈X HS (GnS (x), GS (x)) −→ 0. Now, the only work left is to show that the α-core of = {I, (Xi )i∈I , (Pi )i∈I , (GS )S∈I } is nonempty. From the nonemptiness of α-core of Γ n , we have that there Γ n ). Since X is compact, without loss of generality, we assume that xn −→ x ∈ X. The proof exists xn ∈ C (Γ of x ∈ C (Γ) is omitted here since it is similar to the proof of Theorem 3.5. 2 Next, using Lemmas 4.1 and 4.2, we have the following results. Theorem 4.1. For any M ∈ {Mp , Msc }, there exists a residual subset Q of M such that every member of Q is essential. Theorem 4.2. There exists a dense residual subset Q of Mg such that every member of Q is essential. Note that Mp and Msc are not complete metric spaces, see the following counterexample. Example 4.1. Let X = [0, 1] and ⎧ ⎪ ⎨0 f (x) = 0 ⎪ ⎩1
2
⎧ ⎪ ⎪ ⎪ ⎨
x=0 x=1 x ∈ (0, 1),
0 0 f n (x) = 1 1 ⎪ + ⎪ 2 nx ⎪ ⎩−1x + 1 + n n
1 2
x=0 x=1 x ∈ [0, 12 ] x ∈ ( 12 , 1).
It is easy to check that ρ(f n , f ) −→ 0 as n → ∞, each f n is pseudocontinuous on [0, 1] and the one-person game (X, f ) is not strongly coalitionally C-secure. So, this example shows that the metric spaces Mp and Msc are not complete. The following examples show that C (·) is not necessarily lower semicontinuous for every member of M ∈ {Mp , Msc , Mg }. Example 4.2. Let I = {1, 2}, X1 = X2 = [0, 1], X = [0, 1] × [0, 1], f1 (x1 , x2 ) = 1, f2 (x1 , x2 ) = 2, GS (x) = XS , ∀x ∈ X, ∀S ∈ I. Then the game Γ = (X1 , X2 , f1 , f2 , (GS )S∈I ) ∈ Mp ∩ Msc and C (Γ) = [0, 1] × [0, 1]. For each n, define f1n , f2n : X −→ R by f1n (x1 , x2 ) = 1 − f2n (x1 , x2 ) = 2 +
1 x1 , n
1 x2 , ∀x ∈ X. n
Then the game Γn = (X1 , X2 , f1n , f2n , (GS )S∈I ) ∈ Mp ∩ Msc , C (Γn ) = {0} × {1} and ρ(Γn , Γ) −→ 0. Thus, C : Mp ⇒ X and C : Msc ⇒ X are not lower semicontinuous at Γ. Example 4.3. Let I = {1, 2}, X1 = X2 = [0, 1], X = [0, 1] × [0, 1], the preference correspondences P1 , P2 : X ⇒ X be defined as
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P1 (x1 , x2 ) = P2 (x1 , x2 ) = ∅, GS (x) = XS , ∀x ∈ X, ∀S ∈ I. = (X1 , X2 , P1 , P2 , (GS )S∈I ) ∈ Mg and C (Γ) = [0, 1] × [0, 1]. For each n, Then the abstract economy Γ n n define the correspondences P1 , P2 : X ⇒ X by P1n (x1 , x2 )
=
P2n (x1 , x2 )
=
1 ], x2 ∈ [0, 1] [0, x1 ) × [0, 1] x1 ∈ [0, 2n 1 1 [0, 2n ) × [0, 1] x1 ∈ ( 2n , 1], x2 ∈ [0, 1] 1 ] [0, 1] × [0, x2 ) x1 ∈ [0, 1], x2 ∈ [0, 2n 1 1 [0, 1] × [0, 2n ) x1 ∈ [0, 1], x2 ∈ ( 2n , 1].
n = (X1 , X2 , P n , P n , (GS )S∈I ) ∈ Mg , C (Γ −→ 0. n ) = {0} × {0} and ρ(Γ n , Γ) Then the abstract economy Γ 1 2 Thus, C : Mg ⇒ X is not lower semicontinuous at Γ. Remark 4.1. Yang [26] studied only the essential stability of the α-core for qualitative games. We make the following extensions in the present paper. (1) We obtain the essential stability of the α-core for abstract economies. (2) Using payoff perturbations, we obtain the essential stability of the α-core for the classes Mp and Msc . 5. Conclusions Nash equilibrium and α-core are two major solution concepts for normal form games. While both the existence and continuity of Nash equilibria have been studied extensively in the literature, the results on the α-core are only about its existence/nonemptiness (see [2,4,10,24]). Little attention has been devoted to the continuity of the α-core. In this paper, we fill this gap by analyzing the Hadmard well-posedness and essential stability of the α-core. We show that the Hadmard well-posedness property holds for the following three models: (i) games with pseudocontinuous payoffs and continuous feasible strategy correspondences; (ii) strongly coalitionally C-secure games; (iii) abstract economies with nonordered preferences. Also, we study the essential stability of the α-core by showing that every point of the α-core is essential for every game in a residual (dense) subset of the space of games. References [1] C. Aliprantis, K.C. Border, Infinite Dimensional Analysis, 3rd edn., Springer, Berlin, 2006. [2] R.J. Aumann, The core of a cooperative game without side payments, Trans. Amer. Math. Soc. 98 (1961) 539–552. [3] P. Barelli, I. Meneghel, A note on the equilibrium existence problem in discontinuous games, Econometrica 81 (2013) 813–824. [4] K.C. Border, A core existence theorem for games without ordered preferences, Econometrica 52 (1984) 1537–1542. [5] O. Carbonell-Nicolau, Essential equilibria in normal-form games, J. Econom. Theory 145 (2010) 421–431. [6] O. Carbonell-Nicolau, Further results on essential Nash equilibria in normal-form games, Econom. Theory 59 (2015) 277–300. [7] P. Dasgupta, E. Maskin, The existence of equilibrium in discontinuous economic games. Part I: theory, Rev. Econ. Stud. 53 (1986) 1–26. [8] G. Debreu, A social equilibrium existence theorem, Proc. Nat. Acad. Sci. 38 (1952) 886–893. [9] M.K. Fort, Points of continuity of semicontinuous functions, Publ. Math. Debrecen 2 (1951) 100–102. [10] A. Kajii, A generalization of Scarf’s theorem: an α-core existence theorem without transitivity or completeness, J. Econom. Theory 56 (1992) 194–205. [11] A. McLennan, P.K. Monteiro, R. Tourky, Games with discontinuous payoffs: a strengthening of Reny’s existence theorem, Econometrica 79 (2011) 1643–1664. [12] V. Martins-da-Rocha, N. Yannelis, Nonemptiness of the alpha core, working paper, Manchester School of Social Sciences, University of Manchester, 2011. [13] J. Morgan, V. Scalzo, Discontinuous but well-posed optimization problems, SIAM J. Optim. 17 (3) (2006) 861–870. [14] J. Morgan, V. Scalzo, Pseudocontinuous functions and existence of Nash equilibria, J. Math. Econom. 43 (2) (2007) 174–183.
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AID:21235 /FLA
Doctopic: Optimization and Control
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