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Equilibrium existence in games: Slight single deviation property and Ky Fan minimax inequality
Journal of Mathematical Economics 82 (2019) 197–201
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Equilibrium existence in games: Slight single deviation property and Ky Fan minimax inequality Vincenzo Scalzo Department of Economics and Statistics (DISES), University of Naples Federico II, via Cinthia 21, 80126 Napoli, Italy
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Article history: Received 16 July 2018 Received in revised form 26 February 2019 Accepted 27 February 2019 Available online 5 March 2019 Keywords: Discontinuous ordinal games Ky Fan minimax inequality Existence of Nash equilibria
a b s t r a c t In this paper, we exhibit a relation between the Nash equilibrium existence problem in ordinal games and the Ky Fan minimax inequality. In this way, we obtain new sufficient conditions for the existence of equilibria. The main tools used in the paper are a necessary condition introduced in Nassah and Tian (2016) for normal form games and a generalization of the single deviation property. Examples compare our result with the previous ones. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Consider a finite set N of players and, for any i ∈ N, let Xi be a non-empty ∏ subset of a Hausdorff topological vector space. Denoted by X = j∈N Xj the set of strategy profiles and, for each i ∈ N, let Pi be a mapping from X to Xi . The list G = ⟨Xi , Pi ⟩i∈N is called ordinal game, and a Nash equilibrium of G (equilibrium briefly) is an element x∗ ∈ X so that Pi (x∗ ) = ∅ for all i ∈ N (EG denotes the set of Nash equilibria of G). The existence of Nash equilibria of G can be investigated by means of the Ky Fan minimax inequality. Given a real-valued function Φ defined on X × X , the corresponding Ky Fan minimax inequality (Fan, 1972) is the following problem: find x∗ ∈ X such that Φ (x, x∗ ) ≤ 0 for all x ∈ X (in such a case, x∗ is said to be a solution). Now, for every x and z which belong to X , define ΦG (x, z) as below:
ΦG (x, z) =
∑
˜ Pi (xi , z)
(1)
i∈N
where ˜ Pi (xi , z) = 1 whenever xi ∈ Pi (z) and ˜ Pi (xi , z) = 0 otherwise. Obviously, x∗ is an equilibrium of G if and only if it is a solution to the Ky Fan minimax inequality corresponding to ΦG , that is: ΦG (x, x∗ ) ≤ 0 for all x ∈ X . Conditions which guarantee the existence of solutions to the Ky Fan minimax inequality allow to identify classes of games endowed with Nash equilibria. So, it is interesting to investigate if the Ky Fan minimax inequality approach leads to new sufficient conditions for the existence of Nash equilibria in ordinal games. In this paper, we show that this is possible. Using Tian (1993, Lemma 1), we obtain the existence of solutions to the Ky Fan E-mail address: [email protected]. https://doi.org/10.1016/j.jmateco.2019.02.008 0304-4068/© 2019 Elsevier B.V. All rights reserved.
minimax inequality when the function Φ is slightly diagonally transfer continuous and diagonally transfer quasi-concave: see Proposition 1. So, given an ordinal game G, we relate the slight diagonal transfer continuity of the function ΦG defined by (1) to a generalization of the single deviation property (see Reny, 2009, Nassah and Tian, 2016), that we call slight single deviation property. On the other hand, the diagonal transfer quasi-concavity of ΦG corresponds to a necessary condition for the existence of Nash equilibria called transfer uniform quasi-concavity (see Nassah and Tian, 2016 for the normal form games). Hence, we prove that the slight single deviation property and the transfer uniform quasi-concavity guarantee the existence of Nash equilibria: see Theorem. Let us note that the single deviation property (and the generalization considered in the present paper) is not sufficient to guarantee the existence of equilibria under the standard condition (∗) xi ∈ / coPi (x) for all x ∈ X and all i ∈ N: see Reny (2009, Section 3.1).1 Moreover, we provide examples to show that the transfer uniform quasi-concavity is not connected with condition (∗) and our Theorem applies to situations which are not covered by the previous literature on the existence of Nash equilibria in ordinal games: see Shafer and Sonnenschein (1975), Yannelis and Prabhakar (1983), Wu and Shen (1996), Scalzo (2015), Carmona and Podczeck (2016), He and Yannelis (2016), Nassah and Tian (2016) and Reny (2016). Finally, we consider normal form games G = ⟨Xi , ui ⟩i∈N : in ⟨ ⟩ this case, we associate G with the ordinal game ˆ G = Xi , ˆ Pi i∈N 1 Given a set A, coA denotes the convex hull of A. Condition (∗) has been widely used in the previous literature on the existence of equilibria in ordinal games: among the others, see Shafer and Sonnenschein (1975), Yannelis and Prabhakar (1983), Wu and Shen (1996), Scalzo (2015), He and Yannelis (2016) and Reny (2016).
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where, for any player i, ˆ Pi (x) = {yi ∈ Xi : ui (yi , x−i ) > ui (x)}. In this setting, Nassah and Tian (2016, Theorem 6) states that Nash equilibria exist when the single deviation property and the transfer uniform quasi-concavity hold.2 The result given in the present paper is a slight improvement due to the generalization of the single deviation property. Moreover, we compare our Theorem with the well known Baye et al. (1993, Theorem 1): we prove that they are independent. Furthermore, we remark that our result cannot be deduced from the previous ones given in Dasgupta and Maskin (1986), Reny (1999), Reny (2009), Bagh and Jofre (2006), Carmona (2009), Nessah (2011), Barelli and Meneghel (2013) and Prokopovych (2013). 2. The result Assume that Xi is a non-empty, convex and compact subset of a Hausdorff topological vector space for any i ∈ N. Let Φ be a real-valued function defined on X × X and SΦ be the solution set to the Ky Fan minimax inequality corresponding to Φ . Definition 1. Φ is said to be slightly diagonally transfer continuous if Φ (x, z) > 0 implies that there exist an open neighborhood Oz of z and x′ ∈ X such that Φ (x′ , z ′ ) > 0 for all z ′ ∈ Oz \SΦ .3 Definition 2 (Baye et al., 1993). Φ is said to be diagonally transfer quasi-concave if, for any {x1 , . . . , xk } ⊂ X there exists {z 1 , . . . , z k } ⊂ X , where z h corresponds to xh with ∑hl = 1h, . . . , k, such that, j with λ > 0 for for each z ∈ co{z 1 , . . . , z k } and z = j j=1 λj z h1 hl j = 1, . . . , l, there is x ∈ {x , . . . , x } so that Φ (x, z) ≤ 0. Proposition 1. Assume that X is a non-empty, convex and compact subset of a Hausdorff topological vector space. If Φ is slightly diagonally transfer continuous and diagonally transfer quasi-concave, then SΦ is non-empty.4 The following definition (also considered in Prokopovych (2013)) generalizes the single deviation property introduced in Reny (2009) and Nassah and Tian (2016).5 Definition 3. A game G is said to satisfy the slight single deviation property if, whenever z ∈ / EG , there exist an open neighborhood Oz of z and x′ ∈ X such that for all z ′ ∈ Oz \EG there is a player i for whom x′i ∈ Pi (z ′ ). In order to obtain sufficient conditions for the existence of Nash equilibria in games satisfying the slight single deviation property, in light of Reny (2009, Section 3.1), we need to modify the standard assumptions (∗). So, we introduce (Nassah and Tian, 2016, Definition 16) in the setting of ordinal games: Definition 4. We say that G is transfer uniformly quasi-concave if, for any {x1 , . . . , xk } ⊂ X , there exists {z 1 , . . . , z k } ⊂ X , where z h corresponds to xh with ∑lh = 1h, . . . , k, such that, for each z ∈ j with λ > 0 for j = 1, . . . , l, co{z 1 , . . . , z k } and z = j j=1 λj z there exists x ∈ {xh1 , . . . , xhl } so that xi ∈ / Pi (z) for all i ∈ N. Transfer uniform quasi-concavity is a necessary condition for the existence of Nash equilibria: indeed, if x∗ is a Nash equilibrium, it is sufficient to set {z 1 , . . . , z k } = {x∗ } for any {x1 , . . . , xk }. Obviously, we have: 2 In Nassah and Tian (2016), the single deviation property and the transfer uniform quasi-concavity are called, respectively, weak transfer quasi-continuity and strong diagonal transfer quasi-concavity. 3 With respect to the diagonal transfer continuity introduced in Baye et al. (1993), Definition 1 allows that Oz ∩ SΦ ̸ = ∅. 4 The proof is given in the Appendix.
5 The difference between the single deviation property lies in the fact that Definition 3 allows Oz ∩ EG ̸ = ∅.
Proposition 2. G is transfer uniformly quasi-concave if and only if ΦG is diagonally transfer quasi-concave. Remark 1. The transfer uniform quasi-concavity is not connected with (∗). In fact, let us consider the ordinal game ˆ G associated to the following duopoly G (see Baye et al., 1993, Example 1): two firms set prices xi , xj ∈ [0, T ] and get profits as below, where i ̸ = j and 0 < c < T
{ ui (xi , xj ) =
xi
xi − c
if xi ≤ xj otherwise .
In this situation, if one looks at the associated ordinal game, condition (∗) is equivalent to the quasi-concavity, that is: ui (·, x−i ) is quasi-concave for all x−i ∈ X−i and all i ∈ N.6 Now, the duopoly is not quasi-concave. On the other hand, following (Baye et al., 1993, Example 1), given any set {x1 , . . . , xk } of strategy profiles, fix {y1 , . . . , yk } = {z } with yh1 = yh2 = max{xt1 , xt2 : t = 1, . . . , k}
for any h ∈ {1, . . . , k}. It is easy to see that xi ∈ /ˆ Pi (z) for i = 1, 2 and for any x ∈ {x1 , . . . , xk }, that is: ˆ G is transfer uniformly quasi-concave. Moreover, ˆ G satisfies the single deviation property. In fact, assume that z ∈ / EG = {(x1 , x2 ) : x1 = x2 ∈ [T − c , T ]}. If z1 > z2 , let ε > 0 be such that Oz =]z1 − ε, z1 + ε[×]z2 − ε, z2 + ε[ is below the diagonal of [0, T ]2 and set x′ = (z1 − ε, z1 − ε ). For every z ′ ∈ Oz , player 2 has a profitable deviation by using x′2 , P2 (z ′ ) for all z ′ ∈ Oz . Similarly if because z2′ < x′2 < z1′ ; so, x′2 ∈ ˆ z ∈ / EG and z1 < z2 (in this case, player 1 deviates). If z ∈ / EG and z1 = z2 , the single deviation property is satisfied for any ε > 0 such that ]z1 − ε, z1 + ε[×]z2 − ε, z2 + ε[⊂ [0, T − c [ 2 and x′ = (T , T ). Remark 2. Let us note that, because of Reny (2009, Section 3.1) and Theorem below, it is not true that the standard condition (∗) implies the transfer uniform quasi-concavity. In fact, the ordinal game associated to the normal form game considered in Reny (2009, Section 3.1) satisfies the (slight) single deviation property and (∗), but it does not have Nash equilibria. Connections between the transfer uniform quasi-concavity and other quasiconcavity like properties for normal form games are given in the next section. It is easy to prove the following: Proposition 3. G satisfies the slight single deviation property if and only if ΦG defined by (1) is slightly diagonally transfer continuous. Finally, in light of Propositions 1–3, we obtain a Nash equilibrium existence result: Theorem. Assume that G satisfies the slight single deviation property. Then, EG is non-empty if and only if G is transfer uniformly quasi-concave. Let us point out that, in the framework of normal form games, our Theorem slightly improve (Nassah and Tian, 2016, Theorem 6) (Example 2 provides a game where our result applies but not that in Nassah and Tian (2016)). 6 Assume that u (·, x ) is quasi-concave for all x ∈ X and all i ∈ N, i −i −i −i and let xi ∈ coˆ Pi (x) for some x ∈ X and i ∈ N. So, xi ∈ co{x1i , . . . , xki } j
with ui (xi , x−i ) > ui (x) for j = 1, . . . , k, and one gets a contradiction. On the other hand, suppose that (∗) holds and let x−i ∈ X−i . If ui (·, x−i ) is not quasij concave, we have ui (xi , x−i ) < ui (xi , x−i ), with j = 1, . . . , k, for at least one subset {x1i , . . . , xki } ⊂ Xi and xi ∈ co{x1i , . . . , xki }, which implies xi ∈ coˆ Pi (x), a contradiction.
V. Scalzo / Journal of Mathematical Economics 82 (2019) 197–201
3. Comparison with previous Nash equilibrium existence results In view of the counterexample from Reny (2009, Section 3.1) and Remark 1, we have that our Theorem is not connected with the equilibrium existence results for ordinal and normal form games which satisfy either the property (∗) or the quasi-concavity of ui (·, x−i ): see, among the others, Shafer and Sonnenschein (1975), Yannelis and Prabhakar (1983), Wu and Shen (1996), Scalzo (2015), He and Yannelis (2016) and Reny (2016) for what concern ordinal games and Dasgupta and Maskin (1986), Reny (1999), Reny (2009), Bagh and Jofre (2006), Carmona (2009) and Nessah (2011) for normal form games. Conditions like (∗) have been considered in Carmona and Podczeck (2016) for ordinal games and in Barelli and Meneghel (2013) for normal form games; in these papers, properties like (∗) and generalizations of continuity are merged in only one assumption on the game. The following examples present games satisfying the assumptions of our Theorem but not those of the equilibrium existence results given in Carmona and Podczeck (2016) and Barelli and Meneghel (2013). Example 1. Consider the{( ordinal ) game G = ⟨Xi , Pi ⟩i=1},2 with X1 = 2 1 X2 = [0, 1]. Let K1 = , : n is odd and n > 4 and K2 = n }n {( 1 3 ) 7 , : n is even and n ≥ 4 . P 1 and P2 are given as below : n n P2 (x) = {1}
if x ∈ K1
P1 (x) = {1}
if x ∈ K2
P1 (x) = P2 (x) = {1} P1 (x) = P2 (x) = ∅
if x = (0, 0) otherwise .
G is transfer uniformly quasi-concave because EG = X \[K1 ∪ K2 ∪ {(0, 0)}]. Moreover, it is easy to see that the slight single deviation property is satisfied: for all non equilibrium z, fix Oz = X and x′ = (1, 1). So, the assumptions of our Theorem hold true. Now, starting from G, we build up an ordinal game as considered in Carmona and Podczeck (2016), 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. Consider the non-equilibrium strategy profile x = (0, 0) and the compact set K = K1 ∪ K2 ∪ {(0, 0)}. The condition introduced in Carmona and Podczeck (2016, Defini′ tion that, for at least { } one player i, the set Fi (x ) i = ⋃ 3) requires ′ w ∈ X : ( w , x ) ≻ v be non-empty for some π : i i i i −i v∈π i (x′ ) [0, 1]2 ⇒ [0, 1]2 and for all x′ ∈ Ox ∩ K , where Ox is a suitable open neighborhood of x = (0, 0). Now, take a map π i : [0, 1]2 ⇒ [0, 1]2 for each player i = 1, 2 and an open neighborhood O of x = (0, 0). Consider x′ ∈ O ∩ K1 . Thus, x′2 = 1/n with n odd. For each v ∈ π 1 (x′ ), (w1 , x′2 ) ≻1 v requires that v2 = x′2 and w1 ∈ P1 (v ) (see the definitions of ≻1 and P1 ), which is possible only if w1 = 1 and v ∈ K2 . So, (w1 , x′2 ) ≻1 v implies x′2 = 3/k for some k even, k ≥ 4, and we get n/2 = k/2 − n ∈ N, that is: n is even, a contradiction. Therefore, F1 (x′ ) = ∅. If x′ ∈ O ∩ K2 , similarly to the previous case, we obtain F2 (x′ ) = ∅. Finally, the equilibrium existence results from Carmona and Podczeck (2016) cannot be applied to the game ⟨Xi , ≻i ⟩i=1,2 . Example 2. Let G = ⟨Xi , ui ⟩i=1,2 be such that X1 = X2 = [0, 1] and the functions u1 and u2 are defined as below: u1 (x) = 0 and u2 (x) = 1 if x1 < x2 and x ∈ /B 7 The author thank an anonymous referee for having simplified the original version of this example.
199
u1 (x) = 1 and u2 (x) = 0 if x1 > x2 and x ∈ /B u1 (x) = u2 (x) = 0 if x = (0, 0) u1 (x) = u2 (x) = 1 if x ∈ B , where B = {x : x1 = x2 > 0} ∪ ({0}×]0, 1]) ∪ (]0, 1] × {0}). We have EG = B, and it is clear that G is transfer uniformly quasi-concave. Moreover, G satisfies the slight single deviation property: for all z ∈ / EG , given any open neighborhood of z, it is sufficient to set x′ = (1, 1). So, all the assumptions of our Theorem are satisfied. On the other hand, Barelli and Meneghel (2013, Definition 2.1) fails at z = (0, 0) because none of the players can ensure profitable unilateral deviation for all non-equilibrium close to z. So, Barelli and Meneghel (2013, Theorem 2) cannot be applied in this situation. Finally, since the non-equilibrium (0, 0) is approached by a sequence of equilibria, G does not satisfy the assumptions of Nassah and Tian (2016, Theorem 6). Remark 3. Consider the Nash equilibrium existence result (Nassah and Tian, 2016, Theorem 4). As observed by the authors at page 527, if one starts from an ordinal game ⟨Xi , ⪰i ⟩i∈N where players compare strategy profiles by means of binary relations, in order to apply (Nassah and Tian, 2016, Theorem 4), one has to assume that each ⪰i is complete, reflexive and transitive. Indeed, Nassah and Tian (2016, Theorem 4) is useful to derive several Nash equilibrium existence results when the players have payoff functions: see Nassah and Tian (2016), Corollaries from 1 to 7. Here, the mappings Pi do not need to be derived from binary relations; so, our setting is different. However, even if we assume Pi (x) = {yi ∈ Xi : (yi , x−i ) ≻i x} for each player i, where ≻i is the asymmetric part of the binary relation ⪰i , our Theorem requires neither completeness nor transitivity. So, the ⪰ corresponding mappings Pi defined at page 527 of Nassah and Tian (2016) do not necessarily verify the assumption of Nassah and Tian (2016, Theorem 4). An early result on the existence of Nash equilibria in games satisfying the property given in Definition 3 has been obtained in Prokopovych (2013, Theorem 2). Our Theorem differs from Prokopovych (2013, Theorem 2) because of the quasi-concavity assumption. We use the transfer uniform quasi-concavity, which is a necessary condition for the existence of Nash equilibria and it is not related with the single deviation property, in the sense that the transfer uniform quasi-concavity holds in games where the slight single deviation property fails: for instance, consider the 1-person game with payoff function defined by u(x) = x if x ∈ [0, 1[ and u(1) = 0. The quasi-concavity assumption introduced in Prokopovych (2013, Theorem 2) supposes that the game satisfies the property given in Definition 3. More precisely, for any z ∈ / EG , let xz and Oz be, respectively, the strategy profile and the open neighborhood associated to z by the slight single deviation property. The condition considered in Prokopovych (2013) is the following: (a) for all {z 1 , . . . , z k } with {z 1 , . . . , z k } ∩ EG { = ∅ and all z ′ ∈ ∩}kh=1 Oz h , h
there exists i ∈ N such that zi′ ̸ ∈ co xzi : h = 1, . . . , k . So, it
is not possible to separate this condition from the slight single deviation property. Moreover, since the transfer uniform quasi-concavity is a necessary condition for the existence of Nash equilibria, in light of Prokopovych (2013, Theorem 2), we have that (a) implies the uniform transfer quasi-concavity in every game which satisfies the slight single deviation property. On the other hands, the transfer uniform quasi-concavity holds in games which satisfy the slight single deviation property but
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not (a): in fact, consider the 1-player game with the payoff function u such that u(x) = 1 if x ∈ [0, 1] ∩ Q and u(x) = 0 if x ∈ [0, 1]\Q. In this example, all the assumptions of our Theorem hold but (a) fails. So, our result improves that given in Prokopovych (2013).8 Finally, we compare our result with Baye et al. (1993, Theorem 1), where the existence of Nash equilibria has been obtained for normal form games G = ⟨Xi , ui ⟩i∈N such that the aggregator ΨG defined by:
ΨG (x, z) =
∑
[ui (xi , z−i ) − ui (z)]
∀ (x, z) ∈ X × X
i∈N
is diagonally transfer continuous and diagonally transfer quasiconcave: indeed, x∗ ∈ EG if and only if x∗ is a solution to the Ky Fan minimax inequality corresponding to ΨG .9 It is easy to see that the uniform transfer quasi-concavity of G implies the diagonal transfer quasi-concavity of ΨG , but the converse does not hold, as the following example shows. More precisely, the example below proves that the transfer uniform quasi-concavity in our Theorem cannot be replaced by the diagonal transfer quasiconcavity on ΨG . In particular, it is presented a normal form game which satisfies the single deviation property and ΨG is diagonally transfer quasi-concave, but the set of Nash equilibria is empty. Example 3 (See Sion and Wolfe, 1957). Let G be the 2-player game where X = [0, 1] × [0, 1],
u1 (x1 , x2 ) =
1
⎧ ⎪ ⎪ ⎨−1
if x1 < x2 < x1 +
0 ⎪ ⎪ ⎩
if x1 = x2 or x2 = x1 +
1
2
otherwise
1 2
and u2 (x1 , x2 ) = −u1 (x1 , x2 ) . The game does not have Nash equilibria (see Sion and Wolfe, 1957 for the details). G is better reply secure (see Reny, 1999), which implies the single deviation property (see footnote 7 at page 556 in Reny, 2016).10 Now, let F be a finite subset of strategy profiles. We set F ′ = {z }, where (z1 , z2 ) = (x2 , x1 ) and x = (x1 , x2 ) belongs to F . So, we have:
ΨG (x, z) = u1 (x1 , z2 ) + u2 (z1 , x2 ) = u1 (x1 , x1 ) + u2 (x2 , x2 ) = 0 , that is: diagonal transfer quasi-concavity holds on ΨG . Summing up, we have:
ΨG is diagonally transfer continuous H⇒ G has the slight single deviation property G is transfer uniformly quasi-concave H⇒ ΨG is diagonally transfer quasi-concave and, in the light of Proposition 1, Baye et al. (1993, Theorem 1), our Theorem and Example 3, we deduce: G has the slight single deviation property ̸H⇒ ΨG is diagonally transfer continuous
ΨG is diagonally transfer quasi-concave ̸H⇒ G is uniformly transfer quasi-concave. 8 The same arguments apply to Prokopovych (2016, Theorem 2). 9 The proof of Proposition 1 uses the argument employed in the proof of Baye et al. (1993, Theorem 1), that is Tian (1993, Lemma 1). Another Nash equilibrium existence result using the Ky Fan minimax inequality corresponding the aggregator ΨG is given in Scalzo (2013). 10 The better reply security of G follows from the proof of Carmona (2005, Proposition 4) and Reny (1999, Proposition 3.2).
4. Conclusions In this paper, we have considered ordinal games where players do not necessarily have complete or transitive binary relations. We have related the Nash equilibria with the solutions to a suitable Ky Fan minimax inequality. Therefore, a new equilibrium existence result have been obtained. Tools for our result have been a slight generalization of the single deviation property (see Reny, 2009; Nassah and Tian, 2016), and a new necessary condition for the existence of equilibria (see Nassah and Tian, 2016 for the normal form games). Our result is given for both ordinal and normal form games. Examples have showed that our Theorem is new and cannot be deduced from the previous ones. Acknowledgments The author is grateful to the reviewers for their valuable comments and the editor for his support in the review process. Appendix Assume that X is a non-empty, convex and compact subset of a Hausdorff topological vector space. A mapping F : X ⇒ X is said to be (see Tian, 1993): (i) transfer closed-valued if z ∈ / F (x) implies that z ∈ / clF (x′ ) for ′ 11 some x ∈ X ; (ii) transfer FS-convex if for any {x1 , . . . , xk } ⊂ X there exists h {z 1 , . . . , z k } ⊂ X , where z h corresponds to ⋃lx for heach h ∈ h1 hl j {1, . . . , k}, such that co{z , . . . , z } ⊆ j=1 F (x ) for all h1 hl 1 k {z , . . . , z } ⊆ {z , . . . , z }. One has the following result: Lemma 1 in Tian ⋂(1993). If F is transfer closed-valued and transfer FS-convex, then x∈X F (x) is non-empty and compact. Proof of Proposition 1. Define the map F : X ⇒ X ⋂ by F (x) = {z ∈ X : Φ (x, z) ≤ 0} for all x ∈ X . It is clear that SΦ = x∈X F (x). Suppose that SΦ = ∅. So, the function Φ is diagonally transfer continuous (see Baye et al., 1993), that is: Φ (x, z) > 0 implies that there exists x′ ∈ X so that Φ (x′ , z ′ ) > 0 for all z ′ in some open neighborhood of z. Let z ∈ / F (x) for some x and z. One has Φ (x, z) > 0 and, because of the diagonal transfer continuity, z ∈ / clF (x′ ) for a suitable x′ ∈ X , that is: F is transfer closedvalued. On the other hand, given any {x1 , . . . , xk } ⊂ X , since Φ is diagonally transfer quasi-concave, we find a corresponding {z 1 , . . . , z k } ⊂ X ∑ such that, for each {z h1 , . . . , z hl } ⊆ {z 1 , . . . , z k } l hj and each z = with λj > 0 for j = 1, . . . , l, one j=1 λj z gets Φ (x, z) ≤ 0 for at least one x ∈ {xh1 , . . . , xhl }, which gives ⋃l hj z ∈ F (x) ⊆ j=1 F (x ), and it is proved that F is transfer FSconvex. Finally, Lemma ⋂ 1 in Tian (1993) applies, and we get the contradiction SΦ = x∈X F (x) ̸ = ∅. □ References Bagh, A., Jofre, A., 2006. Reciprocal upper semicontinuity and better reply secure games: a comment. Econometrica 74, 1715–1721. Barelli, P., Meneghel, I., 2013. A note on the equilibrium existence problem in discontinuous games. Econometrica 81, 813–824. Baye, M.R., Tian, G., Zhou, J., 1993. Characterizations of the existence of equilibria in games with discontinuous and non-quasiconcave payoffs. Rev. Econom. Stud. 60, 935–948. Carmona, G., 2005. On the existence of equilibria in discontinuous games: three counterexamples. Internat. J. Game Theory 33, 181–187.
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Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco
Equilibrium existence in games: Slight single deviation property and Ky Fan minimax inequality Vincenzo Scalzo Department of Economics and Statistics (DISES), University of Naples Federico II, via Cinthia 21, 80126 Napoli, Italy
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Article history: Received 16 July 2018 Received in revised form 26 February 2019 Accepted 27 February 2019 Available online 5 March 2019 Keywords: Discontinuous ordinal games Ky Fan minimax inequality Existence of Nash equilibria
a b s t r a c t In this paper, we exhibit a relation between the Nash equilibrium existence problem in ordinal games and the Ky Fan minimax inequality. In this way, we obtain new sufficient conditions for the existence of equilibria. The main tools used in the paper are a necessary condition introduced in Nassah and Tian (2016) for normal form games and a generalization of the single deviation property. Examples compare our result with the previous ones. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Consider a finite set N of players and, for any i ∈ N, let Xi be a non-empty ∏ subset of a Hausdorff topological vector space. Denoted by X = j∈N Xj the set of strategy profiles and, for each i ∈ N, let Pi be a mapping from X to Xi . The list G = ⟨Xi , Pi ⟩i∈N is called ordinal game, and a Nash equilibrium of G (equilibrium briefly) is an element x∗ ∈ X so that Pi (x∗ ) = ∅ for all i ∈ N (EG denotes the set of Nash equilibria of G). The existence of Nash equilibria of G can be investigated by means of the Ky Fan minimax inequality. Given a real-valued function Φ defined on X × X , the corresponding Ky Fan minimax inequality (Fan, 1972) is the following problem: find x∗ ∈ X such that Φ (x, x∗ ) ≤ 0 for all x ∈ X (in such a case, x∗ is said to be a solution). Now, for every x and z which belong to X , define ΦG (x, z) as below:
ΦG (x, z) =
∑
˜ Pi (xi , z)
(1)
i∈N
where ˜ Pi (xi , z) = 1 whenever xi ∈ Pi (z) and ˜ Pi (xi , z) = 0 otherwise. Obviously, x∗ is an equilibrium of G if and only if it is a solution to the Ky Fan minimax inequality corresponding to ΦG , that is: ΦG (x, x∗ ) ≤ 0 for all x ∈ X . Conditions which guarantee the existence of solutions to the Ky Fan minimax inequality allow to identify classes of games endowed with Nash equilibria. So, it is interesting to investigate if the Ky Fan minimax inequality approach leads to new sufficient conditions for the existence of Nash equilibria in ordinal games. In this paper, we show that this is possible. Using Tian (1993, Lemma 1), we obtain the existence of solutions to the Ky Fan E-mail address: [email protected]. https://doi.org/10.1016/j.jmateco.2019.02.008 0304-4068/© 2019 Elsevier B.V. All rights reserved.
minimax inequality when the function Φ is slightly diagonally transfer continuous and diagonally transfer quasi-concave: see Proposition 1. So, given an ordinal game G, we relate the slight diagonal transfer continuity of the function ΦG defined by (1) to a generalization of the single deviation property (see Reny, 2009, Nassah and Tian, 2016), that we call slight single deviation property. On the other hand, the diagonal transfer quasi-concavity of ΦG corresponds to a necessary condition for the existence of Nash equilibria called transfer uniform quasi-concavity (see Nassah and Tian, 2016 for the normal form games). Hence, we prove that the slight single deviation property and the transfer uniform quasi-concavity guarantee the existence of Nash equilibria: see Theorem. Let us note that the single deviation property (and the generalization considered in the present paper) is not sufficient to guarantee the existence of equilibria under the standard condition (∗) xi ∈ / coPi (x) for all x ∈ X and all i ∈ N: see Reny (2009, Section 3.1).1 Moreover, we provide examples to show that the transfer uniform quasi-concavity is not connected with condition (∗) and our Theorem applies to situations which are not covered by the previous literature on the existence of Nash equilibria in ordinal games: see Shafer and Sonnenschein (1975), Yannelis and Prabhakar (1983), Wu and Shen (1996), Scalzo (2015), Carmona and Podczeck (2016), He and Yannelis (2016), Nassah and Tian (2016) and Reny (2016). Finally, we consider normal form games G = ⟨Xi , ui ⟩i∈N : in ⟨ ⟩ this case, we associate G with the ordinal game ˆ G = Xi , ˆ Pi i∈N 1 Given a set A, coA denotes the convex hull of A. Condition (∗) has been widely used in the previous literature on the existence of equilibria in ordinal games: among the others, see Shafer and Sonnenschein (1975), Yannelis and Prabhakar (1983), Wu and Shen (1996), Scalzo (2015), He and Yannelis (2016) and Reny (2016).
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where, for any player i, ˆ Pi (x) = {yi ∈ Xi : ui (yi , x−i ) > ui (x)}. In this setting, Nassah and Tian (2016, Theorem 6) states that Nash equilibria exist when the single deviation property and the transfer uniform quasi-concavity hold.2 The result given in the present paper is a slight improvement due to the generalization of the single deviation property. Moreover, we compare our Theorem with the well known Baye et al. (1993, Theorem 1): we prove that they are independent. Furthermore, we remark that our result cannot be deduced from the previous ones given in Dasgupta and Maskin (1986), Reny (1999), Reny (2009), Bagh and Jofre (2006), Carmona (2009), Nessah (2011), Barelli and Meneghel (2013) and Prokopovych (2013). 2. The result Assume that Xi is a non-empty, convex and compact subset of a Hausdorff topological vector space for any i ∈ N. Let Φ be a real-valued function defined on X × X and SΦ be the solution set to the Ky Fan minimax inequality corresponding to Φ . Definition 1. Φ is said to be slightly diagonally transfer continuous if Φ (x, z) > 0 implies that there exist an open neighborhood Oz of z and x′ ∈ X such that Φ (x′ , z ′ ) > 0 for all z ′ ∈ Oz \SΦ .3 Definition 2 (Baye et al., 1993). Φ is said to be diagonally transfer quasi-concave if, for any {x1 , . . . , xk } ⊂ X there exists {z 1 , . . . , z k } ⊂ X , where z h corresponds to xh with ∑hl = 1h, . . . , k, such that, j with λ > 0 for for each z ∈ co{z 1 , . . . , z k } and z = j j=1 λj z h1 hl j = 1, . . . , l, there is x ∈ {x , . . . , x } so that Φ (x, z) ≤ 0. Proposition 1. Assume that X is a non-empty, convex and compact subset of a Hausdorff topological vector space. If Φ is slightly diagonally transfer continuous and diagonally transfer quasi-concave, then SΦ is non-empty.4 The following definition (also considered in Prokopovych (2013)) generalizes the single deviation property introduced in Reny (2009) and Nassah and Tian (2016).5 Definition 3. A game G is said to satisfy the slight single deviation property if, whenever z ∈ / EG , there exist an open neighborhood Oz of z and x′ ∈ X such that for all z ′ ∈ Oz \EG there is a player i for whom x′i ∈ Pi (z ′ ). In order to obtain sufficient conditions for the existence of Nash equilibria in games satisfying the slight single deviation property, in light of Reny (2009, Section 3.1), we need to modify the standard assumptions (∗). So, we introduce (Nassah and Tian, 2016, Definition 16) in the setting of ordinal games: Definition 4. We say that G is transfer uniformly quasi-concave if, for any {x1 , . . . , xk } ⊂ X , there exists {z 1 , . . . , z k } ⊂ X , where z h corresponds to xh with ∑lh = 1h, . . . , k, such that, for each z ∈ j with λ > 0 for j = 1, . . . , l, co{z 1 , . . . , z k } and z = j j=1 λj z there exists x ∈ {xh1 , . . . , xhl } so that xi ∈ / Pi (z) for all i ∈ N. Transfer uniform quasi-concavity is a necessary condition for the existence of Nash equilibria: indeed, if x∗ is a Nash equilibrium, it is sufficient to set {z 1 , . . . , z k } = {x∗ } for any {x1 , . . . , xk }. Obviously, we have: 2 In Nassah and Tian (2016), the single deviation property and the transfer uniform quasi-concavity are called, respectively, weak transfer quasi-continuity and strong diagonal transfer quasi-concavity. 3 With respect to the diagonal transfer continuity introduced in Baye et al. (1993), Definition 1 allows that Oz ∩ SΦ ̸ = ∅. 4 The proof is given in the Appendix.
5 The difference between the single deviation property lies in the fact that Definition 3 allows Oz ∩ EG ̸ = ∅.
Proposition 2. G is transfer uniformly quasi-concave if and only if ΦG is diagonally transfer quasi-concave. Remark 1. The transfer uniform quasi-concavity is not connected with (∗). In fact, let us consider the ordinal game ˆ G associated to the following duopoly G (see Baye et al., 1993, Example 1): two firms set prices xi , xj ∈ [0, T ] and get profits as below, where i ̸ = j and 0 < c < T
{ ui (xi , xj ) =
xi
xi − c
if xi ≤ xj otherwise .
In this situation, if one looks at the associated ordinal game, condition (∗) is equivalent to the quasi-concavity, that is: ui (·, x−i ) is quasi-concave for all x−i ∈ X−i and all i ∈ N.6 Now, the duopoly is not quasi-concave. On the other hand, following (Baye et al., 1993, Example 1), given any set {x1 , . . . , xk } of strategy profiles, fix {y1 , . . . , yk } = {z } with yh1 = yh2 = max{xt1 , xt2 : t = 1, . . . , k}
for any h ∈ {1, . . . , k}. It is easy to see that xi ∈ /ˆ Pi (z) for i = 1, 2 and for any x ∈ {x1 , . . . , xk }, that is: ˆ G is transfer uniformly quasi-concave. Moreover, ˆ G satisfies the single deviation property. In fact, assume that z ∈ / EG = {(x1 , x2 ) : x1 = x2 ∈ [T − c , T ]}. If z1 > z2 , let ε > 0 be such that Oz =]z1 − ε, z1 + ε[×]z2 − ε, z2 + ε[ is below the diagonal of [0, T ]2 and set x′ = (z1 − ε, z1 − ε ). For every z ′ ∈ Oz , player 2 has a profitable deviation by using x′2 , P2 (z ′ ) for all z ′ ∈ Oz . Similarly if because z2′ < x′2 < z1′ ; so, x′2 ∈ ˆ z ∈ / EG and z1 < z2 (in this case, player 1 deviates). If z ∈ / EG and z1 = z2 , the single deviation property is satisfied for any ε > 0 such that ]z1 − ε, z1 + ε[×]z2 − ε, z2 + ε[⊂ [0, T − c [ 2 and x′ = (T , T ). Remark 2. Let us note that, because of Reny (2009, Section 3.1) and Theorem below, it is not true that the standard condition (∗) implies the transfer uniform quasi-concavity. In fact, the ordinal game associated to the normal form game considered in Reny (2009, Section 3.1) satisfies the (slight) single deviation property and (∗), but it does not have Nash equilibria. Connections between the transfer uniform quasi-concavity and other quasiconcavity like properties for normal form games are given in the next section. It is easy to prove the following: Proposition 3. G satisfies the slight single deviation property if and only if ΦG defined by (1) is slightly diagonally transfer continuous. Finally, in light of Propositions 1–3, we obtain a Nash equilibrium existence result: Theorem. Assume that G satisfies the slight single deviation property. Then, EG is non-empty if and only if G is transfer uniformly quasi-concave. Let us point out that, in the framework of normal form games, our Theorem slightly improve (Nassah and Tian, 2016, Theorem 6) (Example 2 provides a game where our result applies but not that in Nassah and Tian (2016)). 6 Assume that u (·, x ) is quasi-concave for all x ∈ X and all i ∈ N, i −i −i −i and let xi ∈ coˆ Pi (x) for some x ∈ X and i ∈ N. So, xi ∈ co{x1i , . . . , xki } j
with ui (xi , x−i ) > ui (x) for j = 1, . . . , k, and one gets a contradiction. On the other hand, suppose that (∗) holds and let x−i ∈ X−i . If ui (·, x−i ) is not quasij concave, we have ui (xi , x−i ) < ui (xi , x−i ), with j = 1, . . . , k, for at least one subset {x1i , . . . , xki } ⊂ Xi and xi ∈ co{x1i , . . . , xki }, which implies xi ∈ coˆ Pi (x), a contradiction.
V. Scalzo / Journal of Mathematical Economics 82 (2019) 197–201
3. Comparison with previous Nash equilibrium existence results In view of the counterexample from Reny (2009, Section 3.1) and Remark 1, we have that our Theorem is not connected with the equilibrium existence results for ordinal and normal form games which satisfy either the property (∗) or the quasi-concavity of ui (·, x−i ): see, among the others, Shafer and Sonnenschein (1975), Yannelis and Prabhakar (1983), Wu and Shen (1996), Scalzo (2015), He and Yannelis (2016) and Reny (2016) for what concern ordinal games and Dasgupta and Maskin (1986), Reny (1999), Reny (2009), Bagh and Jofre (2006), Carmona (2009) and Nessah (2011) for normal form games. Conditions like (∗) have been considered in Carmona and Podczeck (2016) for ordinal games and in Barelli and Meneghel (2013) for normal form games; in these papers, properties like (∗) and generalizations of continuity are merged in only one assumption on the game. The following examples present games satisfying the assumptions of our Theorem but not those of the equilibrium existence results given in Carmona and Podczeck (2016) and Barelli and Meneghel (2013). Example 1. Consider the{( ordinal ) game G = ⟨Xi , Pi ⟩i=1},2 with X1 = 2 1 X2 = [0, 1]. Let K1 = , : n is odd and n > 4 and K2 = n }n {( 1 3 ) 7 , : n is even and n ≥ 4 . P 1 and P2 are given as below : n n P2 (x) = {1}
if x ∈ K1
P1 (x) = {1}
if x ∈ K2
P1 (x) = P2 (x) = {1} P1 (x) = P2 (x) = ∅
if x = (0, 0) otherwise .
G is transfer uniformly quasi-concave because EG = X \[K1 ∪ K2 ∪ {(0, 0)}]. Moreover, it is easy to see that the slight single deviation property is satisfied: for all non equilibrium z, fix Oz = X and x′ = (1, 1). So, the assumptions of our Theorem hold true. Now, starting from G, we build up an ordinal game as considered in Carmona and Podczeck (2016), 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. Consider the non-equilibrium strategy profile x = (0, 0) and the compact set K = K1 ∪ K2 ∪ {(0, 0)}. The condition introduced in Carmona and Podczeck (2016, Defini′ tion that, for at least { } one player i, the set Fi (x ) i = ⋃ 3) requires ′ w ∈ X : ( w , x ) ≻ v be non-empty for some π : i i i i −i v∈π i (x′ ) [0, 1]2 ⇒ [0, 1]2 and for all x′ ∈ Ox ∩ K , where Ox is a suitable open neighborhood of x = (0, 0). Now, take a map π i : [0, 1]2 ⇒ [0, 1]2 for each player i = 1, 2 and an open neighborhood O of x = (0, 0). Consider x′ ∈ O ∩ K1 . Thus, x′2 = 1/n with n odd. For each v ∈ π 1 (x′ ), (w1 , x′2 ) ≻1 v requires that v2 = x′2 and w1 ∈ P1 (v ) (see the definitions of ≻1 and P1 ), which is possible only if w1 = 1 and v ∈ K2 . So, (w1 , x′2 ) ≻1 v implies x′2 = 3/k for some k even, k ≥ 4, and we get n/2 = k/2 − n ∈ N, that is: n is even, a contradiction. Therefore, F1 (x′ ) = ∅. If x′ ∈ O ∩ K2 , similarly to the previous case, we obtain F2 (x′ ) = ∅. Finally, the equilibrium existence results from Carmona and Podczeck (2016) cannot be applied to the game ⟨Xi , ≻i ⟩i=1,2 . Example 2. Let G = ⟨Xi , ui ⟩i=1,2 be such that X1 = X2 = [0, 1] and the functions u1 and u2 are defined as below: u1 (x) = 0 and u2 (x) = 1 if x1 < x2 and x ∈ /B 7 The author thank an anonymous referee for having simplified the original version of this example.
199
u1 (x) = 1 and u2 (x) = 0 if x1 > x2 and x ∈ /B u1 (x) = u2 (x) = 0 if x = (0, 0) u1 (x) = u2 (x) = 1 if x ∈ B , where B = {x : x1 = x2 > 0} ∪ ({0}×]0, 1]) ∪ (]0, 1] × {0}). We have EG = B, and it is clear that G is transfer uniformly quasi-concave. Moreover, G satisfies the slight single deviation property: for all z ∈ / EG , given any open neighborhood of z, it is sufficient to set x′ = (1, 1). So, all the assumptions of our Theorem are satisfied. On the other hand, Barelli and Meneghel (2013, Definition 2.1) fails at z = (0, 0) because none of the players can ensure profitable unilateral deviation for all non-equilibrium close to z. So, Barelli and Meneghel (2013, Theorem 2) cannot be applied in this situation. Finally, since the non-equilibrium (0, 0) is approached by a sequence of equilibria, G does not satisfy the assumptions of Nassah and Tian (2016, Theorem 6). Remark 3. Consider the Nash equilibrium existence result (Nassah and Tian, 2016, Theorem 4). As observed by the authors at page 527, if one starts from an ordinal game ⟨Xi , ⪰i ⟩i∈N where players compare strategy profiles by means of binary relations, in order to apply (Nassah and Tian, 2016, Theorem 4), one has to assume that each ⪰i is complete, reflexive and transitive. Indeed, Nassah and Tian (2016, Theorem 4) is useful to derive several Nash equilibrium existence results when the players have payoff functions: see Nassah and Tian (2016), Corollaries from 1 to 7. Here, the mappings Pi do not need to be derived from binary relations; so, our setting is different. However, even if we assume Pi (x) = {yi ∈ Xi : (yi , x−i ) ≻i x} for each player i, where ≻i is the asymmetric part of the binary relation ⪰i , our Theorem requires neither completeness nor transitivity. So, the ⪰ corresponding mappings Pi defined at page 527 of Nassah and Tian (2016) do not necessarily verify the assumption of Nassah and Tian (2016, Theorem 4). An early result on the existence of Nash equilibria in games satisfying the property given in Definition 3 has been obtained in Prokopovych (2013, Theorem 2). Our Theorem differs from Prokopovych (2013, Theorem 2) because of the quasi-concavity assumption. We use the transfer uniform quasi-concavity, which is a necessary condition for the existence of Nash equilibria and it is not related with the single deviation property, in the sense that the transfer uniform quasi-concavity holds in games where the slight single deviation property fails: for instance, consider the 1-person game with payoff function defined by u(x) = x if x ∈ [0, 1[ and u(1) = 0. The quasi-concavity assumption introduced in Prokopovych (2013, Theorem 2) supposes that the game satisfies the property given in Definition 3. More precisely, for any z ∈ / EG , let xz and Oz be, respectively, the strategy profile and the open neighborhood associated to z by the slight single deviation property. The condition considered in Prokopovych (2013) is the following: (a) for all {z 1 , . . . , z k } with {z 1 , . . . , z k } ∩ EG { = ∅ and all z ′ ∈ ∩}kh=1 Oz h , h
there exists i ∈ N such that zi′ ̸ ∈ co xzi : h = 1, . . . , k . So, it
is not possible to separate this condition from the slight single deviation property. Moreover, since the transfer uniform quasi-concavity is a necessary condition for the existence of Nash equilibria, in light of Prokopovych (2013, Theorem 2), we have that (a) implies the uniform transfer quasi-concavity in every game which satisfies the slight single deviation property. On the other hands, the transfer uniform quasi-concavity holds in games which satisfy the slight single deviation property but
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not (a): in fact, consider the 1-player game with the payoff function u such that u(x) = 1 if x ∈ [0, 1] ∩ Q and u(x) = 0 if x ∈ [0, 1]\Q. In this example, all the assumptions of our Theorem hold but (a) fails. So, our result improves that given in Prokopovych (2013).8 Finally, we compare our result with Baye et al. (1993, Theorem 1), where the existence of Nash equilibria has been obtained for normal form games G = ⟨Xi , ui ⟩i∈N such that the aggregator ΨG defined by:
ΨG (x, z) =
∑
[ui (xi , z−i ) − ui (z)]
∀ (x, z) ∈ X × X
i∈N
is diagonally transfer continuous and diagonally transfer quasiconcave: indeed, x∗ ∈ EG if and only if x∗ is a solution to the Ky Fan minimax inequality corresponding to ΨG .9 It is easy to see that the uniform transfer quasi-concavity of G implies the diagonal transfer quasi-concavity of ΨG , but the converse does not hold, as the following example shows. More precisely, the example below proves that the transfer uniform quasi-concavity in our Theorem cannot be replaced by the diagonal transfer quasiconcavity on ΨG . In particular, it is presented a normal form game which satisfies the single deviation property and ΨG is diagonally transfer quasi-concave, but the set of Nash equilibria is empty. Example 3 (See Sion and Wolfe, 1957). Let G be the 2-player game where X = [0, 1] × [0, 1],
u1 (x1 , x2 ) =
1
⎧ ⎪ ⎪ ⎨−1
if x1 < x2 < x1 +
0 ⎪ ⎪ ⎩
if x1 = x2 or x2 = x1 +
1
2
otherwise
1 2
and u2 (x1 , x2 ) = −u1 (x1 , x2 ) . The game does not have Nash equilibria (see Sion and Wolfe, 1957 for the details). G is better reply secure (see Reny, 1999), which implies the single deviation property (see footnote 7 at page 556 in Reny, 2016).10 Now, let F be a finite subset of strategy profiles. We set F ′ = {z }, where (z1 , z2 ) = (x2 , x1 ) and x = (x1 , x2 ) belongs to F . So, we have:
ΨG (x, z) = u1 (x1 , z2 ) + u2 (z1 , x2 ) = u1 (x1 , x1 ) + u2 (x2 , x2 ) = 0 , that is: diagonal transfer quasi-concavity holds on ΨG . Summing up, we have:
ΨG is diagonally transfer continuous H⇒ G has the slight single deviation property G is transfer uniformly quasi-concave H⇒ ΨG is diagonally transfer quasi-concave and, in the light of Proposition 1, Baye et al. (1993, Theorem 1), our Theorem and Example 3, we deduce: G has the slight single deviation property ̸H⇒ ΨG is diagonally transfer continuous
ΨG is diagonally transfer quasi-concave ̸H⇒ G is uniformly transfer quasi-concave. 8 The same arguments apply to Prokopovych (2016, Theorem 2). 9 The proof of Proposition 1 uses the argument employed in the proof of Baye et al. (1993, Theorem 1), that is Tian (1993, Lemma 1). Another Nash equilibrium existence result using the Ky Fan minimax inequality corresponding the aggregator ΨG is given in Scalzo (2013). 10 The better reply security of G follows from the proof of Carmona (2005, Proposition 4) and Reny (1999, Proposition 3.2).
4. Conclusions In this paper, we have considered ordinal games where players do not necessarily have complete or transitive binary relations. We have related the Nash equilibria with the solutions to a suitable Ky Fan minimax inequality. Therefore, a new equilibrium existence result have been obtained. Tools for our result have been a slight generalization of the single deviation property (see Reny, 2009; Nassah and Tian, 2016), and a new necessary condition for the existence of equilibria (see Nassah and Tian, 2016 for the normal form games). Our result is given for both ordinal and normal form games. Examples have showed that our Theorem is new and cannot be deduced from the previous ones. Acknowledgments The author is grateful to the reviewers for their valuable comments and the editor for his support in the review process. Appendix Assume that X is a non-empty, convex and compact subset of a Hausdorff topological vector space. A mapping F : X ⇒ X is said to be (see Tian, 1993): (i) transfer closed-valued if z ∈ / F (x) implies that z ∈ / clF (x′ ) for ′ 11 some x ∈ X ; (ii) transfer FS-convex if for any {x1 , . . . , xk } ⊂ X there exists h {z 1 , . . . , z k } ⊂ X , where z h corresponds to ⋃lx for heach h ∈ h1 hl j {1, . . . , k}, such that co{z , . . . , z } ⊆ j=1 F (x ) for all h1 hl 1 k {z , . . . , z } ⊆ {z , . . . , z }. One has the following result: Lemma 1 in Tian ⋂(1993). If F is transfer closed-valued and transfer FS-convex, then x∈X F (x) is non-empty and compact. Proof of Proposition 1. Define the map F : X ⇒ X ⋂ by F (x) = {z ∈ X : Φ (x, z) ≤ 0} for all x ∈ X . It is clear that SΦ = x∈X F (x). Suppose that SΦ = ∅. So, the function Φ is diagonally transfer continuous (see Baye et al., 1993), that is: Φ (x, z) > 0 implies that there exists x′ ∈ X so that Φ (x′ , z ′ ) > 0 for all z ′ in some open neighborhood of z. Let z ∈ / F (x) for some x and z. One has Φ (x, z) > 0 and, because of the diagonal transfer continuity, z ∈ / clF (x′ ) for a suitable x′ ∈ X , that is: F is transfer closedvalued. On the other hand, given any {x1 , . . . , xk } ⊂ X , since Φ is diagonally transfer quasi-concave, we find a corresponding {z 1 , . . . , z k } ⊂ X ∑ such that, for each {z h1 , . . . , z hl } ⊆ {z 1 , . . . , z k } l hj and each z = with λj > 0 for j = 1, . . . , l, one j=1 λj z gets Φ (x, z) ≤ 0 for at least one x ∈ {xh1 , . . . , xhl }, which gives ⋃l hj z ∈ F (x) ⊆ j=1 F (x ), and it is proved that F is transfer FSconvex. Finally, Lemma ⋂ 1 in Tian (1993) applies, and we get the contradiction SΦ = x∈X F (x) ̸ = ∅. □ References Bagh, A., Jofre, A., 2006. Reciprocal upper semicontinuity and better reply secure games: a comment. Econometrica 74, 1715–1721. Barelli, P., Meneghel, I., 2013. A note on the equilibrium existence problem in discontinuous games. Econometrica 81, 813–824. Baye, M.R., Tian, G., Zhou, J., 1993. Characterizations of the existence of equilibria in games with discontinuous and non-quasiconcave payoffs. Rev. Econom. Stud. 60, 935–948. Carmona, G., 2005. On the existence of equilibria in discontinuous games: three counterexamples. Internat. J. Game Theory 33, 181–187.
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