On the existence of essential and trembling-hand perfect equilibria in discontinuous games

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1 Econ Theory Bull (2014) 2:1 12 DOI /s RESEARCH ARTICLE On the exstence of essental and tremblng-hand perfect equlbra n dscontnuous games Vncenzo Scalzo Receved: 31 August 2013 / Accepted: 19 November 2013 / Publshed onlne: 5 December 2013 SAET 2013 Abstract We dentfy classes of dscontnuous games wth nfntely many pure strateges where, for every class and every game n a dense subset, any mxed-strategy equlbrum s essental. Moreover, n some cases, we prove that the essental mxedstrategy equlbra are tremblng-hand perfect and each stable set of equlbra contans only one element. Keywords Dscontnuous nfnte games Essental equlbra Tremblng-hand perfect equlbra Strategc stablty JEL Classfcaton C72 1 Introducton In some recent papers, stablty propertes of Nash equlbra under perturbatons on the payoffs and on the mxed-strategy profles have been studed n classes of games where the payoffs are not necessarly contnuous functons (henceforth dscontnuous games). Carbonell-Ncolau (2010) and Scalzo (2013) dentfy classes of dscontnuous games where, n the generc case, any pure-strategy Nash equlbrum s essental (see Wu and Jang 1962). Regardng the mxed extenson of a game, Carbonell-Ncolau (2011a,b) gave suffcent condtons for the exstence of tremblng-hand perfect (see Selten 1975) Nash equlbra and stable sets of equlbra (see Kohlberg and Mertens 1986; Al-Najjar 1995). V. Scalzo (B) Department of Economcs and Statstcs (DISES), Unversty of Napol Federco II, va Cntha (Monte S. Angelo), Napol, Italy e-mal: vncenzo.scalzo@unna.t; scalzo@unna.t

2 2 V. Scalzo In ths paper, we present classes of dscontnuous games, denoted by g 1, g 2 and g 3, where, for each game n a dense subset, every mxed-strategy Nash equlbrum s essental. We prove that each essental mxed-strategy Nash equlbrum of any game n g 3 s tremblng-hand perfect and every stable set of equlbra contans only one element. The same property holds also n a subspace of g 2, denoted by g 4, when there are only two players. The paper s organzed as follows: Secton 2 ntroduces the settng. In Sect. 3, we present the spaces of games g wth = 1, 2, 3, and prove some of ther propertes. In partcular, we show that g 3 g 2 g 1 and, n each of such classes of games, there exsts a dense subset where, for every game, any mxed-strategy Nash equlbrum s essental. Furthermore, for each game n g 3 and n g 4, we prove that any essental mxed-strategy Nash equlbrum s tremblng-hand perfect. Examples show that g 1, g 2 and g 4 are not ncluded n the spaces of games consdered n Corollary 1 by Carbonell- Ncolau (2010) and n Theorem 1 (and 4) by Carbonell-Ncolau (2011a). Fnally (see Remark 7), we note that our result on the exstence of essental and tremblng-hand perfect equlbra n g 3 s new even f g 3 s ncluded n the space of games consdered n Theorem 1 by Carbonell-Ncolau (2011a). 2 The settng We consder strategc form games (n short games) G =, u I where the set of players I s fnte and, for any I, s a non-empty and compact subset of a locally convex metrzable topologcal vector space. Any payoff u s a bounded Borel measurable real-valued functon defned on the set of strategy profles = j I j; we denote a strategy profle by x = (x, x ), where x and x = j = j.letg b be the space of such games endowed wth the sup-norm metrc ρ: ρ(g, G ) = I sup u (x) u (x). (1) x It s easy to see that g b s a complete metrc space. For each I, we denote by M the set of Borel probablty measures on endowed wth the weak* topology; note that M s compact, convex, locally convex and metrzable (see, for example, Alprants and Border 1999). Let M = I M be equpped wth the product topology. A measure μ M s denoted by (μ,μ ), where μ M and μ j = M j. Because Fubn s theorem holds n our settng, for any μ M and any I,we have: U (μ) = u (x)dμ(x) = dμ (x ) u (x, x )dμ (x ). The game G = M, U I s the mxed extenson of G and the elements of M are called mxed strateges of the player (the elements of are sad pure strateges). A Nash equlbrum of G s called a mxed-strategy equlbrum of G.

3 On the exstence of essental and tremblng-hand perfect equlbra 3 Defnton 1 Let g g b and G g. A mxed-strategy profle μ s sad to be an essental equlbrum of G relatve to g f t s a mxed-strategy equlbrum of G and, for any open neghborhood N μ of μ, there exsts δ>0 such that any game G g wth ρ(g, G )<δhas at least one mxed-strategy equlbrum whch belongs to N μ. An essental equlbrum s stable under slght perturbatons on the payoffs of the game. Now, we consder perturbatons on the set of mxed-strategy profles (see Selten 1975). We recall that μ M s sad strctly postve f μ (O )>0 for any open subset O of. The subset of strctly postve measure of M s denoted by M 0. For any ν M 0 = I M0 and any θ [0, 1[ I I denotes the number of players, we set M θ ν ={μ M : μ θ ν } and M θν = I Mθ ν.ifthe,wehavethegame mxed strateges of any player are restrcted to the set M θ ν G θν = M θ ν, U M θν I, (2) whch s called a Selten perturbaton of G. Defnton 2 A mxed-strategy profle μ of a game G =, u I s sad to be tremblng-hand perfect f there exst sequences (ν n ) n M 0, (θ n ) n [0, 1[ I and (μ n ) n M such that: θ n 0, μ n μ and, for any n, μ n s a Nash equlbrum of the perturbaton G θ n ν n. For any I and any μ M,letμ θ ν = (1 θ )μ + θ ν. It s easy to see that M θ ν ={μ θ ν : μ M } and M θν ={μ θν : μ M}, where μ θν = (μ θ ν ) I (see Zhou et al. 2007). Let G θν =, u θν I be the perturbaton of the game G such that the payoff functons are defned as below (δ x denotes the Drac measure wth support {x}): u θν (x) = u (y)d(δ x θν)(y) x and I, (3) and consder the mxed extenson of G θν G θν = M, U θν I. It s not hard to prove that U θν (μ) = U (μ θν) for any μ M and any I. So, f μ s a Nash equlbrum of G θν, then μ θν s a Nash equlbrum of the Selten perturbaton G θν. For any game G,let G be the functon defned on as below: G (x, y) = I [ u (x, y ) u (y) ]. (4) It s clear that a pure-strategy profle x s a Nash equlbrum of G f and only f G (x, x) 0 for all x. Smlarly, we defne the functon ϒ G on the mxed extenson of G as follows:

4 4 V. Scalzo ϒ G (μ, σ ) = I [ U (μ,σ ) U (σ ) ] (μ, σ ) M M, (5) and μ s a mxed-strategy equlbrum of G f and only f ϒ G (μ, μ) 0 for all μ M. 1 Fnally, we recall that a correspondence F : Y s sad to be upper semcontnuous f, for any y Y and any open set O whch ncludes F(y), there exsts a neghborhood U of y such that F(y ) O for all y U; lower semcontnuous f, for any y Y and any open set O such that O F(y) =, there exsts a neghborhood U of y such that O F(y ) = for all y U. We note that, f Y and are Hausdorff spaces and s compact, F s upper semcontnuous f and only f Graph(F) ={(y, x) : x F(y)} s a closed subset (see Alprants and Border 1999). To obtan condtons whch guarantee the exstence of essental equlbra, useful s the followng result of Fort (1949): Lemma 1 Let Y and be metrc spaces and Y be complete. Assume that F : Y s an upper semcontnuous set-valued functon wth non-empty and compact values. Then, there exsts a dense G δ subset Q of Y such that F s lower semcontnuous at any pont belongng to Q. 2 3 Essental and tremblng-hand perfect equlbra In ths secton, we present classes of dscontnuous games wth the followng property: for each class, there exsts a dense subset where, for every game, every mxed-strategy equlbrum s essental (we also say that essental mxed-strategy equlbra exst n the generc case). Moreover, we show that n one of these classes, every essental equlbrum s tremblng-hand perfect and every stable set of equlbra contan only one element. 3.1 Some classes of dscontnuous games We recall that, n our settng, the pure-strategy set of any player s a non-empty and compact subset of a metrzable and locally convex topologcal vector space and M denotes the set of mxed strateges, that s the set of Borel probablty measures on. 1 We recall that, f Y s a subset of a topologcal vector space and f F s a real-valued functon defned on Y Y, the problem fnd ȳ Y such that F(y, ȳ) 0 y Y s called the Ky Fan s mnmax nequalty (Fan 1972). Exstence results on the Ky Fan s mnmax nequalty can be found n Fan (1972), Yannels (1991), Baye et al. (1993), Chang (2010), Scalzo (2013) and references theren. 2 We recall that a subset of a topologcal space s a G δ -set (n short G δ ) f t s a countable ntersecton of open subsets. We note that the theorem of Fort holds also n the more general case n whch Y s a Bare topologcal space. A Bare s space s a topologcal space where any countable ntersecton of open dense subsets s a dense subset. Examples of Bare spaces are the complete metrc spaces (see Alprants and Border 1999).

5 On the exstence of essental and tremblng-hand perfect equlbra 5 Therefore, M s compact, convex, locally convex and metrzable. Let us ntroduce the followng spaces of dscontnuous games: g 1 s the space of games where, for every game G, the functon ϒ G defned by (5) s postvely quas-transfer contnuous; 3 g 2 s the space of games where, for every game G, G (x, ) s lower semcontnuous for each x, where G s the functon defned by (4); g 3 s the space of games where, for every game, the sum of the payoff functons s upper semcontnuous and u (x, ) s lower semcontnuous for every x and every I. 4 g 2 s strctly ncluded n g 1 (see Proposton 1 and Example 1), and g 3 s strctly ncluded n g 2 (t s easy to see that g 3 g 2 ; and Example 2 shows that the ncluson s strct). For any game G, let us denote by E(G) the set of mxed-strategy equlbra of G. We have that E(G) s non-empty and compact for any G g 1 (see Proposton 2byScalzo 2013). Remark 1 In a recent paper, Prokopovych and Yannels (2012) ntroduced a condton on G, called unform dagonal securty, whch guarantees (n our settng) the exstence of mxed-strategy Nash equlbra, that s: for every x and every ε>0 there exsts x such that, for each y, G (x, y )> G (x, y) ε for any y n some neghborhood of y. The followng Proposton 2 shows that the class of unformly dagonally secure games s a subset of g 1. Remark 2 Any game G g 3 has the followng property: ϒ G (μ, ) s lower semcontnuous for any μ M. In fact, snce I u s upper semcontnuous, n lght of Theorem 14.5 by Alprants and Border (1999), we have that the functon λ I u dλ s lower semcontnuous. Smlarly, because u (x, ) s lower semcontnuous for any I and any x, the functon λ u (x, )dλ s lower semcontnuous. Let (σ n ) n be a sequence convergng to σ n the weak* topology. Therefore, lm nf u (x, )dσ n u (x, )dσ for any I and any x, whch mples lm nf I u dμ dσ n I lm nf u dμ dσ n I u dμ dσ, that s: the functon λ I u dμ dλ s lower semcontnuous. Fnally, we have that ϒ(μ, ) s the sum of lower semcontnuous functons. Proposton 1 g 2 s ncluded n g 1. Proof Let G g 2 and ϒ G (μ, σ ) > t, where μ and σ belong to M and t > 0. Snce μ s a probablty measure, there exsts x such that ϒ G (δ x,σ)>t (we recall that δ x denotes the Drac measure wth support {x}). Snce G (x, ) s lower semcontnuous, we have that the functon: λ M G (x, y)dλ(y) = ϒ G (δ x,λ) 3 Let Y be a topologcal vector space. A functon F : Y Y R s sad to be postvely quas-transfer contnuous f, whenever F(x, y) >t for some (x, y) Y Y and t > 0, there exsts a neghborhood O y of y and x Y such that F(x, y )>t for any y O y (see Scalzo 2013). 4 See also Yu 1999.

6 6 V. Scalzo s lower semcontnuous on M wth respect to the weak* topology (see Theorem 14.5 by Alprants and Border 1999). Therefore, there exsts a neghborhood N σ of σ such that: ϒ G (μ,σ )>t σ N σ, where μ = δ x. Ths proves that ϒ G s postvely quas-transfer contnuous. Proposton 2 Every unformly dagonally secure game belongs to g 1. Proof Let G be a unformly dagonally secure game and suppose that ϒ G (μ, σ ) > t wth some μ and σ belongng to M and t > 0. As n the prevous proof, we have ϒ G (δ x,σ) > t for some x. Letε = ϒ G (δ x,σ) t. Snce G s unformly dagonally secure, there exsts x such that: lm nf G(x, y )> G (x, y) ε y. y y The functon y lm nf y y G (x, y ) s lower semcontnuous and G (x, y) lm nf y y G (x, y ) for each y. For any λ M, wehave: G (x, y)dλ(y) lm nf G(x, y )dλ(y) > G (x, y)dλ(y) ε y y and Theorem 14.5 by Alprants and Border (1999) mples that the functon λ M lm nf y y G (x, y )dλ(y) s lower semcontnuous. So, we obtan: G (x, y)dσ (y) lm nf G(x, y )dσ (y) > y y for all σ n some neghborhood N σ of σ, that s: ϒ G (μ,σ )>ϒ G (δ x,σ) ϒ G (δ x,σ)+ t σ N σ, G (x, y)dσ(y) ε where μ = δ x, and the proof s concluded. Example 1 Let G =, u =1,2 be the verson of the rent-seekng game (see Tullock 1980) consdered by Prokopovych and Yannels (2012), wth 1 = 2 =[0, 2] and the payoff of any player s defned by u (x, x ) = 2π (x, x ) x, where: 1 4 f x = x = 0 1 π (x, x ) = 2 f x > x = 0 x 3 otherwse. x 3 +x3

7 On the exstence of essental and tremblng-hand perfect equlbra 7 G has no pure-strategy Nash equlbra (see Baye et al. 1994), but t s unformly dagonally secure (see Prokopovych and Yannels 2012). Therefore, n lght of Proposton 2, G belongs to g 1. On the other hand, snce u ( 1k, 1 k ) = 1 1 k for any k N and u 1 (0, 0) + u 2 (0, 0) = 1, we have that the sum of the payoff functons s not upper semcontnuous. Moreover, let x = ( 1 4, 0 ) and y = (0, 0); wehave G (x, y) = 1 4 and, for the sequence defned by yk = ( 1 k, 0 ) for any k N, one has G (x, y k ) = k 1 4. Hence, G(x, ) s not lower semcontnuous, that s: G does not belong to g 2. Example 2 Let G =, u =1,2 be the game where 1 = 2 =[0, 1], u 2 (x 1, x 2 ) = 0 for each (x 1, x 2 ) [0, 1] [0, 1] and u 1 s defned as follows: u 1 (x 1, x 2 ) = { x2 f x 2 [0, 1[ 0 f x 2 = 1. Snce G (x, y) = 0 for every (x, y), we have that G g 2. Let us prove that G s not payoff secure 5. Consder x = (1, 1) and let ε ]0, 1[.Wehaveu 1 (1, 1) = 0 and u 1 ( x 1, 1 1 k ) = k for any x 1 1 and any k N. It follows that lm u 1 ( x 1, 1 1 k ) = 1 < ε. Therefore, G s not payoff secure at (1, 1). Snce any game n g 3 s payoff secure, we have that g 3 s strctly ncluded n g Essental and tremblng-hand perfect equlbra In ths secton, frst we prove that for any g {g 1, g 2, g 3 } there exsts a subset q dense ng such that the mxed-strategy equlbra of every game n q are essental equlbra relatve to g. Then, we show that, f G g 3, the perturbed game G θν defned by (3) belongs to g 3. Ths property of g 3 mples that every mxed-strategy equlbrum of every game n a dense subset s essental and tremblng-hand perfect. In the followng, we assume that g 1, g 2 and g 3 are endowed wth the metrc defned by (1). Proposton 3 g 1, g 2 and g 3 are complete metrc spaces and the correspondence E : G g 1 E(G) s upper semcontnuous. Proof The completeness of g 2 and g 3 s obvous; concernng g 1, t s suffcent to use the arguments of the proof of Proposton 3 by Scalzo (2013) and to note that, snce ρ(g n, G) 0 and G s bounded, the Domnated Convergence Theorem (see, for example, Theorem by Alprants and Border 1999) apples and one has: 5 AgameG s payoff secure (Reny 1999) f, for any x and any ε>0, every player has a strategy x such that u (x, x )>u (x) ε for any x n some neghborhood of x.

8 8 V. Scalzo lm n I u n (x, x )dμ (x )dσ (x ) u n (x)dσ(x) = u (x, x )dμ (x )dσ (x ) u (x)dσ(x). I The property of correspondence E follows from Proposton 4 by Scalzo (2013). Snce a mxed-strategy equlbrum of a game G g s essental relatve to g f and only f the correspondence E restrcted to g s lower semcontnuous at G, from Proposton 3 and Lemma 1, wehave: Theorem 1 For any g {g 1, g 2, g 3 }, there exsts a dense G δ subset q of g such that, for every game n q, every mxed-strategy equlbrum s an essental equlbrum relatve to g. Remark 3 When g {g 1, g 2 }, n lght of Example 2, the exstence of essentally mxed-strategy equlbra n the generc case n g cannot be deduced from Corollary 1byCarbonell-Ncolau (2010). Remark 4 In lght of Example 1 by Zhou et al. (2007), we know that the set q s strctly ncluded n g for each g {g 1, g 2, g 3 }.LetG g and E be a non-empty and compact subset of E(G). We recall that E s an essental set (relatve to g) ffor any open set O whch ncludes E there exsts δ>0 such that every game G g wth ρ(g, G) <δhas at least one mxed-strategy equlbrum whch belongs to O. Snce E s upper semcontnuous wth non-empty and compact values, we have that L G ={E E(G) : E s an essental set} s non-empty (ndeed E(G) L G ). Now, let L G be ordered by the set ncluson. It s easy to see that every chan has a lower bound and, n lght of Zorn s Lemma, there exsts a mnmal element of L G, that we denote by m(g) and call a mnmal essental set of G. Fnally, we obtan that, for each g {g 1, g 2, g 3 },everygameng has a mnmal essental set. Remark 5 Let G g 3. Snce E(G) s compact, there exsts a famly C ={C α : α A} of parwse dsjont non-empty compact and connected sets such that E(G) = {C α : α A} (see Dugundj 1966). The elements of C are called components of G, and C α s sad to be an essental component of G f t s an essental set. Now, snce ϒ G (μ, ) s lower semcontnuous for any μ M and any G g 3 (see Remark 2), n lght of Lemma 3 by Zhou et al. (2007), we obtan that every game n g 3 has at least one essental component (see pp 500 and 501 by Zhou et al. 2007). Fnally, we obtan the exstence of tremblng-hand perfect mxed-strategy equlbra n some classes of dscontnuous games. Let g 4 be the space of two-player games such that, for every G g 4, G (x, ) s lower semcontnuous for any x and, for every player, u (, z ) s upper semcontnuous for any z.

9 On the exstence of essental and tremblng-hand perfect equlbra 9 Remark 6 It s easy to see that g 4 s a complete metrc space and the correspondence G g 4 E(G) s upper semcontnuous wth non-empty and compact values. Therefore, n lght of Lemma 1, there exsts a dense subset q of g 4 such that, for every G q, each mxed-strategy equlbrum of G s essental relatve to g 4. Theorem 2 Let g {g 3, g 4 }. There exsts a dense G δ subset q of g such that, for every game n q, every mxed-strategy equlbrum s essental and tremblng-hand perfect. Proof Let g {g 3, g 4 } and G g. For any n N, assume that ν n M 0 and θ n = (θ n,...,θ n ) [0, 1[ I wth θ n 0. Consder the sequence of perturbed games (G θ n ν n ) n where G θ n ν n =, u θ n ν n I and the payoffs u θ n ν n are defned as n (3). It s easy to see that ρ(g θ n ν n, G) 0; ndeed u θ n ν n (x) u (x) θ n K for all x, where K > 0. Now, f we have that the perturbed games G θ n ν n belong to g and f μ s a mxed-strategy essental equlbrum of G relatve to g, we obtan that μ s the lmt of a sequence (μ n ) n where, for each n, μ n s a mxed-strategy equlbrum of G θ n ν n. In partcular, μ n θ n ν n s a Nash equlbrum of the Selten perturbaton G θ n ν n for any n and μ n θ n ν n μ, that s: μ s a tremblng-hand perfect equlbrum of G. Therefore, because essental mxed-strategy equlbra exst n a dense subset of g (see Theorem 1 and Remark 6), the theorem s proved f we show that, for each g {g 3, g 4 }, G θν g for any G g. Let G g = g 3.If(x n ) n s a sequence convergng to x n, one has that (δ x n ) n converges to δ x. So, snce the functon λ u (y, )dλ s lower semcontnuous (see Theorem 14.5 by Alprants and Border 1999), we obtan lm nf u (y, )dδ xn θν u (y, )dδ x θν y, whch mples: lm nf u dδ x θν dδ x n θν u dδ x θν dδ x θν. Ths proves that u θν (x, ) s lower semcontnuous for any x and any I. Smlarly, one can prove that I uθν s upper semcontnuous, and we have G θν g 3. Let G g = g 4. Assume that θ = θ = θ. For any and any (x, z ), we obtan: u θν (x, z ) = u d(δ x θν,δ z θν ) = dδ z θν u dδ x θν = (1 θ) u (x, )dδ z θν + θ u dν dδ z θν = (1 θ) 2 u (x, z ) + θ(1 θ) u (x, )dν +θ(1 θ) u dν dδ z + θ 2 u dν. (6)

10 10 V. Scalzo So, we have: u θν (x, z ) u θν (z, z ) = (1 θ) 2 [ u (x, z ) u (z, z ) ] + θ(1 θ) [u (x, y ) u (z, y ) ] dν (y ), and G θν(x, z) = α(θ) G (x, z) + β(θ)k G (x, z) for any x, z, (7) where lm θ 0 α(θ) = 1, lm θ 0 β(θ) = 0 + and 2 K G (x, z) = =1 2 u (x, y )dν (y ) =1 u (z, y )dν (y ). The functon K G (x, ) s lower semcontnuous for any x. In fact, assume that z n z. For each player, Fatou s lemma (see, for example, Alprants and Border 1999) and u (, y ) upper semcontnuous for any y mply: lm sup u (z n, y )dν (y ) lm sup u (z n, y )dν (y ) u (z, y )dν (y ). (8) Therefore, the functon: 2 z =1 u (z, y )dν (y ) s lower semcontnuous. Snce G (x, ) s lower semcontnuous, we have that G θν(x, ) s lower semcontnuous for any x. On the other hand, from (8) we know that the functon x u (x, y )dν (y ) s upper semcontnuous. Therefore, n lght of (6), we obtan that u θν (, z ) s upper semcontnuous for any z. Fnally, G θν g 4. Remark 7 The exstence of essental and tremblng-hand perfect equlbra n the generc case was obtaned by Carbonell-Ncolau (2011a) n the space of games, denoted by g, where any game satsfes a strengthenng of the payoff securty and the sum of the payoff functons s upper semcontnuous. Example 2 shows a twoplayer game that belongs to g 4, but not to g. Therefore, the exstence of essental and tremblng-hand perfect equlbra n the space g 4 cannot be deduced from Carbonell- Ncolau s paper. On the other hand, t s clear that g 3 g.letg g 3 and μ be an essental mxed-strategy equlbrum of G relatve to g 3. Even f one has that μ s also an essental equlbrum relatve to g, snce any perturbaton G θν of G belongs g 3, to prove that μ s tremblng-hand perfect, we do not need Carbonell-Ncolau s result.

11 On the exstence of essental and tremblng-hand perfect equlbra 11 Remark 8 In the second part of Theorem 2, t has been crucal to prove that G θν(x, ) s lower semcontnuous for every x under the assumpton that u (, z ) s upper semcontnuous for any z and any N. Ths s the reason for whch we restrcted to two-player games. In fact, f we consder the case of a three-player game, n the formula that gves G θν(x, z), we have terms such as γ u 1 (x 1, z 2, )dν 3 γ u 1 (z 1, z 2, )dν 3, where γ>0. Now, to prove the lower semcontnuty at z of the functon G θν(x, ), we need to gve addtonal propertes on the payoffs. In partcular for player 1, we should assume that u 1 (x 1,, y 3 ) s lower semcontnuous for all x 1 1 and all y 3 3. Al-Najjar (1995) extended the strategc stablty by Kohlberg and Mertens (1986) to games wth nfntely many strateges and obtaned the exstence of stable sets of equlbra n contnuous payoffs games. Let G be a game and A be a non-empty and compact subset of E(G). A s sad to be pre-stable f for any open set O whch ncludes A there exsts δ>0 such that, for every θ [0,δ[ I and every ν M 0, the Selten perturbaton G θν has at least one equlbrum whch belongs to O. A non-empty and compact set of equlbra s sad to be stable f t s a mnmal element of the class of all pre-stable set of G ordered by the set ncluson. Now, let G q, where q s as n Theorem 2. Therefore, every mxed-strategy equlbrum of G sessentallyrelatveto g. Assume that μ E(G) and let O be an open set whch ncludes μ. Snce G θν g for every θ [0, 1[ I and every ν M 0 and ρ(g θν, G) 0 when θ 0, because μ θν s an equlbrum of the Selten perturbaton G θν for any μ E(G θν ), we have that there exsts δ>0 such that, for any θ [0,δ[ I, G θν has at least one equlbrum whch belongs to O. Hence, we have a result smlar to Theorem 4 by Carbonell-Ncolau (2011a): Theorem 3 For each G n a dense subset of g {g 3, g 4 }, a set of equlbra A s stable f and only f A ={μ} for some mxed-strategy equlbrum μ of G. References Alprants, C.D., Border, K.C.: Infnte Dmensonal Analyss. Sprnger, Berln/Hedelberg (1999) Al-Najjar, N.: Strategcally stable equlbra n games wth nfntely many pure strateges. Math. Soc. Sc. 29, (1995) Baye, M.R., Tan, G., Zhou, J.: Characterzatons of the exstence of equlbra n games wth dscontnuous and non-quasconcave payoffs. Rev. Econ. Stud. 60, (1993) Baye, M.R., Kovenock, D., de Vres, C.G.: The soluton to the Tullock rent-seekng game when R 2: mxed-strategy equlbra and mean dsspaton rates. Publc Choce 81, (1994) Carbonell-Ncolau, O.: Essental equlbra n nornal-form games. J. Econ. Theory 145, (2010) Carbonell-Ncolau, O.: On strategc stablty n dscontnuous games. Econ. Lett. 113, 120 (2011) Carbonell-Ncolau, O.: The exstence of perfect equlbrum n dscontnuous games. Games 2, (2011) Chang, S.-Y.: Inequaltes and Nash equlbra. Nonlnear Anal. 73, (2010) Dugundj, J.: Topology. Allyn and Bacou, Boston (1966) Fan, K.: A mnmax nequalty and applcatons. In: Shsha, O. (ed.) Inequaltes III. Academc Press, New York (1972) Fort, M.K.: A unfed theory of sem-contnuty. Duke Math. J. 16, (1949) Kohlberg, E., Mertens, J.F.: On the strategc stablty of equlbra. Econometrca 54, (1986) Prokopovych, P., Yannels, N.: On the unform condtons for the exstence of mxed-strategy equlbra, Mmeo (2012)

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