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18 (Area ), the best response is X n : s n ="! x n >0; s n = "! x n =0; for any p; p; v : v " p<v ^ p> v +" (Area 3 ), the best response is X 3 n : s n ="! x n =0; s n = "! x n <0; for any p; p; v : p p+v " ^ p v (Area 4 ), the best response is X 4 n : s n =! x n >0; for any p; p; v : v " p p+v +"^ p+v "<p v +" (Area 5 ), the best response is X 5 n : s n ="! x n >0; s n = "! x n <0; for any p; p; v : v p ^ p> p+v +" (Area 6 ), the best response is X 6 n : s n =! x n <0. Figure 1. Best responses to any out-of-equilibrium pair p; p for arbitrary beliefs v p; p. Now we prove that, given a PBE such that there exists at least a type v V earning (v)< ", this PBE always fails the IC. Given a type v i V earning in equilibrium less than ", where ' 0 ^ < ++, consider an out-of-equilibrium pair p; p : p= p+v i. When p p=", the set BR(V; p; p) contains Xn 5 for sure. Regardless of this set also containing Xn 4 or Xn 6 or both, in Step I type v i s out-of-equilibrium pro ts equal (p; p : p p=" ; Xn 4 ^ Xn 5 ^ Xn; 6 x n ( s n =)=0; v i = p+p ) = " > (v i ). Therefore v i VV(p; b p), and the PBE fails the IC. In fact, regardless of the set BR(VV; b p; p) containing Xn 5 or Xn 4 ^ Xn 5 or Xn 5 ^ Xn 6 or Xn 4 ^ Xn 5 ^ Xn, 6 in Step II type v i s out-of-equilibrium pro ts equal " > (v i ). Unless further proprieties of a speci c PBE are speci ed (that is, specifying neither the equilibrium payo associated to any speci c realization of v, nor whether this realization has a positive probability to occur), no other condition for the IC failure to occur can be identi ed. To show this, let s de ne v l =min v VV(p; b p) and v h =max v VV(p; b p), and study how much a generic type v j earns in Step II out-of-equilibrium in case v j V b V(p; p). No attention in Step II is given any type v V(v l ; v h ) V(p; b p); in fact, since in Step I she does not do better out-of-equilibrium than in equilibrium, in Step II the same result holds. Three relevant classes of out-of-equilibrium pair of prices can be identi ed: (i) 0 p p<"; 17

19 (ii) " p p<"; (iii) " p p. (i.a) if p> p+v h +" (and therefore p> p+v +"; 8), then the set BR(V b V; p; p) contains only X 6 n (uninformed clients sell at price p). Thus, the lower the price p is, the greater type v j s out-of-equilibrium pro ts in Step II are. Speci cally, even in the extreme case in which p=p "+, type v h = p+p " earns (p; p : p p=" ; X 6 n; x n ( s n =)<0; v h = p+p " )= out of equilibrium; when p>p "+, she earns less; any type v j < p+p " earns less as well. (i.b) if p< p+v l " (and therefore p< p+v "; 8), the solution is symmetric to the one in the sub-case above, and the conclusion is identical. (i.c) if p+v l " p p+v h +" (and therefore 9v : p p+v +" ^ p p+v "), then BR(V b V; p; p) contains X 5 n for sure. When uninformed clients satisfy their liquidity needs, even in the extreme case in which p p=" any type v j earns an out-of-equilibrium pro t in Step II not greater than ". (ii.a) if v h <p ^ v h +"<p (and therefore v <p ^ v +"<p; 8), then BR(V b V; p; p) contains only X 6 n. Since v <p, type v j s out-of-equilibrium pro ts in Step II are negative, for all v j. (ii.b) if p<v l " ^ p<v l, then BR(VV; b p; p) contains only Xn. 4 This sub-case is symmetric to the one above, and leads to an identical conclusion. (ii.c) if p v h <p+" (and therefore v "<p; 8 ^ 9v : p v ), then (VV; b p; p) contains Xn 3 for sure. In Step II type v h =p+" earns " out of equilibrium; any type v j <p+" clearly earns less. (ii.d) if v l p<v l +" (and therefore p<v +"; 8^9v : p v ), then (VV; b p; p) contains Xn for sure. This sub-case is symmetric to the one above, and leads to an identical conclusion. (ii.e) if v l +" p^p v h " (and therefore 9v : v +" p^p v "), then BR(VV; b p; p) contains both Xn and Xn 3 for sure. Type v j = p p is the one that in Step II earns the most out of equilibrium. Speci cally, when p p=", she earns ", otherwise less. (ii.f) if v h "<p ^ p<v l +" (and therefore v "<p ^ p<v +"; 8), then BR(VV; b p; p) contains only Xn. 5 Further conditions for the IC failure to occur might be identi ed; however the level of generality in this proof does not allow to assess whether in Step II every v lies on Area 5; this is assessed in Proposition 4. (iii.a) If v h <p (and therefore v <p; 8), then BR(VV; b p; p) contains only Xn. 6 In Step II every type v j earns negative out-of-equilibrium pro ts. (iii.b) If p<v l, then BR(VV; b p; p) contains only Xn. 4 Again, in Step II every type v j earns negative out-of-equilibrium pro ts. (iii.c) If v h "<p (and therefore v "<p; 8), then BR(VV; b p; p) contains Xn 3 for sure. In the extreme scenario in which v h =p+", her out-of-equilibrium pro ts equal (" ); otherwise type v h earns less. Clearly any other type belonging to VV(p; b p) earns less as well. (iii.d) If p<v l +", then set BR(VV; b p; p) contains Xn for sure. This sub-case is symmetric to the one above, and leads to an identical conclusion. (iii.e) If p v h "^v l +" p (and therefore 9v : p v "^v +" p), then BR(VV; b p; p) contains Xn 1 for sure. In Step II every v j earns not more than zero out of equilibrium. Derivation of Proposition 4. From Proposition 3 (sub-case ii.f), recall that a necessary condition for any PBE with dealers earning at least (v)=[ " +\$ (v)], where \$ () [0; "], to fail the IC is that there exists an out-of-equilibrium price pair p; p : p = p + such that BR(V b V; p; p : p = p + ) only contains X 5 n. Conditions for this to happen are listed below. If at the same time 9v : p "<v<p+" ^ (v)<(p; p : p=p+; Xn; 5 x n ( s n =)=0; v)=, it follows that at least a type v VV(p; b p : p = p + ) does inevitably better than in equilibrium the PBE fails the IC. 18

20 First notice the two scenarios below, in which, regardless of what and the uninformed clients beliefs are, the set V and the selection of a pair p; p : p=p+ do not allow X 5 n to be the only reply belonging to BR(VV; b p; p : p=p+). Scenario A 9v : p v<p " ^ p v. Speci cally, since p v, in Step I it follows that 9v : p v ; this implies that BR(V; p; p : p = p + ) also contains Xn, 4 and therefore type v : p v<p " earns more than how much she could ever achieve in equilibrium, being (p; p : p=p+; X 4 n; x n ( s n =)=0; p v<p ")=[(p v) j v<p "] strictly greater than ". Consequently, type v : p v<p " belongs to V b V(p; p : p = p + ). This means that BR(VV; b p; p : p = p + ) also contains Xn. 3 Scenario B 9v : p+"<v p ^ v p. This case is symmetric to the one above, and the conclusion is identical. Given a speci c pair p; p : p = p +, if one of the following six conditions holds, then BR(VV; b p; p : p = p+) contains only Xn. 5 Each condition below accounts for the remaining (relevant) scenarios others than the ones associated to Scenario A and B. In particular, the rst requirement listed in each of the following conditions de nes which expectations of v V given arbitrary posterior beliefs are reasonable, and consequently identi es what BR(V; p; p : p = p + ) contains. Because of what discussed in the beginning (and speci ed in this Proposition), it is implicit that in each of the following six scenarios at least one type v : p "<v<p+" also exists (this is the type that has to do inevitably better than in equilibrium), and therefore both BR(V; p; p : p = p + ) and BR(VV; b p; p : p = p + ) always contain Xn. 5 The remaining requirements guarantee that every type v p " or v p + " cannot do better than in equilibrium by setting an out-of-equilibrium price pair p; p : p = p+, given that liquidity traders play any pair of strategies in BR(V; p; p : p = p+); consequently, none of these types belongs to V b V(p; p : p = p + ). Condition I: requirements R.I and R.a, R.b, R.c, R.d simultaneously hold, where: v p p v : p + " < v p _ p v < p "; (R.I > p : 1 " < 1+ \$ (v) ; (R.a) v < p : 1 " < 1+ \$ (v) ; (R.b) p = p + " : \$ (v) < " ; = p " : \$ (v) < " : (R.d) Regardless of whether type v=p+" and v=p " exist or not, BR(V; p; p) contains X n, X 3 n, X 4 n, X 5 n, and X 6 n. In particular, requirement R.a (or R.b) guarantees that, if type v>p (resp., v<p) exists, then her equilibrium pro ts are not smaller than the maximum she could ever get out of equilibrium, which equals to (p; p : p=p+; X 6 n; x n ( s n =)>0; v>p)=(v p)+(1 )(p v) (resp., (p; p : p=p+; Xn; 4 x n ( s n =)<0; v<p)=(p v)+(1 )(v p)). Requirement R.c (or R.d) ensures that, if type v=p+" (resp., v=p ") exists, in equilibrium this type does not achieve less than (p; p : p=p+; X 6 n; x n ( s n =)=0; v=p+")=" (resp., (p; p : p=p+; Xn; 4 x n ( s n =)=0; v=p ")="), her maximum out-of-equilibrium pro ts. Condition II: requirements R.II and R.e, R.f simultaneously hold, where: p<v v <p; (R.II ) 19

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