Subgame perfection and the rule of k-names

Subgame perfection and the rule of knames∗ Ignacio García-Jurado1, Luciano Méndez-Naya2 1Corresponding author. Departamento de Matemáticas, Universidade da Coruña, 15071 A Coruña, Spain, Phone: +34881011318, E-mail: ignacio.gar[email protected] 2Departamento de Economía Cuantitativa, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain Abstract In this paper we revisit the rule of knames from a game theoretic perspective. This rule can be described as follows. Given a set of candidates for a position, a committee (formed by the proposers) selects kelements of that set using a screening rule; then a single individual from outside the committee (the chooser) chooses for the position one of the kselected candidates. In this context we first give conditions for the existence of a subgame perfect equilibrium. Then we provide conditions for the existence of subgame perfect q-strong equilibria when the screening rule is π-majoritarian. Finally, we show that when the chooser can strategically appoint a delegate to choose on behalf of him, the conditions for the existence of subgame perfect q-strong equilibria are weaker. Key words. Rule of knames, screening rule, subgame perfect equilibrium, strong equilibrium, delegation. 1 Introduction In this paper we revisit the rule of knames first studied from a game theoretic perspective in Barberà and Coelho (2010). This rule can be described as follows. Given a set of candidates for a position, a committee (formed by the proposers) selects kelements of this set; then a single individual from outside the committee (the chooser) chooses for the position one of the kselected candidates. The chooser and also each proposer have their own preferences over the set of candidates. The rule ∗Authors acknowledge the financial support of Ministerio de Economía y Competitividad, Gobierno de España, through project MTM2014-53395-C3-1-P. 1 of knames is in fact a family of rules, because it does not specify the procedure (the screening rule) used by the committee to select the kcandidates; several screening rules associated with the rule of knames are used by different institutions around the world. For instance, Chapter 3 of Colomer (1995) describes how the rule of k-names with a sophisticated screening rule was used to choose Adolfo Suárez for the Presidency of the Spanish Government in July 1976, during the Spanish transition to democracy. The rule of knames has also been analyzed in other papers. For instance, Barberà and Coelho (2008) and Pérez et al. (2012) study several aspects of set valued screening rules, that are one of the main ingredients of the rule of k names. In Barberà and Coelho (2017) a specific family of rules of knames is studied, the v-rules of k-names, and the impact of the choice of parameters vand kupon the distribution of power among the proposers and the chooser is investigated. Barberà and Coelho (2010) provide a game theoretical analysis of the rule of k names using two strategic games: the constrained game in which the chooser is assumed to choose always his best option within the list provided by the committee, and the unconstrained game in which the latter assumption is dropped. They say to offer the analysis of both games (not only of the constrained game, more natural from the point of view of rationality) because the ability of the chooser to commit may be different in different real-life cases. Also, they use the strong equilibrium concept to capture the mix of threats and cooperation, as they write, which is present in these voting situations. They mainly deal with majority screening rules, which are those in which a majority of choosers can impose the election of any set of kcandidates. In this paper we go further in the game theoretical analysis of the rule of k names. We separate from Barberà and Coelho in the following points: •They mainly deal with the strong equilibrium concept. In this paper we use several equilibrium concepts. We stress that the rule of knames is in fact a two-step rule in which the chooser is perfectly informed of the set of kcandidates chosen by the committee in the first step. Consequently, we pay special attention to the subgame perfect equilibrium concept and its variations. •We make a non-cooperative analysis of the rule of knames in a strict sense, i.e., we do not consider any commitment abilities of the chooser or the proposers unless we explicit model them. For instance, in Section 4 we study how the analysis of a k-names game changes when the chooser has a specific commitment ability. For that purpose, we explicitly model this ability defining a particular three-stage game and, then, we analyze the resulting game. •We deal with several families of screening rules, not only with majority or weak majority screening rules. The structure and main results of this paper are as follows. In Section 2 we for2 mally introduce the non-cooperative model that we analyze: the k-names games. We first deal with no-veto screening rules, which are those for which no proposer has the ability of vetoing a candidate. We prove that a screening rule is no-veto if and only if, in each k-names game with this screening rule, there exists a subgame perfect equilibrium whose associated outcome is the best candidate for the chooser. The class of no-veto screening rules is a wide one and includes, for instance, all the majority rules. By the other hand, since the rule of knames is in fact a two-step rule, subgame perfectness is necessary to warrant strategic stability (see, for instance, Selten (1975)). In Section 3 we use a more restrictive equilibrium concept: the subgame perfect q-strong equilibrium, based on the strong equilibrium introduced in Aumann (1959). While the Nash concept of stability defines equilibrium only in terms of unilateral deviations, strong equilibrium allows for deviations by every conceivable coalition; particularly, q-strong equilibrium allows for deviations by every conceivable coalition of proposers. In the context of k-names games it seems to be more appropriate to use the q-strongness equilibrium concept than the strongness equilibrium concept, because while the possible deviation of a group of proponents in the first step is plausible, the possible deviation of a group that includes some proposers and the chooser (in the first and second stages) is not credible unless they have a commitment device. In Section 3 we give a sufficient condition and a necessary condition for the existence of subgame perfect q-strong equilibria in k-names games based on π-majoritarian screening rules, where π∈[1/2, 1)indicates the “degree” of majority required by the rule. We finish this section proving that within the class of majority rules, subgame perfect q-strongness and subgame perfect strongness are somewhat equivalent conditions. Finally, in Section 4 we show that, within the class of majoritarian screening rules, if the chooser can strategically appoint a delegate to choose on behalf of him then the conditions for the existence of subgame perfect q-strong equilibria are weaker (under certain assumptions). According to our result we could say that the capacity of the chooser to appoint a delegate can increase the stability of the game. In Section 5 we enumerate the main conclusions of this paper. 2 The class of k-names games To start with, we introduce the k-names games that we use to analyze the rule of k names. Our model is essentially the same as in Barberà and Coelho (2010) although we stress that we are dealing with the strategic game corresponding to a two-stage game. 3 Definition 2.1. A k-names game Gkis given by the tuple (N,A,X,Sk,X0,{i}i∈N) where: •N={0, 1, . . . , n}is the set of players; 0is the chooser and Np={1 . . . , n}is the set of proposers. •A is the finite set of candidates. We denote by Akthe collection of subsets of A with cardinal k. Since k is the number of candidates selected by the proposers, k can vary from one to |A|. •X is the finite set of strategies of each proposer and, thus, Xnis the set of profiles of strategies of the proposers. We take a general point of view and do not specify the process through which the proposers select their proposals. We just consider that this process can be modeled as a movement in which proposers choose simultaneously strategies of a common set and, depending on the chosen profile of strategies, a screening rule known in advance by the proposers makes a selection in Ak. •Skis the screening rule. So, Sk:Xn→Akselects a set of k candidates Sk(x)for every x= (x1, . . . , xn)∈Xn. •X0is the set of strategies of the chooser. In this two-stage game the chooser is perfectly informed when making his choice and thus he makes a plan for every set in Ak possibly reached after the first stage. Formally, X0={x0:Ak→A|x0(B)∈B,for all B ∈Ak}. •For all i ∈N, iis a binary relation over A that describes the preference of i over A. We assume that iis reflexive, antisymmetric, transitive and complete. The antisymmetry assumption implies that all agents’ preferences are strict. Now, we give the formal definition of Nash equilibrium and subgame perfect equilibrium in this context. As we have already mentioned, since the k-names games can be seen as two-stage games, subgame perfectness is necessary to warrant strategic stability in this context and, thus, in this paper we mainly concentrate on equilibrium concepts that include subgame perfection. Definition 2.2. Take a k-names game Gk= (N,A,X,Sk,X0,{i}i∈N). A Nash equilibrium of Gkis a pair (x,x0)∈Xn×X0such that: 1. x0(Sk(x)) 0x0 0(Sk(x)),for all x0 0∈S0. 2. x0(Sk(x)) ix0(Sk(x−i,x0 i)),for all x0 i∈X and for all i ∈Np, where (x−i,x0 i)denotes the profile (x1, . . . , xi−1,x0 i,xi+1, . . . , xn). Definition 2.3. Take a k-names game Gk= (N,A,X,Sk,X0,{i}i∈N). A subgame perfect equilibrium of Gkis a pair (x,x0)∈Xn×X0such that: 4 1. x0(Sk(¯x)) 0a,for all a ∈Sk(¯x), and for all ¯x ∈Xn. 2. x0(Sk(x)) ix0(Sk(x−i,x0 i)),for all x0 i∈X and for all i ∈Np. Notice that our model allows for all kind of screening rules and, thus, is remarkably general. As a consequence of this, it is easy to see that the set of Nash equilibria of Gkmay be empty. The following example illustrates this fact. Example 2.1. Take G2= (N,A,X,S2,X0,{i}i∈N)with Np={1, 2, 3}, A ={a,b,c}, X={α,β}, where αmeans “I vote for {a,c}” and βmeans “I vote for {b,c}”. To define the screening rule denote by N({a,c},x)the number of votes that {a,c}receives according to x. The screening rule is now given by: S2(x) = ({ a,c} if N({a,c},x)equals 3or 1, { b,c} if N({a,c},x)equals 2or 0. Finally, the preferences of the proposers and of the chooser are as follows: Proposer 1 Proposer 2 Proposer 3 Chooser a b c a b c a b c a b c Let us check that G2does not have Nash equilibria. In fact, it is clear that the result corresponding to a Nash equilibrium cannot be c (because in that case, the chooser is better off by deviating). Suppose that there exists a Nash equilibrium (x,x0)whose corresponding result is a. Then N({a,c},x)equals 3or 1but, in both cases, Proposer 2 would gain by deviating, so (x,x0)cannot be a Nash equilibrium. Suppose now that there exists a Nash equilibrium (x,x0)whose corresponding result is b. Then N({a,c},x)equals 2or 0but, in both cases, Proposer 3 would gain by deviating, so (x,x0)cannot be a Nash equilibrium. The next theorem provides a class of screening rules that guarantee the existence of a subgame perfect equilibrium in all their associated k-names games: the class of no-veto screening rules. In words, a screening rule is said to be no-veto if there does not exist a proposer with the capacity of vetoing a candidate. Moreover, the theorem characterizes this class as the unique one for which the chooser’s best candidate is always a subgame perfect equilibrium outcome. Below we give the formal definition of no-veto rule and the result. Definition 2.4. A screening rule Sk:Xn→Akis said to be no-veto if for every a ∈A there exists x∈Xnsuch that a∈Sk(x−i,y) for all y ∈X and all i ∈Np. 5 Theorem 2.1. Take a screening rule Sk. Skis no-veto if and only if the best candidate of the chooser is a subgame perfect equilibrium outcome of Gk, for all Gkwhose screening rule is Sk. Proof. Take a no-veto Skand Gk= (N,A,X,Sk,X0,{i}i∈N). Denote by x0the strategy that selects for every B∈Akthe best candidate for the chooser in B. Now denote by aGkthe best candidate for the chooser in Gkand take x∈Xnsuch that for all y∈Xit holds that aGk∈Sk(x−i,y)for all i∈Np. It is clear that (x,x0)is a subgame perfect equilibrium of Gkand that x0(Sk(x)) = aGk. Conversely, take a screening rule Sksuch that for all Gk= (N,A,X,Sk,X0,{i}i∈N)there exists (x,x0)a subgame perfect equilibrium of Gkwith x0(Sk(x)) = aGk,aGkbeing the best candidate for the chooser in Gk. Assume that Skfails to be no-veto. Then, there exists a∈Asuch that, for all x∈Xn, a6∈ Sk(x−i,y)(1) for some y∈Xand i∈Np. Notice that, according to Definition 2.4, adoes not depend on {i}i∈N. Now take Gksuch that ais the best candidate for the chooser and ais the worst candidate for all the proposers. Then, in view of (1), there cannot exist (x,x0), a Nash equilibrium of Gk, such that x0(Sk(x)) = a, which is a contradiction (because subgame perfect implies Nash). Remark 2.1. Reading the proof of Theorem 2.1, it is easy to check that its statement also holds when we write Nash equilibrium instead of subgame perfect equilibrium. Definition 2.5 below introduces the class of dictatorial screening rules. In general, these rules fail to be no-veto; however, it is clear that every Gkwith a dictatorial Skhas a subgame perfect equilibrium. Definition 2.5. A screening rule Sk:Xn→Akis said to be dictatorial if there exists i∈Npsuch that for all x,y∈Xnand all z ∈X it holds that Sk(x−i,z) = Sk(y−i,z). Example 2.2. Take Np={1, 2}, A ={a,b,c}, and X =A2. Now consider the dictatorial screening rule S2:X×X→A2given by S2(x) = x1for all x∈X×X. It is clear that S2fails to be no-veto and that every G2whose screening rule is S2has a subgame perfect equilibrium (the first proposer selects his best and his second best candidates and the chooser plans to choose his best candidate between the two selected by the first proposer). Take now G2= (N,A,X,S2,X0,{i}i∈N)with the preferences of the proposers and the chooser as follows: 6 Proposer 1 Proposer 2 Chooser a b c b c a c a b It is easy to check that a is the unique candidate supported by a subgame perfect equilibrium in G2(and also by a Nash equilibrium). Notice that a is not the best candidate for the chooser. The next two classes of screening rules were introduced in Barberà and Coelho (2010).1It is clear that they are included in the class of no-veto rules when n>2. Definition 2.6. A screening rule Sk:Xn→Akis said to be majoritarian if, for every B∈Ak, there exists x∈Xnsuch that for all M ⊂Npwith |M|>n 2and all ¯x ∈Xnit holds that Sk(xM,¯xNp\M) = B.2 Definition 2.7. A screening rule Skis said to be weakly majoritarian if, for every a ∈A, there exists x∈Xnsuch that for all M ⊂Npwith |M|>n 2and all ¯x ∈Xnit holds that a∈Sk(xM,¯xNp\M). 3 Quasi-Strongness We have already seen that the chooser’s best candidate can always be selected in subgame perfect equilibrium when the screening rule is no-veto, and that the class of no-veto screening rules is a wide one; for example it includes the majority rules and the weakly majority rules. However, in the model we are considering, it seems to be appropriate to use an equilibrium concept that is coalition-proof in the first step, in which the proposers can typically discuss before choosing their strategies. The concept we use is the subgame perfect q-strong equilibrium which is a refinement of the q-strong equilibrium. The q-strong equilibrium is a variation of Aumann’s concept of strong equilibrium for the particular problem we are dealing with in this paper. Specifically, a q-strong equilibrium is a strategy profile that is stable against deviations by every possible group of proposers and against unilateral deviations of the chooser. The strong equilibrium is a refinement of the q-strong which also allows for deviations by groups formed by the chooser and some proposers. We believe that q-strongness is a more natural concept than strongness for k-names games; however, at the end of this section we prove for k-names games 1In fact, the classes we define here are more general because Barberà and Coelho deal with symmetric rules and we do not. 2(xM,¯xNp\M)denotes the vector ((xi)i∈M,(¯ xj)j∈Np\M). 7 that, within the class of majority rules, subgame perfect q-strongness and subgame perfect strongness are somewhat equivalent conditions. Definition 3.1. Take a k-names game Gk= (N,A,X,Sk,X0,{i}i∈N). A quasi-strong (abbreviated q-strong) equilibrium of Gkis a pair (x,x0)∈Xn×X0such that: 1. x0(Sk(x)) 0x0 0(Sk(x)),for all x0 0∈S0. 2. For every non-empty M ⊂Npand every x0∈Xn, there exists i ∈M such that: x0(Sk(x)) ix0(Sk(x0M,xNp\M)). Definition 3.2. Take a k-names game Gk= (N,A,X,Sk,X0,{i}i∈N). A subgame perfect q-strong equilibrium of Gkis a pair (x,x0)∈Xn×X0such that: 1. x0(Sk(¯x)) 0a,for all a ∈Sk(¯x), and all ¯x ∈Xn. 2. For every non-empty M ⊂Npand every x0∈Xn, there exists i ∈M such that: x0(Sk(x)) ix0(Sk(x0M,xNp\M)). The next two results provide respectively a necessary condition and a sufficient condition for the existence of subgame perfect q-strong equilibria. Before the results, some previous definitions are needed. Definition 3.3. Take π∈[1/2, 1). A screening rule Sk:Xn→Akis said to be πmajoritarian if, for every B ∈Ak, there exists x∈Xnsuch that for all M ⊂Npwith |M|>πn and all ¯x ∈Xnit holds that Sk(xM,¯xNp\M) = B. It is clear that if a screening rule Skis π-majoritarian, then it is π0-majoritarian for every π0≥π. In view of Definitions 2.6 and 3.3, a majoritarian rule is simply a1 2-majoritarian rule. Notice that π-majoritarian rules with π>1/2 are not rare, since very often a qualified majority is required for making decisions. Definition 3.4. Take a k-names game Gk= (N,A,X,Sk,X0,{i}i∈N)and π∈[1/2, 1). A candidate a ∈B⊂A is the π-Condorcet winner over B if, for all a0∈B\ {a}, it holds that |{i∈Npsuch that a ia0}| >πn. Notice that the π-Condorcet winner over Bmay not exist but, if it exists, then it is unique. Clearly, if a candidate a∈B⊂Ais the π-Condorcet winner over B, then it is the π0-Condorcet winner over Bfor every π0≤πand it is the π-Condorcet winner over B0for all B0⊂Bwith a∈B0. The Condorcet winner over B(a standard concept in social choice) is simply the 1 2-Condorcet winner over B. Definition 3.5. Take a k-names game Gk= (N,A,X,Sk,X0,{i}i∈N)and π∈[1/2, 1). A candidate a ∈B⊂A is π-Condorcet undominated over B if, for all a0∈B\ {a}, it holds that |{i∈Npsuch that a0ia}| ≤ πn. 8 It is clear that if a∈B⊂Ais the π-Condorcet winner over B, then it is πCondorcet undominated over B. It is also easy to check that the reciprocal is not true in general. Moreover, if nis odd, a∈B⊂Ais 1 2-Condorcet undominated over Bif and only if it is the 1 2-Condorcet winner over B. Theorem 3.1. For any π-majoritarian screening rule Sk, if a candidate a ∈A is a subgame perfect q-strong equilibrium outcome of a k-names game Gk= (N,A,X,Sk,X0,{i}i∈N) then a is π-Condorcet undominated over the set of the chooser’s (|A| − k+1)-top candidates. Proof. Assume that candidate ais the outcome of (x,x0), a subgame perfect qstrong equilibrium of Gk. Clearly, ais the best candidate for the chooser in Sk(x). So, ais a chooser’s (|A| − k+1)-top candidate3. Since the screening rule is πmajoritarian, there exists no other chooser’s (|A| − k+1)-top candidate a0that is preferred to aby more than πnchoosers. Otherwise, this coalition could impose the choice of a set where a0is the best candidate for the chooser, which implies that (x,x0)is not a subgame perfect q-strong equilibrium of Gk. Theorem 3.2. For any π-majoritarian screening rule Sk, if a candidate a ∈A is the πCondorcet winner over the set of the chooser’s (|A| − k+1)-top candidates in a k-names game Gk= (N,A,X,Sk,X0,{i}i∈N), then a is a subgame perfect q-strong equilibrium outcome of Gk. Proof. Assume that ais the π-Condorcet winner over the set of the chooser’s (|A| − k+1)-top candidates. Take a set B∈Aksuch that ais the chooser’s best candidate in this set. Since Skis π-majoritarian there exists x∈Xnsuch that for all M⊂ Npwith |M|>πnand all ¯x ∈Xnit holds that Sk(xM,¯xNp\M) = B. Take the strategy x0∈X0that selects for any B∈Akthe best candidate for the chooser in B. It is clear that x0(Sk(x)) = a. Let us check that (x,x0)is a subgame perfect q-strong equilibrium of Gk. Obviously, any coalition (1−π)nproposers or less cannot change the outcome and thus have no incentive to deviate. Notice that any coalition with more than (1−π)nproposers has no incentive to deviate since ais the π-Condorcet winner over the set of the chooser’s (|A| − k+1)-top candidates. Corollary 3.1. For any majoritarian screening rule Skand any k-names game Gk= (N,A,X,Sk,X0,{i}i∈N)with an odd n, a candidate a ∈A is a subgame perfect qstrong equilibrium outcome of Gkif and only if a is the 1 2-Condorcet winner over the set of the chooser’s (|A| − k+1)-top candidates. 3A chooser’s (|A| − k+1)-top candidate is anyone belonging to the set of the |A| − k+1 candidates preferred by the chooser. 9 Since kis odd, a 1 2-Condorcet undominated candidate is the same as a Condorcet winner and, thus, (ii) implies that there exists a delegate ¯ dsuch that a is a Condorcet winner over the set of ¯ d’s (|A| − k+1)-top candidates. Then, Theorem 3.2 implies that: ais a subgame perfect q-strong equilibrium outcome of G¯ d k. (6) Clearly, (5) and (6) imply that ais a subgame perfect q-strong equilibrium outcome of G∗ k. 2⇒1. Assume that ais a subgame perfect q-strong equilibrium outcome of G∗ k. Then: ∀d∈ R(A),∃ba subgame perfect equilibrium outcome of Gd ksuch that a0b, (7) and ∃¯ d∈ R(A)such that ais a q-strong equilibrium outcome of G¯ d k. (8) Now, (7) implies that ais a chooser’s (|A| − k+1)-top candidate. Besides, (8) and Theorem 4.1 imply that ais 1 2-Condorcet undominated over some subset of Awith cardinality (|A| − k+1). Thus, the oddness of kand Theorem 4.2 imply that ais a q-strong equilibrium outcome of Gk. Theorem 4.3 shows that when the chooser can appoint delegates, then the set of candidates that can be supported by a subgame perfect q-strong equilibrium may increase. Let us illustrate it with an example. Example 4.1. Take G2= (N,A,X,S2,X0,{i}i∈N)with Np={1, 2, 3}and A = {a,b,c,d}. S2is a majoritarian screening rule. The preferences of the proposers are as follows: Proposer 1 Proposer 2 Proposer 3 a d b b a d c b a d c c Notice that a is the Condorcet winner over {a,b,c}, b is the Condorcet winner over {b,c,d}, and d is the Condorcet winner over {a,c,d}. We consider now three cases. Case 1: a 0b0c0d. Then Corollary 3.1, Corollary 4.1 and Theorem 4.3 imply that a is the unique subgame perfect q-strong equilibrium outcome of G2and that the set of subgame perfect q-strong equilibrium outcomes of G∗ 2is {a,b}. Since a 0b, in this case the chooser’s ability of appointing delegates increases the set of subgame perfect q-strong 16 equilibrium outcomes, but it cannot be beneficial for the chooser. Case 2: b 0a0c0d. Then Corollary 3.1, Corollary 4.1 and Theorem 4.3 imply that a is the unique subgame perfect q-strong equilibrium outcome of G2and that the set of subgame perfect q-strong equilibrium outcomes of G∗ 2is {a,b}. Since b 0a, in this case the chooser’s ability of appointing delegates increases the set of subgame perfect q-strong equilibrium outcomes, and it can be beneficial for the chooser. Case 3: a 0b0d0c. Then Corollary 3.1, Corollary 4.1 and Theorem 4.3 imply that G2does not have subgame perfect q-strong equilibria and that the set of subgame perfect q-strong equilibrium outcomes of G∗ 2is {a,b,d}. In this case chooser’s ability of appointing delegates increases the set of subgame perfect q-strong equilibrium outcomes, and it is beneficial for the chooser. To finish this section we provide an example that shows that the statement of Theorem 4.3 is not true if we drop the condition that Skis majoritarian. Example 4.2. Take G3= (N,A,X,S3,X0,{i}i∈N)with Np={1, 2, 3}and A = {a,b,c,d,e,f}. In order to define the set X and the screening rule S3consider the following subsets of A: B ={a,b,d}, C ={c,d,e}, D ={a,b,e}, E ={d,e,f}. Now define X={β,γ,δ}, where βmeans “I vote for B”, γmeans “I vote for C”, and δmeans “I vote for D”. The screening rule is given by: S3(x) =            B if at least two proposers choose β, D if at least two proposers choose δ, C if the three proposers choose different strategies, E in any other case. The preferences of the proposers and the chooser are as follows: Proposer 1 Proposer 2 Proposer 3 Chooser d d e a a b f b c c c c b a d d e e a e f f b f Clearly, S3is not majoritarian. Let us check now that c is a q-strong equilibrium outcome of G3. From Corollary 4.1, it is enough to prove that c is a chooser’s (|A| − k+1)-top candidate and a Condorcet winner over some subset of A with cardinality (|A| − k+1). Here (|A| − k+1) = 4; observe that c is in fact a chooser’s 4-top candidate and that c is a Condorcet winner over {a,b,c,e,f}. We next prove by contradiction that c is not a 17 subgame perfect q-strong equilibrium outcome of G∗ 3; this implies that Theorem 4.3 is not true if we drop the condition that Skis majoritarian. Assume that (u,(xv,xv o)v∈R(A))is a subgame perfect q-strong equilibrium outcome of G∗ 3with xu o(S3(xu)) = c; notice that, in particular, this implies that u ∈ R(A), i.e., that u is a reflexive, antisymmetric, transitive and complete binary relation over A. Since xu o(S3(xu)) = c, then S3(xu) = C and xu o(C) = c. Thus the three proposers have chosen different strategies. Let us consider now three cases: 1. xu o(D) = a. In this case xu 1=δ(denote xu= (xu 1,xu 2,xu 3)); otherwise Proposer 1 would prefer deviating to δ, because then S3would select D instead of C and the final result would be a instead of c. Now two subcases: (a) xu o(B)∈ {a,d}. Then Proposer 1 would prefer deviating to β, because then S3 would select B instead of C and the final result would be a or d instead of c. Then, xu o(B)∈ {a,d}is not possible. (b) xu o(B) = b. This would imply that bua (i.e., b is preferred to a by u), but this is in contradiction with xu o(D) = a (that implies aub), because u is antisymmetric. So, this case is impossible. 2. xu o(D) = b. In this case xu 2=δ; otherwise Proposer 2 would prefer deviating to δ, because then S3would select D instead of C and the final result would be b instead of c. Again, two subcases: (a) xu o(B)∈ {b,d}. Then Proposer 2 would prefer deviating to β, because then S3 would select B instead of C and the final result would be b or d instead of c. Then, xu o(B)∈ {b,d}is not possible. (b) xu o(B) = a. This would imply that aub, but this is in contradiction with xu o(D) = b (that implies bua), because u is antisymmetric. So, this case is impossible. 3. xu o(D) = e. In this case xu 3=δ; otherwise Proposer 3 would prefer deviating to δ, because then S3would select D instead of C and the final result would be e instead of c. But then Proposer 3 would prefer deviating to γunless xu o(E) = d (because then S3would select E instead of C and the final result would be an element of E instead of c; the unique element of E which is not preferred to c by Proposer 3 is d). Thus, xu 3=δand xu o(E) = d, but then the proposer choosing β(Proposer 1 or Proposer 2) would prefer deviating to γ, because in that case S3would select E instead of C and the final result would be d instead of c. So, this case is impossible. Hence, it is impossible that (u,(xv,xv o)v∈R(A))is a subgame perfect q-strong equilibrium outcome of G∗ 3with xu o(S3(xu)) = c. 18 5 Conclusions The main conclusions of this paper are the following ones. •By definition, a screening rule is no-veto if and only if no proposer is able to veto one of the candidates. The class of no-veto screening rules is a specially important one; in fact, a screening rule Skis no-veto if and only if for every k-names game Gkwith screening rule Skthere exists a subgame perfect equilibrium whose associated outcome is the best candidate for the chooser. •We give a sufficient condition and a necessary condition for the existence of subgame perfect q-strong equilibria in k-names games based on π-majoritarian screening rules, where π∈[1/2, 1)indicates the “degree” of majority required by the rule. •For the class of majority rules, subgame perfect q-strongness and subgame perfect strongness are somewhat equivalent conditions. •Within the class of majoritarian screening rules, if the chooser can strategically appoint a delegate to choose on behalf of him, then a candidate is a q-strong equilibrium outcome of the k-names game if and only if it is a subgame perfect q-strong equilibrium outcome of the k-names with delegates. 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