4.2 At least four alternatives
Throughout this subsection, \(m \ge 4\). We will exhibit a large class of self-implementable social choice correspondences, containing both the top and the Pareto correspondences.
For \(B {\subseteq } A\) we define the social choice correspondence \(H_B\) by \(H_B(R^N) = TC(R^N) \cup (PC(R^N) \cap B)\) for every \(R^N \in L^N\). In words, \(H_B\) assigns to each preference profile all top alternatives, and additionally all Pareto optimal alternatives that are in B. Then \(H_B\) is unanimous and anonymous. Note that \(H_B = TC\) if \(B={\emptyset }\), and \(H_B=PC\) if \(B=A\). The main result of this subsection will be that \(H_B\) is self-implementable for every \(B {\subseteq } A\).
We first outline the main features of the proof of this result. If x is a top alternative of a profile \(R^N\), then pick an agent, say i, with that alternative at top, and consider the profile with i having x at top and the remaining alternatives in the order \(x_1,\ldots ,x_m\), and the other agents having x at top and the remaining alternatives in the order \(x_m,\ldots ,x_1\). To this profile x is assigned, also still if one of the agents other than i reports a different preference. Hence, this is a Nash equilibrium. Next, if \(x\in B\) is Pareto optimal but not a top alternative at \(R^N\), then we partition the remaining alternatives in k subsets for k different agents, and let each of these k agents have x at top, the alternatives in that agent’s part of the partition at bottom in the order \(x_1,\ldots ,x_m\), and the remaining alternatives in between, in the order \(x_1,\ldots ,x_m\). All the other agents report x at top and the other alternatives in the order \(x_m,\ldots ,x_1\). To this profile x is assigned; an agent in the first category may deviate but can only obtain an alternative from that agent’s bottom part or x, while an agent of the second category can deviate but the chosen outcome will still be x. This is going to be a Nash equilibrium if the partition is chosen in the right way, ensuring that each agent in the first category prefers x to the bottom set of alternatives—this is possible since x is Pareto optimal, hence for each alternative there is an agent preferring x to that alternative. All other cases are taken care of by picking from the reported top alternatives by modulo counting using the numbers \(\nu (R^N)\), and will never result in a Nash equilibrium.
We now proceed with the formal treatment. If a subset B of A is used in the notation of a preference R, e.g., \(R = \ldots B \ldots\), this means that each alternative of \(A {\setminus } B\) is ranked either above or below the alternatives of B, and that the alternatives of B are ranked consistently with the order \(x_1\ldots x_m\): that is, if \(x_j,x_k \in B\) and \(j < k\) then \(x_j R x_k\). The notation \(R = \ldots \overline{B} \ldots\) has a similar meaning except that now the alternatives of B are ranked in reverse order: if \(x_j,x_k \in B\) and \(j < k\) then \(x_k R x_j\).
For each \(x\in A\) let \(P_x = x\,A{\setminus }\{x\} \in L\) and \(\overline{P}_x = x\,\overline{A{\setminus }\{x\}} \in L\). Thus, \(P_x\) has x as top alternative and ranks the remaining alternatives in the order \(x_1,\ldots ,x_m\). Also \(\overline{P}_x\) has x as top alternative, but ranks the remaining alternatives in reverse order.
Let
\(x \in A\),
\({\emptyset } \ne K {\subseteq } N\), and let
\((B_i)_{i\in K}\) be a partition of
\(A{\setminus }\{x\}\), i.e.,
\(B_i\ne {\emptyset }\) for every
\(i\in K\),
\(\cup _{i\in K} B_i = A{\setminus }\{x\}\) and
\(B_i \cap B_j = {\emptyset }\) for all distinct
\(i,j\in K\). We call
\(R^N\in L^N\) an
\((x,(B_i)_{i\in K})\)-
profile if
-
\(R^i = x\, (A{\setminus }(B_i \cup \{x\}))\, B_i\) for every \(i\in K\), and
-
\(R^i = \overline{P}_x\) for every \(i \in N{\setminus } K\).
In words, for an agent
\(i \in K\), in
\(R^i\) the top alternative is
x, next the alternatives that are not in
\(B_i\) are ranked in the order
\(x_1\ldots x_m\), and finally the alternatives in
\(B_i\) are ranked in the same order. Any agent not in
K ranks
x again at top, but all remaining alternatives in reverse order.
We first show that in an \((x,(B_i)_{i\in K})\)-profile the set K and the sets \(B_i\) are uniquely determined.
In view of Lemma
4.2, if
\(R^N\) is an
\((x,(B_i)_{i\in K})\)-profile, we can call it
the \((x,(B_i)_{i\in K})\)-profile.
Fix
\(B {\subseteq } A\). We define the anonymous selection
\(F_B\) from
\(H_B\) (actually, from the top correspondence) as follows. Let
\(R^N \in L^N\).
(1.1)
If there are \(x \in A\) and distinct \(i,j\in N\) such that \(R^i = P_x\) and \(R^k = \overline{P}_x\) for all \(k \in N {\setminus } \{i,j\}\), then \(F_B(R^N)=x\).
(1.2)
If there are \(x\in B\), \(K {\subseteq } N\) with \(|K|\ge 2\), a partition \((B_i)_{i\in K}\) of \(A {\setminus } \{x\}\), and \(j \notin K\) such that \((R^{N{\setminus }\{j\}},\overline{P}_x)\) is the \((x,(B_i)_{i\in K})\)-profile, then \(F_B(R^N)=x\).
(1.3)
If there are
\(x\in B\),
\(K {\subseteq } N\) with
\(|K|\ge 2\), a partition
\((B_i)_{i\in K}\) of
\(A {\setminus } \{x\}\), and
\(j \in K\) such that
\((R^{N{\setminus }\{j\}},x\, (A{\setminus }(B_j \cup \{x\}))\, B_j)\) is the
\((x,(B_i)_{i\in K})\)-profile, then
$$\begin{aligned}F_B(R^N) = \left\{ \begin{array}{ll} t(R^j) &{} \hbox {if }t(R^j)\in B_j \\ x &{} \hbox {otherwise.} \end{array} \right. \end{aligned}$$
(1.4)
Otherwise, let \(TC(R^N) = \{x_{j_1},\ldots ,x_{j_t}\}\) with \(j_1< \ldots < j_t\), then \(F_B(R^N) = x_{j_s}\) where \(s = 1 + [\nu (R^N)\!\!\mod t ]\).
Observe that the four cases in the definition of
\(F_B\) are mutually exclusive and that
\(F_B\) is a unanimous and anonymous selection from the top correspondence and therefore also from
\(H_B\). Also observe that (1.1) treats the case of an
\((x,(B_i)_{i\in K})\)-profile for
\(|K|=1\), but has been defined separately both for clarity and since it applies also for
\(x \notin B\). Note that if, in the situation of (1.1), player
i deviates, then (1.4) applies, so that player
i can achieve any alternative other than
x as well.
It is also important to observe that, if (1.4) applies, then any player
i can achieve any alternative by putting it on top and choosing the ‘right’ preference, as explained in Sect.
5.1: there are
\((m-1)!\) possibilities, and
\((m-1)! > m\) since
\(m\ge 4\). To illustrate in more detail how this works, we consider an (arbitrary) example with four agents and four alternatives
\(x_1=a\),
\(x_2=b\),
\(x_3=c\), and
\(x_4=d\). Let the preference profile be
$$\begin{aligned} \begin{array}{cccc} 1 &{} 2&{} 3 &{} 4 \\ \hline c &{} a &{} b &{} b \\ \cdot &{} d &{} a &{} d \\ \cdot &{} c &{} d &{} a \\ \cdot &{} b &{} c &{} c \end{array} \end{aligned}$$
and suppose that agent 1 wants
c to be chosen from the top elements
a,
b,
c. The preferences of agents 2, 3, and 4 have labels 6, 8, and 11, respectively, according to the formula in Sect.
4.1, hence in total 25. In order to achieve
c, the third top element of the profile according to the order
abcd, we need in (1.4) that
\(3 = 1 + [\nu (R^N)\!\!\mod 3]\). By choosing the preference
cabd, which has label 13, we obtain
\(1+[38\!\!\mod 3] = 3\), as desired. In fact,
cbda, which has label 16, would also work.
Our main result for this case is as follows.
As already announced, by taking
\(B={\emptyset }\) or
\(B=A\) we obtain the following corollary of Theorem
4.3.
It is an open problem whether there are self-implementable social choice correspondences (necessarily between Top and Pareto) which are not of the form \(H_B\).