Suppose we have a society of 100 individuals, and they are asked to express their preferences over alternatives and . The result is the following:
51 individuals report .
49 individuals report .
Which alternative should the society adopt? If we presume that all individuals are equal, and there are no legal, ethical, etc, reasons to a priori prefer to , it seems difficult to justify any outcome but . In a manner that can be made precise (May, 1952), majority voting is the only reasonable way to pick a winner out of two alternatives. But what if we modify the preceding example by inserting a third alternative and obtain the following:
51 individuals report .
49 individuals report .
Now, the solution is no longer obvious, and is very much contingent on just who these “individuals” are, and what they are trying to achieve. If, for example, each preference order represents the results of a sporting event then the case can be made that should still be the winner – the fact that is a more consistent competitor does not change the fact that, put face to face, has a tendency to beat . Likewise, if these are legislators’ preferences over bills, is the only bill that has the support of a parliamentary majority, and is thus unlikely to be overturned in the future. But what if the preferences are over government budgets? The locations of a school? The soundtrack of a busy shopping centre? Are we willing to write off the interests of almost half the population to satisfy the majority?
Should we decide that in our context the minority matters, a natural approach is to turn the problem on its head: rather than ask which alternative to elect, we ask which should we prevent from winning. Mueller (1978) considers the procedure of voting by veto to select a public good, with the explicit aim of protecting an individual from unfair treatment. The alternatives consist of one proposal from each voter, as well as the status quo. An order is formed over the voters and each, in turn, strikes off the alternative they like the least. Exactly one outcome is selected by this procedure, but this procedure, like others of its kind, is not anonymous.
The strategic properties of this procedure, and its variants, have been studied extensively. The sincere outcome of voting by veto has been shown to correspond to a strong equilibrium solution (Peleg, 1978), dominance solution (Moulin, 1980), and maxmin behaviour (Moulin, 1981b); the general message seems to be that while truth-telling may not be an equilibrium, among equally sophisticated voters the same alternative can be expected to be elected.
Moulin (1981a) extended the concept of voting by veto from individuals to coalitions, and studied the core of the resulting cooperative game. The core as a solution to a voting situation is problematic as the ideal number of winners is one, but the core could be larger or smaller, but Moulin has shown that endowing each coalition with the power to veto a number of alternatives proportional to the coalition’s size guarantees that the core is non-empty, and is the smallest possible core out of all such rules. Moreover, such veto power is necessary for implementation via strong equilibrium (Moulin, 1982).
Kondratev and Nesterov (2020) investigated this proportional veto core in the context of balancing the rights of minorities and majorities, and their main criticism of the rule was that it is unclear how the core is to be computed – the naïve algorithm is exponential. In this work we demonstrate that a polynomial solution exists.
Let be the set of voters and the set of alternatives. Every voter has strict preferences over , which we denote by .
The proportional veto power of a coalition is given by:
We say that an alternative is blocked by a coalition just if there exists a set of alternatives such that:
Condition (1) means that every voter in considers every alternative in to be strictly better than , and condition (2) means that the coalition can guarantee that the winner will be among by blocking all the other alternatives.
The set of all alternatives that are not blocked is called the proportional veto core.
Moulin (1981a) showed that the proportional veto core is always non-empty, and is thus a function mapping profiles to non-empty sets of alternatives – in other words, it can be viewed as a voting rule, albeit with a high incidence of ties.
Consider a profile with 5 alternatives and four voters with the following preferences:
In the case of the veto power of a coalition simplifies to . In other words, voters can block exactly alternatives.
Alternative is blocked by the singleton coalition (among others), with ; is blocked by , with ; is not blocked by any singletons, but is blocked by , with ; is blocked by the coalition with . Thus the unique alternative in the core is .
Now let us add a fifth voter:
With , the veto power simplifies to . A singleton coalition can no longer block anything, but the coalition can block with . However, no other alternative is blocked, so the core is .
3 The algorithm
We will need the following lemma of Moulin (1981a)
If , where is the greatest common divisor of and , , then for :
From we obtain .
Case one: is an integer. In this case:
Case two: . In this case , and:
The equality follows because it is clear that . ∎
The proportional veto core can be computed in polynomial time.
We shall show that we can check whether a fixed alternative is blocked in polynomial time – we can then find the core by running the algorithm below on every alternative.
Consider a veto instance with voters and alternatives. Using the Extended Euclidean algorithm we can find such that , where , and in polynomial time. It is easy to see that with , we have and . By Lemma 3.1, the veto power of a coalition of size is .
Construct a bipartite graph with vertices on the left and vertices on the right. Every voter is associated with vertices on the left and every alternative but with vertices on the right. Add an edge between every voter vertex and every alternative vertex that voter considers to be better than .
We can find the maximum for which there exists a complete bipartite subgraph in polynomial time (Dawande et al., 2001)111See the discussion in the appendix.. We claim that is blocked if and only if .
Suppose that is blocked by coalition of size . This means there exists a , , such that the voters in strictly prefer everything in to . Observe that the vertices associated with and form a complete bipartite subgraph with vertices. Since , it follows that and the graph has at least vertices.
Now suppose there exists a complete bipartite subgraph with . Without loss of generality, we can assume . That is, there are voters who all agree that alternatives are better than . We must show that the coalition has enough veto power to force this outcome: . We know that:
Since is an integer it follows that . ∎
Appendix A The maximum vertex biclique problem
The maximum vertex biclique problem is the following, and it can be solved in polynomial time:
Input: A bipartite graph , an integer .
Question: Does there exist a complete bipartite graph , with ?
All the provisos are necessary – the problem becomes NP-hard if is allowed to be a general graph (Yannakakis, 1978), if we search for a subgraph with (Garey and Johnson, 1979), or with (Peeters, 2003).
The commonly cited argument for the polynomial-time solvability of the vertex maximum biclique problem harkens back to Garey and Johnson (1979), and runs along the lines of:
By Kőnig’s theorem (Kőnig, 1931), the size of the maximum matching in a bipartite graph is equal to the size of the smallest vertex cover. The complement of the smallest vertex cover is the maximum independent set. The maximum independent set is a biclique in the bipartite-complement of the graph. Therefore take , construct its bipartite-complement and find the size of the maximum matching, . The size of the maximum independent set in is then , and for the maximum biclique in .
While this is clean and elegant, it is not entirely correct. In particular, if the graph contains a perfect matching then the maximum independent set is of size , or one partition of a graph. The complement of this is not a biclique, unless we admit bicliques of the form .
However it can still be seen that the problem is solvable in polynomial time via the linear programming formulation ofDawande et al. (2001).
Let be a bipartite graph with vertices and . Construct the following integer program:
The variables denote whether the vertex is in the biclique. The objective function thus maximises the number of vertices, the first constraint forbids adding non-adjacent vertices to the biclique, and the last constraint ensures a vertex is either in or out of the biclique.
Take the linear programming relaxation of this program by allowing . In the case of totally unimodular constraint matrices the integral optimum is equal to its linear relaxation (Schrijver, 1998, p. 267, Corollary 19.1a), so let us verify that the constraints are indeed totally unimodular.
The following conditions are sufficient for a matrix to be totally unimodular (Heller and Tompkins, 1956):
Every row of contains at most two non-zero entries.
Every entry of is , or 1.
It is possible to partition the rows of into , such that:
If two non-zero entries of have the same sign, one is in and one in .
If two non-zero entries of have different signs, both are in or both are in .
The rows of the constraint matrix consist of 1s and 0s (condition 2), and contain at most two 1s: for non-adjacent and (condition 1). If we partition the columns into the set and the set , then we will have a 1 for and a 1 for only if and (condition 3).
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