A Policy-Based Rationalization of Collective Rules: Dimensionality, Specialized Houses, and Decentralized Authority

We o⁄er a policy basis for interpreting, justifying, and designing (3 ; 3) -political rules, a large class of collective rules analogous to those governing the selection of papers in peer-reviewed journals, where each referee chooses to accept, reject, or invite a resubmission of a paper, and an editor aggregates his own and referees(cid:146) opinions into one of these three recommendations. We prove that any such rule is a weighted multicameral rule: a policy is collectively approved at a given level if and only if it is approved by a minimal number of chambers- the dimension of the rule-, where each chamber evaluates a di⁄erent aspect of the policy using a weighted rule, with each evaluator(cid:146)s weight or authority possibly varying across chambers depending on his area(s) of expertise. Conversely, it is always possible to design a rule under which a policy is collectively approved at a given level if and only if it meets a certain number of prede(cid:133)ned criteria, so that one can set the standards for policies (cid:133)rst, and then design the rules that justify the passage of policies meeting those standards. These results imply that a given rule is only suitable for evaluating (cid:133)nite-dimensional policies whose dimension corresponds to that of the rule, and they provide a rationale for using di⁄erent rules to pass di⁄erent policies even within the same organization. We further introduce the concept of compatibility with a rule, and use it to propose a method to construct integer weights corresponding to evaluators(cid:146) possible judgments under a given rule, which are more intuitive and easier to interpret for policymakers. Our (cid:133)ndings shed light on multicameralism in political institutions and multi-criteria group decision-making in the (cid:133)rm. We provide applications to peer review politics, rating systems, and real-world organizations.


Introduction
Collective organizational rules are central to the governance of countries, collectivities, clubs, and corporations. They de…ne the procedure by which collective decisions are made, and contain mechanisms for how the preferences of di¤erent individuals are aggregated to yield a collective outcome. The simplest collective rules are weighted rules, where each individual has a certain number of votes (or seats), and a policy proposal is adopted if and only if the total number of votes it receives surpasses a certain threshold or quota. In the real world, however, many organizations use more complicated rules to make decisions.
An important class of such rules is the class of (3; 3)-political rules, which are rules analogous to those governing the selection of papers in peer-reviewed journals, where each referee chooses to accept, reject, or invite a resubmission of a paper, and an editor aggregates his own and referees' opinions into one of these three recommendations. In this paper, we show that any such rule, no matter its complexity, can be written as a collection of perfectly complementary weighted rules, which has important policy implications for the interpretation, design and suitability of collective decision-making mechanisms.
The class of (3; 3)-political rules generalizes well-known classes of rules, including (2; 2)-political rules under which each voter either supports or opposes a policy proposal, the collective outcome being either the adoption of the proposal or its failure. These latter rules, which were introduced by von Neumann and Morgenstern (1944), have served as the cornerstone of the analysis of group decision-making in a broad range of important studies in game theory (e.g., Shapley (1953), Arrow (1963), Peleg (1978Peleg ( , 1984, Taylor and Zwicker (1999), Ray (2007), Laruelle and Valenciano (2008)), political economy (e.g., Barberà and Jackson (2006), Brams (1975), Acemoglu et al. (2011)), and corporate governance (e.g., Leech (1988Leech ( , 2003). But despite their in ‡uence, scholars have argued that (2; 2)-political rules are restrictive, as more than two levels of individual and collective approval are generally observed in real-life decisions (Fishburn (1973), Machover (1997, 1998), Zwicker (2003, 2009 Rubinstein (1980), Hsiao and Raghavan (1993)). For instance, in most realworld elections, individuals might vote for or against any of the candidates, or might abstain, a situation forcelluly brought to scholarly awareness by Machover (1997, 1998). This compelling argument partly motivates our focus on more realistic collective decision-making mechanisms such as (3; 3)-political rules.
In this paper, we o¤er a policy basis for interpreting and designing collective rules. We state our main …ndings and discuss their policy implications.
1. First, we show that any (3; 3)-political rule can be decomposed into a hierarchical system of two weighted multicameral legislatures: the …rst legislature determines whether a proposal should be collectively approved at the highest level, such as accepting a paper, whereas the second legislature determines whether it should be collectively approved at the intermediate level, such as inviting a resubmission of a paper after revision. A proposal that fails in each of the two legislatures is simply collectively rejected. Each legislature consists of several specialized houses or chambers, and a policy proposal is adopted if and only if it is approved by each house, with each house using a weighted rule in that each decision-maker is assigned a (vector) weight that measures his political in ‡uence in that house, and a proposal passes the vote in that house if and only if the sum of points representing voters' opinions exceeds a predetermined quota. If we denote by d 1 the minimum number of houses of the …rst legislature, and by d 2 the minimum number of houses of the second legislature, we say that the corresponding political rule is of 2-dimension (d 1 ; d 2 ).
A practical implication of such a weighted multicameral representation of a (3; 3)-political rule is that such a rule can be interpreted as a multi-criteria decision-making rule that explicitly asks decision-makers to evaluate di¤erent aspects of a proposal to be voted on. Each house evaluates one aspect, with each decision-maker's in ‡uence or authority possibly varying across houses depending on his area(s) of expertise. Importantly, since di¤erent rules generally have di¤erent dimensions, it follows that each rule is best suited for evaluating …nite-dimensional policies whose dimension (number of aspects to be evaluated) corresponds to that of the rule, and that not all policies can be evaluated using the same rule. For instance, a one-dimensional policy such as tax rate should be evaluated using a di¤erent rule than policies of two or more dimensions.
2. Second, we show that any (3; 3)-political rule can be written as the minimum of a …nite number of (3; 3) "quasi-weighted" political rules (in a sense to be de…ned). This allows us to introduce the concept of quasi-dimension of a political rule, which generalizes the traditional concept of political dimension introduced by Taylor and Zwicker (1993).
3. Third, for any given pair of naturals (d 1 ; d 2 ), one can always design a (3; 3)-political rule of 2dimension (d 1 ; d 2 ). This …nding has important implications for the design of constitutions that value the inputs of di¤erent experts for the passage of a …nite-dimensional policy. For instance, if one wants to construct a rule under which a proposal is collectively approved at a given level if and only if it satis…es a …nite set of criteria, this is always possible. In fact, the number of criteria determines d 1 and d 2 . One can therefore set the standards for policies …rst, and then design the rules that would justify the passage of policies meeting those standards or criteria. For instance, one can always design a rule to select the winner of the election of Miss Universe where each contestant is judged regarding predetermined criteria such as beauty, self-con…dence and ability to communicate.
This …nding theoretically provides a rationale for using di¤erent rules to pass di¤erent policies even within the same organization.
4. Fourth, based on a newly de…ned concept of compatibility with a rule, we show that there exist in…nitely many weighted multicameral rules that are compatible with a (3; 3)-political rule, and prove that these rules constitute a topologically open set. Further applying topological concepts, we propose a method to construct integer weights to record the possible judgments of an expert, which are more intuitive and much easier to interpret for policymakers who vote on a regular basis.
To illustrate our main …ndings, let us consider this "thought" process of selecting a paper for publication in a journal. Suppose that a paper has two parts, one theoretical and the other empirical. The editor (E) invites four scholars, two theorists (T 1 and T 2) and two empiricists (E1 and E2), to evaluate each aspect of the paper. We imagine a collective decision-making rule de…ned as follows: The paper is accepted if the following two situations both occur: -At least one of the referees with expertise in theory …nds that the paper makes a theoretical contribution, and similarly, at least one of the empirical referees judges the paper to make an empirical contribution.
-None of the four referees rejects the paper.
The paper is rejected if it is judged by the two theoretical referees to make no theoretical contribution or it is judged by the two empirical referees to make no empirical contribution.
The paper is invited to be resubmitted after revision in all other cases.
If we let C 1 = fT 1; T 2g and C 2 = fE1; E2g, we can summarize this (3; 3)-political rule by the following characteristic function V: is a vote pro…le in which X 1 , X 2 and X 3 are respectively the sets of evaluators accepting the paper, inviting a resubmission, and rejecting the paper, and jX 1 j, for instance, is the cardinality of X 1 .
We note that in the example above, the editor could be one of the referees (e.g., and if he has expertise in both aspects of the paper, he could play the role of two referees, which would allow him to express possibly di¤erent views on the theoretical and empirical contributions of the paper. 2 As we can see, the formalization of this rule by its characteristic function V is economical, but a bit complex. However, we show that it can simply be represented as a pair of weighted two-house legislatures as in the following The …rst legislature (Legislature 1) determines whether the paper will be accepted. In this legislature, the theoretical contribution of the paper is evaluated in the …rst house (called Theory), and its empirical contribution in the second house (called Empirics). In each house, each voter has a three-component vector weight corresponding to the three possible judgments of the paper (accept, resubmit paper, reject). In the …rst house, each theoretical referee is assigned a vector weight of (2,1,0), which means that the paper receives 2 points if accepted by a theoretical referee, 1 point if invited to be revised and resubmitted, and 0 points otherwise. Each empirical referee is assigned a weight vector of (0,0,0), which means that their opinion on the theoretical aspect of the paper does not count. The paper passes the theoretical test if the sum of points received by the paper after each decision-maker evaluates its theoretical contribution is at least equal to 3. In the second house, each theoretical referee is assigned a weight vector of (0,0,0), and each empirical referee a weight vector of (2,1,0), and the paper passes the empirical test if it receives at least 3 points. So only the empirical referees wield power in the second house. The second legislature (Legislature 2) determines whether the paper will be invited to be revised and resubmitted. In this legislature, each evaluator has the same weight as in the …rst legislature; however, the paper can be resubmitted if and only if it receives at least 1 point in each of the two houses. The 2-dimension of this peer review rule is (2; 2).
Like the characteristic function V, most collective rules or constitutions are unfortunately simply de…ned as a distribution of voting power among the di¤erent subgroups of voters. In general, there is no rationale for the nature of policies that can be evaluated under these rules, which are also silent on the precise role of each voter in making decisions. A decomposition such as the one performed for V, however, addresses those shortcomings, as it shows that V is best suited for the evaluation of two-dimensional policy proposals, where each dimension is evaluated using a weighted rule. It also clearly describes the weight given to the opinion of each voter along each dimension. Such an approach further suggests a transparent and policy-driven method to design rules. Indeed, instead of de…ning a rule by its characteristic function as it is usually done, one can simply …x the number of criteria to be met by a policy to be collectively adopted under such a rule, and de…ne a weighted rule for each criterion, which is a simple exercise. Such a method would clearly value the opinion of experts, as the weight given to the judgment of a voter can be made to vary across the di¤erent dimensions.
Given that the class of (3; 3)-political rules generalizes well-known classes of rules, including (2; 2)- deliberative assemblies with four houses (the nobility, the clergy, the burghers, and the peasants), and the Council of the International Seabed Authority (ISA) with four houses, each representing the interest of a speci…c group of agents (consumers, investors, producers of minerals, developing countries and some other countries). 3 Similarly, a …rm's decision is sometimes determined by the di¤erent opinions expressed by its di¤erent departments as discussed in Baucells and Sarin (2003). In this case, a department (e.g., the marketing or the R&D department) plays the role of a specialized house.
Our paper also generalizes classical results on multicameral representation of (2; 2)-political rules (Taylor and Zwicker (1992, 1999). Taylor and Zwicker were the …rst to show that these latter rules have a weighted multicameral representation. They also introduced the algebraic notion of dimension, and argued that it is a "measure of the complexity" of a voting mechanism. They further showed that In addition to generalizing the literature by focusing on rules with more than two inputs and outputs, our analysis yields new theoretical …ndings that have important policy implications for the interpretation and design of political rules. Following basic de…nitions in Section 2, we show that a (3; 3)-political rule is a hierarchical system of two legislatures in Section 3. Building on this result, in Section 4, we introduce two new theories of dimension, namely the concepts of 2-dimension and quasi-dimension, which fully capture the complexity of (3; 3)-political rules. These theories imply that most constitutions are implicit multi-criteria decision-making rules under which each voter can be seen as an expert who has the ability to judge of the pertinence of a proposal only in his area(s) of expertise. We also show that one can always design a decision-making rule that values the views of specialized experts, and that is only suited for the evaluation of policies of a given dimension. In section 5, we introduce the concept of compatibility with a rule, and show that every (3; 3)-political rule has a weighted multicameral representation with integer weights and quotas. Our proofs are new. Section 6 provides some applications to real-world organizations, and Section 7 concludes.

Preliminary De…nitions
N = f1; 2; :::; ng is a non-empty set of evaluators, players, or voters. Subsets of N are coalitions. For any …nite set S, we denote by jSj the number of elements of S.
Voters are invited to vote on a proposal a. Each voter may vote for or against a, or may express an intermediate level of approval such as abstaining. 4 We denote by X 1 the set of voters who vote for, X 2 the set of voters who express the intermediate of approval, and X 3 the set of voters who vote against.
We call X = (X 1 ; X 2 ; X 3 ) a tripartition of N or a vote pro…le, and we denote by N 3 the set of all vote pro…les of N .

(3; 3)-Political Rules
where V is a function that maps each vote pro…le of N into one of three possible collective outcomes in the set f0; 1; 2g. 2 means that the proposal voted on is collectively accepted, 1 means that it is adjourned, and 0 means that it is rejected. When no confusion is possible, V will be used for (N; V). This paper uses the standard de…nition of a (3; 2)-political rule as a rule that maps each vote pro…le of N into one of two possible collective outcomes in the set f0; 1g. In this case, 0 means the proposal has failed, and 1 means the proposal is accepted.

Monotonicity
One of the most intuitive properties of political rules is the monotonicity property. A (3; 3)-political rule is monotonic if any increase in an individual's level of support for a proposal, ceteris paribus, cannot decrease the collectivity's su¤rage for that proposal. To be more speci…c, let us consider two vote pro…les X and Y on a proposal a. We say that X is a sub-tripartition of Y or Y is a super-tripartition of X ( and we write X 3 Y ) to convey that each voter's level of support for a is weakly greater in Y than X.
Formaly, X is a sub-tripartition of Y if either X = Y or X can be transformed into Y by moving one or more voters to a higher level of approval, everything else being equal. On the other hand, if X is a sub-tripartition of Y , distinct to Y , we write X 3 Y .
We de…ne X j " as the set of the voters who choose a level of approval higher than j , which means We say that a (3; 3)-political rule V is monotonic if for any vote pro…les X, Y such that X is a sub-partition of Y , V(X) V(Y ). This means that if X can make a proposal approved at a given level, so does Y . This is because voters have weakly higher level of support for the proposal in Y than in X.

Minimal Winning and Maximal Losing Vote Pro…les
As stated earlier, a (3; 2)-political rule G = (N; V) is a voting rule with only two possible outcomes (such as accepting or rejecting a proposal). Vote pro…les that lead to the approval of the proposal are called winning vote pro…les or winning tripartitions ( V(X) = 1), and those that lead to the rejection of the proposal are losing vote pro…les or losing tripartitions (V(X) = 0). A vote pro…le X is said to be minimal winning if every sub-tripartition of X is losing, and X is said to be maximal losing if every super-tripartition of X is winning.

Monotonic Weighted Multicameral (3; 3)-Political Rules
In the following, we provide a de…nition of a weighted multicameral (3; 3)-political rule. Intuitively, a multicameral voting rule is a rule under which the vote takes place simultaneously in a set of houses or chambers, here denoted by C. As more restrictive conditions should be ful…lled to shift a proposal to a higher level of approval, we split C into two disjoint sets of houses C 1 and C 2 , where each set of houses constitutes a legislature. C 2 represents the set of all the houses that must approve the proposal for it to earn the second level of collective approval. However, to earn the highest level of approval, the proposal should be approved in the additional number of houses that constitute C 1 . Let d be the total number of houses (d = jCj). We use the letters i to index the d houses, and j to index the levels of approval of a voter ( 1 i d and 1 j 3). Each house i is characterized by a quota q i ; each legislator p in house i has a three-component vector weight w i (p) = (w i1 (p); w i2 (p); w i3 (p)), meaning that the number of additional points credited to a proposal in that house is w i1 (p) when p votes for it, w i2 (p) when p abstains, and w i3 (p) when p votes against it 5 ; a proposal is approved in house i if the sum of points credited to it after everybody has voted is at least equal to the quota q i . Our formal de…nition is below.
is the sum of points credited to the proposal in house i after each voter has chosen a level of support, resulting in X). if the proposal is approved in all the chambers of C 2 , but is rejected in at least one of the chambers of C 1 (V(X) = 1 if and only if there is a house i in C 1 such that w i (X) < q i and for every house i in C 2 , (3) A vote pro…le X leads to the collective rejection of a proposal if and only if the proposal is rejected in at least one of the chambers of C 2 (V(X) = 0 if and only if there is a house i in C 2 such that w i (X) < q i ).
The table describing the peer review system in the introduction is a simple illustrative example of the de…nition of a weighted multicameral (3; 3)-political rule. In that example, while the sets of houses in the two legislatures are the same (C 1 = C 2 = fT heory; Empiricsg), this is not always the case in general.
Similarly, while the vector weight of a voter does not vary across the two legislatures in that example, a voter's vector weight does vary across legislatures in general.
A political rule is said to be a weighted multicameral political rule if it has a weighted multicameral representation. Note that C 1 = ; corresponds to the particular case of weighted multicameral (3; 2)political rules. If additionnally, C 2 has only one house, the corresponding (3; 2)-political rule is said to be weighted. We will say that a (3; 3)-political rule G is weighted when the decision criteria are unidimensional, which means both C 1 and C 2 are singletons.
Under a weighted multicameral political rule, a voter's weight associated with a given level of approval might vary across houses, which implies that a voter may be more in ‡uential in certain houses than others, as illustrated in the example in the introduction. In each legislature, each theoretical referee has a greater weight in the Theory house than in the Empirics house. Similarly, each empirical referee has a greater weight in the Empirics house than in the Theory house. A house in which a voter has a high level of in ‡uence may be viewed as the house dealing with matters on which that voter is an expert. We therefore think of houses as being specialized, and of voters as experts who have decentralized authority and vote only on speci…c aspects of a proposal under consideration. This means V 1 considers as winning vote pro…les only those that lead to the highest level of collective approval of a proposal, whereas the winning vote pro…les of V 2 are those that lead to the approval level at least equal to the intermediate level.
The decomposition of V as given in Remark 1 yields a unique pair of (3; 2)-political rules (V 1 ; V 2 ) 6 .
We write V = V 1 ? V 2 and we say that V 1 ? V 2 is a (3; 2)-decomposition of V or V is a (3; 3)-composition of V 1 and V 2 . V 1 is of a higher standard than V 2 in the sense that it is more di¢cult for a policy proposal to 6 Mathematically, V 1 and V 2 satisfy pass under V 1 than under V 2 holding voters' opinions …xed 7 . We say that the pair (V 1 ; V 2 ) is ordered 8 .
Remark 1 proves Theorem 1 by establishing a one-to-one mapping between the set of (3; 3)-political rules V and the set of ordered pairs (V 1 ; V 2 ) of (3; 2)-political rules.
One of the implications of Theorem 1 is that some of the properties of (3; 2)-political rules can be extend to the class of (3; 3)-political rules using the (3; 2)-decomposition of (3; 3)-political rules.
The following result provides a necessary and su¢cient condition for a (3; 3)-political rule to be weighted. Consider a pair of weighted (3; 2)-political rules (V 1 , V 2 ). If (V 1 , V 2 ) is ordered, then the weighted is not ordered, the (3; 3)-political rule G 1 + G 2 is not necessarily weighted. In this case, we said that We can notice that a weighted

The Quasi-Dimension of a (3; 3)-Political Rule
Earlier studies have shown that a (2; 2)-political rule is a …nite intersection of weighted (2; 2)-political rules (intersection should be understood as minimum). According to Taylor and Zwicker (1993), the dimension of a (2; 2)-political rule G is de…ned as the minimum integer m such that G can be written as the minimum of m weighted (2; 2)-political rules. In the following, we represent a The above de…nition is an extension of the concept of dimension as de…ned on the class of (2; 2)political rules. In the following, we therefore generalize some results obtained for (2; 2)-political rules.  The proof of this result is straightforward given Theorem 2.

The 2-Dimension of a (3; 3)-Political Rule
We introduce the concept of 2-dimension of a monotonic We note that the notion of 2-dimension generalizes the traditional notion of dimension since the two concepts coincide on the class of (3; 2)-political rules and (2; 2)-political rules. Indeed, a monotonic (3; 2)-political rule of dimension d can be seen as having 2-dimension (d; 0).
The following theorem establishes that the concept of 2-dimension is well de…ned on the class of monotonic (3; 3)-political rules.

Theorem 3 The 2-dimension of a monotonic (3; 3)-political rule always exists.
If one assumes that one-dimensional policies should be evaluated using weighted (or one-dimensional) rules, from a policy perspective, Theorem 3 implies that a given monotonic (3; 3)-political rule is only suited for the evaluation of …nite-dimensional policies whose dimension corresponds to that of the rule, with each aspect of the policy being evaluated in a di¤erent house or chamber using a weighted rule. The fact that any such rule has a weighted multicameral representation also implies that any (3; 3)-political rule is a collection of perfectly complementary weighted rules in the sense that a policy proposal cannot be approved at a given level if it is not approved under each of the weighted rules of the legislature deciding on the approval of the proposal at that level (in fact, the rule used in each legislature is a minimum of weighted rules, which means that if a policy proposal fails under one of these rules thus yielding an outcome of zero for that rule, the minimum will therefore be zero, and so the proposal will fail overall in that legislature).
We also have the following proposition. This result also provides a rationale for using di¤erent rules to evaluate di¤erent policies, even within the same organization.

Useful Topology of Political Rules for Policymakers
In this section, we introduce the concept of compatibility with a rule, which we subsequently use to show that any (3; 3)-political rule has a weighted multicameral representation in which weights and quotas are integers. Weights therefore represent the number of votes, which facilitates the interpretation of political rules for policymakers.

Compatibility
We introduce the concept of compatibility of a real vector with a rule, and topologically characterize the set of weighted multicameral rules that are compatible with a given political rule. We …rst establish our results on the class of (3; 2)-political rules, then extend them to the class of (3; 3)-political rules.
Before de…ning the concept of compatibility, recall that any weighted multicameral representation of a rule ' = (w ij (p); q i ) 1 i d;1 j 3;1 p n can be viewed as a vector of R d (3n+1) (see Appendix F for the vector con…guration). We have the following de…nition.
' is Vcompatible if for every vote pro…le X, the following two conditions are satis…ed: (1) X is winning if and only if the weight of X in every house i is strictly higher than the corresponding quota q i (V(X) = 1 if and only if w i (X) > q i for all 1 i d).
(2) X is losing if and only if the weight of X is strictly lower than the quota in at least one house, su¢ciently small pertubations, the distribution of political power will remain unchanged. This …nding will prove important in the next section where we show that any (3; 3)-political rule has a weighted multicameral representation whose weights and quotas are integers, and propose a method to construct such a representation.

Integer Weights and Quotas
We show that any (3; 3)-political rule has a weighted multicameral representation in which weights and quota are integers. Such a representation is more intuitive, as it implies that the weights w ij (p) can now be interpreted as the number of votes held by voter p in the house i when he chooses the level of approval j. Such an interpretation is also more likely to be understood by policymakers, lawmakers and shareholders who vote on a regular basis, and to shed light on the real structure of power in an organization.
Theorem 6 Let G = (N; V) be a (3; 3)-political rule. G has a weighted multicameral representation whose weights and quotas are integers.
This result implies that one can always represent any (3; 3)-political rule in a way that is understandable to any policymaker. An example is our example in the introduction where all weights and quotas are integers. See Section 6 for other examples as well.
We now extend below the concept of the weight monotonicity requirement introduced by Freixas and Zwicker (2003) for single-house rules. It says that the higher the level of approval of a voter for a given proposal, the higher should be his contribution to make that proposal approved by the society.
De…nition 5 Let G = (N; V) be a monotonic (3; 3)-political rule. A weighted multicameral representation (w; q) of G satis…es the "weight monotonicity requirement" if : for every voter p 2 N and for every We show below that any (3; 3)-political rule admits a weighted multicameral representation that satis…es the weight-monotonicity requirement.

Applications
In this section, we apply our theoretical …ndings to some real-life organizations. For each organization, we de…ne its decision-making rule and provide a weighted multicameral representation of that rule with integer weights and quotas, and satisfying the weight-monotonicity requirement.

The Council of the International Seabed Authority
The International Seabed Authority is an intergovernmental organization created in 1994 to monitor mineral-related activities in the international seabed territories outside of states regulated jurisdiction.
The ISA has two principal organs: an assembly, which consists of all the ISA members, and a 36-member council, elected by the assembly. The council members are chosen to ensure equitable representation of countries from various groups. As described by Bräuninger (2003), the members of the council are distributed into four houses denoted C 1 ; C 2 ; C 3 ; C 4 as follows: C 1 : four states elected from among the largest consumers of the minerals in question C 2 : four states elected from among the largest investors in deep-sea mining C 3 : four states from among the largest net exporters C 4 : six developing countries and eighteen additional states to ensure a balanced geographical distribution of seats in the council.
In the ISA council, a decision on most issues requires a two-thirds majority approval of its members on the proviso that such decision is not opposed by a majority in any of the four houses. This voting rule can be modeled as a (3; 2)-political rule G = (N; V) as follows: For any vote pro…le X = (X 1 ; X 2 ; X 3 ), jX 3 \ C t j < 1 2 jC t j ; 1 t 4 This decision-making rule was studied by Bräuninger (2003), who proved that it is not of dimension one. In the following, we propose a weighted multicameral representation of the vote within the Council of ISA . In the matrix of weights below, the …rst, the second , the third and the fourth columns represent the house of consumers, investors, exporters and the fourth house, respectively. The last column represents the entire council. The vote in the ISA council is a monotonic (3; 2)-political rule of dimension 5 (the 2-dimension is (5; 0) and the quasi-dimension is 5).

A Voting Rule in the United States Senate
We study the voting rule for electing the Vice-President of the United States when the Electoral College fails to do so. We show that the quasi-dimension of this rule is 1. We argue that the quasi-dimension of this rule is 1 and its 2-dimension is (1; 1). Let G 1 and G 2 be two monotonic (3; 2)-political rules de…ned as follows: V 1 (X 1 ; X 2 ; X 3 ) = 1 if and only if jX 1 j 51 V 2 (X 1 ; X 2 ; X 3 ) = 1 if and only if (jX 1 j 51) or (jX 1 j < 51 and jX 3 j < 51) A weighted representation of G 1 and G 2 is respectively given by: Both G 1 and G 2 are of dimension 1. Since G 1 and G 2 provide a (3; 2)-decomposition of G, the quasi-dimension of G is 1 and the 2-dimension of G is (1; 1).

Core Examination of Graduate Students in United States Universities
In some U.S. universities, Masters' students have two options when completing their Masters' degree.
They can either write a thesis or take an exam. Students who choose the exam option are pooled with Ph.D. candidates to write a core examination. The core examination usually combines several subjects. Suppose that the core examination in a department of Economics is based on three subjects: microeconomics, macroeconomics and econometrics.
Assume that the core examination committee consists of 6 members divided into 3 subcommittees as follows: the microeconomics committee (2 members), the macroeconomics committee (2 members), and the econometrics committee (2 members).
To each student taking the core examination, each committee member assigns three decisions corresponding to their judgment on the student's performance on each of the three subjects. There are three possible decisions for each subject: pass, marginal pass, and fail. There are also three possible …nal outcomes for each student. A student can pass the exam both at the Master and Ph.D. levels, pass the exam only at the Master level, or fail the exam both at the Master and Ph.D. levels. Passing the core examination at a given level requires obtaining a certain grade on each subject. Assume that a student must have at least one "pass" and one "marginal pass" to pass a subject at the Ph.D. level, and at least two "marginal pass" or one "pass" to pass a subject at the Master's level.
The above described rule is a monotonic (3; 3)-political rule. We denote the examination outcomes as follows: 2 = Ph.D., 1 = Master, 0 = fail. For any vote pro…le X = (X 1 ; X 2 ; X 3 ), X 1 , X 2 and X 3 are respectively the set of professors who vote "pass", "marginal pass" and "fail". Denote by C 1 , C 2 , C 3 the microeconomics committee, the macroeconomics committee, and the econometrics committee, respectively. On a given subject, only the corresponding subcommittee has the full competence to judge the performance of a student. Therefore, we assume that the decisions of the members outside of a subcommittee do not a¤ect the students' outcome for that subject. The above described core examination can be modeled as a (3; 3)-political rule G = (N; V) as follows: For any vote pro…le X = (X 1 ; X 2 ; X 3 ), Ph.D. if (jX 1 \ C t j = 2 or (jX 1 \ C t j = 1 and jX 2 \ C t j = 1) for all 1 t 3 Master if (jX 2 \ C t j = 2) or (jX 1 \ C t j = 1) for all 1 t 3 Fail if 91 t 3; (jX 3 \ C t j 1) and (jX 1 \ C t j = 0).
Let G 1 (resp. G 2 ) be the monotonic (3; 2)-political rule that determines whether a student will pass the core examination at the Ph.D. (resp. at least at the Master level ). G 1 and G 2 can be de…ned as follows: for any vote pro…le X = (X 1 ; X 2 ; X 3 ), The weighted representations of G 1 and G 2 are given by: Therefore, both G 1 and G 2 are of dimension 3. Since G 1 and G 2 provide a decomposition of G as in Remark 1, the 2-dimension of G is (3; 3).
One might want to modify the examination system to avoid situations where Masters' students are evaluated on econometrics as is the case in certain universities. Assume, for example, that every student is now required to have at least two "marginal pass" outcomes to pass microeconomics and macroeconomics.
Ph.D. students are required to have one "pass" or one "marginal pass" to pass econometrics. The modi…ed rule has the following weighted multicameral representation:

Concluding Remarks
We have provided a policy basis for rationalizing, that is, interpreting, justifying and designing ( We have also shown that it is always possible to design a (3; 3)-political rule of any given 2-dimension.
Practically, this …nding implies that it is always possible to design a rule under which a proposal is collectively approved at a given level if and only if it satis…es a certain number of "prede…ned criteria", so that one can set the criteria for policies …rst, and then design the rules that rationalize the passage of policies meeting those criteria. This shows that rules can be designed so as to be suitable only for the evaluation of certain policies, and suggests a rationale for using di¤erent rules to pass di¤erent policies. Accordingly, a unidimensional policy should be evaluated using a (3; 3)-political rule that has only one house in each legislature, since only one aspect of the policy will be evaluated. However, multidimensional policy proposals should be evaluated under weighted multicameral legislatures, where each house examines one aspect of the policy, and each legislator is given more weight in the house evaluating the aspect that pertains to his area of expertise. We note that a legislator may be expert in many areas, in which case he should have in ‡uence in di¤erent houses.
Designing a collective rule based on a set of criteria to be met by an ideal policy is not only transparent, but practical. Indeed, it might be cumbersome to try to write a decision-making rule as we did in our example in the introduction. A simpler way of writing the same rule would be to consider the matrix of weights, which clearly show how much in ‡uence each evaluator wields along a given criterion. If one wants to change the rule at some point, perhaps because one wants to increase the standards of policies to be adopted, or because one wants to get more evaluators involved along each criterion, one can simply modify the weights and/or the quotas, which is simple (although this might not be so easy in practice).
For instance, one might change the weight matrix of the two legislatures in that example as follows: We note that under the modi…ed rule, it is harder to get a paper accepted, or considered for resubmission, compared to the original rule. Under the modi…ed rule, while experts still have more power in their area of expertise, their only opinion is no longer su¢cient to determine that a paper passes the test in their area. For instance, while under the …rst rule, the coalition of theoretical referees fT 1; T 2g has dictatorial power on acceptance along the theoretical dimension, it only has a veto right under the modi…ed rule. Similarly, as regarding the empirical assessment of the paper, the coalition of empirical referees fE1; E2g has only a veto right for acceptance under the modi…ed rule, whereas they had dictatorial power under the …rst rule. However, each expert still has a veto right along his area of expertise since the paper cannot get accepted (even after revision) without him strongly backing the decision.
These two rules might also be used by two di¤erent journals or organizations. One lesson to be learned here is that writing such a rule explicitly, as we did in the introduction, could be very confusing in some instances and could subject the written rule to di¤erent interpretations. Similarly, writing the characteristic function of a rule may not let policymakers know whether the rule is really appropriate for evaluating a given proposal, and it may not be explicit on the precise role of each evaluator. The table, on the contrary, clearly shows the power structure among the di¤erent evaluators along each dimension of a potential proposal, and should therefore be preferred to a written rule. It also shows that the original and the modi…ed rules are most appropriate for the evaluation of two-dimensional policies.
We have also proposed two new concepts of dimension (the quasi-dimension and the 2-dimension), and have shown that they generalize the literature. While the traditional theory is easily interpreted within our framework, we show that the 2-dimension measure does not only capture the number of possible outputs yielded by a (3; 3)-political rule, but it also takes into account the hierarchy in these outputs. Furthermore, we have introduced the concept of compatibility of a vector with a rule, which we have used to show that there exist in…nitely many weighted rules that are strictly compatible with a (3; 3)-political rule, and that these rules form a topologically open set. We have also proposed a method to construct integer weights and quotas, which are more intuitive, more relevant for rule design and policy-making, and more transparent for policymakers. In sum, our results rationalize multicameralism in real-world organizations and multi-criteria group decision-making in market organizations (Baucells and Sarin (2003)), and suggest a simple and rational policy-driven approach to the design of collective rules.

Appendices Appendix A: Proof of Proposition 1
If G is weighted, then fw ij ; i 2 C 1 g and fw ij ; i 2 C 2 g are the systems of weights of G 1 and G 2 , respectively.
Conversely, if fw 1j g 1 j 3 and fw 2j g 1 j 3 are respectively the systems of weights of the weighted rules G 1 and G 2 , de…ne the system of weights of G, fw ij g, as w ij = w 1j if i 2 C 1 and w ij = w 2j if i 2 C 2 with C 1 = f1g and C 2 = f2g.
Appendix B: Proof of Theorem 2 The following lemma is needed to prove our result.
The quota for each house i is q i = 1; 1 i d

We have:
V(X) = 1 i¤ for all 1 i d; X * 3 Y i i¤ for all 1 i d; there exists 1 j 3, X j "* Y i j " i¤ for all 1 i d, there exist 1 j 3 and p 2 X j " such that p = 2 Y i j " i¤ for all 1 i d, there exist 1 j 3 and p 2 X j " such that w ij (p) = 1 i¤ for all 1 i d; w i (X) q i : Therefore, w = (w ij ) 1 i d;1 j 3 and q = (q i ) 1 i d de…ne a weighted multicameral representation of G.
For each 1 i d, let G i = (N; V i ) be the (monotonic) weighted (3; 2)-political rule represented by (w ij ; q i ). It can be easily shown that for every X 2 N 3 , V(X) = min fV i (X); 1 i dg.
Below is a two-step proof that the dimension of the above-described monotonic (3; 2)-political rule is m.
Step 1 : First, we show that G is a minimum of m weighted monotonic (3; 2)-political rules. Let X 2 N 3 be a winning vote pro…le.
X 2 V 1 f1g i¤ X 1 \ f2i 1; 2ig 6 = ;, for all 1 i m i¤ there is p 2 N such that w ij (p) = 1, for all 1 i m i¤ w i (X) q i , for all 1 i m i¤ X 2 V 1 i f1g , 81 i m i¤ X 2 \ V 1 i f1g j 1 i m .
Thus V 1 f1g = \ V 1 i f1g j 1 i m . step 2: Our next goal is to show that G is not the minimum of m 1 weighted monotonic (3; 2)political rules. Assume on the contrary that G can be written as the minimum of m 1 weighted monotonic (3; 2)-political rules G Lemma 3 Let G = (N; V) be a (3; 2)-political rule. The set (G) of all the vectors ' 2 R d (3n+1) that are V-compatible is the intersection of U G and V G .
Proof: Let ' 2 R d (3n+1) a vector that is V-compatible and X a vote pro…le of N . If X is a winning vote pro…le, then w i (X) > q i for all 1 i d. Therefore ' 2 U X . This implies that ' 2 U G = \fU X ; X winning vote pro…le g If X is a losing vote pro…le, then there exists 1 i d such that w i (X) < q i . Therefore ' 2 V X . This implies that ' 2 V G = \fV X ; X losing vote pro…leg.
Therefore, (G) is the intersection of U G and V G . Now, we can proof our main result.

Appendix H: Proof of Theorem 6
To proof our main result, we will need the following lemmas that are also interesting on their own. Lemma 4 Let G = (N; V) be a (3; 2)-political rule. If ' = (w ij (p); q i ) 2 R d (3n+1) is a weighted multicameral representation of G, then there exists q 0 2 R d such that ' 0 = (w ij (p); q 0 i ) is V-compatible.
Proof : De…ne i as the lower bound of the set fw i (X)= w i (X) q i ; X 2 N 3 g and i as the upper bound of the set fw i (X)= w i (X) < q i ; X 2 N 3 g. Notice that i < q i i . De…ne q 0 i = i + i 2 and consider a vote pro…le X.
If X is a winning vote pro…le, then w i (X) i > q 0 i ; for all 1 i d. If X is a losing vote pro…le, then there exists 1 i d such that w i (X) i < q 0 i . Thus ' 0 = (w ij (p); q 0 i ) is V-compatible . It follows that a V-compatible vector can be obtained from a weighted multicameral representation of a (3; 2)-political rule G. Altering the weights of voters is unnecessary. A judicious alteration of the quotas of G is su¢cient. A corollary to the previous lemma is the following.
Corollary 1 Let G be a (3; 2)-political rule. G is a weighted multicameral rule if and only if there exists ' 2 R d (3n+1) that is V-compatible.
Lemma 5 Let G be a (3; 2)-political rule. G has a weighted multicameral representation if and only if there exists ' 2 Q d (3n+1) that is V-compatible.
Lemma 6 Let G be (3; 2)-political rule. G has a weighted multicameral representation if and only if there exists ' 2 Z d (3n+1) that is V-compatible.
The converse is trivial.

Proof of Theorem 6
Let G 1 ? G 2 the (3; 2)-decomposition of a (3; 3)-political rule G. The monotonicity of G insures the monotonicity of G 1 and G 2 . We can …nd weighted multicameral representations of G 1 and G 2 with integer weights and quotas (Lemma 6). Use these systems of weights to construct a weighted multicameral representation of G with integer weights and quotas . For the method of construction, see the proof of Proposition 2.
Appendix I: Proof of Theorem 7 Let w = (w ij ; q i ) 1 i d;1 j 3 be a weighted multicameral representation of G that violates the weightmonotonocity requirement. This implies there is a voter x 2 N and a house t such that w tl (x) < w tl+1 (x), with 1 t d; 1 l 2.
Construct the new weighted multicameral representation u of G as follows: u ij (p) = 8 > < > : w tl+1 (p) if p = x; i = t and j = l w tl (p) if p = x; i = t and j = l + 1 w ij (p) otherwise: We want to show that both u and w identically distribute the elements of N 3 into the group of winning and losing vote pro…les. Let X be any vote pro…le with x 2 X l+1 and Y the vote pro…le derived from X by shifting voter x to X l : We see that: 1. X V Y (by monotonicity), and for every 1 i d , w i (X) > w i (Y ): 2. u i (X) = w i (Y ) and u i (Y ) = w i (X): Therefore, for every 1 i d, w i (X) q i if and only if w i (Y ) q i (from (1)), which happens if and only if u i (X) q i (from (2)).
The technique may be repeated for each voter whose weights violate the weight-monotonicity requirement.