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¹⁸ A distance measure is a function that has the properties that for points a, b and c in W: d(a, b) = 0⇔a = b, d(a, b) = d(b, a) and d(a, b) + d(b, c) ≥ d(a, c).

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²² We say that a player, i∈N, has generalized Euclidean preferences over points in if there exists an and a semipositive definite matrix A_i such that for all , x x′⇔(y_i − x)′A(y_i − x)≤(y_i − x′)′A(y_i − x′). Euclidean preferences (with uniform weights) obtain if A = I_{n × n}.

²³ It is important to emphasize that the assumption of a Euclidean cognitive metric to measure distance is quite distinct from the assumption of a quadratic loss function, despite the use of quadratic terms in each. The metric assumption is a cognitive assumption about how agents trade differences off between stimuli on distinct dimensions of difference. The loss function is a cognitive assumption about how these distances, calculated after having made these trade-offs, map into agent utility. Thus, it is quite consistent to apply quadratic or exponential loss functions to city block distances.

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²⁵ As opposed to some particular pathological configuration. Formally a collection of points in is in ‘general position’ if, for any k≤m, no more than m ideal points lie on any m − 1 dimensional affine manifold.

²⁶ Consider, for example, a case in which five players have ideals given by (1,1,0), (1,0,1) (0,1,1), (−1,−1,−1), (−1,−1,−1). In this case (1,1,1) is preferred by the first three agents to (0,0,0).

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³¹ To see this, fix i and let k be given by k = (i + 1)(mod(m)). Define and for j = 1,2,…,m by and . Note that and are elements of x, moreover, y_i∈closure and y_k∈closure but that . Hence implies , and so y_i and y_k lie in opposite halfspaces.

³² Assume to the contrary that a rugged halfspace contained only a minority of ideal points, then a majority of ideals would lie in the interior of . But such a majority would contain two (neighbouring) points that are not in opposite halfspaces.

³³ In the simulations presented in Table 1, players have ideals drawn independently from a multivariate normal distribution with μ = 0 and Σ_{m × m} = I_{m × m}. For each draw the DDM is subjected to 5,000 alternatives within 0.001 of the DDM on each dimension.

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³⁵ The largest two-party rival to a winning coalition between the largest and smallest parties is a coalition between the second and third largest parties, which must be losing since it is in the complement of the winning coalition between largest and smallest parties. Some non-cooperative game theorists refer to a system-dominant party as an ‘apex player’.

³⁶ More generally still, we note that the probability that any party in a weighted majority-rule voting game is median on an arbitrary policy dimension is its Shapley value in the game. The Shapley value is the probability a party is pivotal in a random ordering; in this context, we consider the probability that a party is pivotal (median) in a random policy ordering of parties on an arbitrary dimension.

³⁷ For one third of the cases, one player is at the DDM and no MRE exists; for the remaining two thirds, only one sixth have weights that guarantee an MRE.

³⁸ These inequalities ensure that each player cares (relatively) more about the dimension that she is median on than do the other players. This makes it difficult for two players to strike a deal involving changes in either of the (two) dimensions on which they are median.

³⁹ A proof of this claim is available from the authors.

⁴⁰ In particular, one of three agents is necessarily median on two dimensions, hence with ideals in general position, neither of the remaining two agents is median on those two dimensions.

⁴¹ The Minkowski distance of order p between two points, x and y is given by for p≥1 The Euclidean and city block metrics are special cases with p = 2 and p = 1 respectively. The ∞− norm distance between two stimuli is the distance between them on the dimension on which they are farthest apart.

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⁴⁶ Which of these notions of distance is used matters substantively: a small deviation from city block may produce a set of points that beat the DDM, which, though small in measure, includes points that are distant from the DDM. As an example, consider two ideal points in , (1,0) and (0,1). Now consider the set P = {(x,x):x∈(0,1)}. With Minkowski exponent p = 1, no point in P is preferred by both players to the origin. But for any p∈(1,2), all points in P are preferred to the origin.

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⁴⁹ Note that a similar argument can be made for small deviations from Plott conditions.

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