Abstract
We extend the single-crossing property to tree networks to facilitate its application in network games. It is equivalent to intermediate preferences and order restriction (also extended to networks). Moreover, to facilitate broader applications in real world cases and simulations, we develop algorithms that answer the following two questions. Given a preference profile, can we construct a tree graph that supports single-crossing? Given a set of alternatives, can we generate single-crossing preference profiles with associated tree graphs?
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Notes
Let \(>\) be a linear order on the set \(N\) of individuals. Preference profile \(\succeq \) satisfies single-crossing on linear order \(>\), if there is a linear order \(>_{X}\) on \(X\) such that for any pair \( x,y\in X\) with \(x>_{X}y\) and for all \(i, j\in N\) with \(i>j\), we have \(x\succeq _{i}y\) implies \(x\succeq _{j}y\) and \(x\succ _{i}y\) implies \(x\succ _{j}y\) . Preference profile \(\succeq \) is intermediate on linear order \(>\), if for any pair \(x, y\in X\) for all \(i, j, k\in N\) with \(i<j<k\), (i) [\(x\succ _{i}y\) and \(x\succ _{k}y\)] implies \(x\succ _{j}y\), and (ii) [\(x\succeq _{i}y \) and \(x\succeq _{k}y\)] implies \(x\succeq _{j}y\). Preference profile \(\succeq \) satisfies order restriction on linear order \(>\), if for any pair \(x,y\in X\), either \(\left\{ i\mid x\succ _{i}y\right\} >\left\{ i\mid x\sim _{i}y\right\} >\left\{ i\mid y\succ _{i}x\right\} \), or \(\left\{ i\mid x\succ _{i}y\right\} <\left\{ i\mid x\sim _{i}y\right\} <\left\{ i\mid y\succ _{i}x\right\} \).
Saporiti and Tohmé (2006) use a version of single-crossing where the alternatives are real numbers, hence have a preexisting order over individuals.
After the following modification on top-monotonicity, Theorem 1 of Barberà and Moreno (2011) generalizes to tree graphs immediately: Preference profile \(\succeq \) is top—monotonic on \(G\), if for any \(i,j\in G\), there is a linear order \(>_{X}\) on \(X\), such that (i) \(t_{i}\left( A\right) \) is a finite union of closed intervals for all \(i\in p\left( ij\right) \), and (ii) For all \(S\in A\left( \succeq \right) \), for all \(m,n\in p\left( ij\right) \), all \(x\in t_{m}\left( S\right) \), all \(y\in t_{n}\left( S\right) \), and any \(z\in S\), we have \(\left[ x>_{X}y>_{X}z\ \text {or}\ z>_{X}y>_{X}x\right] \) implies (a) \(y\succeq _{m}z\ \)if \(z\in t_{m}\left( S\right) \cup t_{n}\left( S\right) \), and (b) \(y\succ _{m}z\) if \( z\not \in t_{m}\left( S\right) \cup t_{n}\left( S\right) \).
Donder (2013) examines single-crossing satisfied separately inside each among many voter groups. His space of alternatives is, however, a common real interval, thus implies a common order. On the other hand, the order of alternatives in our model is not predetermined and can differ for each chosen path of individuals.
Demange (1994) uses this property as the definition of intermediate preferences.
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Kung, FC. Sorting out single-crossing preferences on networks. Soc Choice Welf 44, 663–672 (2015). https://doi.org/10.1007/s00355-014-0852-5
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DOI: https://doi.org/10.1007/s00355-014-0852-5