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Copy file name to clipboardExpand all lines: lectures/match_transport.md
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## Introduction
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This notebook presents Python code for solving **composite sorting** problems of the kind
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studied in *Composite Sorting* by Job Boerma, Aleh Tsyvinski, Ruodo Wang,
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and Zhenyuan Zhang {cite}`boerma2023composite`.
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studied in the August 2023 paper*Composite Sorting* by Job Boerma, Aleh Tsyvinski, Ruodo Wang,
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and Zhenyuan Zhang.
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+++ {"user_expressions": []}
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@@ -487,11 +487,7 @@ Suppose that agents $i$ of type $z_i$ and $j$ of type $z_j$, with $z_i < z_j,$ a
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Then there is an equal number of agents from each side in $\{i+1, \dots, j-1\},$ if this set is not empty.
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Indeed, if this were not the case, then some agent $k \in \{i+1,j-1\}$ would be matched with some agent $\ell$ with $\ell \notin \{i,\dots, j\},$ i.e., there would be types
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$$
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z_i < z_k < z_j < z_\ell
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$$
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$$z_i < z_k < z_j < z_\ell$$
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with matches $(z_i,z_j)$ and $(z_k, z_\ell),$ violating the no intersecting pairs property.
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We conclude that we can define a binary relation on $[N]$ such that $i \sim j$ if there is an equal number of agents of each side in $\{i,i+1,\dots, j\}$ (or if this set is empty).
We will now present two key results in the context of OT with concave type costs.
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We refer {cite}`boerma2023composite` and {\cite}`delon2011minimum` for proofs.
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We refer to the original papers XXXX (can cite both Boerma et al (2023) and [Delon, Salomon, Sobolevski (2011)](https://link.springer.com/article/10.1007/s10958-012-0714-6))
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XXXX for proofs.
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Consider the problem faced within a layer, i.e., types from $Y \sqcup X$
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The dual solution $(u,v)$ of $W_D$ can be interpreted as equilibrium utilities of the agents, which include the individual specific amenities and equilibrium shadow costs.
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{cite}`boerma2023composite` propose an efficient method to compute the dual variables from the optimal matching (primal solution) in the case of composite sorting.
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The authors propose an efficient method to compute the dual variables from the optimal matching (primal solution) in the case of composite sorting.
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Let's generate an instance and compute the optimal matching.
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