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Update mathematical expressions in match_transport.md#317

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mmcky merged 1 commit intoQuantEcon:mainfrom
enzodipasquale:enzodipasquale-patch-1
Apr 2, 2026
Merged

Update mathematical expressions in match_transport.md#317
mmcky merged 1 commit intoQuantEcon:mainfrom
enzodipasquale:enzodipasquale-patch-1

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netlify Bot commented Apr 1, 2026

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mmcky commented Apr 1, 2026

I've reviewed the mathematical fix in this PR and confirm it is correct.

The original proof had two bugs:

  1. Wrong alpha formula: The original alpha = (|x-y| + |x'-y|) / (|x-y'| - |x'-y|) is not generally in (0,1). A quick check with x=0, x'=2, y'=5, y=6 gives alpha = 10, and the denominator can even be zero.

  2. Reversed convex decomposition: The original wrote |x-y| and |x'-y'| as combinations of |x-y'| and |x'-y|, but the proof needs the opposite direction -- expressing the crossed distances as combinations of the uncrossed ones -- so that strict concavity of h yields the desired inequality.

The fix is sound:

  • The new alpha = (|x-y'| - |x'-y'|) / (|x-y| - |x'-y'|) simplifies to (x'-x) / ((y-y') + (x'-x)), which is always in (0,1).
  • The convex combination identities hold and are verified algebraically.
  • Applying strict concavity of h and adding the two resulting inequalities correctly yields h(|x-y'|) + h(|x'-y|) > h(|x-y|) + h(|x'-y'|), proving uncrossing strictly reduces cost.

The presentation is also improved by breaking the final inequality chain into clear, individually justified steps.

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mmcky commented Apr 2, 2026

thanks @enzodipasquale for this update.

It looks great. We really appreciate the review and feedback.

Thanks @thomassargent30 and @HumphreyYang for approvals and feedback.

@mmcky mmcky merged commit 0f8291c into QuantEcon:main Apr 2, 2026
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