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Adjusted Winner: Extensions to 3 or More People

Adjusted Winner: Extensions to 3 or More People. Allison M. Pacelli Williams College David Craft Mass General Hospital. Example. Annie: Townhouse, Convertible. 65 points. Ben: Piano, Red Sox Tickets. 55 points.

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Adjusted Winner: Extensions to 3 or More People

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  1. Adjusted Winner:Extensions to 3 or More People Allison M. Pacelli Williams College David Craft Mass General Hospital

  2. Example Annie: Townhouse, Convertible 65 points Ben: Piano, Red Sox Tickets 55 points Equitability Adjustment: transfer part of townhouse 20 + 45x = 55 + 35(1 - x)  x = 7/8 Annie and Ben each receive a total of 59.375 points.

  3. What if there are three or more parties? • An efficient, equitable, envy-free solution may not exist. J.H. Reijnierse and J.A.M. Potters example. Only efficient, equitable solution: Annie gets 1, Ben gets 2, Chris gets 3 Ben envies Annie though, so the allocation is not envy-free.

  4. Maximize Tj = Integer Programming • With integer programming, we can always find the efficient, equitable, and envy-free solution, if it exists. • If not, can find efficient/equitable or equitable/envy-free; still working on efficient/envy-free. • New parameter: d = # items allowed to be split • (if d = n, then linear programming) • n = # items, p = # parties • xij = fraction of item i to person j • vij = value of item i to person j

  5. Maximize Tj = Model

  6. Equitable/Envy-free/Efficient: 1 item split Annie: .32 of item 1, all of item 3 Ben: .64 of item 1 Chris: .04 of item 1, .all of item 2 Each gets ~46.08 points. More Examples

  7. With 1 item split: Efficient/Envy-free Efficient/Envy-free/Equitable: 2 items split Annie: .8 of item 1 (41.6 points) Ben: all of item 3 (43 points) Chris: .2 of item 1, all of item 2 (42 points) Annie: .826 of item 1 Ben: item .174 of item 1, .882 of item 3 Chris: all of item 2, .118 of item 1 Each gets ~42.95 points. More Examples

  8. Efficient, Envy-free. Efficient, Envy-free Equitable.

  9. Questions • Efficient and Envy-free Solution? • Windows around point allocations? • Equitability window? • Three people: sufficient conditions for all three?

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