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Class 15: Stable Matching cs2102: Discrete Mathematics | F16 uvacs2102.github.io David Evans | University of Virginia Storage fees on Exam 1 double after today! Can pick up after class today No class Thursday

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Plan Stable Matching Gale-Shapley Algorithm Proving Correctness Secure Stable Matching No class Thursday PS6 is due Friday

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Stable Matching

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Formal Problem Definition Input: Output:

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Formal Problem Definition Input: Output: Stable matching between and , , ≺% , ≺& A matching, = { , , , , . , . , … , 0 , 0 } is a stable matching if there is no pair (3 , 4 ) where 3 ≺67 4 and 4 ≺89 3 .

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Stable Matching Anna Elsa Hans Kristoff

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Stable Matching Anna Elsa Hans Kristoff Kristoff ≺:;;?

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Stable Matches? Anna Elsa Hans Kristoff Kristoff ≺:;;?

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Stable Matches? Anna Elsa Olaf Kristoff Olfa ≺:;;?

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Gale- Shapley Algorithm Lloyd Shapley (1923-2016) accepting Nobel Prize (2012) (co-awarded to Alvin Roth)

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David Gale (1921 – 2008)

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Olfa ≺:;;< Kristoff Kristoff ≺=>?< Olaf Anna ≺G>

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Simplified Code (Non-Defensive) Do not write programs like this: assertions (especially) and comments are valuable (but don’t fit on slides)

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Modeling the Program How should we model the program as a state machine?

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= { , | = { , ∈ , ∈ , no duplicate occurences of or = , , . , … , 0 where 0 ≤ 3 ≤ }

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= { , | = { , ∈ , ∈ , no duplicate occurences of or = , , . , … , 0 where 0 ≤ 3 ≤ } = , ⟶ g, g All cases:

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= { , | = { , ∈ , ∈ , no duplicate occurences of or = , , . , … , 0 where 0 ≤ 3 ≤ } = , ⟶ g, g All cases: g = , , . , … , 3 + 1, … , 0 j is the 3 -ranked match for 3 Case 1:

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= { , | = { , ∈ , ∈ , no duplicate occurences of or = , , . , … , 0 where 0 ≤ 3 ≤ } = , ⟶ g, g All cases: g = , , . , … , 3 + 1, … , 0 j is the 3 -ranked match for 3 Case 1: new pairing ∀l ∈ . l , j ∉ g = ∪ 3 , j Case 2: better pairing ∃l ∈ . l , j ∈ , j prefers 3 to j g = ∪ 3 , j Case 3: rejected proposal ∃l ∈ . l , j ∈ , j prefers j to 3 g =

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Proving Termination For all valid inputs, gale_shapley eventually terminates (returns a pairing). Each step, one of the proposals counts increases by 1 (and all others stay the same). After .steps, must be , , … , so no further transitions possible.

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Proving Correctness For all valid inputs, gale_shapley returns a stable matching of its inputs. A matching, = { , , , … , 0 , 0 } is a stable matching if there is no pair (3 , 4 ) where 3 ≺67 4 and 4 ≺89 3 .

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For all valid inputs, gale_shapley returns a stable matching of its inputs. A matching, = { , , , … , 0 , 0 } is a stable matching if there is no pair (3 , 4 ) where 3 ≺67 4 and 4 ≺89 3 . = , ⟶ g, g All cases: g = , , . , …, 3 + 1, …, 0 jis the 3-ranked match for 3 Case 1: new pairing ∀l ∈ . l , j ∉ g = ∪ 3 , j Case 2: better pairing ∃l ∈ . l , j ∈ , j prefers 3 to j g = ∪ 3 , j Case 3: rejected proposal ∃l ∈ . l , j ∈ , j prefers j to 3 g = Left as problem for PS7

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Charge No class Thursday Problem Set 6 due Friday