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CMU 15-896 Fair division 4: Indivisible goods Teacher: Ariel Procaccia Indivisible goods Set of goods Each good is indivisible Players have arbitrary valuations for bundles of goods Envy-freeness and proportionality are

  1. CMU 15-896 Fair division 4: Indivisible goods Teacher: Ariel Procaccia

  2. Indivisible goods • Set of goods • Each good is indivisible • Players have arbitrary valuations � for bundles of goods • Envy-freeness and proportionality are infeasible! 15896 Spring 2016: Lecture 9 2

  3. Minimizing envy • Given allocation , denote �� � � � � �� • Theorem [Nisan and Segal 2002]: Every protocol that finds an allocation minimizing must use an exponential number of bits of communication in the worst case 15896 Spring 2016: Lecture 9 3

  4. Communication complexity • Protocol defined by a 1 binary tree 0 1 • Complexity is the 1 2 height of the tree 0 1 0 1 1 2 1 • Complexity of a 1 0 1 0 0 1 problem is the height of the shortest tree 15896 Spring 2016: Lecture 9 4

  5. Proof of theorem • Let is a set of functions s.t. for all • , � � • � 15896 Spring 2016: Lecture 9 5

  6. Proof of theorem • Suppose , and denote a valuation � profile by • Lemma: Suppose , then the sequence of bits transmitted on input is different from the sequence transmitted on • Assume the lemma is true, then there must be at least sequences, and the height of � the tree must be at least � 15896 Spring 2016: Lecture 9 6

  7. Proof of lemma • Assume not; then and generate the same sequence 1 1 0 1 0 1 ��, �� ��, �� 1 1 2 2 0 1 0 1 0 1 0 1 1 1 2 1 2 1 0 1 0 1 0 1 0 1 0 1 0 1 15896 Spring 2016: Lecture 9 7

  8. 1 1 0 1 0 1 ��, �� ��, �� 1 1 2 2 0 1 0 1 0 1 0 1 1 1 2 1 2 1 1 0 1 1 0 1 0 0 1 0 0 1 1 0 1 ��, �� 1 2 0 1 0 1 1 2 1 0 1 0 1 0 1 15896 Spring 2016: Lecture 9 8

  9. Proof of lemma • If , such that , • The allocation is EF for , is EF for • Given , protocol produces an EF , is also returned on , but is • not EF 15896 Spring 2016: Lecture 9 9

  10. Approximate EF • Define the maximum marginal utility � � • Theorem [Lipton et al. 2004]: An allocation with can be found in polynomial time • Note: we are still not assuming anything about the valuation functions! 15896 Spring 2016: Lecture 9 10

  11. Proof of Theorem • Given allocation , we have an edge in its envy graph if envies • Lemma: Given partial allocation with envy graph , can find allocation with acyclic envy graph s.t. 15896 Spring 2016: Lecture 9 11

  12. Proof of lemma � � � � • If has a cycle , shift allocations along to obtain ; � clearly � � � � • #edges in envy graph of decreased: Same edges between � ∖ � o � � � � Edges from � ∖ � to � shifted o Edges from � to � ∖ � can only o decrease Edges inside C decreased � � � � o • Iteratively remove cycles 15896 Spring 2016: Lecture 9 12

  13. Proof of theorem • Maintain envy and acyclic graph • In round 1, allocate good � to arbitrary agent ��� are allocated in acyclic • � • Derive by allocating � to source • �� �� • Use lemma to eliminate cycles 15896 Spring 2016: Lecture 9 13

  14. EF cake cutting, revisited • Want to get -EF cake division � � • Agent makes marks � such ⁄ � � � � that for every � � ��� • If intervals between consecutive marks are indivisible goods then • Now we can apply the theorem � • Need cut queries and eval queries 15896 Spring 2016: Lecture 9 14

  15. An even simpler solution • Relies on additive valuations • Create the “indivisible goods” like before • Agents choose pieces in a round-robin fashion: • Each good chosen by agent is preferred to the next good chosen by agent • This may not account for the first good chosen by , but � 15896 Spring 2016: Lecture 9 15

  16. Maximin share guarantee • Let us focus on indivisible goods and additive valuations • Communication complexity is not an issue • But computational complexity is • Observation: Deciding whether there exists an EF allocation is NP-hard, even for two players with identical additive valuations 15896 Spring 2016: Lecture 9 16

  17. Maximin share guarantee Total: Total: Total: $30 $50 $20 $30 $50 $2 $5 $5 $3 $5 $2 $10 $5 $20 $20 $3 $40 Total: Total: Total: $30 $30 $40 15896 Spring 2016: Lecture 9 17

  18. • Maximin share (MMS) guarantee [Budish, 2011] of player : � � � � ,…,� � � • Theorem [P & Wang, 2014]: there exist additive valuation functions that do not admit an MMS allocation 15896 Spring 2016: Lecture 9 18

  19. Counterexample for 17 17 17 25 12 1 17 25 12 1 17 25 12 1 25 25 12 12 1 1 2 22 3 28 2 22 3 28 2 22 3 28 2 22 3 28 2 22 3 28 11 0 21 23 11 0 21 23 11 0 21 23 11 0 21 23 11 0 21 23 15896 Spring 2016: Lecture 9 19

  20. Counterexample for 1 1 1 1 17 25 12 1 1 1 1 1 2 22 3 28 1 1 1 1 11 0 21 23 3 -1 -1 -1 3 -1 0 0 3 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 0 -1 Player 1 Player 2 Player 3 15896 Spring 2016: Lecture 9 20

  21. • Maximin share (MMS) guarantee [Budish, 2011] of player : � � � � ,…,� � � • Theorem [P & Wang, 2014]: there exist additive valuation functions that do not admit an MMS allocation • Theorem [P & Wang, 2014]: It is always possible to guarantee each player of his MMS guarantee 15896 Spring 2016: Lecture 9 21

  22. 15896 Spring 2016: Lecture 9

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