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Fair Division of Indivisible Goods Thomas Kalinowski (Newcastle)

Fair Division of Indivisible Goods Thomas Kalinowski (Newcastle) Nina Naroditskaya, Toby Walsh (NICTA, UNSW) Lirong Xia (Harvard). Decentralized protocol. Found in school playgrounds around the world … Nominate two captains They take turns in choosing players. Decentralized protocol.

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Fair Division of Indivisible Goods Thomas Kalinowski (Newcastle)

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  1. Fair Division of Indivisible Goods Thomas Kalinowski (Newcastle) Nina Naroditskaya, Toby Walsh (NICTA, UNSW) Lirong Xia (Harvard)

  2. Decentralized protocol • Found in school playgrounds around the world … • Nominate two captains • They take turns in choosing players

  3. Decentralized protocol • Studied in [Bouveret, Lang IJCAI 2011] • Avoids elicitation of preferences • Used to assign courses to students at Harvard Business School • Simple model with additive utilities • Utility(S)=ΣsεS score(s) • Borda, lexicographical, quasi-indifferent scores, …

  4. Decentralized protocol • Captain1 • Captain2

  5. Decentralized protocol • Captain1 • Captain2

  6. Decentralized protocol • Captain1 • Captain2

  7. Decentralized protocol • Captain1 • Captain2

  8. Decentralized protocol • Captain1 • Captain2

  9. Decentralized protocol • Captain1 • Captain2

  10. Decentralized protocol • Captain1 • Captain2

  11. Decentralized protocol • But Captain1 has some advantage • We generalize this to any picking order • Alternating policy: 12121212.. • Reverse policy: 12211221..

  12. “Optimal” policy • Utilitarian standpoint • Expected sum of utilities • Individual utility: Borda score, lex score …

  13. “Optimal” policy • Utilitarian standpoint • Expected sum of utilities • Individual utility: Borda score, lex score … • Assume all preference profiles equally likely • [Bouveret & Lang IJCAI 2011] conjecture that alternating policy 1212… is optimal for Borda scoring • Based on computer simulation with 12 or fewer items

  14. “Optimal” policy • Egalitarian standpoint • [Bouveret & Lang IJCAI 2011] somewhat strangely look at minimum of expected utilities of different agents • More conventional to look at expected minimum utility, or minimum utility

  15. “Optimal” policy • Egalitarian standpoint • Protocol A: toss coin, if heads all item to agent1 otherwise all items to agent2

  16. “Optimal” policy • Egalitarian standpoint • Protocol A: toss coin, if heads all item to agent1 otherwise all items to agent2 • Protocol B: toss coin, if heads then next item to agent1 otherwise next item to agent2

  17. “Optimal” policy • Egalitarian standpoint • Protocol A: toss coin, if heads all item to agent1 otherwise all items to agent2 • Protocol B: toss coin, if heads then next item to agent1 otherwise next item to agent2 • Arguably B more egalitarian than A as each agent gets ½ items on average?

  18. “Optimal” policy • Egalitarian standpoint • Protocol A: toss coin, if heads all item to agent1 otherwise all items to agent2 • Protocol B: toss coin, if heads then next item to agent1 otherwise next item to agent2 • MinExpUtil(A) = MinExpUtil(B) • But ExpMinUtil(A)=0, ExpMinUtil(B)=max/2 • And MinUtil(A)=0, MinUtil(B)=0

  19. “Optimal” policy • Egalitarian standpoint • Protocol A: toss coin, if heads all item to agent1 otherwise all items to agent2 • Protocol B: toss coin, if heads then next item to agent1 otherwise next item to agent2 • MinExpUtil(A) = MinExpUtil(B) • But ExpMinUtil(A)=0, ExpMinUtil(B)=max/2 • And MinUtil(A)=0, MinUtil(B)=0

  20. “Optimal” policy • Egalitarian standpoint • Protocol A: toss coin, if heads all item to agent1 otherwise all items to agent2 • Protocol B: toss coin, if heads then next item to agent1 otherwise next item to agent2 • MinExpUtil(A) = MinExpUtil(B) • But ExpMinUtil(A)=0, ExpMinUtil(B)=max/2 • And MinUtil(A)=0, MinUtil(B)=0

  21. “Optimal” policy • Egalitarian standpoint • [Bouveret & Lang IJCAI 2011] somewhat strangely look at minimum of expected utilities of different agents • We considered expected minimum utility, and minimum utility • Computed optimal policies by simulation

  22. “Optimal” policy • Egalitarian standpoint, Borda scores

  23. Other properties • This mechanism is Pareto efficient • We can't swap players between teams and have both captains remain happy • Supposing captains picked teams truthfully • This mechanism is not envy free • One agent might prefer items allocated to other agent

  24. Strategic play • This mechanism is not strategy proof • Captain1 can get a better team by picking players out of order • No need for Captian1 to pick early on a player that he likes but Captain2 dislikes • And vice versa

  25. Strategic play • What is equilibrium behaviour? • Nash equilibrium: no captain can do better by deviating from this strategy • Subgame perfect Nash equilibrium: at each move of this repeated game, play Nash equilibrium

  26. Strategic play • With 2 agents • There is unique subgame perfect Nash equilibrium • It can be found in linear time • Even though there is an exponential number of possible partitions to consider!

  27. Strategic play • With 2 agents • There is unique subgame perfect Nash equilibrium • It can be found in linear time SPNE(P1,P2,policy) = allocate(rev(P1),rev(P2), rev(policy))

  28. Strategic play • With k agents • There can be multiple subgame perfect Nash equilibrium • Deciding if utility of an agent is larger than some threshold T in any SPNE is PSPACE complete

  29. “Optimal” policy • Supposing agents are strategic, lex scores

  30. Disposal of items • Other protocols possible • E.g. captains pick a player for the other team • Addresses an inefficiency of previous protocol • One captain may pick player in early round that the other captain would happily give away

  31. Disposal of items • Borda scores

  32. Conclusions • Many other possible protocols • TwoByTwo: Agent1 picks a pair of items, Agent2 picks the one he prefers, Agent1 gets the other • TakeThat: Agent1 picks an item, Agent2 can accept it (if they are under quota in #items) or lets Agent1 take it • … • Many open questions • How to compute SPNE with disposal of items? • How to deal with non-additive utilities?

  33. Questions? • PS I’m hiring! • Two postdoc positions @ Sydney • 3 years (in 1st instance)

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