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

NICTA Copyright 2011From imagination to impact Decentralized protocol Found in school playgrounds around the world … Nominate two captains They take turns in choosing players

NICTA Copyright 2011From imagination to impact 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)=Σ sεS score(s) Borda, lexicographical, quasi-indifferent scores, …

NICTA Copyright 2011From imagination to impact Decentralized protocol Captain1 Captain2

NICTA Copyright 2011From imagination to impact Decentralized protocol Captain1 Captain2

NICTA Copyright 2011From imagination to impact Decentralized protocol Captain1 Captain2

NICTA Copyright 2011From imagination to impact Decentralized protocol Captain1 Captain2

NICTA Copyright 2011From imagination to impact Decentralized protocol Captain1 Captain2

NICTA Copyright 2011From imagination to impact Decentralized protocol Captain1 Captain2

NICTA Copyright 2011From imagination to impact Decentralized protocol Captain1 Captain2

NICTA Copyright 2011From imagination to impact Decentralized protocol But Captain1 has some advantage –We generalize this to any picking order –Alternating policy: –Reverse policy:

NICTA Copyright 2011From imagination to impact “Optimal” policy Utilitarian standpoint –Expected sum of utilities –Individual utility: Borda score, lex score …

NICTA Copyright 2011From imagination to impact “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

NICTA Copyright 2011From imagination to impact “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

NICTA Copyright 2011From imagination to impact “Optimal” policy Egalitarian standpoint –Protocol A: toss coin, if heads all item to agent1 otherwise all items to agent2

NICTA Copyright 2011From imagination to impact “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

NICTA Copyright 2011From imagination to impact “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?

NICTA Copyright 2011From imagination to impact “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

NICTA Copyright 2011From imagination to impact “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

NICTA Copyright 2011From imagination to impact “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

NICTA Copyright 2011From imagination to impact “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

NICTA Copyright 2011From imagination to impact “Optimal” policy Egalitarian standpoint, Borda scores MinExpUtilExpMinUtilMinUtil , , , ,..

NICTA Copyright 2011From imagination to impact 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

NICTA Copyright 2011From imagination to impact 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

NICTA Copyright 2011From imagination to impact 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

NICTA Copyright 2011From imagination to impact 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!

NICTA Copyright 2011From imagination to impact 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))

NICTA Copyright 2011From imagination to impact 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

NICTA Copyright 2011From imagination to impact “Optimal” policy Supposing agents are strategic, lex scores ExpSumUtilExpMinUtilMinUtil ,

NICTA Copyright 2011From imagination to impact 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

NICTA Copyright 2011From imagination to impact Disposal of items Borda scores ExpSumUtilExpMinUtil , , 1-121, 1-222, , , , , , , , , ,

NICTA Copyright 2011From imagination to impact 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?

NICTA Copyright 2011From imagination to impact Questions? PS I’m hiring! Two postdoc Sydney 3 years (in 1 st instance)