Fair Division of Indivisible Goods Thomas Kalinowski (Newcastle) Nina Naroditskaya, Toby Walsh...

Fair Division of Indivisible Goods

Thomas Kalinowski (Newcastle)

Nina Naroditskaya, Toby Walsh (NICTA, UNSW)

Lirong Xia (Harvard)

NICTA Copyright 2011 From imagination to impact

Decentralized protocol

• Found in school playgrounds around the world …

• Nominate two captains• They take turns in choosing players



• 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, …



Captain1

Captain2



• But Captain1 has some advantage– We generalize this

to any picking order– Alternating policy:

12121212..– Reverse policy:

12211221..


“Optimal” policy

• Utilitarian standpoint– Expected sum of utilities– Individual utility: Borda score, lex score …



• 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



• 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



• Egalitarian standpoint– Protocol A: toss coin, if heads all item to

agent1 otherwise all items to agent2




agent1 otherwise all items to agent2– Protocol B: toss coin, if heads then next item

to agent1 otherwise next item to agent2





to agent1 otherwise next item to agent2– Arguably B more egalitarian than A as each

agent gets ½ items on average?





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



• 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



• Egalitarian standpoint, Borda scores

MinExpUtil ExpMinUtil MinUtil

12 12 12

122 122 122

1221 1221 1221

11222 12122 12122,..

121221 121221 121221,..

1122122 1212122 1212212,..

12212112 12122121 12212112,..


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


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


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


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!


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))


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



• Supposing agents are strategic, lex scores

ExpSumUtil ExpMinUtil MinUtil

12 12 12

121 122 122

1212, 1221 1221 1222

12122 12122 12222

122112 122121 122222

1212122 1221122 1222222

12211221 12212211 12222222


Disposal of items

• Other protocols possibleE.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


Disposal of items

• Borda scores

ExpSumUtil ExpMinUtil

12 12

121, -121 122

1-121 1221, 1-121, 1-222, -1211

12-212, -12-212 12122, -1-1-212

1-12121, 1-1-2-121 1-2-1-121, -121121

1212-212, 1-2-12-212, .. 12-1-1-212

1-1212121, 1-121-2-121, .. 1-2-1-1-2-121, -12112121


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?


Questions?

PS I’m hiring!

Two postdoc positions @ Sydney

3 years (in 1st instance)

Fair Division of Indivisible Goods Thomas Kalinowski (Newcastle) Nina Naroditskaya, Toby Walsh...

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