Fair Division of Indivisible Goods Thomas Kalinowski (Newcastle) Nina Naroditskaya, Toby Walsh (NICTA, UNSW) Lirong Xia (Harvard)
Dec 16, 2015
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
NICTA Copyright 2011 From 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)=ΣsεS score(s)
Borda, lexicographical, quasi-indifferent scores, …
NICTA Copyright 2011 From imagination to impact
Decentralized protocol
Captain1
Captain2
NICTA Copyright 2011 From imagination to impact
Decentralized protocol
Captain1
Captain2
NICTA Copyright 2011 From imagination to impact
Decentralized protocol
Captain1
Captain2
NICTA Copyright 2011 From imagination to impact
Decentralized protocol
Captain1
Captain2
NICTA Copyright 2011 From imagination to impact
Decentralized protocol
Captain1
Captain2
NICTA Copyright 2011 From imagination to impact
Decentralized protocol
Captain1
Captain2
NICTA Copyright 2011 From imagination to impact
Decentralized protocol
Captain1
Captain2
NICTA Copyright 2011 From imagination to impact
Decentralized protocol
• But Captain1 has some advantage– We generalize this
to any picking order– Alternating policy:
12121212..– Reverse policy:
12211221..
NICTA Copyright 2011 From imagination to impact
“Optimal” policy
• Utilitarian standpoint– Expected sum of utilities– Individual utility: Borda score, lex score …
NICTA Copyright 2011 From 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 2011 From 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 2011 From imagination to impact
“Optimal” policy
• Egalitarian standpoint– Protocol A: toss coin, if heads all item to
agent1 otherwise all items to agent2
NICTA Copyright 2011 From 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 2011 From 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 2011 From 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 2011 From 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 2011 From 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 2011 From 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 2011 From imagination to impact
“Optimal” policy
• 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,..
NICTA Copyright 2011 From 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 2011 From 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 2011 From 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 2011 From 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 2011 From 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 2011 From 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 2011 From imagination to impact
“Optimal” policy
• 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
NICTA Copyright 2011 From imagination to impact
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
NICTA Copyright 2011 From imagination to impact
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
NICTA Copyright 2011 From 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 2011 From imagination to impact
Questions?
PS I’m hiring!
Two postdoc positions @ Sydney
3 years (in 1st instance)