Online Cake Cutting
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Transcript of Online Cake Cutting
Online Cake Cutting
Toby WalshNICTA and UNSWSydney, Australia
Algorithmic Decision Theory• Apply algorithmic
ideas to decision theory– e.g. apply online
algorithms to fair division
Outline
• Online cake cutting– Definition of the problem
• Axiomatic properties– Definition of fairness, etc.
• Some example procedures– Online versions of cut-and-choose,
moving knife and mark-and-choose• Conclusions
Cake cutting• Dividing [0,1]
between n players• Each player has a
valuation function– Unknown to
other players• Players are risk
averse– Maximize
minimum value of cake they receive
Online cake cutting• Dividing [0,1]
between n players• Each player has a
valuation function• Players are risk
averse• Some schedule for
arrival & departure of players
Birthday example• Congratulations
– It's your birthday• You bring a cake into
the office– People arrive (and
depart)• You need a
procedure to share the cake
Axiomatic properties
• Offline properties– Proportionality– Envy freeness– Equitability– Efficiency– Strategy proofness
Axiomatic properties
• Online properties– Proportionality– Envy freeness– Equitability– Efficiency– Strategy proofness– Order monotonicity– ...
Proportionality
• Offline– Each player assigns at least 1/k total to
their piece
Proportionality• Offline
– Each player assigns at least 1/k total to their piece
• Online– May be impossible (e.g. suppose you
only like the iced part of the cake)– Forward proportional: each player
assigns at least 1/j of the value that remains where j is #players to be allocated cake
Envy freeness
• Offline– No player envies the cake allocated to
another– Implies proportionality
Envy freeness• Offline
– No player envies the cake allocated to another
• Online– Again may be impossible– Forward envy free: no player envies the
cake allocated to a later arriving player– Immediately envy free: no player envies
the cake allocated to a player after their arrival and before their departure
Equitability
• Offline– All players assign the same value to their
cake– For 3 or more players, equitability and
envy freeness can be incompatible
Equitability• Offline
– All players assign the same value to their cake
– For 3 or more players, equitability and envy freeness can be incompatible
• Online– Little point to consider weaker versions– Either players assign same value or they
don't
Efficiency
• Offline– Pareto optimality: no other allocation
that is more valuable to one player and at least as valuable to others
– weak Pareto optimality: no other allocation that is more valuable for all players
Efficiency• Offline
– Pareto optimality: no other allocation that is more valuable to one player and at least as valuable to others
– weak Pareto optimality: no other allocation that is more valuable for all players
• Online– Again little point to consider weaker
versions
Strategy proofness
• Offline– Weakly truthful: for all valuations a
player will do at least as well by telling the truth
– i.e. a risk averse player will not lie– Truthful: there do not exist valuations
where a player profits by lying– i.e. even a risky player will not lie
Order monotonicity
• Online property– A player's valuation of their allocation
does not decrease when they move earlier in the arrival order
– +ve: players have an incentive to arrive early
– -ve: arriving late can hurt you
(Im)possibility theorems
• Impossibility– No online cake cutting procedure is
proportional, envy free or equitable• Possibility
– There exist online cake cutting procedures which are forward proportional, forward envy free, weakly Pareto optimal, truthful, order monotonic
Online cut-and-choose
• First player to arrive cuts a slice
• Either next player to arrive chooses slice and departs
• Or first player takes slice
• Repeat
Online moving knife
First k players to arrive perform a moving knife procedure
A knife is moved from one end of the cake
Anyone can shout “stop” and take the slice
Repeat
Note: k can change over course of procedure
Online mark-and-chooseFirst player marks cake
into k slicesk is #unallocated
playersNext player chooses
slice for first player to have
RepeatHas advantage that
players depart quickly
Properties
• Thm: all these procedures are forward proportional, immediately envy free, and weakly truthful
Properties
• Thm: all these procedures are forward proportional, immediately envy free, and weakly truthful
• Thm: none of these procedures are proportional, (forward) envy free, equitable, (weakly) Pareto optimal, truthful or order monotonic.
Competitive analysis
• Theoretical tool used to study online algorithms– Ratio between offline performance & online
performance– Performance:
• Egalitarian: smallest value assigned by agent• Utilitarian: sum of values assigned by agents
Competitive analysis
• Egalitarian performance:– Even with 3 agents, competitive ration can be
unbounded• Utilitarian performance:
– Online cut-and-choose and moving knife procedures have competitive ratio that is O(n2)
– Hence only competitive if n bounded!
Auckland, Feb 19th 2010
Experimental analysis
Auckland, Feb 19th 2010
Extensions
• Information about total number of players– e.g. upper bounded, unknown, ...
• Information about arrival order– e.g. players don't know when they are in
the arrivale order• Informations about players' valuation
functions
Conclusions• ADT can profit from considering online
problems• Still much to be done for online fair division
– Indivisible goods– Information about players' valuation
functions– Undesirable goods (e.g. chores) where we
want as little as possible...