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...