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...