Online Cake Cutting Toby Walsh NICTA and UNSW Sydney, 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-choose First player marks cake into k slices k is #unallocated players Next player chooses slice for first player to have Repeat Has 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(n 2 ) – Hence only competitive if n bounded! Auckland, Feb 19 th 2010

Experimental analysis Auckland, Feb 19 th 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...