Online Cake Cutting

1. Toby Walsh NICTA and UNSW Sydney, Australia Online Cake Cutting

2. Algorithmic Decision Theory • Apply algorithmic ideas to decision theory • e.g. apply online algorithms to fair division

3. 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

4. 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

5. 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

6. 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

7. Axiomatic properties • Offline properties • Proportionality • Envy freeness • Equitability • Efficiency • Strategy proofness

8. Axiomatic properties • Online properties • Proportionality • Envy freeness • Equitability • Efficiency • Strategy proofness • Order monotonicity • ...

9. Proportionality • Offline • Each player assigns at least 1/k total to their piece

10. 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

11. Envy freeness • Offline • No player envies the cake allocated to another • Implies proportionality

12. 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

13. Equitability • Offline • All players assign the same value to their cake • For 3 or more players, equitability and envy freeness can be incompatible

14. 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

15. 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

16. 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

17. 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

18. 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

19. (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

20. 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

21. 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

22. 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

23. Properties • Thm: all these procedures are forward proportional, immediately envy free, and weakly truthful

24. 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.

25. 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

26. 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

27. Experimental analysis Auckland, Feb 19th 2010

28. 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

29. 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 ...