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

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(n 2 ) – Hence only competitive if n bounded! Auckland, Feb 19 th 2010

27. Experimental analysis Auckland, Feb 19 th 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...