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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• 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
• 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
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