Computationally-Efficient Approximation Mechanisms
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Transcript of Computationally-Efficient Approximation Mechanisms
Computationally-Efficient Approximation Mechanisms
Monotonicity and Implementability
Computationally-Efficient Approximation Mechanisms
Algorithms in Computer Science, and Mechanisms in Game Theory, are remarkably similar objects.
But the resulting two sets of properties are completely different.
We would like to merge them – to simultaneously exhibit “good” game theoretic properties as well as “good” computational properties.
Outline:
Computationally-Efficient Approximation Mechanisms
• A social choice setting – reminder• Two monotonicity conditions• Cyclic monotonicity
• representation graph of a social choice function
• Weak monotonicity• Weak monotonicity in Order-based domain
• Single-Dimensional Domains and Job Scheduling• Scheduling related machines• single-dimensional linear domains
• Summary
Reminder: A social choice setting
Computationally-Efficient Approximation Mechanisms
A finite set .Each player has a type (valuation function)
Goal: find dominant strategy:
social choice function: Requirement: price function: s.t:
¿
Outline:
Computationally-Efficient Approximation Mechanisms
• A social choice setting – reminder• Two monotonicity conditions• Cyclic monotonicity
• representation graph of a social choice function
• Weak monotonicity• Weak monotonicity in Order-based domain
• Single-Dimensional Domains and Job Scheduling• Scheduling related machines• single-dimensional linear domains
• Summary
Two monotonicity conditions
Computationally-Efficient Approximation Mechanisms
• Fix a player and
• Assume w.l.o.g is onto
• Dominant strategy:
Prices in are now constants:
Cyclic monotonicity
Two monotonicity conditions
Computationally-Efficient Approximation Mechanisms
• Need to find s.t
Definition:
Motivation: If we’ll show that then
Cyclic monotonicity
Computationally-Efficient Approximation Mechanisms
Definition: Representation graph
The representation graph of a social choice function is a directed weighted graph where and . The weight of an edge (for ) is
This can easily solved by looking at the representation graph
Two monotonicity conditionsCyclic monotonicity∀𝑎 ,𝑏∈ 𝐴 𝛿𝑎 ,𝑏≥𝑝𝑎−𝑝𝑏
Computationally-Efficient Approximation Mechanisms
Representation graph example:Single Player
Lets build the representation graph:
1 20 23 1
Two monotonicity conditionsCyclic monotonicity
Computationally-Efficient Approximation Mechanisms
Representation graph example:
and .
𝑎𝑏
Calculating:
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-1
Two monotonicity conditionsCyclic monotonicity
Computationally-Efficient Approximation Mechanisms
Representation graph example:
and .
𝑎𝑏
Calculating:
-1
2
1 20 23 1
Two monotonicity conditionsCyclic monotonicity
Computationally-Efficient Approximation Mechanisms
Proposition:There exists a feasible assignment to
the representation graph has no negative-length cycles.
Assignment: set to the length of the shortest path from to some arbitrary fixed node .
Two monotonicity conditionsCyclic monotonicity
Computationally-Efficient Approximation Mechanisms
no negative-length cycles.
Suppose is a negative cycle, i.e.
and
Proof:
Two monotonicity conditionsCyclic monotonicity
𝑎1
𝑎2𝑎3
𝑎4𝑎k −1
Computationally-Efficient Approximation Mechanisms
no negative-length cycles.
Suppose every cycle is non-negative. Fix arbitrary and set = length of the shortest path from to (well defined).
The shortest path from to (= ) is no longer than + the shortest path from to (= ).i.e.
Proof cont:
𝑎𝑏
𝑎∗𝑝𝑎
𝑝𝑏𝑤𝑎 ,𝑏
Two monotonicity conditionsCyclic monotonicity
Computationally-Efficient Approximation Mechanisms
Definition: Cyclic monotonicityA social choice function satisfies cyclic monotonicity if for every player , some integer and Where for and
Proposition: satisfies Cyclic monotonicity the representation graph of has no negative cycles
Two monotonicity conditionsCyclic monotonicity
Computationally-Efficient Approximation Mechanisms
• satisfies cyclic monotonicity
Proof:
definition no negative cycles
Two monotonicity conditionsCyclic monotonicity
Computationally-Efficient Approximation Mechanisms
• satisfies cyclic monotonicity Suppose is a negative cycle, i.e. and <0Define to be the that gives the inf value for .Therefore, () - () is a negative cycle, hence:<0 Therefore, violates cyclic monotonicity .
Proof cont:
Two monotonicity conditionsCyclic monotonicity
Computationally-Efficient Approximation Mechanisms
Corollary:A social choice function is dominant-strategy implementable it satisfies cyclic monotonicity
Going back to our example:
Two monotonicity conditionsCyclic monotonicity
Computationally-Efficient Approximation Mechanisms
Example:
Should check:
𝑎𝑏
Set .The shortest path from to The shortest path from to
-1
2
1 20 23 1
1−2 0−22−0 2−03−1 0−2
𝑣 𝑖(𝑎)−𝑣 𝑖(𝑏)≥𝑝𝑎−𝑝𝑏
Two monotonicity conditionsCyclic monotonicity
Outline:
Computationally-Efficient Approximation Mechanisms
• A social choice setting – reminder• Two monotonicity conditions• Cyclic monotonicity
• representation graph of a social choice function
• Weak monotonicity• Weak monotonicity in Order-based domain
• Single-Dimensional Domains and Job Scheduling• Scheduling related machines• single-dimensional linear domains
• Summary
Computationally-Efficient Approximation Mechanisms
Definition: Weak monotonicity (W-MON)A social choice function satisfies W-MON if for every player , and , and with
Cyclic monotonicity: We found condition on involves only the properties of , without existential price qualifiers.Only:
It is quite complex. k could be large, and a “shorter” condition would have been nicer.
Two monotonicity conditionsWeak monotonicity
Computationally-Efficient Approximation Mechanisms
Definition: Weak monotonicity (W-MON)A social choice function satisfies W-MON if for every player , and , and with
If the outcome changes from to when changes hertype from to , then ’s value for has increased at least as ’s value for in the transition to .
Note: W-MON is a special case of Cyclic monotonicity when
Two monotonicity conditionsWeak monotonicity
Computationally-Efficient Approximation Mechanisms
W-MON is necessary for truthfulness. When is it also sufficient?
Theorem:If the domain is convex, then any social choice function that satisfies W-MON is dominant-strategy implementable.
We will prove it for special case: “base-order” domains.
Fix player , some .W.l.o.g: : (otherwise we remove from for player )
Two monotonicity conditionsWeak monotonicity
Computationally-Efficient Approximation Mechanisms
Definition: Order-based domain A domain is “order-based” if there exists a partial order over the set s.t : with .
Example: = {chocolate, banana, apple}≻́𝒊 ≻́𝒊c b a
∉𝑉 𝑖
∈𝑉 𝑖
Two monotonicity conditionsWeak monotonicity
Computationally-Efficient Approximation Mechanisms
Theorem:If the domain is ordered-based then any social choice function that satisfies W-MON is dominant-strategy implementable.
Open problem: Exactly characterize the domains for which W-MON is sufficient for implementability.
Two monotonicity conditionsWeak monotonicity
Outline:
Computationally-Efficient Approximation Mechanisms
• A social choice setting – reminder• Two monotonicity conditions• Cyclic monotonicity
• representation graph of a social choice function
• Weak monotonicity• Weak monotonicity in Order-based domain
• Single-Dimensional Domains and Job Scheduling• Scheduling related machines• single-dimensional linear domains
• Summary
Computationally-Efficient Approximation Mechanisms
23−8176
𝑥𝑛+𝑦𝑛=𝑧𝑛 ,𝑛>2
2458
n jobs m machines1+100:00:01
00:00:07 00:31:08
00:00:4299:99:99
99:99:9920:99:98
1066 MHz3060 MHz
3000 MHz
2 Hz
Single-Dimensional Domains and Job SchedulingScheduling related machines
Computationally-Efficient Approximation Mechanisms
jobs are to be assigned to machines, where job consumes time-units, and machine has speed .
Thus machine requires time-units to complete job.Let be the load on machine .
Goal: minimize (the makespan).
Single-Dimensional Domains and Job SchedulingScheduling related machines
Computationally-Efficient Approximation Mechanisms
1066 MHz3060 MHz
3000 MHz
2 Hz
Each machine is selfish entity.
Utility of a machine with a load and a payment :
Single-Dimensional Domains and Job SchedulingScheduling related machines
Computationally-Efficient Approximation Mechanisms
Disclosing of player gives us the entire valuation vector.
Machine scheduling is single-dimensional linear domain:For each , , is the load of machine according to
Definition: single-dimensional linear domainsA domain of player is a single-dimensional linear domain if: (loads) s.t (cost) s.t
Single-Dimensional Domains and Job Schedulingsingle-dimensional linear domains
Computationally-Efficient Approximation Mechanisms
Goal: design a computationally-efficient approximation algorithm, that is also implementable.
Can we use VCG?
No: we have min-max and not minimize of sum of costs.
We have convex domain we need a W-MON
algorithm
Single-Dimensional Domains and Job Schedulingsingle-dimensional linear domains
Computationally-Efficient Approximation Mechanisms
Definition: Weak monotonicity (W-MON)A social choice function satisfies W-MON if for every player , and , and with
Assume
W-MON:
Remember:
𝑐 𝑐 ′
Single-Dimensional Domains and Job Schedulingsingle-dimensional linear domains
Computationally-Efficient Approximation Mechanisms
Theorem:If the domain is ordered-based then any social choice function that satisfies W-MON is dominant-strategy implementable.
Remember:
We got W-MON
Such an algorithm is implementable its load functions are monotone non-
increasing.
Single-Dimensional Domains and Job Schedulingsingle-dimensional linear domains
Computationally-Efficient Approximation Mechanisms
Theorem:An algorithm for a single-dimensional linear domain is implementable load functions are non-increasing. Furthermore,if this is the case then charging from every player a price
From here one can show:
Finaly one can show A monotone algorithm for the job scheduling problem.
Single-Dimensional Domains and Job Schedulingsingle-dimensional linear domains
Summary:
Computationally-Efficient Approximation Mechanisms
• A social choice setting – reminder• Two monotonicity conditions• Cyclic monotonicity
• representation graph of a social choice function
• Weak monotonicity• Weak monotonicity in Order-based domain
• Single-Dimensional Domains and Job Scheduling• Scheduling related machines• single-dimensional linear domains
• Summary