Computationally-Efficient Approximation Mechanisms

Computationally-Efficient Approximation Mechanisms

Monotonicity and Implementability


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:


• 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


A finite set .Each player has a type (valuation function)

Goal: find dominant strategy:

social choice function: Requirement: price function: s.t:

¿

Outline:






• Summary

Two monotonicity conditions


• Fix a player and

• Assume w.l.o.g is onto

• Dominant strategy:

Prices in are now constants:

Cyclic monotonicity

Two monotonicity conditions


• Need to find s.t

Definition:

Motivation: If we’ll show that then

Cyclic monotonicity


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


Representation graph example:Single Player

Lets build the representation graph:

1 20 23 1

Two monotonicity conditionsCyclic monotonicity


Representation graph example:

and .

𝑎𝑏

Calculating:

1 20 23 1

-1



Representation graph example:

and .

𝑎𝑏

Calculating:

-1

2

1 20 23 1



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 .



no negative-length cycles.

Suppose is a negative cycle, i.e.

and

Proof:


𝑎1

𝑎2𝑎3

𝑎4𝑎k −1


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



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



• satisfies cyclic monotonicity

Proof:

definition no negative cycles



• 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:



Corollary:A social choice function is dominant-strategy implementable it satisfies cyclic monotonicity

Going back to our example:



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


Outline:






• Summary


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



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



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 )



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



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.


Outline:






• Summary


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


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).



1066 MHz3060 MHz

3000 MHz

2 Hz

Each machine is selfish entity.

Utility of a machine with a load and a payment :



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


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




Assume

W-MON:

Remember:

𝑐 𝑐 ′



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.



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.


Summary:






• Summary

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Computationally-Efficient Approximation Mechanisms