Prophet Secretary for Combinatorial Auctions - PowerPoint Presentation

Slide1

Prophet Secretaryfor Combinatorial Auctionsand Matroids

Soheil EhsaniJanuary 2018

Joint work with M.

Hajiaghayi

, T.

Kesselheim

, S.

SinglaSlide2

The problem consists of an initial setting and a

sequence of events.We have to take particular actions

during time.The goal is maximize expected reward or minimize expected cost.

Two important fundamental problems are

Secretary problem and Prophet Inequality.

Online Decision MakingSlide3

Secretary Problem

There is an

unknown

set of numbers

and an

unknown

random permutation

.In an online fashion and at each step we observe the value of .The goal is to select one value , such that is maximized.First introduced by Dynkin in 1960’s and first studied in mechanism design by Hajiaghayi, Kleinberg and Parkes, EC’04 (introduced the value version v.s. the classic rank version and obtained the first bound for the general multi-choice version).

 Slide4

Prophet Inequalities

Given

known

probability

distributions

In an online fashion and at each step

,

we observe a value .The goal is to select one value , upon its arrival, such that is maximized.First introduced by Krengel and Sucheston in 1970’s and first studied in computer science by Hajiaghayi, Kleinberg and Sandholm, AAAI’07 (introduced the first bound for the general multi-choice version). Slide5

Compute a threshold (virtual price)

.

Select the first item

where

.

The above method

-approximation (guarantees

).And is tight! A Simple Tight Algorithm! with probability   otherwise  

with probability

Worst Order

Best Order

 Slide6

Prophet Secretary

Similar to prophet inequality, but the boxes arrive in a random permutation.

Given known

distributions

and

unknown

random permutation

.At each time , we observe and randomly draw .The goal is to make an irrevocable and immediate selection such that is maximized.Can we achieve an approximation factor better than for prophet secretary?There is a approximation algorithm using different thresholds [EHLM ESA’15].There is a -approximation algorithm for i.i.d.

and large market instances [AEEHKL STOC’17].

 Slide7

Our Contribution

Prophet Inequality has been generalized to many settings:

½-approximation for matroids [KW STOC’12].½-approximation for combinatorial auctions

[FGL SODA’15].

-approximation for best-order

matroids [YAN SODA’11

].

Is there a better than ½-approximation for prophet inequality in random order? Combinatorial Auctions: we show a approximation algorithm (incentive-compatible for unit demand).Matroids: we show a approximation algorithm.Single Items: we show a single threshold approximation algorithm (tight). YESSlide8

Generalization: Combinatorial Auctions

Combinatorial Auctions:There is a set

of items and of buyers, each with a valuation function

.

We

want

allocate items to buyers to

maximize social welfare, i.e. Valuation functions are XOS, i.e. for some additive functions .Prophet Inequality Variant:We know distributions for buyers’ valuations.Buyers arrive one by one and we have to allocate a bundle to them immediately. Slide9

Generalization: Matroids

Matroids:Set of elements

such that

where

there exists such that .E.g. graphic matroids, transversal matroids.Prophet Inequality:Given a set of elements, their probability distribution on their values, and a matroid .At each step we draw and should decide weather to collect it (add to set ).The goal is to collect an independent set A of matroid with maximum value, i.e. . Slide10

Our Approach

First we convert the problem into a continuous-time setting.

Assume each buyer arrives at time

(independently at random).

Then we focus on designing dynamic-pricing algorithms: if we serve buyer

with assignment

Her gain is

Her virtual payment is This approach allows us to split the objective function into revenue and utility:

Achieving

for

Prophet Secretary

time-dependent

discount function

time-independent

base priceSlide11

Lower bounding Social Welfare

The goal is to maximize social welfare, i.e.

We define a

residual function

to capture the expected remaining value at time

.

Residual function is defined for a given instance of prophet secretary and a dynamic-pricing algorithm.Definition: a residual function has following properties:

 Slide12

Proof Template

Let

be a prophet secretary instance

Propose a dynamic pricing algorithm

based on

and

 Show there exists a residual function for , , and any  Using a good discount function prove is approximation Slide13

Tuning the Discount Function

Proof:

Lemma: If an algorithm has a residual function then s

etting

results in a

-approximation algorithm.

Second property of residual functions says: .From integration by parts we know: .Therefore,

Now by adding Revenue and Utility we get:

 Slide14

Algorithm: Combinatorial Auctions

Preprocess:Set a base price for each item based on its contribution to the optimum,

.

Draw

numbers

independently

and uniformly at

random from .Assignments:Buyer selects each remaining item if . Lemma: The following function is a residual for the algorithm: . Slide15

Algorithm: Single Item

Algorithm:Preprocess the base price

.

Draw

values

for arrival times.

The item is assigned to the first buyer

such that .The existence of a residual function in conjunction with the above tuning of the discount function results in a approximation algorithm.We need to show is a residual function.

 Slide16

Analysis: Single Items

First property

:

Second property

: note that

is

probability

of selling item at time (for price ).Third property:

For the conditional utility of buyer

given arrival time

we have

Since

we have

 Slide17

Conclusion

Present

approximation algorithms for most important generalizations of single item.Can we beat this barrier? [ACK arXiv 2017] improve by

for single item.

What about the i.i.d. case? [AEEHKL STOC’17] present 0.74 approximation algorithm for single item.

Mohamad

Hajiaghayi (UMD)Sahil Singla (CMU)

Thomas

Kesselheim

(TUD)

Thanks to my co-authors!