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Economics of Age of Information (AoI) Management:Competition and Pricing

Presenter: Lingjie Duan

Engineering Systems and Design PillarSingapore University of Technology and Design (SUTD)

May 4, 2019

Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 1 / 61

Acknowledgement

This is a joint work withShugang Hao (PhD student)

Xuehe Wang (Postdoc fellow)

Parts of results here will appear in ACM MobiHoc 2019 and IEEEISIT 2019 Symposia.

S. Hao and L. Duan, “Economics of Age of Information Management under

Network Externalities,” to appear in the Twentieth International Symposiumon Mobile Ad Hoc Networking and Computing (ACM MobiHoc), 2019.[Online]. Available: https://arxiv.org/pdf/1904.01841.pdf

X. Wang and L. Duan, “Dynamic pricing for controlling age of information,”

to appear in IEEE International Symposium on Information Theory (ISIT),

2019. [Online]. Available: https://arxiv.org/pdf/1904.01185.pdf

Background: Who care about AoI?

Age of Information (AoI): duration from the moment that the latest

content was generated to current reception time.

Today many customers do not want to lose any breaking news or

useful information in smartphone even if in minute.

Online content platforms (such as navigation and shopping

applications) aim to keep their information update fresh.

Background: Crowdsourcing for reducing AoI

Crowdsourcing: To keep high sampling rate, platforms invite and pay

crowd to collect information updates, introducing large sampling cost.

Background: Crowdsourcing for reducing AoI

Crowdsourcing: To keep high sampling rate, platforms invite and pay

crowd to collect information updates, introducing large sampling cost.

Research Questions

Economic Issues on AoI was largely overlooked in the literature.

Information supply: Platform crowdsourcing incur large sampling cost.

Tradeo↵ between AoI reduction & sampling cost hasn’t been studied.

Information delivery: More than one platform selfishly shares the

content delivery network

Updates of platforms may preempt or jam each other.

Negative network externalities between platforms.

Research Questions

content delivery network.

How to best tradeo↵ between AoI reduction and sampling cost?

How bad is platform competition and how to enforce e�cient cooperation

between selfish platforms?

Related Works on AoI

Queueing analysis on average AoI estimation

Single link: Costa et al. (2016), Huang et al. (2015), Kaul et al.

(2012) and Sun et al. (2017).

Multi-source LCFS queue with preemption: Kaul et al. (2012).

Such work do not consider sampling cost or the tradeo↵ between AoI

reduction and sampling cost.

Scheduling broadcast channel among multiple sources for AoI

Hsu et al. (2017). Bedewy et al. (2017).

Such work assumes sources/platforms will follow recommendation,

and do not consider selfish sources’ update competition over the

Related Works on AoI

Queueing analysis on average AoI estimation

Single link: Costa et al. (2016), Huang et al. (2015), Kaul et al.

(2012) and Sun et al. (2017).

Multi-source LCFS queue with preemption: Kaul et al. (2012).

Such work do not consider sampling cost or the tradeo↵ between AoI

reduction and sampling cost.

Scheduling broadcast channel among multiple sources for AoI

Hsu et al. (2017). Bedewy et al. (2017).

Such work assumes sources/platforms will follow recommendation,

and do not consider selfish sources’ update competition over the

Related Work on Platform Competition

Selfish sharing under negative network externalities

Duopoly competition in network sharing: Gibbens et al. (2000),

Roughgarden et al. (2002), Duan et al. (2015).

Mechanism design to mitigate competition:

Direct pricing as penalty: Courcoubetis (2003), Varian (2004), though

di�cult to enforce additional penalty on platforms here.

Repeated games approach with indirect punishment: Treust et al.

(2010), Xiao et al. (2012), Lanctot et al. (2017), which requires

complete information and su�ciently large discount factor to work.

We will investigate how to regulate platform competition under

incomplete information, and the proposed non-monetary approach

works for any discount factor.

Part I: Competition regulation for AoI-driven platforms in long run

Part II: Dynamic pricing to control AoI in short-term

System Model on Platforms

Two platforms (Groupon and Waze) need to decide how many samples to

buy from their own crowdsourcing pools with sampling rates �1

and �2

and then update through the delivery network of bandwidth µ.

System Model on AoI

Consider each platform:

We consider LCFS M/M/1 queue with preemption (Kaul et al.

(2012)).

Preemption happen within and between platform(s).

Status update age = completion time - generation time.

Average AoI for a single platform in long run is

System Model on AoI

(2012)).

System Model on AoI

(2012)).

Average AoI for Duopoly Platforms

Average AoI of platform 1 and platform 2 under network sharing:

+ �2

decreases with sampling rate �1

and bandwidth µ, and increases

with the other platform’s sampling rate �2

Negative network externalities due to competition on µ.

+ �2

Complete Information Scenario

System Model under Complete Information

Model ci as unit cost per sampling rate. Sampling cost is �ici wheninviting sensors of density �i to contribute.

Both platforms have full information on their sampling costs.

Platform 1’s cost function:

) = �

) + c1

implying the tradeo↵ between AoI and sampling cost by deciding �1

) = �

) + c2

,Social cost function:

⇡(�1

) = ⇡1

) + ⇡2

) = �

) + c1

) = �

) + c2

⇡(�1

) = ⇡1

) + ⇡2

) = �

) + c1

) = �

) + c2

⇡(�1

) = ⇡1

) + ⇡2

) = �

) + c1

) = �

) + c2

Social cost function:

⇡(�1

) = ⇡1

) + ⇡2

) = �

) + c1

) = �

) + c2

⇡(�1

) = ⇡1

) + ⇡2

Non-cooperative Static Game under Complete Information

Non-cooperative game with equilibrium (�⇤1

,�⇤2

Min-social-cost problem: centrally follow social optimizers (�⇤⇤1

,�⇤⇤2

⇡(�1

Question: equilibrium (�⇤1

,�⇤2

) versus optimal (�⇤⇤1

,�⇤⇤2

,�⇤2

,�⇤⇤2

⇡(�1

,�⇤2

,�⇤⇤2

,�⇤2

,�⇤⇤2

⇡(�1

,�⇤2

,�⇤⇤2

Competition Equilibrium and Social Optimizers

Proposition 1 (Equilibrium vs Social Optimizers under complete information)

Under complete information, the competition equilibrium (�⇤1

,�⇤2

) are the uniquesolutions to

� 1�2

(1 +�2

µ) + c

� 1�2

(1 +�1

µ) + c

= 0. (1)

Di↵erently, the social optimizers (�⇤⇤1

,�⇤⇤2

), are the unique solutions to

� 1�2

(1 +�2

µ) + c

µ= 0,

� 1�2

(1 +�1

µ) + c

µ= 0. (2)

By comparing (1) and (2), we conclude competition leads over-sampling (�⇤i � �⇤⇤

i fori = 1, 2) at the equilibrium.

Direct pricing approach: charge price

�⇤⇤j µ per sampling rate from platform i to

reach (�⇤⇤1

,�⇤⇤2

).Yet our focus is non-monetary approach!

,�⇤2

� 1�2

(1 +�2

µ) + c

� 1�2

(1 +�1

µ) + c

= 0. (1)

,�⇤⇤2

� 1�2

(1 +�2

µ) + c

µ= 0,

� 1�2

(1 +�1

µ) + c

µ= 0. (2)

reach (�⇤⇤1

,�⇤⇤2

Yet our focus is non-monetary approach!

,�⇤2

� 1�2

(1 +�2

µ) + c

� 1�2

(1 +�1

µ) + c

= 0. (1)

,�⇤⇤2

� 1�2

(1 +�2

µ) + c

µ= 0,

� 1�2

(1 +�1

µ) + c

µ= 0. (2)

reach (�⇤⇤1

,�⇤⇤2

).Yet our focus is non-monetary approach!

Equilibrium under Complete Informatiion

Corollary 1

Equilibrium �⇤1

increases with �⇤2

, and decreases with c1

, µ, respectively.

�⇤1

: competition to preempt and occupy µ.

�⇤1

decreases with µ: less competition given ample bandwidth.

�⇤1

decreases with c1

: avoid high samping cost.

�⇤1

decreases with c2

: �⇤2

decreases with c2

Corollary 1

Equilibrium �⇤1

, µ, respectively.

�⇤1

decreases with c1

�⇤1

decreases with c2

: �⇤2

decreases with c2

Corollary 1

Equilibrium �⇤1

, µ, respectively.

�⇤1

decreases with c1

�⇤1

decreases with c2

: �⇤2

decreases with c2

Corollary 1

Equilibrium �⇤1

, µ, respectively.

�⇤1

decreases with c1

�⇤1

decreases with c2

: �⇤2

decreases with c2

Corollary 1

Equilibrium �⇤1

, µ, respectively.

�⇤1

decreases with c1

�⇤1

decreases with c2

: �⇤2

decreases with c2

E�ciency loss with Competition: versus optimum

Price of Anarchy (PoA):

PoA = max

⇡(�⇤1

,�⇤2

⇡(�⇤⇤1

,�⇤⇤2

� 1.

We use PoA � 1 to tell e�ciency loss due to competition in the worst case.

Proposition 2 (Huge e�ciency loss under complete information)

Price of anarchy under complete information is PoA = 1, which is

achieved when platform 1’s sampling cost c1

is infinitesimal.

When c1

equilibrium �⇤1

! 1, �⇤2

optimal �⇤⇤1

< 1, �⇤⇤2

Need non-monetary mechanism to remedy the huge e�ciency loss!

PoA = max

⇡(�⇤1

,�⇤2

⇡(�⇤⇤1

,�⇤⇤2

� 1.

is infinitesimal.

When c1

equilibrium �⇤1

! 1, �⇤2

optimal �⇤⇤1

< 1, �⇤⇤2

PoA = max

⇡(�⇤1

,�⇤2

⇡(�⇤⇤1

,�⇤⇤2

� 1.

is infinitesimal.

When c1

equilibrium �⇤1

! 1, �⇤2

optimal �⇤⇤1

< 1, �⇤⇤2

PoA = max

⇡(�⇤1

,�⇤2

⇡(�⇤⇤1

,�⇤⇤2

� 1.

is infinitesimal.

When c1

equilibrium �⇤1

! 1, �⇤2

optimal �⇤⇤1

< 1, �⇤⇤2

PoA = max

⇡(�⇤1

,�⇤2

⇡(�⇤⇤1

,�⇤⇤2

� 1.

is infinitesimal.

When c1

equilibrium �⇤1

! 1, �⇤2

optimal �⇤⇤1

< 1, �⇤⇤2

Trigger-and-punishment Mechanism under Complete Information

Our Repeated Game Approach

We consider how to regulate competition in long run in repeated

games with discount factor ⇢ < 1.

Each time slot in the repeated games is long enough for each

platform i ’s AoI statistic to converge to its average value �i .

Definition 1 (Non-forgiving trigger mechanism of punishment undercomplete information)

In each round, recommend cooperation profile (

˜�1

(⇢), ˜�2

(⇢)) tofollow, if neither was found to deviate in the past.

Once a deviation was found in the past, the two platforms will keepplaying the punishment/equilibrium profile (�⇤

,�⇤2

) forever.

How to detect deviation and trigger punishment?

Platform 1 can identify the other platform’s sampling rate �2

from its

own average AoI experience �

) and �1

The key is how to design (

˜�1

(⇢), ˜�2

(⇢)) to never trigger punishment.

˜�1

(⇢), ˜�2

,�⇤2

) forever.

from its

) and �1

˜�1

(⇢), ˜�2

˜�1

(⇢), ˜�2

,�⇤2

) forever.

from its

) and �1

˜�1

(⇢), ˜�2

˜�1

(⇢), ˜�2

,�⇤2

) forever.

from its

) and �1

˜�1

(⇢), ˜�2

˜�1

(⇢), ˜�2

,�⇤2

) forever.

from its

) and �1

˜�1

(⇢), ˜�2

Condition for No Deviation for Both Platforms

Ideally, we want to ensure no deviation of each platform from

˜�1

(⇢), ˜�2

(⇢)) = (�⇤⇤1

,�⇤⇤2

Platform 1’s long-term cost over all time stages without deviation:

= ⇡1

(�⇤⇤1

,�⇤⇤2

) + ⇢⇡1

(�⇤⇤1

,�⇤⇤2

) + ⇢2⇡1

(�⇤⇤1

,�⇤⇤2

) + · · · ,

1� ⇢⇡1

(�⇤⇤1

,�⇤⇤2

Platform 1’s long-term cost over all time stages by deviating in the

first round with best response �1

q1+�⇤⇤

= ⇡1

✓s1 + �⇤⇤

,�⇤⇤2

◆+ ⇢⇡

(�⇤1

,�⇤2

) + ⇢2⇡1

(�⇤1

,�⇤2

) + · · · ,

= ⇡1

✓s1 + �⇤⇤

,�⇤⇤2

1� ⇢⇡1

(�⇤1

,�⇤2

˜�1

(⇢), ˜�2

(⇢)) = (�⇤⇤1

,�⇤⇤2

= ⇡1

(�⇤⇤1

,�⇤⇤2

) + ⇢⇡1

(�⇤⇤1

,�⇤⇤2

) + ⇢2⇡1

(�⇤⇤1

,�⇤⇤2

) + · · · ,

1� ⇢⇡1

(�⇤⇤1

,�⇤⇤2

q1+�⇤⇤

= ⇡1

✓s1 + �⇤⇤

,�⇤⇤2

◆+ ⇢⇡

(�⇤1

,�⇤2

) + ⇢2⇡1

(�⇤1

,�⇤2

) + · · · ,

= ⇡1

✓s1 + �⇤⇤

,�⇤⇤2

1� ⇢⇡1

(�⇤1

,�⇤2

˜�1

(⇢), ˜�2

(⇢)) = (�⇤⇤1

,�⇤⇤2

= ⇡1

(�⇤⇤1

,�⇤⇤2

) + ⇢⇡1

(�⇤⇤1

,�⇤⇤2

) + ⇢2⇡1

(�⇤⇤1

,�⇤⇤2

) + · · · ,

1� ⇢⇡1

(�⇤⇤1

,�⇤⇤2

q1+�⇤⇤

= ⇡1

✓s1 + �⇤⇤

,�⇤⇤2

◆+ ⇢⇡

(�⇤1

,�⇤2

) + ⇢2⇡1

(�⇤1

,�⇤2

) + · · · ,

= ⇡1

✓s1 + �⇤⇤

,�⇤⇤2

1� ⇢⇡1

(�⇤1

,�⇤2

˜�1

(⇢), ˜�2

(⇢)) = (�⇤⇤1

,�⇤⇤2

= ⇡1

(�⇤⇤1

,�⇤⇤2

) + ⇢⇡1

(�⇤⇤1

,�⇤⇤2

) + ⇢2⇡1

(�⇤⇤1

,�⇤⇤2

) + · · · ,

1� ⇢⇡1

(�⇤⇤1

,�⇤⇤2

q1+�⇤⇤

= ⇡1

✓s1 + �⇤⇤

,�⇤⇤2

◆+ ⇢⇡

(�⇤1

,�⇤2

) + ⇢2⇡1

(�⇤1

,�⇤2

) + · · · ,

= ⇡1

✓s1 + �⇤⇤

,�⇤⇤2

1� ⇢⇡1

(�⇤1

,�⇤2

˜�1

(⇢), ˜�2

(⇢)) = (�⇤⇤1

,�⇤⇤2

= ⇡1

(�⇤⇤1

,�⇤⇤2

) + ⇢⇡1

(�⇤⇤1

,�⇤⇤2

) + ⇢2⇡1

(�⇤⇤1

,�⇤⇤2

) + · · · ,

1� ⇢⇡1

(�⇤⇤1

,�⇤⇤2

q1+�⇤⇤

= ⇡1

✓s1 + �⇤⇤

,�⇤⇤2

◆+ ⇢⇡

(�⇤1

,�⇤2

) + ⇢2⇡1

(�⇤1

,�⇤2

) + · · · ,

= ⇡1

✓s1 + �⇤⇤

,�⇤⇤2

1� ⇢⇡1

(�⇤1

,�⇤2

Condition for No Deviation for Both Platforms (Cont.)

No deviation for platform 1: ⇧

is equivalent to

discount factor ⇢ � ⇢th1

✓r1+�⇤⇤

�⇤⇤2

1+�⇤⇤2

2�⇤⇤1

✓�⇤1

1+�⇤⇤2

Similarly, no deviation for platform 2:

✓r1+�⇤⇤

�⇤⇤1

1+�⇤⇤1

2�⇤⇤2

✓�⇤2

1+�⇤⇤1

We assume c1

. Which platform is more likely to deviate?

Platform 1 is more likely to deviate with ⇢th1

� ⇢th2

is equivalent to

✓r1+�⇤⇤

�⇤⇤2

1+�⇤⇤2

2�⇤⇤1

✓�⇤1

1+�⇤⇤2

✓r1+�⇤⇤

�⇤⇤1

1+�⇤⇤1

2�⇤⇤2

✓�⇤2

1+�⇤⇤1

We assume c1

� ⇢th2

is equivalent to

✓r1+�⇤⇤

�⇤⇤2

1+�⇤⇤2

2�⇤⇤1

✓�⇤1

1+�⇤⇤2

✓r1+�⇤⇤

�⇤⇤1

1+�⇤⇤1

2�⇤⇤2

✓�⇤2

1+�⇤⇤1

We assume c1

� ⇢th2

is equivalent to

✓r1+�⇤⇤

�⇤⇤2

1+�⇤⇤2

2�⇤⇤1

✓�⇤1

1+�⇤⇤2

✓r1+�⇤⇤

�⇤⇤1

1+�⇤⇤1

2�⇤⇤2

✓�⇤2

1+�⇤⇤1

We assume c1

� ⇢th2

Cooperation Profile for Large ⇢ Regime

Large ⇢ Regime: ⇢ � max{⇢th1

, ⇢th2

} = ⇢th1

. Both will never deviate.

Proposition 3 (Large ⇢ Regime)

Under complete information, if ⇢ � ⇢th1

, both platforms will follow the

perfect cooperation profile (

˜�1

(⇢), ˜�2

(⇢)) = (�⇤⇤1

,�⇤⇤2

) all the time without

triggering the punishment profile (�⇤1

,�⇤2

Corollary 2 (symmetric ⇢th1

= ⇢th2

versus µ, c1

Given c1

under complete information, ⇢th1

= ⇢th2

decreases with µand c

µ 1/4 and then increases with µ and c1

µ > 1/4.

Small µ: scarce bandwidth causes intense competition, and as µincreases, competition is mitigated.

Large µ: ample bandwidth allures both platforms keep their AoI

small, and as µ increases, both sample more aggressively.

, ⇢th2

} = ⇢th1

˜�1

(⇢), ˜�2

(⇢)) = (�⇤⇤1

,�⇤⇤2

,�⇤2

= ⇢th2

versus µ, c1

Given c1

= ⇢th2

µ > 1/4.

, ⇢th2

} = ⇢th1

˜�1

(⇢), ˜�2

(⇢)) = (�⇤⇤1

,�⇤⇤2

,�⇤2

= ⇢th2

versus µ, c1

Given c1

= ⇢th2

µ > 1/4.

, ⇢th2

} = ⇢th1

˜�1

(⇢), ˜�2

(⇢)) = (�⇤⇤1

,�⇤⇤2

,�⇤2

= ⇢th2

versus µ, c1

Given c1

= ⇢th2

µ > 1/4.

, ⇢th2

} = ⇢th1

˜�1

(⇢), ˜�2

(⇢)) = (�⇤⇤1

,�⇤⇤2

,�⇤2

= ⇢th2

versus µ, c1

Given c1

= ⇢th2

µ > 1/4.

Cooperation Profile Design for Medium ⇢ Regime

⇢ < max{⇢th1

, ⇢th2

}, we cannot use (

˜�1

(⇢), ˜�2

(⇢)) = (�⇤⇤1

,�⇤⇤2

cooperation profile.

⇢th2

⇢ < ⇢th1

Platform 2 will still follow social optimizer �⇤⇤2

Platform 1 will deviate and we should redesign

˜�1

(⇢) to satisfy

˜�1

(⇢),�⇤⇤2

(�⇤1

,�⇤2

) or simply,

˜�1

(⇢),�⇤⇤2

)� ⇡1

✓q1+�⇤⇤

,�⇤⇤2

(�⇤1

,�⇤2

)� ⇡1

✓q1+�⇤⇤

,�⇤⇤2

⇢ < max{⇢th1

, ⇢th2

}, we cannot use (

˜�1

(⇢), ˜�2

(⇢)) = (�⇤⇤1

,�⇤⇤2

⇢th2

⇢ < ⇢th1

˜�1

(⇢) to satisfy

˜�1

(⇢),�⇤⇤2

(�⇤1

,�⇤2

) or simply,

˜�1

(⇢),�⇤⇤2

)� ⇡1

✓q1+�⇤⇤

,�⇤⇤2

(�⇤1

,�⇤2

)� ⇡1

✓q1+�⇤⇤

,�⇤⇤2

⇢ < max{⇢th1

, ⇢th2

}, we cannot use (

˜�1

(⇢), ˜�2

(⇢)) = (�⇤⇤1

,�⇤⇤2

⇢th2

⇢ < ⇢th1

˜�1

(⇢) to satisfy

˜�1

(⇢),�⇤⇤2

(�⇤1

,�⇤2

) or simply,

˜�1

(⇢),�⇤⇤2

)� ⇡1

✓q1+�⇤⇤

,�⇤⇤2

(�⇤1

,�⇤2

)� ⇡1

✓q1+�⇤⇤

,�⇤⇤2

⇢ < max{⇢th1

, ⇢th2

}, we cannot use (

˜�1

(⇢), ˜�2

(⇢)) = (�⇤⇤1

,�⇤⇤2

⇢th2

⇢ < ⇢th1

˜�1

(⇢) to satisfy

˜�1

(⇢),�⇤⇤2

(�⇤1

,�⇤2

) or simply,

˜�1

(⇢),�⇤⇤2

)� ⇡1

✓q1+�⇤⇤

,�⇤⇤2

(�⇤1

,�⇤2

)� ⇡1

✓q1+�⇤⇤

,�⇤⇤2

⇢ < max{⇢th1

, ⇢th2

}, we cannot use (

˜�1

(⇢), ˜�2

(⇢)) = (�⇤⇤1

,�⇤⇤2

⇢th2

⇢ < ⇢th1

˜�1

(⇢) to satisfy

˜�1

(⇢),�⇤⇤2

(�⇤1

,�⇤2

) or simply,

˜�1

(⇢),�⇤⇤2

)� ⇡1

✓q1+�⇤⇤

,�⇤⇤2

(�⇤1

,�⇤2

)� ⇡1

✓q1+�⇤⇤

,�⇤⇤2

Cooperation Profile Design for Medium ⇢ Regime (Cont.)

Proposition 4 (Medium ⇢ Regime)

In the repeated game under complete information, if ⇢th2

⇢ < ⇢th1

, both

platforms will always follow the cooperation profile below without

deviating to trigger punishment (�⇤1

,�⇤2

˜�1

(⇢), ˜�2

(⇢)) =

✓⇢�⇤

+ (1� ⇢)

�⇤⇤2

vuut✓⇢�⇤

+ (1� ⇢)

�⇤⇤2

�1 +

�⇤⇤2

,�⇤⇤2

�⇤⇤1

< ˜�1

(⇢) < �⇤1

˜�1

(⇢) decreases with ⇢ 2 [⇢th2

, ⇢th1

) and eventually

˜�1

(⇢) ! �⇤⇤1

Cooperation Profile Design for Medium ⇢ Regime (Cont.)

Proposition 4 (Medium ⇢ Regime)

In the repeated game under complete information, if ⇢th2

⇢ < ⇢th1

, both

platforms will always follow the cooperation profile below without

deviating to trigger punishment (�⇤1

,�⇤2

˜�1

(⇢), ˜�2

(⇢)) =

✓⇢�⇤

+ (1� ⇢)

�⇤⇤2

vuut✓⇢�⇤

+ (1� ⇢)

�⇤⇤2

�1 +

�⇤⇤2

,�⇤⇤2

�⇤⇤1

< ˜�1

(⇢) < �⇤1

˜�1

(⇢) decreases with ⇢ 2 [⇢th2

, ⇢th1

) and eventually

˜�1

(⇢) ! �⇤⇤1

Cooperation Profile Design for Small ⇢ Regime

What if ⇢ < min{⇢th1

, ⇢th2

} = ⇢th2

Neither platforms will follow the social optimizers (�⇤⇤1

,�⇤⇤2

We redesign (

˜�1

(⇢), ˜�1

(⇢)) jointly such that

˜�1

(⇢), ˜�2

(⇢)) = ˆ

(�⇤1

,�⇤2

) for platform 1 and

˜�1

(⇢), ˜�2

(⇢)) = ˆ

(�⇤1

,�⇤2

) for platform 2.

We jointly solve (

˜�1

(⇢), ˜�1

(⇢)) according to

˜�1

(⇢), ˜�2

(⇢))� ⇡1

✓q1+

˜�2

(⇢)/µc1

, ˜�2

(�⇤1

,�⇤2

)� ⇡1

✓q1+

˜�2

(⇢)/µc1

, ˜�2

˜�1

(⇢), ˜�2

(⇢))� ⇡2

✓˜�1

(⇢),q

˜�1

(⇢)/µc2

(�⇤1

,�⇤2

)� ⇡2

✓˜�1

(⇢),q

˜�1

(⇢)/µc2

, ⇢th2

} = ⇢th2

,�⇤⇤2

We redesign (

˜�1

(⇢), ˜�1

˜�1

(⇢), ˜�2

(⇢)) = ˆ

(�⇤1

,�⇤2

˜�1

(⇢), ˜�2

(⇢)) = ˆ

(�⇤1

,�⇤2

) for platform 2.

We jointly solve (

˜�1

(⇢), ˜�1

(⇢)) according to

˜�1

(⇢), ˜�2

(⇢))� ⇡1

✓q1+

˜�2

(⇢)/µc1

, ˜�2

(�⇤1

,�⇤2

)� ⇡1

✓q1+

˜�2

(⇢)/µc1

, ˜�2

˜�1

(⇢), ˜�2

(⇢))� ⇡2

✓˜�1

(⇢),q

˜�1

(⇢)/µc2

(�⇤1

,�⇤2

)� ⇡2

✓˜�1

(⇢),q

˜�1

(⇢)/µc2

, ⇢th2

} = ⇢th2

,�⇤⇤2

We redesign (

˜�1

(⇢), ˜�1

˜�1

(⇢), ˜�2

(⇢)) = ˆ

(�⇤1

,�⇤2

˜�1

(⇢), ˜�2

(⇢)) = ˆ

(�⇤1

,�⇤2

) for platform 2.

We jointly solve (

˜�1

(⇢), ˜�1

(⇢)) according to

˜�1

(⇢), ˜�2

(⇢))� ⇡1

✓q1+

˜�2

(⇢)/µc1

, ˜�2

(�⇤1

,�⇤2

)� ⇡1

✓q1+

˜�2

(⇢)/µc1

, ˜�2

˜�1

(⇢), ˜�2

(⇢))� ⇡2

✓˜�1

(⇢),q

˜�1

(⇢)/µc2

(�⇤1

,�⇤2

)� ⇡2

✓˜�1

(⇢),q

˜�1

(⇢)/µc2

, ⇢th2

} = ⇢th2

,�⇤⇤2

We redesign (

˜�1

(⇢), ˜�1

˜�1

(⇢), ˜�2

(⇢)) = ˆ

(�⇤1

,�⇤2

˜�1

(⇢), ˜�2

(⇢)) = ˆ

(�⇤1

,�⇤2

) for platform 2.

We jointly solve (

˜�1

(⇢), ˜�1

(⇢)) according to

˜�1

(⇢), ˜�2

(⇢))� ⇡1

✓q1+

˜�2

(⇢)/µc1

, ˜�2

(�⇤1

,�⇤2

)� ⇡1

✓q1+

˜�2

(⇢)/µc1

, ˜�2

˜�1

(⇢), ˜�2

(⇢))� ⇡2

✓˜�1

(⇢),q

˜�1

(⇢)/µc2

(�⇤1

,�⇤2

)� ⇡2

✓˜�1

(⇢),q

˜�1

(⇢)/µc2

Cooperation Profile Design for Small ⇢ Regime (Cont.)

Proposition 5 (Small ⇢ Regime)In the repeated game under complete information, if ⇢ < ⇢th

, the two platforms will always

follow the optimal cooperation profile (

˜�1

(⇢), ˜�2

(⇢)) below as unique solutions to

˜�1

(⇢)� ⇢�⇤1

� (1� ⇢)

vuut1 +

˜�2

(⇢)µ

vuuut✓⇢�⇤

+ (1� ⇢)

vuut1 +

˜�2

(⇢)µ

�1 +

˜�2

(⇢)µ

˜�2

(⇢)� ⇢�⇤2

� (1� ⇢)

vuut1 +

˜�1

(⇢)µ

vuuut✓⇢�⇤

+ (1� ⇢)

vuut1 +

˜�1

(⇢)µ

�1 +

˜�1

(⇢)µ

Proved properties:

�⇤⇤1

< ˜�1

(⇢) �⇤1

and �⇤⇤2

< ˜�2

(⇢) �⇤2

As ⇢ ! 0, (

˜�1

(⇢), ˜�2

(⇢)) approach competition equilibrium (�⇤1

,�⇤2

and the repeated game degenerates to one-shot static game.

As ⇢ increases in the low regime (0, ⇢th2

), both

˜�1

(⇢) and ˜�2

(⇢)decrease and the duopoly competition mitigates.

Proved properties:

�⇤⇤1

< ˜�1

(⇢) �⇤1

and �⇤⇤2

< ˜�2

(⇢) �⇤2

As ⇢ ! 0, (

˜�1

(⇢), ˜�2

,�⇤2

), both

˜�1

(⇢) and ˜�2

Proved properties:

�⇤⇤1

< ˜�1

(⇢) �⇤1

and �⇤⇤2

< ˜�2

(⇢) �⇤2

As ⇢ ! 0, (

˜�1

(⇢), ˜�2

,�⇤2

), both

˜�1

(⇢) and ˜�2

Numerical Results

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Discount Factor

rofile

Low ⇢ regime: 0 - 0.3, Medium ⇢ regime: 0.3 - 0.7, High ⇢ regime: 0.7 - 1.

Cooperation profile (

˜�1

(⇢), ˜�2

(⇢)) decrease with ⇢ and converge to

social optimizers (�⇤⇤1

,�⇤⇤2

Numerical Results (Cont.)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Discount Factor

Fix c1

= 1, c2

= 1.5, and alter µ and ⇢.Compare the social costs under recommendation (

˜�1

(⇢), ˜�2

(⇢)) versussocial optimum (�⇤⇤

,�⇤⇤2

Social cost under our trigger-and-punishment mechanism

˜�1

(⇢), ˜�2

(⇢)) decreases with ⇢ and converges to social optimum.

Social cost ratio under (

˜�1

(⇢), ˜�2

(⇢)) decreases to 1 as µ increases.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Discount Factor

Fix c1

= 1, c2

˜�1

(⇢), ˜�2

,�⇤⇤2

˜�1

(⇢), ˜�2

˜�1

(⇢), ˜�2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Discount Factor

Fix c1

= 1, c2

˜�1

(⇢), ˜�2

,�⇤⇤2

˜�1

(⇢), ˜�2

˜�1

(⇢), ˜�2

Incomplete Information Scenario

System model under one-sided incomplete information

Platform 1’s cost function when c1

= cH :

(cH),�2

(cH) + �2

◆+ cH�1

(cL),�2

(cL) + �2

◆+ cL�1

= cH :

(cH),�2

(cH) + �2

◆+ cH�1

(cL),�2

(cL) + �2

◆+ cL�1

= cH :

(cH),�2

(cH) + �2

◆+ cH�1

(cL),�2

(cL) + �2

◆+ cL�1

System Model under Incomplete Information

Indi↵erent between cH and cL instances, platform 2’s cost function:

((�1

(cH),�1

(cL)),�2

) = pH ·✓�1

(cH) + �2

◆◆

+ (1� pH) ·✓�1

(cL) + �2

◆◆+ c

Social cost function in average sense:

⇡((�1

(cH),�1

(cL)),�2

) =pH⇡1

(cH),�2

) + (1� pH)⇡1

(cL),�2

+ ⇡2

((�1

(cH),�1

(cL)),�2

System Model under Incomplete Information

Indi↵erent between cH and cL instances, platform 2’s cost function:

((�1

(cH),�1

(cL)),�2

) = pH ·✓�1

(cH) + �2

◆◆

+ (1� pH) ·✓�1

(cL) + �2

◆◆+ c

Social cost function in average sense:

⇡((�1

(cH),�1

(cL)),�2

) =pH⇡1

(cH),�2

) + (1� pH)⇡1

(cL),�2

+ ⇡2

((�1

(cH),�1

(cL)),�2

Bayesian Game under Incomplete Information

Non-cooperative Bayesian game with equilibrium

((�⇤1

(cH),�⇤1

(cL)),�⇤2

(cH)>0

(cH),�2

(cL)>0

(cL),�2

((�1

(cH),�1

(cL)),�2

Min-social-cost problem with social optimizers

((�⇤⇤1

(cH),�⇤⇤1

(cL)),�⇤⇤2

(cH),�1

(cL),�2

⇡((�1

(cH),�1

(cL)),�2

Bayesian Game under Incomplete Information

Non-cooperative Bayesian game with equilibrium

((�⇤1

(cH),�⇤1

(cL)),�⇤2

(cH)>0

(cH),�2

(cL)>0

(cL),�2

((�1

(cH),�1

(cL)),�2

Min-social-cost problem with social optimizers

((�⇤⇤1

(cH),�⇤⇤1

(cL)),�⇤⇤2

(cH),�1

(cL),�2

⇡((�1

(cH),�1

(cL)),�2

Proposition 6 (Equilibrium vs social optimizers under incomplete information)

The competition equilibrium ((�⇤1

(cH),�⇤1

(cL)),�⇤2

) are the unique solutions to

✓1 +

◆+ cH = 0,

✓1 +

◆+ cL = 0,

✓1 +

◆�

1 � pH

✓1 +

◆+ c

Social optimizers ((�⇤⇤1

(cH),�⇤⇤1

(cL)),�⇤⇤2

) are the unique solutions to

✓1 +

◆+ cH +

µ= 0,

✓1 +

◆+ cL +

µ= 0,

✓�

✓1 +

◆+ c

(cH )µ

◆+ (1 � pH )

✓�

✓1 +

◆+ c

(cL)µ

◆= 0.

Both platforms will over-sample at equilibrium, i.e., �⇤1

(cH) � �⇤⇤1

�⇤1

(cL) � �⇤⇤1

(cL) and �⇤2

� �⇤⇤2

. Additionally, �⇤1

(cH)/�⇤1

(cL) =pcL/cH .

Ine�ciency of Competition Equilibrium

PoA = max

cL,cH ,c2,µ,pH

⇡((�⇤1

(cH),�⇤1

(cL)),�⇤2

⇡((�⇤⇤1

(cH),�⇤⇤1

(cL)),�⇤⇤2

� 1.

Proposition 7

Price of anarchy under incomplete information is PoA = 1, which is

achieved when platform 1’s smaller sampling cost cL is infinitesimal.

cL ! 0, �⇤1

(cL) ! 1, �⇤2

((�⇤⇤1

(cH),�⇤⇤1

(cL)),�⇤⇤2

) < 1.

Need non-monetary mechanism to remedy huge e�ciency loss!

PoA = max

cL,cH ,c2,µ,pH

⇡((�⇤1

(cH),�⇤1

(cL)),�⇤2

⇡((�⇤⇤1

(cH),�⇤⇤1

(cL)),�⇤⇤2

� 1.

Proposition 7

cL ! 0, �⇤1

(cL) ! 1, �⇤2

((�⇤⇤1

(cH),�⇤⇤1

(cL)),�⇤⇤2

) < 1.

PoA = max

cL,cH ,c2,µ,pH

⇡((�⇤1

(cH),�⇤1

(cL)),�⇤2

⇡((�⇤⇤1

(cH),�⇤⇤1

(cL)),�⇤⇤2

� 1.

Proposition 7

cL ! 0, �⇤1

(cL) ! 1, �⇤2

((�⇤⇤1

(cH),�⇤⇤1

(cL)),�⇤⇤2

) < 1.

PoA = max

cL,cH ,c2,µ,pH

⇡((�⇤1

(cH),�⇤1

(cL)),�⇤2

⇡((�⇤⇤1

(cH),�⇤⇤1

(cL)),�⇤⇤2

� 1.

Proposition 7

cL ! 0, �⇤1

(cL) ! 1, �⇤2

((�⇤⇤1

(cH),�⇤⇤1

(cL)),�⇤⇤2

) < 1.

PoA = max

cL,cH ,c2,µ,pH

⇡((�⇤1

(cH),�⇤1

(cL)),�⇤2

⇡((�⇤⇤1

(cH),�⇤⇤1

(cL)),�⇤⇤2

� 1.

Proposition 7

cL ! 0, �⇤1

(cL) ! 1, �⇤2

((�⇤⇤1

(cH),�⇤⇤1

(cL)),�⇤⇤2

) < 1.

Hurt with More Information for Platform 1

Question: Does platform 1 take advantage from knowing more information

about the sampling costs of both platforms?

Answer: Not exactly even in average sense!

Hurt with More Information for Platform 1

Question: Does platform 1 take advantage from knowing more information

about the sampling costs of both platforms?

Answer: Not exactly even in average sense!

Hurt with More Information for Platform 1 (Cont.)

Proposition 8

Under incomplete information, the cost objective of platform 1 under each

= cH realization is greater than that under complete information, and

becomes smaller under each c1

= cL realization.

Perhaps surprisingly, its average cost

pH⇡1(�⇤1

(cH),�⇤2

) + (1� pH)⇡1(�⇤1

(cL),�⇤2

) becomes greater once

pH �pcL(

¯�2

(cL)/µ�p1 + �⇤

/µ)pcL(

¯�2

(cL)/µ�p

1 + �⇤2

/µ) +pcH(

p1 + �⇤

/µ�p

¯�2

(cH)/µ)

Platform 2 cannot identify c1

= cH or c1

= cL, and its over-sampling

when c1

= cH forces platform 1 to over-sample.

When pH is large, this happens more often and platform 1 loses in

average sense.

Proposition 8

= cL realization.

pH⇡1(�⇤1

(cH),�⇤2

) + (1� pH)⇡1(�⇤1

(cL),�⇤2

pH �pcL(

¯�2

(cL)/µ�p1 + �⇤

/µ)pcL(

¯�2

(cL)/µ�p

1 + �⇤2

/µ) +pcH(

p1 + �⇤

/µ�p

¯�2

(cH)/µ)

= cH or c1

when c1

average sense.

Proposition 8

= cL realization.

pH⇡1(�⇤1

(cH),�⇤2

) + (1� pH)⇡1(�⇤1

(cL),�⇤2

pH �pcL(

¯�2

(cL)/µ�p1 + �⇤

/µ)pcL(

¯�2

(cL)/µ�p

1 + �⇤2

/µ) +pcH(

p1 + �⇤

/µ�p

¯�2

(cH)/µ)

= cH or c1

when c1

average sense.

Proposition 8

= cL realization.

pH⇡1(�⇤1

(cH),�⇤2

) + (1� pH)⇡1(�⇤1

(cL),�⇤2

pH �pcL(

¯�2

(cL)/µ�p1 + �⇤

/µ)pcL(

¯�2

(cL)/µ�p

1 + �⇤2

/µ) +pcH(

p1 + �⇤

/µ�p

¯�2

(cH)/µ)

= cH or c1

when c1

average sense.

Approximate Mechanism under Incomplete Information

New Challenge for Mechanism Design under IncompleteInformation

Even if ⇢ is large enough, can we still use social optimizers

((�⇤⇤1

(cL),�⇤⇤1

(cH)),�⇤⇤2

) as in complete information scenario?

Lemma 5.1.

Given the cooperation profile (�⇤⇤1

(cL),�⇤⇤1

(cH)) for platform 1 under

su�ciently large ⇢, platform 1 will not deviate from �⇤⇤1

(cL) when c1

= cLbut may deviate from �⇤⇤

(cH) when c1

= cH .

Platform 2 over-samples when c1

= cH compared to complete

information scenario.

Platform 1 benefits from choosing �⇤⇤1

(cL) and sampling more than

�⇤⇤1

When sampling according to larger �⇤⇤1

(cL), he will not be caught.

Need to design new ((

˜�1

(cH , ⇢), ˜�1

(cL, ⇢)), ˜�2

(⇢)) even for large ⇢ regime!

((�⇤⇤1

(cL),�⇤⇤1

(cH)),�⇤⇤2

Lemma 5.1.

(cL),�⇤⇤1

(cL) when c1

(cH) when c1

= cH .

�⇤⇤1

˜�1

(cH , ⇢), ˜�1

(cL, ⇢)), ˜�2

((�⇤⇤1

(cL),�⇤⇤1

(cH)),�⇤⇤2

Lemma 5.1.

(cL),�⇤⇤1

(cL) when c1

(cH) when c1

= cH .

�⇤⇤1

˜�1

(cH , ⇢), ˜�1

(cL, ⇢)), ˜�2

((�⇤⇤1

(cL),�⇤⇤1

(cH)),�⇤⇤2

Lemma 5.1.

(cL),�⇤⇤1

(cL) when c1

(cH) when c1

= cH .

�⇤⇤1

˜�1

(cH , ⇢), ˜�1

(cL, ⇢)), ˜�2

((�⇤⇤1

(cL),�⇤⇤1

(cH)),�⇤⇤2

Lemma 5.1.

(cL),�⇤⇤1

(cL) when c1

(cH) when c1

= cH .

�⇤⇤1

˜�1

(cH , ⇢), ˜�1

(cL, ⇢)), ˜�2

Approximate Profile Design under Incomplete Information

New idea: recommend platform 1 to behave indi↵erently no matter

= cH or c1

= cL. That is, ˜�1

(cH , ⇢) = ˜�1

(cL, ⇢) = ˜�1

(⇢).

Definition 2 (Approximate trigger mechanism of punishment underincomplete information)

In each round, two platforms follow approximate cooperation profile(

˜�1

(⇢), ˜�2

(⇢)) if neither was detected to deviate from tits profile inthe past.

Once a deviation was found in the past, the two platforms will keepplaying the equilibrium punishment profile ((�⇤

(cH), �⇤1

(cL)), �⇤2

forever.

What is the best design for (

˜�1

(⇢), ˜�2

(⇢))?

= cH or c1

(cH , ⇢) = ˜�1

(cL, ⇢) = ˜�1

(⇢).

˜�1

(⇢), ˜�2

(cH), �⇤1

(cL)), �⇤2

forever.

˜�1

(⇢), ˜�2

(⇢))?

= cH or c1

(cH , ⇢) = ˜�1

(cL, ⇢) = ˜�1

(⇢).

˜�1

(⇢), ˜�2

(cH), �⇤1

(cL)), �⇤2

forever.

˜�1

(⇢), ˜�2

(⇢))?

Approximate Cooperation Profile

New min-social cost problem with optimal profile (

˜�1

(⇢), ˜�2

(⇢)):

(cH ,⇢),�1

(cL,⇢),�2

(⇢)>0

⇡((�1

(cH , ⇢),�1

(cL, ⇢)),�2

(⇢))

s.t. �1

(cH , ⇢) = �1

(cL, ⇢) := ˜�1

Optimal approximate profile:

˜�1

(⇢) =

vuut 1 +

˜�2

(⇢)/µ

pHcH + (1� pH)cL +1

˜�2

(⇢)µ

˜�2

(⇢) =

vuut1 +

˜�1

(⇢)/µ

˜�1

(⇢)µ

which proves to provide at most 2-approximation of minimum social cost.

Approximate Cooperation Profile

New min-social cost problem with optimal profile (

˜�1

(⇢), ˜�2

(⇢)):

(cH ,⇢),�1

(cL,⇢),�2

(⇢)>0

⇡((�1

(cH , ⇢),�1

(cL, ⇢)),�2

(⇢))

s.t. �1

(cH , ⇢) = �1

(cL, ⇢) := ˜�1

Optimal approximate profile:

˜�1

(⇢) =

vuut 1 +

˜�2

(⇢)/µ

pHcH + (1� pH)cL +1

˜�2

(⇢)µ

˜�2

(⇢) =

vuut1 +

˜�1

(⇢)/µ

˜�1

(⇢)µ

which proves to provide at most 2-approximation of minimum social cost.

Approximate Mechanism Design under IncompleteInformation

Derive ⇢th1

and ⇢th2

similarly as under complete information.

Divide profile design into three di↵erent ⇢ regimes (low, medium and

high):

High ⇢ regime (⇢ � ⇢th1): both platforms follow optimal

recommendation.

Medium ⇢ regime (⇢th2 ⇢ < ⇢th1): only one platform follows optimal

recommendation.

Low ⇢ regime (⇢ < ⇢th2): neither follows optimal recommendation.

Approximate Mechanism Design under IncompleteInformation

Derive ⇢th1

and ⇢th2

similarly as under complete information.

Divide profile design into three di↵erent ⇢ regimes (low, medium and

high):

High ⇢ regime (⇢ � ⇢th1): both platforms follow optimal

recommendation.

Medium ⇢ regime (⇢th2 ⇢ < ⇢th1): only one platform follows optimal

recommendation.

Low ⇢ regime (⇢ < ⇢th2): neither follows optimal recommendation.

Numerical Results

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Discount Factor

rofile

Low ⇢ regime: 0 - 0.3, Medium ⇢ regime: 0.3 - 0.7, High ⇢ regime: 0.7 - 1

˜�1

(⇢), ˜�2

optimal recommended profile.

Numerical Results

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Discount Factor

rofile

Low ⇢ regime: 0 - 0.3, Medium ⇢ regime: 0.3 - 0.7, High ⇢ regime: 0.7 - 1

˜�1

(⇢), ˜�2

optimal recommended profile.

Conclusion

The first work to analyze tradeo↵ between AoI reduction and

sampling cost for online content platforms in the long run.

Competition between platforms when co-using content delivery

network can lead to huge e�ciency loss (PoA ! 1) under both

complete and incomplete info.

Under complete information, propose repeated games mechanism

with the threat of future punishment to enforce e�cient cooperation

under any discount factor.

Under incomplete information, propose approximate mechanism to

negate the platform with information advantage.

Conclusion

Long-term tradeo↵ between average AoI and sampling cost:

min�1

) = �1

) + c1

Require to examine AoI evolution over time t

Use age-dependent pricing p(t) instead of fixed c1

to motivatetime-varying sampling

Complex even for a single platform’s problem

Long-term tradeo↵ between average AoI and sampling cost:

min�1

) = �1

) + c1

Require to examine AoI evolution over time t

Use age-dependent pricing p(t) instead of fixed c1

to motivatetime-varying sampling

Complex even for a single platform’s problem

System model: dynamic pricing to control AoI

A price p(t) is announced at time t to motivate a user arrival to

sample and contribute information in a discrete time horizon.

A user may arrive randomly (s(t) = 1) with probability ↵ in this time

slot and (if so) he further decides to sample or not based on the price

p(t) and its private sampling cost ⇡ ⇠ F (·).The sensor data (e.g., about tra�c and road condition) is transmitted

with fixed delay A0

to finally post in the platform.

p(t) and its private sampling cost ⇡ ⇠ F (·).

The sensor data (e.g., about tra�c and road condition) is transmitted

with fixed delay A0

p(t) and its private sampling cost ⇡ ⇠ F (·).The sensor data (e.g., about tra�c and road condition) is transmitted

with fixed delay A0

System model: AoI dynamics over time

The probability that a user will appear and accept the price is

Pr(s(t) = 1)Pr(⇡ < p(t)) = ↵F (p(t)) to reduce A(t + 1) to A0

otherwise, age increases by 1 unit of time.

E.g., for uniform cost distribution ⇡ ⇠ [0, b], update probability is

↵F (p(t)) = ↵p(t)b and average payment is

↵p2(t)b .

System model: AoI dynamics over time

The probability that a user will appear and accept the price is

Pr(s(t) = 1)Pr(⇡ < p(t)) = ↵F (p(t)) to reduce A(t + 1) to A0

otherwise, age increases by 1 unit of time.

E.g., for uniform cost distribution ⇡ ⇠ [0, b], update probability is

↵F (p(t)) = ↵p(t)b and average payment is

↵p2(t)b .

Dynamic pricing problem formulation

Dynamic of expected age at time t + 1 is

The platform’s problem formulation to tradeo↵ between AoI and

sample cost in T time slots is

where ⇢ is the discount factor.

This is a constrained nonlinear dynamic program, challenging to solve

analytically due to curse of dimensionality. Complexity is O((

b✏ )

Approximate Dynamic pricing

We reformulate the nonlinear age dynamic into linear term:

where � is a time-average term to approximate the dynamic age

reduction A(t)� A0

over time.

We can show the best estimator of � is

1� ⇢

1� ⇢T

T�1X

⇢t(A(t)� A0

a↵ected by all A(t) over any time t, which will in turn a↵ect A(t).

over time.

1� ⇢

1� ⇢T

T�1X

⇢t(A(t)� A0

over time.

1� ⇢

1� ⇢T

T�1X

⇢t(A(t)� A0

Approximate Dynamic pricing: Analysis of pricing

We want to obtain the structure of value function V (A(t), t) w.r.t. A(t):

Backward induction from time T : by minimizing A2

(T ) +

↵b p

optimal price p(T ) = 0 without considering future age.

Back to time T � 1:

Iteratively, we can prove that optimal p(t) is a linear function of A(t)!Value function is in the quadratic structure of A(t):

V (A(t), t) = QtA2