The Allocation of Value For Jointly Provided Services

ByP. Linhart, R. Radner, K. G. Ramkrishnan, R. Steinberg

Telecommunication Systems, Vol. 4, 1995

Presented By :Matulya Bansal

Outline

Introduction The Telephone Carrier Network The Problem of Allocating Values Co-operative Game Theory Shapley Value Solving the Caller ID Problem Example Conclusion

Introduction

The Caller ID Service Revenue allocation is currently simplistic An allocation mechanism is needed when the

service is provided by more than one carrier

The Telephone Carrier Network

Geographically distributed into Local Access and Transport Areas (LATAs)

Local Exchange Carriers (LECs) operate in LATAs

e.g. Regional Bell Operating Companies (RBOCs)

Long Distance Carriers or InterExchange Carriers (IXCs) provide InterLATA connectivity

e.g. AT&T, MCI, Sprint

The Domestic Telecom Market

Local Calls (involve one LEC) IntraLATA Toll Calls (involve 2 LECs) InterLATA Long Distance Calls (invlove 1 or 2

LECs and 1 IXC)

Basic Problem

How should the Caller ID Service Revenues be divided among the participating companies?

Or equivalently, what should be the payoff of players participating in this collaborative game?

Desirable Properties

Stability : Players have an incentive to participate in the coalition (A solution which is stable is said to be in the core)

Fairness : The allocation should be perceived as in some sense fair

These considerations suggest the use of cooperative game theory

The Core

Let N (1, 2, 3, …, n) be a set of players. Let v(N) be the value generated by the

coalition of all players participating in the Caller ID Game.

Let v(S) be the value generated by any subset S of players where v(S) >= 0 and v() = 0.

The Core

Let N (1, 2, 3, …, n) be a set of players. Let v(N) be the value generated by the

coalition of all players participating in the Caller ID Game.

Let v(S) be the value generated by any subset S of players where v(S) >= 0 and v() = 0.

Let x be an allocation of total value among the players and

),....,,( 21 nxxx

Nii Nvx

)(

The Core

Let N (1, 2, 3, …, n) be a set of players. Let v(N) be the value generated by the

coalition of all players participating in the Caller ID Game.

Let v(S) be the value generated by any subset S of players where v(S) >= 0 and v() = 0.

Let x be an allocation of total value among the players and .

A coalition N shall fall apart unless for every S N.

),....,,( 21 nxxx

Nii Nvx

)(

Nii Svx )(

Shapley Value

One of the most-popular fairness criterion. Introduced by L. S. Shapley in 1953. Has been used for allocation of aircraft landing

fees, cost of public goods & services, water resources costs and depreciation.

Shapley Value : Axioms

Domain Axiom – The allocation depends only on the values that can be earned by all possible combinations of one or more players acting in coalition.

Anonymity Axiom – The allocation does not depend on the players’ labels

Dummy Axiom – A player who adds nothing to the value of the coalition is allocated nothing

Additivity Axiom – If two allocation problems are combined by adding the characteristic function, then for each player the new allocation is just the sum of the earlier ones.

The Shapley Formula

The Shapley Value is given by :

)]()([!

)!()!1(iSvSv

n

snsxi

The Shapley Formula

The Shapley Value is given by :

The equation may be interpreted probabilistically as the expected marginal contribution of player I, assuming that the coalitions form randomly and that each coalition is equiprobable.

)]()([!

)!()!1(iSvSv

n

snsxi

Shapley Value

In general, the Shapley Value need not be in the core. However, if the game is convex, the Shapley Value is in the core.

Shapley Value

In general, the Shapley Value need not be in the core. However, if the game is convex, the Shapley Value is in the core.

Our game is convex : if a RBOC joins in a coalition, it adds value to itself as well as to the coalition.

where

)(}){()(}){( TvaTvSvaSv

TSaNTS },{,

Subtracting the Singletons

The revenue an LEC generates on account of its IntraLATA traffic is really not up for negotiation.

So, we define a new characteristic function w(S) = v(S) – v’(S) where v’(S) is the total value generated due to

intraLATA calls. The additive property ensures that if v(S) is in

the core of game v, then w(S) is in the core of game w.

Solving the Caller ID Problem

We need to calculate the characteristic function to

- determine if the allocation is in the core - calculate the Shapley value To do this in the absence of actual

experiments with all possible coalition structures, we require a demand model for the Caller ID.

The Demand Model

Q : subset of subscribers that have the Caller ID facility (for Q)

: i’s willingness to pay (wtp) for this service

We assume that a subscriber’s wtp is a linear function of the number of calls received.

: number of calls from j to i : total number of calls received by I : average number of calls received The wtp function is assumed to be of the form

)(QLi

ijm

Qj

ijiji mzQL )(

im

)(Qm

The Demand Model (contd …)

Define . Let F(x) be the probability that a subscriber I

drawn at random from Q will have an not exceeding x.

Hence, the total revenue is given by where

We wish to determine the price that maximizes revenue.

This gives the characteristic function to be

)(

)(

Qm

QLX i

iX

)](1[)( uFuQmR

)(Qm

pu

)]([)](1[)( ** SQmuFuSv

Deriving the characteristic function

Assume that there are only 11 players (8 RBOCs and 3 IXCs)

Given : RBOC to RBOC AT&T Traffic

From/To

1 2 3

1 932961 485323

2 368220

3

Approximating InterLATA Traffic

Approximate AT&T market share by RBOC Approximate interLATA traffic for MCI and

Sprint.

Approximating AT&T market share by RBOC

AT&T’s market share in a RBOC in current year is given by the multiplying AT&T’s market share in the RBOC in a recent year with the ratio of AT&T’s US market share in current year to AT&T’s US market share in the recent year.

So, if in 1990 AT&T’s US Market Share was 60 million and it grew to 80 million in 1991 and AT&T’s share in a RBOC was 3 million, it grows to 4 million.

Approximating InterLATA Traffic

The MCI (resp. Sprint) interLATA message volume for calls originating in a given RBOC can be approximated from the AT&T interLATA message volumes by multiplying by a proportionality factor.

1 2 3

MCI .194 .222

Sprint .132

The Zero Normalized Function

X : set of Interexchange Carriers I : set of RBOCs : number of calls originating in l and

terminating in m. A denotes AT&T. : denotes the current market share for

in RBOC , divided by the current market

share for AT&T in that RBOC.

),,( AiiR ml

k

l

XiP kX

li

The Zero Normalized Function

The zero normalized function w can be derived from the characteristic function by subtracting out the singleton coalition values representing IntraLATA calls. It is given by

0),...,( 1 NXXw

0),...,( 1 niiw

)],,([),...,;,...,( '

1111 AiiRPXXiiw

ll

N

k

Xi

n

lNn

k

l

RBOC to RBOC AT&T Traffic

From/To 1 2 3

1 932961 485323

2 368220

3

Proportionality Factors

1 2 3

MCI .194 .222

Sprint .132

Calculating w

w(1,A) = R(1,1,A) = 932961

Calculating w

w(1,A) = R(1,1,A) = 932961 w(1,M) = . R(1,1,A) = .194 * 932961 =

180944

MP1

Calculating w

w(1,A) = R(1,1,A) = 932961 w(1,M) = . R(1,1,A) = .194 * 932961 =

180944 w(1,2,A) = R(1,1,A) + R(1,2,A) + R(2,2,A) +

R(2,1,A)

MP1

Calculating w

w(1,A) = R(1,1,A) = 932961 w(1,M) = . R(1,1,A) = .194 * 932961 =

180944 w(1,2,A) = R(1,1,A) + R(1,2,A) + R(2,2,A) +

R(2,1,A) w(1,2,M) = [R(1,1,A) + R(1,2,A)] + [R(2,2,A) + R(2,1,A)]

MP1

MP1MP2

Calculating Shapley Values

The Shapley values can now be calculated using

)]()([!

)!()!1(iSwSw

n

snsxi

Other Notions of Fairness

Nucleolus- Tries to make the least happy player as happy as possible- Not monotone with respect to value

Other Notions of Fairness

Nucleolus- Tries to make the least happy player as happy as possible- Not monotone with respect to value

Incremental Recording - Allocates points on a per call basis - Simple, but doesn’t guarantee fairness

Conclusion

Two desirable properties for allocation of revenues for jointly provided services are Stability & Fairness

In general, the core contains several solutions Shapley value provides a solution that is stable

and fair. It also ensures marginality and anonymity.

The Caller ID Problem (and in general more allocation problems) can be solved by applying Cooperative Game Theory.

References

“The allocation of value for jointly provided services”, P. Linhart et. al., Telecommunication Systems, Vol. 4, 1995.

“A value for n-person games”, L. S. Shapley, Contributions to the Theory of Games, Vol. 2, 1953.

Thanks !

Presentation ByMatulya Bansal