The allocation of value for jointly provided services
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The Allocation of Value For Jointly Provided Services. By P. 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

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The Allocation of Value For Jointly Provided Services

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The Allocation of Value For Jointly Provided Services

By

P. 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


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.


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 :


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.


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


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


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


Deriving the characteristic function

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

  • Given : RBOC to RBOC AT&T Traffic


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.


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.


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


RBOC to RBOC AT&T Traffic


Proportionality Factors


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


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)


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)]


Calculating Shapley Values

  • The Shapley values can now be calculated using


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 By

Matulya Bansal


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