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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

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

- The Caller ID Service
- Revenue allocation is currently simplistic
- An allocation mechanism is needed when the service is provided by more than one carrier

- 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

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

- 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?

- 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

- 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 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

- 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.

- 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.

- 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 Value is given by :

- 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.

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

- 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

- 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.

- 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.

- 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

- 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

- Assume that there are only 11 players (8 RBOCs and 3 IXCs)
- Given : RBOC to RBOC AT&T Traffic

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

- 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.

- 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.

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

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

- w(1,A) = R(1,1,A) = 932961
- w(1,M) = . R(1,1,A) = .194 * 932961 = 180944

- 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,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)]

- The Shapley values can now be calculated using

- Nucleolus
- Tries to make the least happy player as happy as possible

- Not monotone with respect to value

- 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

- 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.

- “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.

Presentation By

Matulya Bansal