Tài liệu Pricing communication networks P7 docx

The service provider wants to share the total cost of providing the services amongst the customers in a manner that they think is fair.. It may be possible to check 7.5 by computing and

Part C

Pricing

Pricing Communication Networks: Economics, Technology and Modelling.

Costas Courcoubetis and Richard Weber Copyright  2003 John Wiley & Sons, Ltd.

ISBN: 0-470-85130-9

Cost-based Pricing

This chapter is about prices that are directly related to cost We begin with theproblem of finding cost-based prices that are fair or stable under potential competition(Sections 7.1 –7.2) We look for types of prices that can protect an incumbent against entry

by potential competitors, or against bypass by customers who might find it cheaper to supplythemselves We explain the notions of subsidy-free and sustainable prices Such prices arerobust against bypass Similar notions are addressed by the idea of the second-best core Theaim now differs from that of maximizing economic efficiency We see that Ramsey prices,which are efficient subject to the constraint of cost recovery, may fail sustainability tests

In Section 7.3 we take a different approach and look at practical issues of constructingcost-based prices Now we emphasize necessary and simplicity Prices are to be computedfrom quantities that can be easily measured and for which accounting data is readilyavailable An approach that has found much favour with regulators is that of FullyDistributed Cost pricing (FDC) This is a top-down approach, in which costs are attributed

to services using the firm’s existing cost accounting records It ignores economic efficiency,but has the great advantage of simplicity

Section 7.3.5 concerns the Long-Run Incremental Cost approach (LRIC) This is abottom-up approach, in which the costs of the services are computed using an optimizedmodel for the network and the service production technologies It can come close toimplementing subsidy-free prices We compare FDC and LRIC in Section 7.4, from theviewpoint of the regulator, who wishes to balance the aims of encouraging efficiency andcompetition, and of the monopolist who would like to set sustainable prices The regulatormay prefer the accounting-based approach of FDC pricing because it is ‘automatic’ andauditable However, it may obscure old and inefficient production technology or the factthat the network has been wrongly dimensioned These problems can be remedied by theLRIC approach, but it is more costly to implement

Flat rate pricing is the subject of Section 7.5 In this type of pricing a customer’s chargedoes not depend on the actual quantity of services he consumes Rather, he is charged theaverage cost of other customers in the same customer group We discuss the incentives thatsuch a scheme provides and their effects on the market

7.1 Foundations of cost-based pricing

In Chapters 5 and 6 we considered the problem of pricing in a context in which socialwelfare maximization is the overall aim We posed optimization problems with unique

Costas Courcoubetis and Richard Weber Copyright  2003 John Wiley & Sons, Ltd.

ISBN: 0-470-85130-9

164 COST-BASED PRICINGsolutions, each achieved by unique sets of prices However, welfare maximization is notthe only thing that matters A firm’s prices must ensure that it is profitable, or at leastthat it covers its costs Cost-based pricing focuses on this consideration Unfortunately, afundamental difficulty in defining cost-based prices is that services are usually producedjointly A large part of the total cost is a common cost, which can be difficult toapportion rationally amongst the different services One can think of several ways to

do it So although cost-based prices may reasonably be expected to satisfy certainnecessary conditions, they differ from welfare-maximizing prices in that they are usuallynot unique

One necessary condition that cost-based prices ought reasonably to satisfy is that offairness Some customers should not find themselves subsidizing the cost of providingservices to other customers If so, these customers are likely to take their business elsewhere

This motivates the idea of subsidy-free prices A second reasonable necessary condition is

that prices should be defensive against competition, discouraging the entry of competitorswho by posting lower prices could capture market share This motivates the idea of

sustainable prices If prices do not reflect actual costs or they hide costs of inefficient

production then they invite competition from other firms Since customers will choose theprovider from whom they believe they get the best deal, a game takes place amongstproviders, as they seek to offer better deals to customers by deploying different costfunctions and operating at different production levels Prices must be subsidy-free and

sustainable if they are to be stable prices, that is, if they are to survive the competition in

this game

Interestingly, the set of necessary conditions that we might like to impose on prices can

be mutually incompatible They can also be in conflict with the aim of maximizing socialwelfare maximization, since they restrict the feasible set of operating points, sometimesreducing it to a single point

7.1.1 Fair Charges

Consider the problem of a single provider who wishes to price his services so that theycover their production cost and are fair in the sense that no customer feels he is subsidizingothers Unfair prices leave him susceptible to competition from another provider, who hasthe same costs, but charges fairly Customers might even become producers of their ownservices

Let N D f1; 2; : : : ; ng denote a set of n customers, each of whom wishes to buy some services For T that is a subset N , and let c.T / denote the minimal cost that could by

incurred by a facility that is optimized to provide precisely the services desired by the set

of customers T We call this the stand-alone cost of providing services to the customers in

T Assume that because of economies of scale and scope this cost function is subadditive That is, for all disjoint sets T and U ,

In the terminology of cooperative games, c Ð/ is called a characteristic function.

The service provider wants to share the total cost of providing the services amongst the

customers in a manner that they think is fair Suppose he charges them amounts c1; : : : ; c n.Let us further suppose that he exactly covers his cost, and soP

i 2N c i Dc N/ The charges are said to subsidy free if they satisfy the following two tests:

ž The charge made to any subset of customers is no more than the stand-alone cost ofproviding services to those customers,

The reason these conditions are interesting is that if either (7.2) or (7.3) is violated, then

a new entrant can attract dissatisfied customers If (7.2) is violated, then a firm producing

only services for T and charging only c.T / could lure away these customers Similarly, if (7.3) is violated, then a firm producing only the services needed by N n T could charge less

for these services than the incumbent firm This happens because the incumbent uses part

of the revenue obtained from selling services to N n T to pay for some of the cost of the services wanted by T Next, we investigate certain variations and refinements of the above

concepts

7.1.2 Subsidy-free, Support and Sustainable Prices

Let reformulate the ideas of the previous section to circumstances in which charges are

computed from prices Suppose that a set of n services is N D f1; : : : ; ng and an incumbent firm sells service i in quantity x i , at price p i , for a total charge of p i x i Suppose that x i

is given and does not depend on p D p1; : : : ; p n / We call p a subsidy-free price if it

satisfies the two tests

Inequalities (7.4) and (7.5) are respectively the stand-alone test and incremental-cost test.

They have natural interpretation similar to (7.2) and (7.3) For instance, if (7.4) is violated

then a new firm could set up to produce only the services in T and sell these at lower prices than the incumbent Note that, by putting T D N , these tests implyP

i p i x i Dc N/.

Thus the producer must operate with zero profit Also, prices must be above marginal cost;

to see this, consider the set T D fi g, imagine that x i is small and apply the incrementalcost test

Example 7.1 (Subsidy-free prices may not exist) Consider a network offering voice and

video services The cost of the basic infrastructure that is common to both services is 10units, while the incremental cost of supplying 100 units of video service is 2 units andthe incremental cost of supplying 1000 units of voice is 1 unit To be subsidy-free, the

revenues r1.100/ and r2.1000/ that are obtained from the video and the voice servicesmust satisfy

2  r1.100/  12; 1  r2.1000/  11; r1.100/ C r2.1000/ D 13

166 COST-BASED PRICINGThus, assuming that there is enough demand for services, possible prices are 0:006 unitsper voice service and 0:07 units per video service Note that such prices are not uniqueand they may not even exist for general cost functions Suppose three services are pro-

duced in unit quantities with a symmetric cost function that satisfies (7.1) Let c.fig/ D 2:5,

c fi; jg/ D 3:5, and c.fi; j; kg/ D 5:5, where i; j; k are distinct members of f1; 2; 3g Then

we must have 2  p i  2:5, for i D 1; 2; 3, but also p1C p2C p3 D 5:5 So thereare no subsidy-free prices The problem is that economies of scope are not increasing, i.e

c fi; j; kg/  c.fi; jg/ > c.fi; jg/  c.fig/.

How can one determine if (7.4) and (7.5) are met in practice? Assume that a firm posts itsprices and makes available its cost accounting records for the services It may be possible

to check (7.5) by computing and then summing the incremental costs of each service in

T (though this only approximates the incremental cost of T because we neglect common cost that is directly attributable to services in T ) Condition (7.4) is hard to check, as it

imagines building from scratch a new facility that is specialized to produce the services in

the set T This cost cannot in general be derived from the cost accounting information of the firm which produces the larger set of services N In practice, one tries to approximate

c T /, as well as possible given the available information.

There is another possible problem with the above tests Although individual outputs maypass the incremental cost test, combinations of outputs may not For example, suppose

N D f1; 2; 3g It is possible that the incremental cost test can be satisfied for every single

good, i.e for T D fi g, for all i , but not for T D f2; 3g This could happen if there is a fixed

common cost associated with services 2 and 3, in addition to their individual incrementalcosts, and each such service is priced at its incremental cost Thus, the tests can be difficult

to verify in practice

In defining subsidy-free prices we assumed that services are sold in large known

quantities (the x is in (7.4) and (7.5)) using uniform prices, as happens when incumbentcommunications firms supply the market In practice, individual customers consume small

parts of each x i and a coalition of customers may feel that it can ‘self-produce’ its servicerequirements at lower cost In this case, it is reasonable to require (7.2) and (7.3) Clearly,such a ‘consumer subsidy-free’ price condition imposes restrictions on the cost function Forinstance, imagine a single service has a cost function with increasing average cost Sellingthe service at its average cost price violates (7.3) if individual customers request less than

the total that is produced, although (7.4) and (7.5) are trivially satisfied for N D f1g An appropriate definition is the following Let us now write c x/ as the cost of providing

services in quantities .x1; : : : ; x n / We say the vector p is a support price for c at x if it

satisfies the two conditions

Note these implyP

i 2N p i x i Dc x/ We can compare them to (7.2) and (7.3) For example,

(7.6) implies that one cannot produce some of the demand for less than it is sold Theyimply (7.4) and (7.5) (but are more general since they deal with arbitrary sub-quantities of

the vector x, instead of looking just at subsets of service types), and hence a support price

has all the nice fairness properties mentioned above A last concern is whether such prices

are achievable in the market, where demand is a function of price Suppose p is the vector

of support prices for x and, moreover, x is precisely the quantity vector that is demanded

at price p We call such prices anonymously equitable prices Clearly, if they exist, these

have a very good theoretical claim for being an intelligent choice of cost-based prices

If prices affect demand

By allowing demand to depend upon price, we introduce subtle complications Customersmay feel badly treated even if the incremental cost test in (7.7) is passed For example,

if two services are substitutes then introducing one of them as a new service can reducethe demand for the other and the revenue it produces Prices may have to increase if weare still to cover costs and this could mean that the price of the pre-existing service has

to increase This runs counter to what we expect: that adding a new service should allowprices of pre-existing services to decrease because of economies of scope in facility andequipment sharing If the prices of pre-existing services increase then customers of theseservices will feel that they are subsidizing the cost of the new service

To see this, let T be a subset of N , and define p0

i D 1, i 2 T , and p0

i D p i , i 62 T Thus, under price vector p0 we do not sell any of the services in T (because their prices are infinite) If services in T are substitutes for those in N n T , then we can have, (recalling

were happy when only services in N n T were offered, rather than only making charges to customers who purchase services in T These former set of customers may feel that they

are subsidizing the later set of customers, and that these new services decrease the overallefficiency of the system We conclude that, as a matter of fairness between customers, thesecond test condition (7.7) should take account of demand, and reason in terms of the netincremental revenue produced by an additional service, taking account of the reduction of

revenue from other services In other words, services are fairly priced if when service i is offered at price p i the customers of the other services feel that they benefit from service i

They are happy because the prices of the services they want to buy decrease This is called

the net incremental revenue test Let us look at an example.

Example 7.2 (Net incremental revenue test) Suppose a facility costs C and there is no

variable cost It initially produces a single service 1 in quantity x1Da at price p1DC =a Then, a new service is added, at no extra cost, and at a price p2that is just a little morethan 0 As a result, demand for service 2 increases at the expense of demand for service

1 To cover the cost, p must increase, making even more customers switch to service 2

168 COST-BASED PRICING

At the end, suppose that an equilibrium is reached where p1 D 10C=a, x1 D 0:1a and

x2 D 0:9a C b Note that, by our previous definition, these prices are subsidy-free, and(almost) all the revenue is collected by charging for service 1 These customers (the onesleft using service 1) are right to complain that they subsidize service 2, since they see theirprices increase after the addition of the new service Indeed, choosing such a low pricefor service 2 results in an overall revenue reduction if prices of existing services are not

allowed to increase A fair price would be to choose p2 in such a way that the overall

net revenue (keeping the other prices, i.e p1, fixed) would increase Then, the zero profitcondition may be achieved by reducing the other prices and hence benefiting the customers

of the other services In our example, suppose that by setting p2 D p1 and keeping p1at

its initial value, x1 becomes a =2 and x2Da =2 C b=2 In other words, half the customers

of service 1 find service 2 to suit them better at the same price, and so switch There arealso new customers that like to use service 2 at that price Then the net revenue increase

becomes p1b=2 > 0; so it is possible to decrease p1 and allow customers of service 1 tobenefit from the addition of service 2

Finally, consider a model of potential competition Imagine an incumbent firm sets prices

to cover costs at the demanded quantities, i.e

X

i 2N

Suppose a competitor having the same cost function as the incumbent tries to take away

part of the incumbent’s market by posting prices p0 which are less for at least one service

Suppose x E p; p0/ is the demand for the services provided by the new entrant when he

and the incumbent post prices p0 and p respectively Suppose that there is no p0 and x0

such that

X

i 2N

p i0x i0½c x0/ ; and p0

i < p i for some i ; and x0 x E p; p0/ (7.9)

That is, there is no way that the potential entrant can post prices that are less than theincumbent’s for some services and then serve all or part of the demand without incurring

loss Prices satisfying this condition are called sustainable prices We have yet one more

‘fairness test’ by which to judge a set of prices

The above model motivates the use of sustainable prices in contestable markets A

market is contestable when low cost ‘hit-and-run’ entry and exit are possible, without

giving enough time to the incumbent to react and adjust his prices or quantities hesells Such low barrier to entry is realized by using new technologies such as wireless,

or when the regulator prescribes that network elements can be leased from incumbents

at cost

In the idea of sustainable prices we again see that price stability is related to efficiency

If prices are sustainable, a new entrant cannot take away market share if his cost function

is greater than that of the incumbent Hence sustainable prices discourage inefficient entry.However, if a new entrant is more efficient than the incumbent, and so has a smallercost function, then he can always take away some of the incumbent’s market share byposting lower prices Thus an incumbent cannot post sustainable prices if he operates withinefficient technologies

It can be shown that for his prices to be sustainable, an incumbent firm must fulfil aminimum of three necessary conditions:

1 He must operate with zero profits.

2 He must be a natural monopoly (exhibit economies of scale) and produce at minimumcost

3 His prices for all subsets of his output must be subsidy free, i.e fulfil the stand-aloneand incremental cost tests

The last remark provides one more motivation to use the subsidy-free price tests to detectpotential problems with a given set of prices

Ramsey prices

Unfortunately, there is no straightforward recipe for constructing sustainable prices.Constructing socially optimal prices that are sustainable is even harder However, underconditions that are frequently encountered in communications, Ramsey prices can besustainable Recall that Ramsey prices maximize social welfare under the constraint ofrecovering cost Again we see a connection between competition and social efficiency: in

a contestable market, i.e under potential competition, incumbents will be motivated to use

prices that maximize social efficiency with no need of regulatory intervention

However, Ramsey prices are not always sustainable They are certainly not sustainable

if any service, say service 1, is priced below its marginal cost and there are economies ofscale To see this, note that revenue from service 1 does not cover its own incremental cost

since by concavity of the cost function x1p1 < x1@c=@x1 < c.x/  c 0; x2; : : : ; x n// So

a supplier who competes on the same set of services and with the same cost function canmore than cover his costs by electing not to produce service 1 After doing this, he canslightly lower the prices of all the services that are priced above their marginal costs, so as

to obtain all that demand for himself and yet still cover his costs

Example 7.3 (Ramsey prices may not be sustainable) Whether or not Ramsey prices

are sustainable can depend on how services share fixed costs, i.e., on the economies ofscope Consider a market in which there are customers for two services The producer’scost function and demand functions for the services are

c x1; x2/ D 25x1 =2

1 C20x1=2

2 CF ; x1.p/ D x2.p/ D 104

.10 C p/2

The Ramsey prices are shown in Table 7.1 When the fixed cost F is 6 the Ramsey prices

are not sustainable even though they exceed marginal cost The revenue from service 2 is169:45 and this is enough to cover the sum of its own variable cost and the entire fixedcost, a total of 162:76 This means that a provider can offer service 2 at a price less than theRamsey price of 2:76 and still cover his costs In fact, he can do this for any price greaterthan 2:62 However, if the fixed cost is 30 this is now great enough that it is impossible tocover costs by providing just one of the services alone at a lower price.1Hence, in this case,the Ramsey prices are sustainable The lesson is that Ramsey prices may be sustainable ifall services are priced above marginal cost and the economies of scope are great enough

1 The other possibility for a new entrant is to provide both services at lower prices But it is impossible to lower both prices and still cover costs If all prices are lower the consumer surplus must increase Since we require the producer surplus to remain nonnegative, and it was zero at our Ramsey prices, this would imply that the social welfare — which is the sum of consumer and producer surpluses — would increase; this means we could not have been at the Ramsey solution.

Example 7.4 (Common cost and sustainability of Ramsey prices) Suppose that two

services are produced with same stand-alone cost function A C bx First, consider the

case in which there is no economy of scope, and hence the total cost is the sum

of the stand-alone cost functions Since both services are produced at equal quantities

x i D x j D x we have x p i Cp j / D 2.A C bx/ which implies xp i < A C bx < xp j

But A C bx is the stand-alone cost for service j , which violates the sustainability

conditions

Now suppose that there are economies of scope and the fixed cost A is common to both services Then x p iCp j / D AC2bx, and since p i > b we obtain xp jCbx < AC2bx This implies x p j < A C bx, which is the stand-alone cost for service j Hence, the existence of

common cost is vital for Ramsey prices to be sustainable Observe that, in this particular

case, any amount of common cost, A, will make Ramsey prices sustainable In general, as suggested by Example 7.3, large values of A ensure sustainability.

7.1.3 Shapley Value

Let us now leave the subject of prices and return to the simple model at the start of the

chapter, in which cost is to be fairly shared amongst n customers The provider’s charging

algorithm could be coded in a vector function
which divides c.N/ as c1; : : : ; c n/ D

he must see similarly that customer j is at least as much disadvantaged Putting this all

together requires

i T / 
i T n f jg/ D
j T / 
j T n fig/ (7.10)

On the other hand, if
j T / 
j T n fig/ < 0, then customer j is better off because customer i is also being served Customer i might feel aggrieved unless he benefits at least as much from the fact that customer j is present But then customer j will feel

aggrieved unless he benefits at least as much from customer i ’s presence So again, we

must have (7.10)

Surprisingly, there is only one function
which satisfies (7.10) for all T  N and

i ; j 2 T It is called the Shapley value, and its value for player i is the expected incremental

cost of providing his service when provision of the services accumulates in random order

It is best to illustrate this with an example

Example 7.5 (Sharing the cost of a runway) Suppose three airplanes A, B, C share a

runway These planes require 1, 2 and 3 km to land So a runway of 3 km must be built.How much should each pay? We take their requirements in the six possible orders Cost ismeasured in units per kilometer

So they should pay for 2=6, 5=6 and 11=6 km, respectively

Note that we would obtain the same answer by a calculation based on sharing commoncost The first kilometer is shared by all three and so its cost should be allocated

as 1=3; 1=3; 1=3/ The second kilometer is shared by two, so its cost is allocated as.0; 1=2; 1=2/ The last kilometer is used only by one and so its cost is allocated as.0; 0; 1/ The sum of these vectors is 2=6; 5=6; 11=6/ This happens generally Supposeeach customer requires some subset of a set of resources If a particular resource is required

by k customers, then (under the Shapley value paradigm) each will pay one-kth of its cost.

The intuition behind the Shapley value is that each customer’s charge depends on theincremental cost for which he is responsible However, it is subtle, in that a customer ischarged the expected extra cost of providing his service, incremental to the cost of firstproviding services to a random set of other customers in which each other customer isequally to appear or not appear

The Shapley value is also the only cost sharing function that satisfies four axioms, namely,(1) all players are treated symmetrically, (2) those whose service costs nothing are chargednothing, (3) the cost allocation is Pareto optimal, and (4) the cost sharing of a sum of costs

is the sum of the cost sharings of the individual costs For example, the cost sharing of

an airport runway and terminal is the cost sharing of the runway plus the cost sharing ofthe terminal The Shapley value also gives answers that are consistent with other efficiencyconcepts such as Nash equilibrium

The Shapley value need not satisfy the stand-alone and incremental cost tests, (7.2) and

(7.3) However, one can show that it does so if c is submodular , i.e if

c T \ U/ C c.T [ U/  c.U/ C c.T / ; for all T; U  N (7.11)The reader can prove this by looking at the definition of the Shapley value and using

an equivalent condition for submodularity, that taking the members of N in any order,

172 COST-BASED PRICING

say i; j; k; : : : ; ` , we must have

c fig/ ½ c.fi; jg/  c.f jg/ ½ c.fi; j; kg/  c.f j; kg/ ½ Ð Ð Ð ½ c.N/  c.N  fig/

Note that choosing T and U disjoint shows that submodularity is consistent with c.Ð/ being

c i Dc N/ and c i c fig/ ; for all i

That is, the provider exactly covers his costs and no customer is charged more than hisstand-alone cost

We now suggest a reasonable condition that the imputation c should satisfy Suppose that for all imputations c0 and subsets T  N such that P

So if a set of customers T prefers an imputation c0(because their total charge is less), then

there is always some other set of customers U who can object because

ž under c0 the total charge they pay is more, i.e.P

i 2U c0

i >Pi 2U c i, and

ž they pay under c0 a greater increment over their stand-alone cost, c U/, than T pays under c over its stand-alone cost, c T /.

so U argues that T should not have a cost-reduction at U ’s expense.

Then c is said to be in the nucleolus (of the coalitional game) It is a theorem that

the nucleolus always exists and is a single point Thus the nucleolus is a good candidatefor being the solution to the cost-sharing problem In the runway-sharing example, thenucleolus is 1=2; 1; 3=2/ Note that it is not the same as the Shapley cost allocation of

c D 2=6; 5=6; 11=6/ The fact that c is not the nucleolus can be seen by taking T D fB; Cg and c0D.3=6; 5=6; 10=6/ There is no U that can object to this.

What would have happened if we had simultaneously tried to satisfy the conditions ofboth the nucleolus and Shapley ‘stories’? The answer is that there would be no solution.The lesson in this is that ‘fair’ allocations of cost cannot be uniquely-defined There aremany definitions we might choose, and our choice should depend on the sort of unfairnessesthat we are trying to avoid We now end this section with a final story

7.1.5 The Second-best Core

Thus far we have mostly been allocating cost without paying attention to the benefit thatcustomers obtain Surely, it is fair that a customer who benefits more should pay more Weend this section with a cost sharing problem that takes account of the benefit that customersobtain

1 2

N

coalition

Figure 7.1 The second-bestž core The monopolist fixes p s.t p>x  c x/ ½ 0, where x is the

x SDP

for all i 2 S We say p is in the second-best core if an entrant has no such possibility.

Suppose any subset of a set of customers N is free to bypass a monopolist by producing

and supplying themselves with goods, at a cost specified by the sub-additive cost function

c (which is the same as the monopolist’s cost function) This subset must choose a price with which to allocate the jointly produced goods amongst its members A price vector p is said to be in the second-best core if there is no strict subset of customers S who can choose prices p0 so that they cover the costs of their demands at price p0 and all members of S have at least the net benefit that they did under p We express this as the requirement that

P

i 2N

P

j p j x i j p/ ½ c
Pi 2N x i p/Ðand there is no S ² N , and p0 such that both

See also, Figure 7.1

We can see that from the way that second-best core prices are constructed that they are

also Ramsey prices They maximize the net benefit of the customers in the set N subject

to cost recovery, which is also what Ramsey prices do However, although Ramsey pricesalways exist for the large coalition, they may be unstable, since smaller coalitions may beable to provide incentives for customers to leave the large coalition Hence second-bestcore prices may not exist

There is a subtle difference in the assumptions underlying sustainable prices and

second-best core In the second-second-best core model a customer who is a member of a coalition S must

buy all his services from the coalition and nothing from the outside So a successful entrant

must be able to completely lure away a subset of customers, S This is in contrast to the

sustainable price model, where a customer may buy services from both the monopolist andthe new entrant

This difference means that sustainable prices are quite different to second-best core prices.Prices that are stable in the sense of the second-best core may not be stable if a customer

is allowed to split his purchases Also, prices that are not sustainable because a competitormay be able to price a particular service at a lesser price may be stable in the second-bestcore sense, since the net profit of customers that switch to the new entrant can be less Inthe second-best core model customers must buy bundles of services and the price of thebundle offered by the entrant could be more

In conclusion to this section, let us say that we have described a number of criteria bywhich to judge whether customers will see a proposed set of costs as fair, and presenting

174 COST-BASED PRICING

no incentive for bypass or self-supply Anonymously equitable prices are attractive, butthey may not exist We would not like to claim that one of these many criteria is themost practical or useful in all circumstances Rather, the reader should think of using thesecriteria as possible ways of checking what problems a proposed set of prices may or maynot be present

7.2 Bargaining games

Another approach to cost-sharing is to let the customers bargain their way to a solution

7.2.1 Nash’s Bargaining Game

Suppose that the cost of supplying x is c x/, x 2 X Here, x is the matrix x D x i j/,

where x i j is the quantity of service j supplied to customer i Customer i is to pay a portion

of the cost, c i Let us code all possible allocations of output and cost as y 2 Y , where

y D x; c1; : : : ; c n /, with x 2 X andPi c i Dc x/ Suppose that, after taking into account the cost he pays, customer i has utility at y of u i y/ The customers are to bargain their way to a choice of point u in the set U D f.u1.y/; : : : ; u n y// : y 2 Y g, which we call the bargaining set It is reasonable to suppose that U is a convex set, since if u and u0 are in

U then the utilities of any point on the line between them can be achieved (in expected value) by randomizing between u and u0

To begin, suppose that there are just two players in the bargaining game Infinite rounds of

bargaining are to take place until a point in U is agreed At the first round, player 1 proposes

that they settle for .u1; u2/ 2 U Player 2 can accept this, or make a counterproposal

.v1; v2/ 2 U at the second round Now, player 1 can accept that proposal, or make a new

proposal at the third round, and so on, until some proposal is accepted We assume that

both players know U Note that only proposals corresponding to points on the northeast boundary of U need be considered, i.e the players should restrict themselves to Pareto efficient points of U

Rounds are s minutes apart Let us penalize procrastination by saying that if bargaining concludes at the nth round, then the utility of player i is reduced by a multiplicative factor of

exp..n1/si/ If 1and2differ then the players have different urgencies to settle Notethat this game is stationary with respect to time, in the sense that at every odd numberedround both players see the same game that they saw at round 1, and at every even numberedround they see the same game that they saw at round 2 Thus player 1 can decide at round

1 what proposal he will make at every odd numbered round and make exactly the sameproposal every time, say.u1; u2/ Similarly, player 2 can decide whether he will ever acceptthis proposal, and if not, what he would propose at the even numbered rounds, say.v1; v2/.Now there is no point in player 1 making a proposal that he knows will not be accepted

So, given v2, he must choose u2 ½ es 2v2 But he need not offer more than necessary

for his proposal to be accepted, and so he does best for himself taking a u such that

u2Des 2v2 Similar reasoning from the viewpoint of player 2 implies thatv1Des 1u1

In summary,

u2Des 2v2 and v1Des 1u1 (7.12)

Let u and v be the two points on the boundary of U for which (7.12) holds A possible

strategy for player 2 is to propose.v1; v2/ and accept player 1’s proposal if and only if he

would get at least u A possible strategy for player 1 is to propose .u ; u / and accept

(u1, u2)

u2

u1U

bargaining solution point

(uˆ1, uˆ2) ( 1, 2)

u1 u2= = constant 1 , 2

Figure 7.2 Nash’s Bargaining game Two players of equal bargaining power are to settle on a

player 2’s proposal if and only if he would get at leastv1 The reader can check that this

is a pair of equilibrium strategies, in the sense that player i can do no better if he changes his strategy while player j ’s strategy remains fixed, j 6D i The equalities in (7.12) also imply that for all s,

u i andvi tend to the same value, saybu i Assuming U is a closed and convex set,bu must

be the point on the boundary of U at which u1= 1

1 u1= 2

2 is maximized Figure 7.2 illustratesthis for 1 D 2 D1 Equivalently, writing wi D1=i , this is where u 2 U maximizes

w1log u1Cw2log u2 Note that if player 1 has less urgency to settle, i.e., 1 < 2, then

he has the stronger bargaining position, which is reflected in log u1 being multiplier by a

greater weight than is log u2 There is a more subtle analysis that one can make of this game

to prove that the solution we have found is also the unique subgame perfect equilibrium

If there are more than two players, then it is reasonable to ask that at the solution point

bu D.bu1; : : : ;bu n /, we should have that for each pair i and j the values of b u i;bu j maximize

u1=i

i u1=j

j subject to u 2 U and u k Dbu k , k 6D i ; j This condition is satisfied if we take b u

as the point in U whereP

iwi log u i is maximized We will meet this again, as ‘weightedproportional fairness’, in Section 10.1

The Nash bargaining solution is usually defined withi the same for all i Additionally,

we suppose that if bargaining breaks down then the players obtain utilities d1; : : : ; d N Thesolution to the Nash bargaining game.d; U/ says that

u should be chosen in U to maximize

N

Y

i D1

The generalization in which u should maximizeQ

i u id i/wi comes from imagining that

if u is chosen then there are actuallywi players who accrue benefit u i Thus the choice of

u i affects wi players and the choice of u j affects wj players If wi > wj there is more

‘bargaining power’ influencing the choice of u i than u j There are several other ways tomotivate the solution (7.13), including the following axiomatic approach

176 COST-BASED PRICING

Let f d; U/ be a function that determines the agreement point of the bargaining game d; U/ That is, u D f d; U/ It is defined as f d; U/ D d if they cannot agree The players might at least agree that f should be consistent with the following ‘rules’ Rather surprisingly, if they do, then one can prove that (7.13) must characterize f :

1 Pareto optimality If f d; U/ D u, then there can be no v 2 U such that v ½ u

andvi > u i for at least one i In other words, the agreement point must be on the boundary of U

2 Symmetry If d1 D Ð Ð Ð Dd N and U is symmetrical about the line u1 D Ð Ð Ð Du N,

then f1.d; U/ D Ð Ð Ð D f N d; U/.

3 Linear invariance If any player, say 1, decides to define a different point as his point

of 0 utility, and/or to linearly rescale the units in which he measures his utility, thenthe bargaining solution is essentially unchanged It becomes transformed in the natural

way That is, if d0D.aCbd1; d2; : : : ; d N / and U0D f.aCbu1; u2; : : : ; u N / : u 2 Ug, then f1.d0; U0/ D a C bf1.d; U/, f j d0; U0/ D f j d; U/, j D 2; : : : ; N.

4 Independence of irrelevant alternatives If U ² U0, f d; U0/ D u and u 2 U, then

f d; U/ D u This says that if the set U is increased to U0and u is the solution within

U0, but u happens to lie in U , then it must also be the solution for the bargaining

game.d; U/.

Example 7.6 (A merger of two firms) Suppose firm 1 is a cable operator who provides

both cable local access and cable TV content Suppose firm 2 is a provider of an Internetportal service Both have customers and they intend to merge, since they expect the merger

of the two businesses to be worth more than they are separately Suppose that separately

they are worth d1 and d2, and together they will be worth d3, where d3> d1Cd2 Howmuch should the value of the new firm be distributed fairly amongst the owners of the two

firms at merger? The Nash bargaining paradigm suggests that they should receive u1; u2,where these maximize.u1d1 /.u2d2 /, subject to u1Cu2Dd3 (assuming both owners

have linear utilities) This gives u i Dd iC.d3d1d2 /=2, i D 1; 2.

For example, suppose d1D10, d2D20 and d3D40 The Nash solution is u D.15; 25/.Each gets half of the added-value Note that this is the same as in the Shapley allocation

Firm 1 would be considered to bring d1 or d3d2 depending on whether he adds value

first or second The average of these is d1C.d3d1d2/=2

7.2.2 Kalai and Smorodinsky’s Bargaining Game

Of course, there are other reasonable axioms that could be agreed Suppose rule 4 of theaxioms specifying the solution of the Nash’s bargaining game is replaced by a monotonicity

condition which says that if U is increased then no one must be worse off More precisely,

use instead the rule

5 Monotonicity Suppose U ² U0, and for all i

supfu i : u 2 U0g Dsupfu i : u 2 U g and for j 6D i

supfu j : u i ½t ; u 2 U0g ½supfu j : u i ½t ; u 2 Ug; for all t Then f d; U0/ ½ f d; U/ for all j 6D i.

This is the Kalai and Smorodinsky bargaining game It turns out that there is precisely one way to satisfy axioms 1–3 and 5 Let m i Dsupfu i : u 2 U g and m D m1; : : : ; m N/.

Then f d; U/ must be the point in U on the line joining d to m whose Euclidean distance

to m is least.

Consider again Example 7.6 The Nash solution is u D 15; 25/ The Kalai and

Smorodinsky solution is u D.16; 24/ Just as we saw in our discussion of Shapley valueand nucleolus solutions, there can be more than one solution concept Note that there is nosolution concept that obeys all the ‘reasonable’ axioms 1–5

7.3 Pricing in practice

Many methodologies have been proposed for assigning costs to services Most of themfollow basic common principles and are motivated by the requirements of fairness andstability that have been mentioned in Section 7.1 They differ in the details of how theydefine and assign costs We start with a brief overview of the practical problems andmethodologies We examine various types of cost, the accounting bases for defining costs,and methods for mapping the costs of input factors to costs of services

7.3.1 Overview

In the previous sections we have characterized the properties that prices should possess ifthey are to be stable under competition However, this has not provided us with a recipefor constructing prices In practice, we do not know the complete cost function That is,

we do not know the cost of producing any arbitrary bundle of services We know onlythe current cost of producing the bundle of services that is presently being sold Anotherpractical difficulty is that most of the cost may be common cost, which cannot be attributed

to any particular service so far as the accounting records show For example, accountingrecords may not show part of a maintenance crew’s cost as attributed to providing a video-conferencing service Usually only a small part of the total cost is comprised of factorsthat can be attributed to a single service This is a major problem when trying to constructcost-based prices

In practice, we can identify some key principles that are closely related to concepts of

fairness These include the principles of cost causation (the cost of a service should be

related as much as possible to the cost of the factors that are consumed by the service),

objectivity (the cost of the service should be related to the cost factors in an objective way), and transparency (the cost of a service should be related to the cost factors in a clear and

formulaic manner, and so that it can be easily checked for possible inconsistencies).The first two of these principles are difficult to implement since, as we have commentedabove, the accounting records usually attribute only a small part of the total cost to individualservices, and so the greatest part of the cost, i.e., the common cost, may be unattributed.One solution is to make each service pay for part of the common cost This is the FullyDistributed Cost (FDC) approach that we investigate in Section 7.3.3 Unfortunately, the

division of the common cost amongst the services is rather ad hoc Since common cost

accounts for a large proportion of the cost, prices can be ‘cooked’ in many ways, makingcertain prices artificially large or small

The definition of subsidy-free prices suggests that a reasonable way to construct theprice of a service (actually a lower bound on the price) is to calculate the incrementalcost of the service This clearly includes the directly attributable cost from the accounting