Optimal Platform Strategies in the Smartphone Market

MASARU UNNO1 and HUA XU21NTT Communications Corporation, Japan

2The University of Tsukuba, Japan

SUMMARY

In a smartphone market, smartphone makers encour-age smartphone application providers (APs) to create morepopular smartphone applications by making a revenue-sharing contract with APs and providing application-pur-chasing support to end users. In this paper, we studyrevenue-sharing and application-purchasing support prob-lem between a risk-averse smartphone maker and a smart-phone application provider. The problem is formulated asthe smartphone makers’ risk-sensitive stochastic controlproblem. The sufficient conditions for the existence of theoptimal revenue-sharing strategy, the optimal application-purchasing support strategy, and the incentive compatibleeffort recommended to APs are obtained. The effects of thesmartphone makers’ risk-sensitivity on the optimal strate-gies are also discussed. A numerical example is solved toshow the computation aspects of the problem. © 2013Wiley Periodicals, Inc. Electron Comm Jpn, 96(7): 1–10,2013; Published online in Wiley Online Library (wileyon-linelibrary.com). DOI 10.1002/ecj.11462

Key words: two-sided market; principal-agentproblem; risk-sensitive stochastic control.

1. Introduction

In recent years, the number of smartphone users hasincreased globally. The smartphone is a type of multifunc-tional mobile phone that comes with applications such asmail, internet connection, calendar, notepad, or camera inaddition to the original calling function. The greatest dif-ference between smartphones and earlier multifunctionalmobile phones is the fact that various applications can beadded after the purchase of the device. Smartphones canadd not only applications provided by phone manufacturers

and wireless service providers, but also applications pro-vided by a third-party application provider (AP).

We can view smartphones as a two-sided market.†

The main members of the smartphone market are smart-phone manufacturers, wireless service providers, APs, andsmartphone users. The manufacturer and the wireless serv-ice provider cooperate to build a platform. The smartphonemanufacturer is more strictly speaking the manufacturer ofthe smartphone operating system. Needless to say, the mostimportant part of the smartphone is the OS, and OS manu-facturers maintain a strong influence on the smartphonemarket. In early 2011, the main OS (and their manufactur-ers) were Symbian (Nokia), Android (Google), Research InMotion (RIM), iOS (Apple), and Windows (Microsoft).Symbian, which had the largest market share among these,has seen steep declines while the newcomer Android hasgrown its share rapidly.

Platforms in the smartphone market form a structureconsisting of several layers. The bottom layer is the wirelessservice provider, who is responsible for building and main-taining wireless networks and obtaining mobile phone us-ers. Smartphone manufacturers are one layer abovewireless network providers and they obtain APs whileserving as the intermediary between smartphone users andAPs. Such cases of unbundled platforms are prevalent inbut not limited to the telecommunications sector. For exam-ple, in the credit card business, we can view the relationshipbetween brand companies such as VISA/MasterCard, is-suer card companies, and acquirer card companies as anunbundled platform as well. Prior to the smartphone, theplatform was bundled with a wireless service provider. Thearrival of smartphones has unbundled the platform that wasformerly dominated by the wireless service providers.

© 2013 Wiley Periodicals, Inc.

Electronics and Communications in Japan, Vol. 96, No. 7, 2013Translated from Denki Gakkai Ronbunshi, Vol. 132-C, No. 3, March 2012, pp. 467–476

†A two-sided market is a market where two distinct customer groups thatcan trade between each other exist, and an intermediary is always involvedin such trade. These intermediaries can also be referred to as platforms. Ine-commerce, Rakuten or Yahoo! Malls would be examples of such plat-forms.

1

When multiple companies within a layered structurecooperate to expand business, there are issues of optimalstrategies for each layer and the optimal strategy for theentire platform, which encompasses multiple layers. Re-garding the latter problem, Unno and Xu [11] have demon-strated that by optimizing the revenue-sharing schemebetween layers it is possible for a global optimum to beattained. Therefore, in this paper, we will analyze the opti-mal strategy for the smartphone manufacturer in the toplayer of the platform.

Manufacturers such as Google and Apple each runapplication stores such as Android Market and the AppStore in order to facilitate the purchase of various applica-tions by smartphone users. These can be viewed as ECMalls that sell applications designed for a specific smart-phone. We will refer to these smartphone manufacturersthat provide these stores as Smartphone Platforms (SPF).

Previous works on two-sided markets (Rochet andTirole, 2003 [8]; Rochet and Tirole, 2006 [9]; Armstrong,2006 [1]; Hagiu, 2009 [6]; etc.) have focused on the pricestructure proposed by a monopolistic platform or competi-tive platforms to APs and smartphone users. These re-searches showed that by proposing an asymmetric pricestructure, the platform is able to collect more APs and users.In reality, many two-sided markets have such asymmetricprice structure.

On the other hand, there are also a considerablenumber of platforms that do not charge a fee for one sideof the market (the smartphone market is among these). Inthe smartphone market, the user pays the SPF only for theapplication and nothing else. The costs of the device andusage are paid to the wireless service provider. In addition,the number of smartphone users is an external condition forSPF and APs since they are obtained by the wireless serviceprovider. In previous works on two-sided market, platformshave been assumed to be able to control the number of usersthrough price structures. Therefore, it is difficult to applythe results of previous researches directly to the smartphonemarket.

In order for the SPFs to increase sales on the applica-tion stores, they must obtain high-quality APs and incen-tivize the APs to develop and sell their applications. At thesame time, they must provide support for smartphone usersto purchase more applications. This kind of support in-cludes recommendations based on the user’s taste or searchfunctionality to help users find the applications that they arelooking for. The problem for the SPF is then to determinethe right amount of incentive for the APs, the revenue-shar-ing structure to be more specific, and the right quality ofpurchase support for smartphone users.

In this paper, we will consider the revenue-sharingproblem between an SPF, which must incur the cost ofpurchase support, and the APs. We consider especially thecase of a risk-averse SPF. Normally, platforms in a two-

sided market are assumed to be risk-neutral because theyare assumed to have higher risk tolerance than other mem-bers of the market. However, it seems difficult to assumerisk-neutrality because the SPF cannot control the numberof smartphone users directly. Therefore, we will formulatethe revenue-sharing problem with a risk-averse SPF as arisk-sensitive stochastic control problem. There have notbeen many studies of the applications of the risk-sensitivestochastic control problem and this paper is also intendedto explore the applications of risk-sensitive stochastic con-trol to two-sided markets. In Section 2, we will formulatethe problem and in Section 3, we will demonstrate that theoptimal revenue-sharing strategy and purchase supportstrategy of the SPF, as well as the effort by the APs, can bedetermined as a solution to a risk-sensitive stochastic con-trol problem, and will obtain a solution through numericalsimulation. In Section 4, we will study the effect of risk-aversion of the SPF on the optimal solution. Section 5 is theconclusion.

2. Problem Formulation

We consider a smartphone market with only one SPF.The APs participating in this market agree to a contract withthe SPF by a revenue-sharing rule proposed by the SPF. Weassume that the APs are homogeneous and that the SPFcontracts with all APs under the same conditions. There-fore, without loss of generality, we can assume that thereexists only one AP.

Let the time 0 represent the time at which the SPFand AP strike a contract, and let N(0) be the number ofsmartphone users in the SPF at time 0. N(0) is known byboth SPF and AP and they strike a contract based on N(0).

The amount of users that purchase applications of theAP depends on the number of users N(0) and the effort bythe AP. As the AP makes more effort to develop applicationsand promotes sales, more users will purchase applicationson average. Both SPF and AP can observe the number ofusers who have purchased applications, but the SPF isunable to observe the AP’s effort. In addition, we assumethat the SPF will provide support to make the purchase ofthe AP’s applications easier. The better the support, thegreater is the average number of users who purchase appli-cations. However, users are fickle and there will be cases inwhich users do not purchase applications that are useful tothem, or purchase applications that are of no use to themwith no regard to the efforts by the AP or the support by theSPF (factors such as inaccurate appraisal of an application’svalue or fads).

With these points in mind, we assume that X(t), thenumber of users purchasing applications at time t, performsBrownian motion:

(1)

2

Here, Z = Z(t), F(t); 0 ≤ t < ∞ is standard Brownian mo-tion, σ is a constant, F(t); 0 < t < ∞ is a filtration onX(t); 0 ≤ t < ∞. a(t) is the development effort expendedby the AP at time t, and µ(t) is the purchase support offeredby the SPF at time t. q(a(t), µ(t)) is the attractiveness of theapplications, which is a function of a(t) and µ(t). There isan upper bound to the effort by the AP, and the domain ofpossible effort is a(t) ∈ [0, a

_]. There is also an upper bound

µ__

to the amount of support offered by the SPF. The qualityfunction q(a(t), µ(t)) is convex, strictly increasing, and con-tinuously differentiable with respect to a(t), µ(t) such that∂q(a, µ) / ∂a = qa(a, µ) < ∞, ∂q(a, µ) / ∂µ = qµ(a, µ) < ∞. Wealso assume that the SPF has knowledge of the shape of thefunction. q(a(t), µ(t)) ∈ [0, 1], (a(t), µ(t)) ∈ [0, a

_] × [0, µ

__],

and q(a(t), µ(t)) = 1 show that the application is attractiveto all users. We will assume that all users who find anapplication attractive will purchase it.

For simplicity, we normalize the price of the applica-tion to 1. Therefore, X(t) represents the cumulative sales upto time t. We assume that the SPF can fully observe thecumulative sales of the AP.†

The revenue X(t) will be distributed according to thecontract conditions between SPF and AP. Letγ(t) ≥ 0, t ∈ [0, ∞) represent the amount distributed to theAP at time t. Since the expected revenue is determined bythe attractiveness q(a(t), µ(t)) and the number of users inthe beginning N(0), and since a rational SPF will distributeonly part of the revenue, the distribution strategy γ(t) has anupper bound γ

_(t), γ(t) ≤ γ

_(t) = N(0).

The AP receives utility u(γ(t)) through distributionγ(t). We assume that the utility function u(γ(t)) is concave,increasing, continuously differentiable, u′(⋅) < ∞ and nor-malized to u(0) = 0. The AP has some reservation utility,but since we assume that there are no external opportunities,we let the reservation utility be 0. Furthermore, the APincurs a development cost h(a(t)) to maintain the develop-ment effort a(t). h(a(t)) is in the same units as u(γ(t)) and isconvex, strictly increasing, continuously differentiable,h′(⋅) < ∞. We assume that the SPF is able to observe theutility function and cost function of the AP.‡

Currently, when an AP expends a development efforta(t), 0 ≤ t < ∞, the net expected utility of the AP is

where r is the discount rate.

However, the SPF will incur a cost βdX(t) based onthe AP’s sales, where β is a constant, as well as the supportcost c(µ(t)). c(µ(t)) is strictly increasing, convex, continu-ously differentiable, and c′(⋅) < ∞. The net expected utilityof the SPF is

The SPF is risk-averse and has the following utility functionon revenue:

Here, r is the discount rate; for simplicity we assume thatit is the same as that of the AP. ρ is a constant that representsthe risk sensitivity of the SPF§ and is private information ofthe SPF.

2.1 Continuation value of AP

The AP will determine the level of development effortto maximize its own utility based on the revenue-sharingproposal of the SPF. The SPF wishes to determine a reve-nue-sharing strategy that defines a recommended level ofdevelopment effort of the AP so as to maximize the utilityof the SPF based on the utility-maximization behavior ofthe AP. It also wishes to ensure that the AP will execute therecommended level of effort.

In order to ensure the execution of the recom-mended level effort, as in Unno and Xu [11], we considervarying the amount of revenue shared based on the totalexpected utility received by the AP from time t ≥ 0onwards. That is, the distribution is varied based onthe continuation value of the AP. When we know theconditions up to time t, 0 ≤ t < ∞, an arbitrary distributionγ = γ(t) : γ(t) ∈ [0, γ

_(t)] and purchase support

µ = µ(t) : µ(t) ∈ [0, µ__

] are determined and when the APtakes an arbitrary development effort strategya = a(t) : a(t) ∈ [0, a

_] with respect to those, the continu-

ation value of AP at time t is

†This assumption is fully satisfied when the SPF is responsible for collect-ing fees from the users on behalf of the AP. Even if this is not the case, thiscondition can be satisfied by including a revenue disclosure clause in thecontract.‡In reality, the SPF can obtain information on financial conditions, feestructures, and the risk appetite of the AP during negotiations and thereforecan measure to some degree its utility function and cost function.

§If we let the utility function be u(⋅), the risk sensitivity isρ = |u′′(⋅) / u′(⋅)|.

3

Ea is the expected value under the probability meas-ure Pa when the AP’s effort is a(t). We will now derivethe optimal effort, optimal distribution, and optimal pur-chase support with the continuation value W(t; γ, µ, a) asthe only state variable. By the incentive compatibility con-straint, AP will act to maximize W(t; γ, µ, a), and the SPFcan let the AP achieve any effort level by varying the futuredistribution. Since W(t; γ, µ, a) contains information on thedevelopment effort up to time t, the development cost up totime t, the revenue X(t) as a result of those, and the historyof distributions to the AP, it is possible to incentivize the APby varying the distribution based on W(t; γ, µ, a).

We assume that the SPF can terminate the contract atany time by paying a termination fee to the AP based on thecurrent distribution scheme regardless of the continuationvalue W(t). The termination fee is determined by the con-tinuation value at the time of termination W(t). We let thevalue funct ion of the SPF at termination beΩ(W(t)) = − δγ(t), where δ is a constant and Ω(0) = 0. Sincethe AP will no longer continue the development effort aftertermination, the revenue of the AP at the time of terminationt is

When W(t) becomes extremely large, the SPF willterminate the contract with the AP. This is because eventhough W(t) increases as the revenue X(t) increases throughAP’s development effort, the development effort has anupper bound. When this bound is reached and no furtherincrease in X(t) can be expected, it can be increased furtherthrough increasing γ(t). However, when γ(t) becomes toolarge, the revenue of the SPF after distribution may becomelower than the cost of the SPF. Therefore, there exists a levelof W(t) such that it is better for the SPF to terminate thecontract through payment of Ω(W(t)). Let this value beW# > 0. If we let W

__# represent the continuation value of the

AP when the distribution is at its upper bound γ_(t), then

W# ≤ W__

#.

2.2 The SPF’s problem

The AP will not strike a contract with the SPF unlessthere exists a nonnegative utility. The SPF will also need toguarantee more utility than the lower bound demanded bythe AP (i.e., the reservation utility). This is called theparticipation constraint. Furthermore, the AP will behaveto maximize its own expected utility and the SPF will needto take into account such behavior by the AP. This is calledthe incentive compatibility constraint. The SPF’s problemis an optimization problem to determine the development

effort a(t); 0 ≤ a(t) < a_

, revenue-sharing strategyγ(t); 0 ≤ γ(t) < γ

_(t), and support strategy

µ(t); 0 ≤ µ(t) < µ__

that can maximize

subject to the incentive compatibility constraint

from the set of (γ(t), a(t)) that satisfies the participationconstraint

3. Optimal Revenue-Sharing and Optimal PurchaseSupport

In this section, we solve the optimization problem ofthe SPF described above. We have the following proposi-tion regarding the AP’s continuation value (for the proof,see Appendix 1).

Proposition 1 When the AP’s continuation value isdefined by Eq. (2), for any revenue-sharing strategyγ = γ(t), support strategy µ = µ(t), and AP’s strategya = a(t) past time t, there exists an F(t)-measurableadapted process Y(t) such that W(t; γ, µ, a) can be ex-pressed as follows:

We also have the following proposition regarding theincentive compatibility constraint on AP’s developmenteffort (for the proof, see Appendix 2).

Proposition 2 Let Y(t) be the adapted process ob-tained from Proposition 1. Then the AP’s strategy a isoptimal almost everywhere if it satisfies

By Proposition 2, if Y(t) is the process that repre-sents W(t; γ, µ, a) with respect to strategy a, then

(3)

(2)

(6)

(5)

(4)

(7)

(8)

(9)

4

Notice that y(a(t), µ(t)) is increasing with respect to a(t). InEq. (7), the σ-normalized Y(t) is the volatility of processW(t; γ, µ, a), so that greater a(t) implies greater risk for theAP.

3.1 Risk-sensitive stochastic control problem

Using the monotonicity of log functions, the SPF willsolve an equivalent maximization problem to utility maxi-mization problem (6) with the following objective function:

This allows us to measure the objective function with thesame units as the value function at termination Ω(⋅). Givenat time t the AP’s continuation value W(t), optimal devel-opment effort a(t), and appropriate γ(t) and optimal pur-chase support µ(t), we formulate the SPF’s utilitymaximization problem that satisfies the AP’s incentivecompatibility constraint as the following risk-sensitive sto-chastic control problem:

subject to

However,

and furthermore let Ψ(W) = ψ(W) − 1. Thus,

We will solve this problem using dynamic programmingand the following Hamilton–Jacobi–Bellman (HJB) equa-tion:

3.2 Solution to the risk-sensitive stochasticcontrol problem

This problem has a unique and bounded solution andwhen Eq. (15) is maximized, the development effort, thepurchase support, and revenue-sharing distribution are thesolution to the SPF’s utility maximization problem. This isguaranteed by the following proposition (for the proof, seeAppendix 3).

Proposition 3 In risk-sensitive stochastic optimalcontrol problem (11)–(12), if the value function Ψ(W(t)) isdefined by Eq. (14), the HJB equation (15) has a uniquebounded solution, and furthermore, if a revenue-sharingdistribution γ(t), purchase support µ(t), and developmenteffort a(t) that satisfy HJB equation (15) are executable int ∈ [0, ∞) for a corresponding continuation value W(t), thenγ(t), µ(t), and a(t) is the solution to the SPF’s utility maxi-mization problem.

By the monotonicity of the log function, Π(W) hasthe same properties as Ψ(W), and therefore has a uniquebounded solution. Thus, Π(W) satisfies the following HJBequation:

Here we use the results

Therefore, the SPF’s problem is reduced to finding thesolution to HJB equation (16).

The solution must also satisfy the following initialcondition, value matching condition, and smooth pastingcondition:

From Eq. (16), the optimal effort of the AP is

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

5

and from the first-order condition

we can obtain the optimal level of effort a(W(t), µ(t)) as afunction of W(t) and µ(t). Here h(a)Π′(W) is the compen-sa t ion for the AP’s development effort and(1 − β)q(a, µ)N(0) is the revenue stream.1 / 2 σ2y(a(t))2Π′′(W) is the risk premium that should bepaid to the AP for carrying on business with risk, but it isdiscounted by 1 / 2 ρ σ2y(a, µ)2(Π′(W))2 in the second term.This discount term will not be present if the SPF is risk-neutral.

Similarly, the optimal revenue-sharing strategy forthe SPF is

and from the first-order conditionΠ′(W) = −[1 / u′(γ)], γ(W(t)) is obtained as a function ofW(t). Π′(W) is the reduction in the SPF’s revenue for eachunit of increase in the AP’s continuation value,1 / u′(γ) (= dγ / du(γ)) is the distribution rate that the SPFneeds to give to the AP in order to increase the AP’s utilityby 1 unit. This means that at optimum, these two valuesmust match. If we let Π′(W∗) = 0 at point W∗, then fromu(γ) ≥ 0, the revenue-sharing strategy that maximizes Eq.(21) on the interval W ≤ W∗ is 0.

The optimal purchase support is then

and the first-order condition

can be used to obtain the optimal support µ(W(t), a(t)) as afunction of W(t), a(t).

The solution Π(W) can be obtained through numeri-cal calculation by applying the effort, revenue-sharing strat-egy, and purchase support which maximizes (20) to (22) toHJB equation (16) and solving it under boundary condi-tions (17) to (19).

Figure 1 shows the value function Π(W) as well asthe optimal effort and optimal revenue-sharing strategywhen the parameters are specified as follows:

The effort and revenue-sharing can be obtained froma = −0.9 / [Π′(W)+ ρ(Π′(W) )2 + Π′′(W)] − 0.5, √γ =−Π′(W) / 2, γ = Π′(W)2

/ 4. In this example,W∗ = 1.155, Π′(W∗) = 0.9653846 and µ was constant at µ =0.09.

In the actual contract, the SPF will propose W∗ anda(W∗) to the AP and will recommend it strongly for the APto achieve that effort. The AP will move to achieveW(t) = W∗ in the shortest possible time because at t = 0, a =0 implies W = 0 and no revenue-sharing occurs within the

(20)

(21)

Fig. 1. SPF’s value function, effort, and optimalrevenue-sharing strategy. [Color figure can be viewed in

the online issue, which is available atwileyonlinelibrary.com.]

(22)

6

periods W(t) ≤ W∗, 0 < t < ∞. Through the AP’s effort, W(t)moves along with Eq. (7), and when W∗ is reached, the APmaintains the effort level a(W∗) and the SPF distributesγ(t) to maintain W∗. At the same time, the SPF will executepurchase support µ(W(t), a(t)) based on W(t) and a(W(t)).

4. Effect of SPF’s Risk-Sensitivity

The risk-sensitivity ρ becomes larger as the SPFbecomes more risk-averse. This section will study the effectof the SPF’s risk-sensitivity on the optimal solution.

To start, at point W∗ where Π′(W∗) = 0, the ρ termsdisappear by Eqs. (20) and (22). Therefore, at point W∗, theoptimal effort and support are

respectively. Notice that to maximize the SPF’s revenue, theAP’s effort and the SPF’s support have no effect regardlessof the risk tolerance of the SPF. Of course, it should also benoted that W∗ may not be the same for different ρ.

The effect of risk sensitivity ρ on the value functionΠ(W) is not evident. Normally, a higher risk-aversion ten-dency does not demand large revenue since the utility doesnot increase much even if the revenue increases. The higherthe risk-sensitivity, the smaller is the revenue required toachieve the same utility. Therefore, we can expect a morerisk-sensitive SPF to be satisfied by revenue equal to or lessthan that of a less risk-sensitive SPF. However, this may notbe entirely correct if the SPF can transfer the risk to anotherparty. We will simulate with the previous set of parameters.

The curves in Fig. 2 show Π(W), a(W), and γ(W)when simulated with ρ = 0.05, 0.1, 0.2. Contrary to thehypothesis, larger ρ indicated larger Π(W∗) on the interval0 ≤ W ≤ W#. Also, larger ρ also showed larger Π′(0) andW∗. This result can be interpreted in the following way.

First, notice that greater Π′(0) implies greatera(W(0)) = a(0). Indeed, at t = 0 we have γ(0) = 0, µ(0) = 0so that Π′(0) is dependent on a(0) based on the definitionof Π. The same result can be observed in the simulation.Therefore, when ρ becomes larger, a(0) and W∗ also be-comes larger.

A risk-averse SPF should recommend larger W∗ tothe AP. This has at least the following benefits:

(i) It can demand a higher level of effort from the APfrom the beginning and can eliminate weaker APs.† (ii)

Since no revenue-sharing takes place until W(t) reachesW∗, a larger W∗ implies a later start of revenue-sharing.

By recommending greater W∗, the AP will be de-manded to provide a higher level of effort from the start,but it also increases the volatility of the AP’s continuationvalue W(t) [see Eq. (9)] and thus increases the risk for theAP. The AP’s risk is compensated by the risk premium paidby the SPF, but this is discounted by the second term in Eq.(20). In short, a greater W∗ allows the SPF to transfer morerisk to the AP.

This study allows us to state that even if the SPF isrisk-averse, it can transfer the risk to the AP by recommend-ing greater W∗ to the AP. Greater W∗ means a higher levelof effort of the AP at the beginning, but it also increasesΠ′(0) and as a result increases Π(W) as well.

†From (7), the AP must spend more effort to achieve greater W∗. When theeffort increases, h(a) increases as well and experiences a larger dW(t) drift.As a result, W(t) likewise increases faster.

Fig. 2. Results of risk sensitivity analysis. [Color figurecan be viewed in the online issue, which is available at

wileyonlinelibrary.com.]

7

5. Conclusions

In this paper, we have formulated the revenue-sharingproblem between a risk-averse SPF and an AP in the smart-phone market as a risk-sensitive stochastic control problemwith the AP’s continuation value as the state variable. Weshowed that this problem has a unique bounded solutionand we were able to derive the HJB equation as well as tofind a specific solution through numerical simulation.There have not been studies of two-sided markets whererisk-sensitive stochastic control have been applied, and thispaper has applied risk-sensitive stochastic control to thestudy of two-sided markets.

In addition, this paper has performed sensitivityanalysis on the SPF’s risk-aversion, and by numerical simu-lation has showed that the greater the risk-sensitivity of theSPF (i.e., the more risk-averse the SPF is), the greater theSPF’s value function. In a typical two-sided market, theplatform establishes the transaction rules and usually has astronger negotiation power over other members of themarket. In addition, due to the network effect, each membertends to become locked into a specific platform. The smart-phone market is an oligopoly of a few wireless serviceproviders in the lower layers of the platform due to thescarcity of radio wave resources. Even the SPFs in thehigher layers are limited to a few companies. Thus, plat-forms have greater dominance over APs in the smartphonemarket and therefore it is easier for platforms to transferrisks to APs. This paper’s results suggest that in a two-sidedmarket where risk transfer to the AP is possible, risk-averseplatforms strike a contract similar to that suggested in thepaper with the APs and actually transfer risks to the APsand obtain greater revenue.

This model has analyzed a monopolistic SPF, but inreality there are cases where multiple SPFs compete overthe same wireless service provider or the same SPF usesmultiple wireless service providers as the lower layer. Inaddition, when there are multiple SPFs in competition, theAP must decide whether or not to use multiple platforms.It will be an interesting problem to see how the differencein risk-sensitivity can affect the SPF’s strategy or the AP’sdecision process. We plan to consider these topics in furtherstudies.

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APPENDIX

1. Proof of Proposition 1

Suppose that the information before time t is given,and the strategies (γ, µ, a) are applied after time t. Then, theAP’s total expected payoff is

It is easy to confirm that V(t) is an Ea-martingale. Since thefiltration F(t) is the same as the σ-algebra induced by thestochastic process dZ(t) = 1 / σ[dX(t) / N(0) − q(a(t))dt],there exists a measurable process Y(t), F(t); 0 ≤ t < ∞such that

by the Martingale Representation Theorem. DifferentiatingEqs. (23) and (24) with respect to t gives

and

(23)

(24)

8

Therefore,

2. Proof of Proposition 2

Consider an arbitrary alternative strategy a~ =a~(t); a~(t) ∈ [0, a

_]. Define

as the AP’s total expected utility at time t when the APfollows the effort a~(t) before time t, and chooses the efforta(t) after time t. Then, the F~(t)-measurable stochastic proc-ess V

^(t) becomes

where σN(0)Y(t)dZ(t) = σN(0)Z~(t) + ∫0

t (q(a~(s), µ(s)) −q(a(s), µ(s)))N(0)ds. If a(t) does not meet (8) on a set ofpositive measure, we can choose a∗(t) that maximizes

Then, the drift of V^(t) is nonnegative, and becomes positive

on a set of positive measure. Therefore, there exists a timet > 0 such that

Ea∗(t)[V

^(t)] is the AP’s total expected utility when the APfollows a∗(t) until t and then switches to a(t). SinceEa

∗(t)[V^(t)] is greater than W(0; γ, µ, a), which is the AP’s

total expected utility when a(t) is chosen from time t = 0,a(t) is not optimal.

Suppose that Eq. (8) holds for the strategy a; thenV^(t) is a supermartingale for any alternative strategy a~.Moreover, since the stochastic process W(t; γ, µ, a) isbounded from the bottom, we can take

as the limit of V^(t).† Therefore,

and the strategy a is at least as good as any alternativestrategy a~.

3. Proof of Proposition 3

From the assumptions of the SPF’s problem and Eq.(9), we can find that:

(i) γ(t), µ(t), a(t) are bounded and closed sets.(ii) [rW − u(γ(t)) + h(a(t))] is bounded and continu-

ously different iable on the domain(W(t), γ(t), a(t)) ∈ [0, W#

___] × [0, γ

_(t)] × [0, a

_] and its first

derivative is also bounded. Moreover,[(1 − β)q(a(t), µ(t))N(0) − γ(t) − c(µ(t))] is bounded andcontinuously differentiable on the domain(γ(t), µ(t), a(t)) ∈ [0, γ

_(t)] × [0, µ

__] × [0, a

_] and its first de-

rivative is bounded.(iii) y(a(t), µ(t)) is continuously differentiable on the

domain (a(t), µ(t)) ∈ [0, a_] × [0, µ

__] and y(a(t), µ(t))2 > 0.

Therefore, by Theorem IV.4.1 (p. 162) of Fleming andSoner [4], there exists a unique solution to HJB equation(15).

Second, suppose that γ∗(t), µ∗(t), a∗(t) is the solutionof Eq. (13). Then, from Eq. (13), we have

†See Problem 3.16 in Ref. 7.

(25)

(26)

(27)

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Here, as ∆t → 0, (∆t)2 and ∆ψ(W(t))∆t will be approaching0 more quickly. Thus, eliminating the corresponding termsin the above gives

Thus, we arrive at the HJB equation

Since Ψ(W(t)) = ψ(W(t)) − 1, Ψ′(W(t)) = ψ′(W(t)),Ψ′′(W(t)) = ψ′′(W(t)), we have

Therefore, γ∗(t), µ∗(t), a∗(t) is the solution of Eq. (15). Thatis, the solution γ∗(t), µ∗(t), a∗(t) of Eq. (28) is the solutionof the SPF’s problem.

AUTHORS (from left to right)

Masaru Unno (nonmember) received a bachelor’s degree from the School of Political Science and Economics, WasedaUniversity, in 1986 and became an employee of NTT Corporation. He received his master’s degree in systems managementfrom the Graduate School of Business Sciences of the University of Tsukuba in 2003. He has been engaged in the developmentof network services to companies.

Hua Xu (member) received his D.Eng. degree in information engineering from Hiroshima University in 1993 and becamea research associate there. He was appointed an associate professor in 1996. Since 1998 he has been affiliated with the GraduateSchool of Business Sciences of the University of Tsukuba, first as an associate professor, and as a professor since 2003. Hiscurrent research interests include dynamic optimization, dynamic games, and their application in business areas. He is a memberof the International Society of Dynamic Games and of several academic societies in Japan.

(28)Using Ito’s Lemma, we have

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