The rapid growth of data-intensive services has resulted in an explosion of the Internet traffic, and it is expected to increase at an even faster rate in the future . To accommodate this increase in traffic, and to provide better Quality of Service (QoS) for end users, Internet service providers (ISPs) need to upgrade their network infrastructure and expand capacity. This development follows the deployment of next generation networks that will induce more business interactions between service providers and content providers. For example, caching technologies has recently received increased attention from industry and academia to be a key solution for next generation networks . As it is specified in this special issue, the economics of caching will be one of important aspect in deciding the monetary interactions between the ISPs and CPs.
In terms of return on investments, the ISPs, especially the ones providing last-mile connectivity, feel that the revenue from end-users (mainly access charges) are often not enough to recoup their investment costs and they propose that the CPs share the risk by sharing part of their revenues. The CPs may also have the incentive to contribute to ISP capacity expansion, as increased capacity and better QoS trigger higher demand for content and help them earn even higher revenues (mainly from subscriptions and advertisements) . For example, in , the authors propose a model in which a Mobile Network Operator leases its edge caches to a Content Provider. This is an increasingly relevant scenario and follows proposals for deploying edge storage resources at mobile 5G networks . A recent announcement by Comcast that it will bundle Netflix subscription in its package is another example of an ISP helping a CP to increase its demand.
If a CP enters into a contract with an ISP to share its revenue in return for ISP putting efforts to improve demand for its content, the CP may want to monitor the efforts level of ISP so that the contract is honored. However such monitoring of efforts levels may not be always feasible in the Internet as there is an inherent asymmetry of information between CPs and ISPs – CPs cannot observe the exact investment (caching effort) made by the ISP, but can only observe the resulting increase in user demand for its content, which is a (random) function of the ISP’s investment. In this situation, an obvious question arises: how can this imperfect information about the ISP’s investment be used to formulate an optimal revenue sharing contract? This is a situation where a privately taken action (investment) by the ISP influences the probability distribution of the outcome (demand) for the CP.
The role of this information asymmetry between the CP and ISP is critical under such agreements, where ISP makes an investment which ultimately benefits the CP, and should be considered in the structure of the contract. Therefore, moral hazard can be applied to propose a contract in which the ISP (agent) knows that CP (principle) will pay to cover its risks, which in turn gives the ISP the incentive to make the (risky) investment. Also, the moral hazard model is proven to induce proper incentives for taking appropriate action , and this may provide fair revenue sharing between CPs and ISPs. In our work, we propose an incentivizing mechanism using the moral hazard approach in which CPs share a part of their revenues with an ISP expecting a better QoS for their content and higher revenue in return from the ISP efforts.
The classical moral hazard problem deals with a single principle and a single agent, whereas we are faced with possibility of multiple principles interacting with multiple agents. In this work, we focus on a monopolistic ISP connecting end users to multiple CPs, i.e., multiple principles and a single agent. Our interest is in determining optimal sharing mechanisms/contracts between each CP and the ISP. We distinguish two cases for the effort (investment) made by the ISP. In one case, we allow the ISP to make different amounts of effort for each CP, and in the other case, the ISP is constrained to put an equal amount of effort for all CPs. The former case corresponds to a ‘non-neutral’ regime where the ISP is allowed to differentiate between CPs, and the latter case corresponds to a ‘neutral’ regime where the ISP cannot differentiate between CPs. We compare the revenue of each players and social utilities under both the regimes and analyze which regime is preferred by the players.
We consider competition between multiple CPs that provide content of a particular type, for example, video, and aim to earn higher revenue by entering into a revenue sharing contract with the ISP. Each CP separately negotiates with the ISP knowing that the other CPs can also enter into similar negotiations with the ISP. We consider linear contracts and analyze the equilibrium sharing contracts in both the neutral and non-neutral regime. We first consider the symmetric case where all the CPs earn same revenue per unit demand. This corresponds to the case where ability of the CPs to monitize their demand is the same. We then study the asymetric case where the CPs capability to monetize their demand could be different. Our contributions and observations can be summarized as follows:
We model the competitive revenue sharing of CPs with an ISP in return for improved QoS in the moral hazard framework with multiple principles and a single agent.
We analyze the equilibrium contracts in a regime where the ISP can put a different level of effort for each CP (non-neutral) and in a regime where it is constrained to put equal efforts for all CPs (neutral).
In the symmetric case we show that all the players prefer the non-neutral regimes as their utilities are higher
In the asymmetric case we show that the CP which can better monetize the demand for its traffic always prefers the non-neutral regime whereas the CP with weaker monetization power can prefer the neutral or non-neutral regime depending on its relative monetization power.
The ISP always prefers the non-neutral regime. Moreover, social utility (defined as the sum of the average earning of all players) is also higher in the non-neutral regime.
This paper is organized as follows. In Section II we discuss the problem setup and define contracts under the neutral and non-neutral regime. We study the equilibrium contracts under neutral and non-neutral regime under the symmetric case in in Section III and under asymetric case in Section IV. Conclusions and future extensions are discussed in Section V. Proofs of all stated results can be found in the appendix.
I-a Related works
Several works [11, 12, 19, 18, 9] study the possibility of content charges by ISPs to recover investment costs. In  and , the authors investigated the feasibility of ISPs charging a content charge to CPs, and evaluate its effect by modeling the Stackelberg game between CPs and ISPs. In  and , a revenue-sharing scheme is proposed when the ISP provides a content piracy monitoring service to CPs for increasing the demand for their content. This work is extended to two ISPs competing with each other in  where only one of them provides the content piracy monitoring service. Several studies considered cooperative settlement between service providers for profit sharing [16, 15, 5] where the mechanisms are derived using the Shapely value concept.
A moral hazard framework is applied in  to study interconnection contracts between ISP and end users in the market for network transport services. However, the contract design problem between ISPs and CPs remained unaddressed in this work. In the present paper, we apply the moral hazard approach to study revenue sharing between an ISP and multiple CPs. The CPs act as the principles that enter into a contract with the ISP (agent) to improve QoS for their content. We thus end up with a multiple principle, single agent problem.
A moral hazard framework where two principals offer a contract to the same agent is studied in  where the principals can only observe correlated noisy signals of the common one-dimensional action taken by the agent, where the principals’ output is linearly increasing with the agent’s action. Our work is different from them as we consider that even though the agent (ISP) is common, it chooses different actions for each principal (CP). We also compare it with the setting when the ISP is forced to choose the same action for both CPs. Also, we consider the demands of CPs to be logarithmic in the efforts of the agent, unlike the linear case considered in . Such a demand function comes from the fact that the delay experienced by end users is exponential in the cache and not linear as assumed in  and .
A moral hazard setup is also used in  for motivating end users to participate in crowd-sourced services in one principal and several agent problem.
Ii Problem Description
We consider multiple Content Providers (CPs) and a single Internet Service Provider (ISP) that connects end users to the content of the CPs. Each CP can enter into a contract with the ISP under which the ISP agrees to offer a better quality of service on the contents of the CPs to the end users by investing in the network infrastructure  and in turn, each CP agrees to share a part of its revenue with the ISP. One example of ISP investing in network infrastructure to improve QoS is caching, where better caching efforts by the ISP for a CP’s content results in higher revenue for the CP. However, such caching decisions (effort or action) of ISP may not be directly visible to the CPs, but each CP can observe the QoS experienced by end users through demand for their contents. Thus, higher the effort (caching) by the ISP, higher will be the revenue for CP because of the price paid for their content or from advertisements (from the click-through rate) . However, the ISP’s profit maximization strategy may not be aligned with the interests of the CPs, and moreover, the ISP effort (that influence CP revenue) is not directly observable by CPs. This scenario gives rise to the Moral Hazard problem.
Let denote the number of CPs and the set of CPs. For each we denote -th CP as . The revenue of from its content is random and denoted as. The amount of efforts put by the ISP to improve the demand for the content of is quantified by a positive number denoted as . shares part of his revenue with the ISP to incentivize ISP investments. This share is determined by the outcome as sharing function , which is called contract or agreement in the moral hazard framework. Specifically, if is the realized revenue in a month for , it gives to the ISP. Then, the net revenue of is . Our goal in this work is first to design optimal sharing functions that maximize the expected net revenues of each CPs taking into account the rational behavior of the ISP and its participation constraints. We assume that the CPs offer a similar type of content and compete with each other to attract more demand. Our model with multiple CPs and a single ISP is depicted in Figure 1. In the terminology of moral hazard, the CPs are the principles, and the ISP is the agent. We thus have multiple principles and a single agent.
Demand and Revenue: The increment in demand for content of depends on the effort by the ISP to improve QoS for content and could be random. Let denote this increment in demand. We assume that the mean of grows logarithmically in (law of diminishing gains) and is given by , where is the random variation in the demand for .
The revenue generated for is proportional to the demand and given by , where is a constant that captures how each unit of demand translates to earnings. For example, could be revenue per click for when is interpreted as total number of clicks on ’s content. The expected revenue for is then .111Our model generalizes trivially to the case where i.e., the average demand scales with a CP-specific multiplicative constant; this constant can simply be absorbed into
Ii-a Utilities and Objective
For a given contract the net revenue for is . Let denote the utility function of and is typically assumed to an increasing concave function of the ’s net revenue. We assume the utility of is linear in its net revenue and set . The utility of the ISP depends on revenue-share it receives from both the CPs and also the cost involved in the efforts it puts for the CPs. The earning for the ISP from the CPs is while it incurs a total cost of , where is a positive constant. The net earnings for the ISP is then . We assume the ISP is risk-averse and set its utility as , where is a risk-averse parameter. This utility is referred to as Constant Absolute Risk Aversion (CARA) in the literature . The ISP enters into agreement with the CPs only if its expected utility is larger than a certain threshold denoted as .
The CPs compete against each other and aim to maximize their utility. The objective of taking into participation constraints of the ISP is given as follows:
Constraint (1) guarantees the ISP minimum expected utility and is called as individual rationality (IR) or the participation constraint. The second constraint (2) is the ISP’s optimization problem and is called an incentive compatibility constraint (IC). It also captures the fact that the CPs cannot observe the effort level of the ISP. Note that the Constraint in (2) may have multiple optima hence objective of each CP is optimized over all of these possibilities. The structure of the optimization problem is hierarchical and can be studied considering a Stackelberg solution concept where the principals (here each CP) can be seen as leaders and the agent (here the ISP) as the follower who plays after observing the action of the principals. In our setting, there is another level of complexity as there are multiple strategic principals. This gives rise to a static game between CPs with shared constraints.
Ii-B Linear Contracts
In the following we consider a specific type of contract where CPs share a fraction of their revenue to ISP. These contracts are of the form and are referred to as linear contracts, where for all . The ISP chooses whether to accept or reject the contract. Linear contracts are shown to be optimal in  particularly when agent has a risk averse utility which is also the case in our setting. Henceforth we denote a linear contract between the ISP and by its parameter . The expected utility of is then:
and the expected utility of the is:
The ISP’s optimization problem is to find an effort level for each CP which maximizes its expected utility. Notice that maximizing expected utility (IC constraint) is equivalent to maximizing over .
Ii-C Neutral vs Non-Neutral regime
We distinguish two scenarios based on differentiation in the efforts by the ISP for each CP. We say that the network is neutral if ISP puts the same amount of efforts for both the CPs irrespective of the revenue-share it can get from them, i.e., ISP always sets . We say that the network is non-neutral if the ISP can put different amount of effort for each CP, i.e., is permitted. Hence in the neutral regime the ISP treats each CP identically, whereas it can differentiate between them in the non-neutral regime.
Under the neutral regime and linear contracts, the IC constraint of the ISP, i.e., , simplifies to . The optimal effort level for a given contracts is given by:
The objective function of in the neutral regime can then be expressed as follows:
Under the non-neutral regime, the IC constraint of the ISP, i.e., , simplifies to . The optimial efforts level for a given contracts are:
Simplified optimization problem for each can then be expressed as the following bi-level optimization problem:
Notice that the ISP has incentive to enter into contract with the CPs only when , otherwise its net earning from the CPs is negative For any value of , the IR constraint make the strategies of the players coupled and the game can have continuum of equilibria as it is the case with general coupled constrained games . However, with continuum of equilibria we will be faced with equilibrium selection problem and a systematic comparison of CP utilities under the two regime is not possible. We thus set under which the IC constraint ensures that the IR constraint always holds and hence the objective of the CPs are no more jointly constrained. As we will see in the subsequent sections, this avoids the continuum of equilibrium. We note that even after relaxing the IR constraint the problem is still challenging to analyze, but makes it possible to compare equilibrium utilities of all players under both the regimes.
We say that a contract profile is an equilibrium if no CP has an incentive for unilateral deviation from its contract. In the following we superscript the quantities computed at equilibrium with NN and N when they are associated with non-neutral regime and neutral regime respectively.
Iii Symmetric Case
In this section we consider the symmetric case where revenue per unit demand for all the CPs is the same, i.e., In other words, the CPs are symmetric with regards to the ability to monetize their content. In this setting, we analyze the equilibrium contracts arising in the neutral as well as non-neutral regime, and the resulting surplus of the CPs and the ISP. Our results highlight, surprisingly, that even when the CPs are symmetric, the imposition of neutrality actually shrinks the surplus of all parties involved. Moreover, this ‘loss of surplus’ becomes more pronounced as the number of CPs grows.
Iii-a Non-neutral regime
In the non-neutral regime, it is easy to see that when , the interactions between each CP and the ISP are decoupled. The optimization problem in (5) for after substituting the optimal effort simplifies to:
Moreover, we note that it is only interesting to consider the case Indeed, since the monetization resulting from ISP effort for CP equals the marginal monetization is at most Thus, if it is not worthwhile for CPs to make any investments to grow the demand.
The following result characterizes the equilibrium contracts between each CP and the ISP. The contracts are expressed in terms of the LambertW function computed on its principle branch, denoted as (see ).
If the equilibrium contract between and the ISP is given by
Since in strictly increasing and it follows that when Moreover, note that equilibrium fraction of CP revenue that is shared with the ISP is a strictly decreasing function of the ratio as might be expected.
Using Theorem 1, one can easily characterize the equilibrium effort of the ISP as well as the surplus of each agent.
Assume The equilibrium effort for each put by the ISP is given by
The equilibrium surplus of is given by
Finally, the equilibrium surplus of the ISP is given by
Note that so long as the equilibrium contracts award each CP and the ISP a positive surplus.
Iii-B Neutral regime
We now consider the neutral regime. The CPs are still assumed to be symmetric, only the ISP is now constrained to make the same investment decision for all CPs, i.e., . The surplus of in this case, after substituting the optimal ISP effort simplifies to:
Since the surplus of each CP in the neutral regime depends on the actions of all CPs, we seek contract profiles that constitute a Nash equilibrium between CPs. These equilibria are characterized completely in the following theorem. As before, the only scenario of interest is
Consider the neutral regime with In this case, only symmetric Nash equilibria exist. When there are two Nash equilibria, and where
When is the only Nash equilibrium, where is given by (10).
Note that when unlike in the non-neutral regime, making no contributions to the ISP, resulting in zero suplus for all parties, is an equilibrium between the CPs. The other equilibrium, given by (10), results in a positive surplus for all parties (as is shown in the following corollary). In the remainder of this section, we will refer to this latter equilibrium as the non-zero equilibrium.
Consider the neutral regime, with Under the non-zero equilibrium:
The effort for , put by the ISP is given by
The surplus of is given by
The surplus of ISP is given by
Iii-C Neutral regime v/s Non-neutral regime
Having now characterized the equilibrium contracts and the surplus of each CP and the ISP under the neutral and the non-neutral regime, we are now in a position to compare the two. As the following result shows, the non-neutral regime is actually better for all parties as compared to the neutral regime.
Suppose and In the symmetric case, at equilibrium, the following statements hold.
CPs share a higher fraction of their revenue with the ISP in the non-neutral regime, i.e.,
The effort by the ISP for each CP is higher in the non-neutral regime, i.e.,
The surplus of each CP is higher in the non-neutral regime, i.e., for all
The surplus of the ISP is higher in the non-neutral regime, i.e., .
The above result highlights that, surprisingly, constraining the ISP to be neutral is actually sub-optimal for all parties, even when the CPs are symmetric. In other words, the non-neutral regime is actually preferable to the ISP as well as the CPs. Intuitively, the reason for this tragedy of the commons
is that the imposition of neutrality skews the payoff landscape for each CP, such that the ‘benefit’ of any additional investment it makes gets ‘shared’ across all CPs. This induces the CPs to commit smaller fractions of their revenues to the ISP, which in turn results in a lower ISP effort, and a lower demand growth for all CPs. Indeed, as we show below, this effect gets further magnified with an increase in the number of CPs.
Iii-D The effect of number of CPs
In the non-neutral regime, the interactions between the different CPs and the ISP are decoupled, implying that the impact of scaling is trivial. Thus, we now study the impact of scaling in the neutral regime, on the equilibrium ISP effort, and the surplus of each agent. Note that when the neutral and the non-neutral regime coincide. Our main result is the following.
Suppose that In the neutral regime, the non-zero equilibrium satisfies the following properties.
is a strictly decreasing function of
The effort by the ISP for each CP () is a strictly decreasing function of even though the total effort () by the ISP is a strictly increasing function of
The surplus of each CP is a strictly decreasing function of and
The surplus of the ISP is eventually strictly decreasing in and
Theorem 4 highlights that an increase in the number of CPs further exacerbates the sub-optimality of the neutral regime for the CPs as well as the ISP. As before, the explanation for this is that with increasing the surplus resulting from an additional contribution by any gets ‘split’ further, thus disincentivising the CPs from offering a significant fraction of their revenues to the ISP. The variation of ISP utility as a function of is depicted in Figure 2 for different values of . In all cases, the utility first increase for some and the decreases thereafter.
Iv Asymmetric case
In this section we study the asymmetric case where monetizing power of all the CPs need not be the same, i.e., for . Our interest in this section is to understand how disparity in the monetizing power influences preference of the players for the neutral and non-neutral regimes. We focus on the case with two CPs () and without loss of generality assume that monetizing power of is more than that of i.e., We refer to as dominant and as non-dominant.
Recall that the objective of the in the non-neutral regime can be expressed as
and in the neutral regime it can be expressed as
As discussed in Section III, the case for all is not interesting as none of the CPs would have the incentive to contribute towards ISP effort. Thus, in this section, we restrict ourselves to the case where for at least one . The following results characterize the equilibrium contracts for the neutral and the non-neutral regime.222The characterization of equilibrium contracts can actually be done for any see Appendix -E.
Iv-a Equilibrium contracts
In the non-neutral regime, the interactions between each CP and the ISP remain decoupled, and thus the equilibrium contracts follow easily from Theorem 1.
In the non-neutral regime, the equilibrium contract is as follows:
Note that when the equilibrium contract between and the ISP is not uniquely defined, since any would result in zero surplus to
Next, we characterize equilibrium contract in the neutral regime.
Consider the neutral regime, with
. If then the equilibrium
contract is given by:
When the equilibrium contract is not unique, though the outcome is that ISP effort equals zero. When the equilibrium contract is unique, and at least one CP (specifically, ) is guaranteed to contribute a constant fraction of her revenue to the ISP. Note that when there is no CP contribution in the neutral regime, even though there is in the non-neutral regime.
To interpret the equilibrium when let denote the value of that satisfies the following relation for a given
For the condition in (14) holds where revenue shared by both the CPs is strictly positive, i.e., for all . For the condition in (14) fails in which only the ’s share is strictly positive and does not share anything, i.e., and . Further, it is easy to verify that is monotonically increasing in and .
Iv-B Comparison between Neutral and Non-neutral regimes
Having characterized the equilibrium contracts in both regimes, we compare and contrast the neutral and non-neutral regimes in the remainder of this section. We begin by comparing the equilibrium contracts, followed by CP/ISP utility, social utility, and finally ISP effort.
The following proposition provides a comparison of the equilibrium contracts in both the regimes.
Fix We have
For , . Moreover, decreases in .
For , . Moreover, decreases in for .
The conclusions of Proposition 1 are summarised in the scatter plot in Fig. (a)a. Note that the non-dominant CP always contributes a smaller fraction of its revenue in the neutral regime. With the dominant CP, the contribution factor is larger in the neutral regime when the revenue rates are highly asymmetric (see the green region in Figure (a)a, and larger in the non-neutral regime when the revenue rates are symmetric (see the red region in Figure (a)a). A sufficient condition for the former is The latter observation is of course consistent with Theorem 3, which dealt with the case of perfect symmetry.
Proposition 1 also establishes monotonicity properties of the sharing contracts of in the neutral regime in for a fixed While descreasess in eventually decreasing in see Figs. (b)b and (c)c. Note that can actually be increasing with respect to when the revenue rates are nearly symmetric, in contrast with the non-neutral setting.
Iv-B2 Utility of CPs
The following proposition characterizes preference of the CPs for the neutral and non-neutral regime.
Fix an . We have
For all , prefers the non-neutral regime.
For all , prefers the neutral regime.
Figure (a)a shows the scatter plot utilities of the players in both the regimes. Note that the dominant CP has higher utility in the non-neutral regime as can be observed from the red and magenta regions. This is because in the neutral regime, the dominant CP is ‘forced’ to pay for capacity investments that also benefit the non-dominant CP. Indeed, note that in the region , the dominant CP shares a smaller fraction of its revenue in the non-neutral regime, but still ends-up with a higher utility. Interestingly, the non-dominant CP obtains a higher utility in the neutral regime when the asymmetric revenue rates are too separated (see the pink region in Fig. (a)a) A sufficient condition for this is This is of course due to the ‘subsidization’ it receives from the dominant CP. On the other hand, when the revenue rates are nearly symmetric, even the non-dominant CP prefers the non-neutral regime, once again consistent with Theorem 3. The above observations further illustrated in Figs. (b)b and (c)c.
Iv-B3 ISP utility
We next compare utility of the ISP in the non-neutral and neutral regime. For simplicity, we take ISP utility to be the expected revenue given as (ignoring the risk-sensitive utility defined before). Its value in the non-neutral regime is given by:
and in the neutral regime for all is given by:
The utility for in the neutral regime is cumbersome and we skip its expression. The following lemma demonstrates the ISPs earnings are higher in the non-neutral regime when monetization power of the dominant CP is much larger than the other, i.e., is much larger than .
There exist , such that for all the ISP’s utility is higher in the non-neutral regime.
A general comparison of ISP utility in the two regimes is not analytically tractable. We give a numerical illustration in Figure 5. As seen in the first figure, utility of ISP in the non-neutral regime is higher than in the neutral regime for all for a given and . Scatter plot in the second figure shows that this observation extends over the entire parameter range.
Iv-B4 Social Utility
The social utility in the non-neutral and neutral regimes are given, respectively, as follows:
As it is not easy to compare the social utilities analytically, we resort to numerical comparison of the utilities in Figure 6. As seen social utility in the non-neutral regime dominates that in the neutral regime for all values of for a given and . The scatter plot in the second part of the figure shows that the observation continue to hold for all parameters.
Iv-B5 Total Effort by ISP
Finally we compare the total effort by ISP for CPs in the non-neutral and neutral regime given, respectively, as follows
There exist a threshold such that total effort by ISP is higher in neutral regime than in the non-neutral. The threshold satisfies the following:
The threshold is given by following equation:
It can be seen from above equation that is monotonically increasing in .
The above lemma implies that total effort by ISP in the neutral regime becomes higher when there is high asymmetry between CP’s revenue per click rates.
V Conclusions and Regulatory Issues
We studied the problem of revenue sharing between multiple CPs and an ISP on the Internet using the moral hazard framework with multiple principles and a single agent. We compared the revenues of each player and the social utility in a regime where the ISP is forced to put equal effort for all the CPs (neutral) with a regime where there are no such restrictions (non-neutral) on the ISP. Our key take-away is that every one is better off and social utility is higher in non-neutral regime when the CPs ability to montize their demand are ‘nearly’ the same. When the there is a large disparity in the monetization power of the CPs, for the case of two CPs we showed that non-neutral regime is preferable from the standpoint of the dominant CP (with higher monetizing power), the ISP, and from the standpoint of social utility. On the other hand, the non-dominant CP is benefited by a neutrality stipulation since it gets to ‘free-ride’ on the contribution made by the dominant CP, the very reason that makes this regime less preferred by the dominat CP.
Our analysis throws up an intriguing dilemma for a regulator –enforcing neutrality brings in parity in the way ISP treat the CPs, but it worsens the social utility and pay-off of all the players compared to the neutral regime if the players act non-cooperatively. It is then interesting to study mechanisms that the regulator can use to induce cooperation among the the players so that the social utility and players pay-off are no worse than in the non-neutral regime.
-a Proof of Theorem 1
From CP optimization problem, it can be observed that for for all . Henc e no CP has an incentive to share a fraction of their revenue with the ISP and is the equilibirum. Now assume . For this case the optimial value of will be such that and the optimization problem of reduces to
The first order optimality condition then gives:
Solving the first order conditions for each CP, we get:
Using the definition of the LamebertW funtion we get
Hence we get equilibrium contract given in (6). ∎
-B Proof of Theorem 2
Recall the objective of
First assume that . In this case for any given , best response of is to set . Thus is an equilibrium.
Next consider the case . Fix an and assume for all . Then the object of simplifies to
and the best response of is to set . Hence for all is an equilibrium. We next look for a non-zero equilibrium. By symmetry, it must be such that . Further, at equilibrium it must be the case that , otherwise CPs have incentive to deviate to make their share zero. Writing the first order condition for the optimaization problem of , i.e.,
Setting we have
Simplyfying the above as earlier in the format of LambertW function we get . ∎
For the case , at equilibrium is not arise, however the equilibrium still holds. This completes the proof.
-C Proof of Theorem 3
Part 1: When and the relation holds trivially. In the range , two equilibria are possible in the neutral regime, or . If is the equilibrium, again the relation holds trivially. Consider the case when is the equilibrium for . Define and .
The limit holds as the equlibrium definition holds for all and is continuous at . Also, is monotonically increasing in forall as
Hence . It holds similarly for the case .
Part 2: Since investment decision by ISP is monotonically increasing in the share in gets fromt the CPs (from Eqns. (7) and (11), by Part it is clear that ISP make more investment in non-neutral regime as compared to neutral regime.
Part 3: In both non-neutral and neutral regime equilibrium effort, for given is .
Now, in both non-neutral and neutral regimes, each CP’s utility at equilibrium is the same function given by which is concave in and from Part 1 we have that . This implies that (seen Fig. 8)
-D Proof of Theorem 4
Part 1: Considering to be continuous variable,
Now, decreases with iff
it holds as .
Part 2: Effort of ISP for each CP is decreasing following directly as is decreasing in . The total effort of ISP is . In the following we show that for any . We have