Contract Design for Composed Services in a Cloud Computing Environment

Abstract

In this paper, we study markets in which sellers and buyers interact with each other via an intermediary. Our motivating example is a market with a cloud infrastructure where single services are flexibly combined to composed services. We address the contract design problem of an intermediary to purchase complementary single services. By using a non-cooperative game-theoretic model, we analyze the incentives for high- and low-quality composed services to be an equilibrium outcome of the market. It turns out that equilibria with low quality can be obtained in the short run and in the long run, whereas those with high quality can only be achieved in the long run. In our analysis we explicitly determine the according discount factors needed in an infinitely repeated game. Furthermore, we derive optimal contracts for the supply of high- and low-quality composed services.

This work was partially supported by the German Research Foundation (DFG) within the Collaborative Research Centre “On-The-Fly Computing” (SFB 901).

A Analytical Example

The client’s demand typically decreases if the price of the composed service increases. As composed services of high quality are more valuable we suppose that for a given price they are demanded in greater quantities than composed services of low quality. The demand functions $D_{\theta ^0}(P)=\frac{1}{\beta _{\theta ^0}^2}\frac{1}{P^2}$ for $\theta ^0\in \{L,H\}$ with $\beta _L>\beta _H>0$ have these properties. The service providers’ costs are typically increasing in the quality they provide. Linear cost functions $C^i_{\theta ^i}\left( D^i\right) =\gamma _{\theta ^i} D^i$ for $\theta ^i\in \{L,H\}$ with $\gamma _H>\gamma _L>0$ for $i=1,2$ describe the service providers’ costs. The profit from producing composed services of quality $\theta ^0$ from the optimal contracts of Proposition 4 and 5 is given by

$$\begin{aligned} PD_{\theta ^0}(P)-\left( 1+\tilde{\varepsilon }_{\theta ^0}\right) \sum _{i=1}^2 C^i_{\theta ^0}\left( D_{\theta ^0}(P)\right)&=\frac{P-2\gamma _{\theta ^0}\left( 1+\tilde{\varepsilon }_{\theta ^0}\right) }{\beta _{\theta ^0}^2 P^2} \end{aligned}$$

(A.1)

with $\tilde{\varepsilon }_L=0$ and $\tilde{\varepsilon }_H>0$. To simplify the calculations we use $\tilde{\varepsilon }_H$ as a multiplicative and not as an additive term (Proposition 5). Profit maximization yields

$$\begin{aligned}&P_{L*}^{T^{L*}*}=P_{H*}^{T^{L*}*}=P_{L*}^{T^{H*}*}=4\gamma _L \quad \text {and} \quad D_L^{T^{L*}*}=D_H^{T^{L*}*}=D_L^{T^{H*}*}=\frac{1}{16\gamma _L^2\beta _L^2} \end{aligned}$$

and for high-quality composed services

$$\begin{aligned}&P_H^{T^{H*}*}=4\left( 1+\tilde{\varepsilon }_H\right) \gamma _H \quad \text {and} \quad D_H^{T^{H*}*}=\frac{1}{16\left( 1+\tilde{\varepsilon }_H\right) ^2\gamma _H^2\beta _H^2}. \end{aligned}$$

By the choice of the parameter $\gamma _L<\gamma _H$ we observe immediately that the price the intermediary charges for high quality is strictly greater than the price he charges for low quality, whereas the effect on the quantity remains ambiguous. Figure 2(a) illustrates the client’s demand functions and Fig. 2(b) the short run profits of the OTF provider for low- and high-quality composed services.

Fig. 2.

Example for $\beta _H=\frac{1}{2}$, $\beta _L=1$, $\gamma _H=\frac{1}{3}$, $\gamma _L=\frac{1}{8}$, $\tilde{\varepsilon }_H=\frac{1}{100}$

Consider the long run quality choice of the intermediary. The conditions derived in Sect. 5.3 and Corollary 1 are for our example

$$\begin{aligned} \frac{1}{\left( 1+\tilde{\varepsilon }_H\right) \gamma _H\beta _H^2}&>\frac{1}{ \gamma _L\beta _L^2}, \end{aligned}$$

(Ex-HIGH)

$$\begin{aligned} \frac{1}{\left( 1+\tilde{\varepsilon }_H\right) \gamma _H\beta _H^2}&<\frac{1}{ \gamma _L\beta _L^2}, \end{aligned}$$

(Ex-LOW)

$$\begin{aligned} \delta _0&\ge \frac{\left( 1+\tilde{\varepsilon }_H\right) \gamma _H-\gamma _L}{\left( 1+\tilde{\varepsilon }_H\right) \gamma _H-\gamma _L+\left( 1+\tilde{\varepsilon }_H\right) ^2\gamma _H^2\beta _H^2\left( \frac{1}{\left( 1+\tilde{\varepsilon }_H\right) \gamma _H\beta _H^2}-\frac{1}{\gamma _L\beta _L^2}\right) } . \end{aligned}$$

(Ex-D0)

The three possible market situations are as follows: If (Ex-HIGH) and (Ex-D0) hold, high-quality composed services are produced. If (Ex-HIGH) holds and (Ex-D0) does not hold, low-quality composed services are produced even if high-quality composed services yield a higher profit for the intermediary if the services are traded once. If (Ex-LOW) holds, low-quality composed services are produced. Inserting the values from Fig. 2 yields that high-quality composed services are produced if $\delta _0\ge 0.66$. This means as soon as future profits are considered to be sufficient attractive high-quality composed service will be the equilibrium outcome of the market.