Serving many masters: an agent and his principals

Abstract

While there is vast literature on principal-agent service contracts in which a principal pools the service capacities of multiple agents for economy of scale, here we focus on the case that exists in practice of an agent pooling multiple principals. Since it is reasonable to presume that an agent of good standing attracts multiple contract offers, his main strategic decision is to select his principals. It is generally known that a principal can extract all economic surplus from a risk-neutral agent while the agent breaks even. However, this is not the case for an agent contracting multiple principals while accounting for their interdependent failure characteristics. In this paper we describe a methodology that enables an agent to calculate the value of each potential principal and therefore to contract a Pareto optimal subset of principals in a market where neither principals’ nor agents’ collusion is allowed. Unfortunately, computational intractability of first order analysis forces us to rely on a Monte Carlo simulation to understand the agent’s choice of the principals. The computation of each principal’s contribution to the agent’s welfare is enabled by a specific cooperative game of independent interest. We show that under certain conditions the agent can do better than break-even and can realize profits convexly increasing in the cardinality of the contracted principals. Our findings not only equip agents with a mathematical instrument for assessment of service contract’s financial viability but also offer agents a holistic perspective for screening principals before accepting contract offers.

Appendix

1.1 A.1 Simulation results with homogeneous principals

The simulation with homogeneous principals is characterized by the revenue rate r, failure rate $\lambda$, and the size of principals set |S|. Note that from A’s perspective, different queuing disciplines (e.g. FCFS and HOLP) have no impact on his expected profit when principals have the same revenue rate r and the same failure rate $\lambda$, because all principals propose the same contract. Therefore we ran simulations for the combinations of different values of r, $\lambda$, and |S| under FCFS queuing discipline. As mentioned in Sect. 1, our principal-agent model can be applied to various revenue generating service. In this section we take wind energy industry as an example, where renewable energy companies operate industrial wind turbines to produce electricity. According to an extensive review of the reliability of wind turbines (Pfaffel et al. 2017) the failure rate of a wind turbine falls below 1 failure per year. Given current (2018) electricity price, which ranges from 7 cents per kilowatt hour (kWh) to nearly 30 cents per kWh across different states¹, a wind turbine with a capacity of 8 to 10 megawatts² (MW) could bring in an annual revenue of $5 million to $25 million³. Therefore in our simulation, we choose the values of failure rate (in number of failures per year) and revenue rate (in million dollars per year) following the corresponding ranges reported by the wind energy industry. Specifically we choose $r\in \lbrace 10,15,20\rbrace$, $\lambda \in \lbrace 0.8,1.0\rbrace$ and $|S|\in \lbrace 2,3,4\rbrace$. The sample size of each simulation run is 30 (For each combination of revenue rate, failure rate, and size of principals set, we solve by numerical simulations the optimization problem (10) independently 30 times.) The sample mean and standard deviation (in parentheses in the tables) of the estimated optimal point (the estimated optimal values of variables $\lbrace \mu _S\rbrace \cup \lbrace \psi _{ij}\vert i<j,i,j\in S\rbrace$) and A’s estimated optimal expected profit rate ${\varPi }^*_S\equiv {\varPi }(\mu ^*_S;\mathbf{w }_S,\mathbf{p }_S,{\varPsi }^*_S)$ are reported in Tables 1 and 2. The estimated optimal values of the dependency coefficients (i.e. $\lbrace \psi ^*_{12}\rbrace$ for $|S|=2$; $\lbrace \psi ^*_{12},\psi ^*_{13},\psi ^*_{23}\rbrace$ for $|S|=3$; $\lbrace \psi ^*_{12},\psi ^*_{13},\psi ^*_{14},\psi ^*_{23},\psi ^*_{24},\psi ^*_{34}\rbrace$ for $|S|=4$) represent the agent’s desired pairwise interdependencies between all principals, and these values exhibit two important characteristics: (1) The means of all dependency coefficients are close to zero; (2) The standard deviations (reported in the parentheses following the means) are large compared to the corresponding means. These characteristics imply that A’s desired dependency coefficients cannot be significantly distinguished from zero. Therefore the simulation results imply that A prefers highly negative interdependencies between all pairs of homogeneous principals in set S (i.e. $\psi _{ij}\sim 0$ for all $i,j\in S$ and $i\ne j$). In essence highly negative interdependency implies that it is unlikely to observe more than one revenue generating unit from set S being inoperative at any time during the infinite contract horizon.

Table 1 Simulation results with homogeneous principals ($\lambda =0.8$, FCFS queuing discipline)

Table 2 Simulation results with homogeneous principals ($\lambda =1.0$, FCFS queuing discipline)

The maximum number of principals in our simulation setting is 4, because as the number of principals increases, the time it takes to numerically solve A’s optimization problem increases exponentially. Table 3 summarize the average time the CPU⁴ spends numerically solving A’s optimization problem with 2, 3, and 4 homogeneous principals under FCFS queuing discipline. In other words, given a pair of values of r and $\lambda$ it takes 60 hours of CPU time to numerically solve A’s optimization problem 30 times with 4 principals.

Table 3 Average CPU time (in minutes) of solving A’s optimization problem with various number of homogeneous principals under FCFS queuing discipline

A’s benefits from pooling the service demands of multiple principals if pooling results in a superadditive profit rate with respect to the subset of principals he contracts with. Due to the assumption of homogeneous revenue rate and failure rate, all principals in S propose the same compensation rate and the same penalty rate (i.e. $w_i=2\sqrt{r\lambda }-\lambda$ and $p_i=r$ for all $i\in S$). Therefore A does not distinguish between different principals in N and his optimal expected profit rate ${\varPi }(\mu ^*_S;\mathbf{w }_S,\mathbf{p }_S,{\varPsi }^*_S)$ only depends on the values of r, $\lambda$ and |S|. We check the superadditivity of A’s estimated optimal expected profit rate for all values of r and $\lambda$ in Tables 1 and  2. For example, for $r=10$ and $\lambda =0.8$, A’s estimated optimal expected profit rates with a set of 2, 3, and 4 homogeneous principals under FCFS queuing discipline is found in the 3rd column in Table 1: ${\varPi }^*_{S,|S|=2}=3.3388$, ${\varPi }^*_{S,|S|=3}=7.1963$ and ${\varPi }^*_{S,|S|=4}=11.2236$. Recall that from Zeng and Dror (2015) we have ${\varPi }^*_{S,|S|=1}=0$. Note that the following inequalities hold:

$$\begin{aligned}&{\varPi }^*_{S,|S|=1}+{\varPi }^*_{S,|S|=1}=0<3.3388={\varPi }^*_{S,|S|=2}\\&{\varPi }^*_{S,|S|=1}+{\varPi }^*_{S,|S|=2}=3.3388<7.1963={\varPi }^*_{S,|S|=3}\\&{\varPi }^*_{S,|S|=1}+{\varPi }^*_{S,|S|=3}=7.1963<11.2236={\varPi }^*_{S,|S|=4}\\&{\varPi }^*_{S,|S|=2}+{\varPi }^*_{S,|S|=2}=6.6776<11.2236={\varPi }^*_{S,|S|=4} \end{aligned}$$

indicating that for $r=10$ and $\lambda =0.8$, A’s estimated optimal expected profit rate is superadditive with respect to the set of principals he contracts with. In a similar way we validate that A’s estimated optimal expected profit rate is superadditive for all values of r and $\lambda$ in Tables 1 and 2.

Given superadditivity of expected profit has been determined we check if A’s estimated optimal expected profit rate is convex with respect to the set of principals he contracts with. Convexity implies superadditivity (Peleg and Sudhölter 2007). We check the convexity of A’s estimated optimal expected profit rate for all values of r and $\lambda$ in Tables 1 and 2. For example, for $r=10$ and $\lambda =0.8$, A’s estimated optimal expected profit rate with a set of 2, 3, and 4 homogeneous principals under FCFS queuing discipline are: ${\varPi }^*_{S,|S|=2}=3.3388$, ${\varPi }^*_{S,|S|=3}=7.1963$ and ${\varPi }^*_{S,|S|=4}=11.2236$, and the following inequalities hold:

$$\begin{aligned}&{\varPi }^*_{S,|S|=2}-{\varPi }^*_{S,|S|=1}=3.3388<3.8575={\varPi }^*_{S,|S|=3}-{\varPi }^*_{S,|S|=2}\\&{\varPi }^*_{S,|S|=3}-{\varPi }^*_{S,|S|=2}=3.8575<4.0273={\varPi }^*_{S,|S|=4}-{\varPi }^*_{S,|S|=3} \end{aligned}$$

indicating that for $r=10$ and $\lambda =0.8$, A’s estimated optimal expected profit rate is convex with respect to the set of principals he contracts with. In a similar way we validate that A’s estimated optimal expected profit rate is convex for all values of r and $\lambda$ in Tables 1 and 2.

Table 4 Simulation results with heterogeneous principals ($r_1=12$, $\lambda _1=0.9$, $r_2=10$, $\lambda _2=0.8$, $r_3=11$, $\lambda _3=1.1$, $r_4=8$, $\lambda _4=1.2$, FCFS queuing discipline)

1.2 A.2 Simulation results with heterogeneous principals

The simulation setting with heterogeneous principals is characterized by the values of revenue rates $r_i$, failure rates $\lambda _i$ and the queuing discipline. We ran simulations for all nonempty subsets of 4 principals with different revenue rates and different failure rates ($r_1=12$, $\lambda _1=0.9$, $r_2=10$, $\lambda _2=0.8$, $r_3=11$, $\lambda _3=1.1$, $r_4=8$ and $\lambda _4=1.2$) and the two queuing disciplines FCFS, HOLP. When the principals are heterogeneous, A distinguishes between different principals he contracts with, and chooses a HOLP queueing discipline for his optimal subset of principals. We assume that A assigns the HOLP priority order in descending order of the value $r_i/\lambda _i$, since a principal with higher revenue rate charges the agent more during her unit’s downtime under the same failure rate (recall that $p_i=r_i$). Recall that in Sect. 3.2 we assume that HOLP priority order is strict with $i<j$ implying that principal i has higher priority than principal j, and it can be verified that the configurations in our simulations satisfy such assumption ($r_1/\lambda _1=13.33>r_2/\lambda _2=12.5>r_3/\lambda _3=10>r_4/\lambda _4=6.67$). (The sample size of each simulation setting is 30, which means for each of the nonempty subsets and queuing discipline we solve by numerical simulations optimization problem (10) independently 30 times.) The sample mean and standard deviation (in parentheses in tables) of the estimated optimal point (the estimated optimal values of variables $\lbrace \mu _S\rbrace \cup \lbrace \psi _{ij}\vert i<j,i,j\in S\rbrace$) and A’s estimated optimal expected profit rate ${\varPi }^*_S\equiv {\varPi }(\mu ^*_S;\mathbf{w }_S,\mathbf{p }_S,{\varPsi }^*_S)$ are reported in Tables 4 and 5.

Table 5 Simulation results with heterogeneous principals ($r_1=12$, $\lambda _1=0.9$, $r_2=10$, $\lambda _2=0.8$, $r_3=11$, $\lambda _3=1.1$, $r_4=8$, $\lambda _4=1.2$, HOLP queuing discipline)

The estimated optimal values of dependency coefficients $\psi _{ij}^*$, $i<j$, $i,j\in S$ represent A’s desired pairwise interdependencies between all principals in subset S. Similar to the simulation results with homogeneous principals, these values exhibit two important characteristics: (1) The means of all dependency coefficients are close to zero; (2) The standard deviations (reported in the parentheses following the means) are large compared to the corresponding means. These characteristics imply that A’s desired dependency coefficients cannot be significantly distinguished from zero, thus the simulation results imply that A prefers highly negative interdependencies between all pairs of heterogeneous principals in set S (i.e. $\psi _{ij}\sim 0$ for all $i,j\in S$ and $i\ne j$).

The maximum number of principals in our simulation setting is 4, because as the number of principals increases, the time it takes to numerically solve A’s optimization problem increases exponentially. Tables 6 and 7 summarize the average time the CPU⁵ spends numerically solving A’s optimization problem with 2, 3, and 4 heterogeneous principals under FCFS and HOLP queuing disciplines. In other words, given the values of r and $\lambda$ of 4 principals, it takes nearly 57 hours of CPU time to numerically solve A’s optimization problem 30 times under FCFS queuing discipline, and nearly 70 hours of CPU time to numerically solve A’a optimization problem 30 times under HOLP queuing discipline.

Table 6 Average CPU time (in minutes) of solving A’s optimization problem with various number of heterogeneous principals under FCFS queuing discipline

Table 7 Average CPU time (in minutes) of solving A’s optimization problem with various number of heterogeneous principals under HOLP queuing discipline

We check if A’s estimated optimal expected profit rate is convex with respect to the set of principals he contracts with. It is trivial if $T_1=T_2\subseteq S\setminus \lbrace i\rbrace$ since ${\varPi }^*_{T_1\cup \lbrace i\rbrace }-{\varPi }^*_{T_1}={\varPi }^*_{T_2\cup \lbrace i\rbrace }-{\varPi }^*_{T_2}$, therefore we focus on cases where $T_1\subset T_2\subseteq S\setminus \lbrace i\rbrace$. For example, we check the convexity of A’s estimated optimal expected profit rate in Table 4 when $i=1$ as follows:

$$\begin{aligned}&{\varPi }^*_{\lbrace 1,2\rbrace }-{\varPi }^*_{\lbrace 2\rbrace }=3.5648<4.3667={\varPi }^*_{\lbrace 1,2,3\rbrace }-{\varPi }^*_{\lbrace 2,3\rbrace }\\&{\varPi }^*_{\lbrace 1,2\rbrace }-{\varPi }^*_{\lbrace 2\rbrace }=3.5648<4.2841={\varPi }^*_{\lbrace 1,2,4\rbrace }-{\varPi }^*_{\lbrace 2,4\rbrace }\\&{\varPi }^*_{\lbrace 1,2\rbrace }-{\varPi }^*_{\lbrace 2\rbrace }=3.5648<4.6022={\varPi }^*_{\lbrace 1,2,3,4\rbrace }-{\varPi }^*_{\lbrace 2,3,4\rbrace }\\&{\varPi }^*_{\lbrace 1,3\rbrace }-{\varPi }^*_{\lbrace 3\rbrace }=4.0189<4.3667={\varPi }^*_{\lbrace 1,2,3\rbrace }-{\varPi }^*_{\lbrace 2,3\rbrace }\\&{\varPi }^*_{\lbrace 1,3\rbrace }-{\varPi }^*_{\lbrace 3\rbrace }=4.0189<4.3178={\varPi }^*_{\lbrace 1,3,4\rbrace }-{\varPi }^*_{\lbrace 3,4\rbrace }\\&{\varPi }^*_{\lbrace 1,3\rbrace }-{\varPi }^*_{\lbrace 3\rbrace }=4.0189<4.6022={\varPi }^*_{\lbrace 1,2,3,4\rbrace }-{\varPi }^*_{\lbrace 2,3,4\rbrace }\\&{\varPi }^*_{\lbrace 1,4\rbrace }-{\varPi }^*_{\lbrace 4\rbrace }=3.7287<4.2841={\varPi }^*_{\lbrace 1,2,4\rbrace }-{\varPi }^*_{\lbrace 2,4\rbrace }\\&{\varPi }^*_{\lbrace 1,4\rbrace }-{\varPi }^*_{\lbrace 4\rbrace }=3.7287<4.5265={\varPi }^*_{\lbrace 1,3,4\rbrace }-{\varPi }^*_{\lbrace 3,4\rbrace }\\&{\varPi }^*_{\lbrace 1,4\rbrace }-{\varPi }^*_{\lbrace 4\rbrace }=3.7287<4.6022={\varPi }^*_{\lbrace 1,2,3,4\rbrace }-{\varPi }^*_{\lbrace 2,3,4\rbrace }\\&{\varPi }^*_{\lbrace 1,2,3\rbrace }-{\varPi }^*_{\lbrace 2,3\rbrace }=4.3667<4.6022={\varPi }^*_{\lbrace 1,2,3,4\rbrace }-{\varPi }^*_{\lbrace 2,3,4\rbrace }\\&{\varPi }^*_{\lbrace 1,2,4\rbrace }-{\varPi }^*_{\lbrace 2,4\rbrace }=4.2841<4.6022={\varPi }^*_{\lbrace 1,2,3,4\rbrace }-{\varPi }^*_{\lbrace 2,3,4\rbrace }\\&{\varPi }^*_{\lbrace 1,3,4\rbrace }-{\varPi }^*_{\lbrace 3,4\rbrace }=4.5265<4.6022={\varPi }^*_{\lbrace 1,2,3,4\rbrace }-{\varPi }^*_{\lbrace 2,3,4\rbrace } \end{aligned}$$

We validate in the same manner that A’s estimated optimal expected profit rates in Tables 4 and  5 for $i=1,2,3,4$ are all convex.

1.3 A.3 Simulation of the Louderback’s value

We resort to numerical simulation to demonstrate that the Louderback’s value has a very high chance to be in the core of $(N_A,v)$ when the principals are heterogeneous. By the definitions of the value function (15) and the Louderback’s value (17) we note that they are functions of the revenue rates $\lbrace r_i,i\in N\rbrace$ and the failure rates $\lbrace \lambda _i,i\in N\rbrace$ of the principals in set N. Therefore the parameters of our simulations are the player set N, the failure rates $\lbrace \lambda _i,i\in N\rbrace$ and the revenue rates $\lbrace r_i,i\in N\rbrace$. We restrict the value of |N| to be the integers in interval [2, 15]. Furthermore note that for any principal $i\in N$, her revenue rate and failure rate always appear as a single term $r_i\lambda _i$. Therefore without loss of generality in each round of our simulation we set $\lambda _i=1$ and $r_i\in {{\mathcal {U}}}(5,25)$ for all $i\in N$, where ${{\mathcal {U}}}(\alpha ,\beta )$ is the uniform distribution with $\alpha$ and $\beta$ as its minimum and maximum values respectively. For each value of $|N|+1$ we run the simulation for 100000 rounds, where in each round we draw a sample of $\lbrace r_i,i\in N\rbrace$ independently from the uniform distribution ${{\mathcal {U}}}(5,25)$, calculate v(S) and $a^L(S)$ for all $S\subseteq N_A$ and check whether $a^L\in {{\mathcal {C}}}(N_A,v)$ by definition. We report the frequency of the Louderback’s value being in the core of $(N_A,v)$ for each value of |N| in Table 8.

Table 8 The frequency of $a^L\in {{\mathcal {C}}}(N_A,v)$ for different values of |N|

1.4 A.4 Proof of Lemmas and Theorems

Lemma A.1

$\sqrt{x}+\sqrt{y}-\sqrt{x+y}>0$, $\forall$$x>0$ and $y>0$.

Proof

Let $x>0$ and $y>0$. $2\sqrt{xy}>0\Leftrightarrow x+2\sqrt{xy}+y>x+y\Leftrightarrow \sqrt{x}+\sqrt{y}-\sqrt{x+y}>0$. $\square$

Lemma A.2

$\sqrt{y+a}-\sqrt{y}>\sqrt{x+a}-\sqrt{x}$, $\forall$$x>y>0$ and $a>0$.

Proof

Let $a>0$. Define $f(t)=\sqrt{t+a}-\sqrt{t}$ for $t>0$. Note that $f'(t)=1/2\sqrt{t+a}-1/2\sqrt{t}<0$. Let $x>y>0$, therefore $\sqrt{y+a}-\sqrt{y}>\sqrt{x+a}-\sqrt{x}$. $\square$

Proof (Theorem 3)

Note that the game $(N_A,v)$ is convex if and only if for all $i\in N_A$ and for all S, T such that $S\subseteq T\subseteq N_A\setminus \lbrace i\rbrace$, we have $v(S\cup \lbrace i\rbrace )-v(S)\le v(T\cup \lbrace i\rbrace )-v(T)$. It is easy to see that given $i\in N_A$ and $S=T\subseteq N_A\setminus \lbrace i\rbrace$, we have $v(S\cup \lbrace i\rbrace )-v(S)=v(T\cup \lbrace i\rbrace )-v(T)$. Therefore we exhaustively examine the remaining cases where $i\in N_A$ and $S\subset T\subseteq N_A\setminus \lbrace i\rbrace$. Table 9 helps illustrate the proof by highlighting the cardinality of set S and T in different cases and subcases.

Table 9 A summary of the cases in the proof of Theorem 3 where $i\in N_A$ and $S\subset T\subseteq N_A\setminus \lbrace i\rbrace$

• Case 1:$\varvec{i\in N_A}$ and $\varvec{S\subset T\subseteq N_A\setminus \lbrace i\rbrace }$ where $\varvec{i\ne A}$, $\varvec{S\not \ni A}$ and $\varvec{T\not \ni A}$. This is the case where i is a principal and both set S and T consist of principals only. From (15) $v(S\cup \lbrace i\rbrace )-v(S)=v(T\cup \lbrace i\rbrace )-v(T)=0$.

• Case 2:$\varvec{i\in N_A}$ and $\varvec{S\subset T\subseteq N_A\setminus \lbrace i\rbrace }$ where $\varvec{i=A}$, $\varvec{S\not \ni A}$ and $\varvec{T\not \ni A}$. This is the case where i is the agent, and set S and set T consist of principals only. We exhaustively examine the following subcases.

  • Subcase 2.1:$\varvec{S=\emptyset }$ and $\varvec{\lbrace j\rbrace =T\subseteq N}$. From (15) $v(S\cup \lbrace i\rbrace )=v(S)=v(T\cup \lbrace i\rbrace )=v(T)=0$, therefore $v(S\cup \lbrace i\rbrace )-v(S)=v(T\cup \lbrace i\rbrace )-v(T)$.

  • Subcase 2.2:$\varvec{S=\emptyset }$ and $\varvec{T\subseteq N}$ where $\varvec{|T|\ge 2}$. From (15) $v(S\cup \lbrace i\rbrace )=v(S)=v(T)=0$ and $v(T\cup \lbrace i\rbrace )=2\left( \sum _{j\in T}\sqrt{r_j\lambda _j}-\sqrt{\sum _{j\in T}r_j\lambda _j}\right)$. Since $r_j,\lambda _j>0$ for $j\in N$, from Lemma A.1, $v(T\cup \lbrace i\rbrace )>0$. Thus $v(S\cup \lbrace i\rbrace )-v(S)<v(T\cup \lbrace i\rbrace )-v(T)$.

  • Subcase 2.3:$\varvec{\lbrace j\rbrace =S\subset T\subseteq N}$. From (15) $v(S\cup \lbrace i\rbrace )=v(S)=v(T)=0$ and $v(T\cup \lbrace i\rbrace )=2\left( \sum _{k\in T}\sqrt{r_k\lambda _k}-\sqrt{\sum _{k\in T}r_k\lambda _k}\right)$. Since $r_k,\lambda _k>0$ for $k\in N$, from Lemma A.1, $v(T\cup \lbrace i\rbrace )>0$. Thus $v(S\cup \lbrace i\rbrace )-v(S)<v(T\cup \lbrace i\rbrace )-v(T)$.

  • Subcase 2.4:$\varvec{S\subset T\subseteq N}$ where $\varvec{|S|\ge 2}$. From (15) $v(S)=v(T)=0$, $v(S\cup \lbrace i\rbrace )=2\left( \sum _{j\in S}\sqrt{r_j\lambda _j}-\sqrt{\sum _{j\in S}r_j\lambda _j}\right)$, and

    $$\begin{aligned} v(T\cup \lbrace i\rbrace )=2\left( \sum _{j\in T}\sqrt{r_j\lambda _j}-\sqrt{\sum _{j\in T}r_j\lambda _j}\right) \end{aligned}$$

    Note that

    $$\begin{aligned}&v(T\cup \lbrace i\rbrace )-v(T)-\left( v(S\cup \lbrace i\rbrace )-v(S)\right) \\&\quad =2\left( \sum _{j\in T\setminus S}\sqrt{r_j\lambda _j}+\sqrt{\sum _{j\in S}r_j\lambda _j}-\sqrt{\sum _{j\in T}r_j\lambda _j}\right) \end{aligned}$$

    Since $r_j,\lambda _j>0$ for $j\in N$, according to Lemma A.1: $\sum _{j\in T\setminus S}\sqrt{r_j\lambda _j}+\sqrt{\sum _{j\in S}r_j\lambda _j}-\sqrt{\sum _{j\in T}r_j\lambda _j}>0\Rightarrow v(T\cup \lbrace i\rbrace )-v(T)>v(S\cup \lbrace i\rbrace )-v(S)$.

• Case 3:$\varvec{i\in N_A}$ and $\varvec{S\subset T\subseteq N_A\setminus \lbrace i\rbrace }$ where $\varvec{i\ne A}$, $\varvec{S\not \ni A}$ and $\varvec{T\ni A}$. This is the case where i is a principal, the subset S consists of principals only but the subset T includes the agent. We exhaustively examine the following subcases.

  • Subcase 3.1:$\varvec{S\subset T\setminus \lbrace A\rbrace =\lbrace j\rbrace \subseteq N\setminus \lbrace i\rbrace }$. From (15) $v(S\cup \lbrace i\rbrace )=v(S)=v(T)=0$ and $v(T\cup \lbrace i\rbrace )=2\left( \sqrt{r_i\lambda _i}+\sqrt{r_j\lambda _j}-\sqrt{r_i\lambda _i+r_j\lambda _j}\right)$. Since $r_i,\lambda _i,r_j,\lambda _j>0$ for $i,j\in N$, let $x=r_i\lambda _i$ and $y=r_j\lambda _j$. According to Lemma A.1: $\sqrt{r_i\lambda _i}+\sqrt{r_j\lambda _j}-\sqrt{r_i\lambda _i+r_j\lambda _j}\Rightarrow v(T\cup \lbrace i\rbrace )-v(T)>v(S\cup \lbrace i\rbrace )-v(S)$.

  • Subcase 3.2:$\varvec{S\subset T\setminus \lbrace A\rbrace \subseteq N\setminus \lbrace i\rbrace }$ where $\varvec{|T\setminus \lbrace A\rbrace |\ge 2}$. From (15) $v(S\cup \lbrace i\rbrace )=v(S)=0$ and

    $$\begin{aligned} v(T)= & {} 2\left( \sum _{j\in T\setminus \lbrace A\rbrace }\sqrt{r_j\lambda _j}-\sqrt{\sum _{j\in T\setminus \lbrace A\rbrace }r_j\lambda _j}\right) \\ v(T\cup \lbrace i\rbrace )= & {} 2\left( \sqrt{r_i\lambda _i}+\sum _{j\in T\setminus \lbrace A\rbrace }\sqrt{r_j\lambda _j}-\sqrt{r_i\lambda _i+\sum _{j\in T\setminus \lbrace A\rbrace }r_j\lambda _j}\right) \end{aligned}$$

    Since $r_i,\lambda _i,r_j,\lambda _j>0$ for $i,j\in N$, according to Lemma A.1: $\sqrt{r_i\lambda _i}+\sqrt{\sum _{j\in T\setminus \lbrace A\rbrace }r_j\lambda _j}-\sqrt{r_i\lambda _i+\sum _{j\in T\setminus \lbrace A\rbrace }r_j\lambda _j}>0\Rightarrow v(T\cup \lbrace i\rbrace )-v(T)>v(S\cup \lbrace i\rbrace )-v(S)$.

• Case 4:$\varvec{i\in N_A}$ and $\varvec{S\subset T\subseteq N_A\setminus \lbrace i\rbrace }$ where $\varvec{i\ne A}$, $\varvec{S\ni A}$ and $\varvec{T\ni A}$. This is the case where i is a principal, and set S and set T both include the agent. We exhaustively examine the following subcases.

  • Subcase 4.1:$\varvec{S\setminus \lbrace A\rbrace =\emptyset }$ and $\varvec{\lbrace j\rbrace =T\setminus \lbrace A\rbrace \subseteq N\setminus \lbrace i\rbrace }$. Note that from (15) $v(S\cup \lbrace i\rbrace )=v(S)=v(T)=0$ and $v(T\cup \lbrace i\rbrace )=2\big (\sqrt{r_i\lambda _i}+\sqrt{r_j\lambda _j}-\sqrt{r_i\lambda _i+r_j\lambda _j}\big )$. Since $r_i,\lambda _i,r_j,\lambda _j>0$ for $i,j\in N$, let $x=r_i\lambda _i$ and $y=r_j\lambda _j$. According to Lemma A.1: $\sqrt{r_i\lambda _i}+\sqrt{r_j\lambda _j}-\sqrt{r_i\lambda _i+r_j\lambda _j}>0\Rightarrow v(T\cup \lbrace i\rbrace )-v(T)>v(S\cup \lbrace i\rbrace )-v(S)$.

  • Subcase 4.2:$\varvec{S\setminus \lbrace A\rbrace =\emptyset }$, $\varvec{T\setminus \lbrace A\rbrace \subseteq N\setminus \lbrace i\rbrace }$ where $\varvec{|T\setminus \lbrace A\rbrace |\ge 2}$. From (15): $v(S\cup \lbrace i\rbrace )=v(S)=0$ and

    $$\begin{aligned} v(T)= & {} 2\left( \sum _{j\in T\setminus \lbrace A\rbrace }\sqrt{r_j\lambda _j}-\sqrt{\sum _{j\in T\setminus \lbrace A\rbrace }r_j\lambda _j}\right) \\ v(T\cup \lbrace i\rbrace )= & {} 2\left( \sum _{j\in T\cup \lbrace i\rbrace \setminus \lbrace A\rbrace }\sqrt{r_j\lambda _j}-\sqrt{\sum _{j\in T\cup \lbrace i\rbrace \setminus \lbrace A\rbrace }r_j\lambda _j}\right) \end{aligned}$$

    Note that

    $$\begin{aligned}&v(T\cup \lbrace i\rbrace )-v(T)-\left( v(S\cup \lbrace i\rbrace )-v(S)\right) \\&\quad =2\left( \sqrt{r_i\lambda _i}+\sqrt{\sum _{j\in T\setminus \lbrace A\rbrace }r_j\lambda _j}-\sqrt{r_i\lambda _i+\sum _{j\in T\setminus \lbrace A\rbrace }r_j\lambda _j}\right) \end{aligned}$$

    Since $r_i,\lambda _i,r_j,\lambda _j>0$ for $i,j,\in N$, let $x=r_i\lambda _i$ and $y=\sum _{j\in T\setminus \lbrace A\rbrace }r_j\lambda _j$. From Lemma A.1: $\sqrt{r_i\lambda _i}+\sqrt{\sum _{j\in T\setminus \lbrace A\rbrace }r_j\lambda _j}-\sqrt{r_i\lambda _i+\sum _{j\in T\setminus \lbrace A\rbrace }r_j\lambda _j}>0\Rightarrow v(T\cup \lbrace i\rbrace )-v(T)>v(S\cup \lbrace i\rbrace )-v(S)$.

  • Subcase 4.3:$\varvec{\lbrace j\rbrace =S\setminus \lbrace A\rbrace \subset T\setminus \lbrace A\rbrace \subseteq N\setminus \lbrace i\rbrace }$. According to (15) $v(S)=0$, $v(T)=2\left( \sum _{k\in T\setminus \lbrace A\rbrace }\sqrt{r_k\lambda _k}-\sqrt{\sum _{k\in T\setminus \lbrace A\rbrace }r_k\lambda _k}\right)$, $v(S\cup \lbrace i\rbrace )=2\left( \sqrt{r_i\lambda _i}+\sqrt{r_j\lambda _j}-\sqrt{r_i\lambda _i+r_j\lambda _j}\right)$, and

    $$\begin{aligned} v(T\cup \lbrace i\rbrace )=2\left( \sum _{k\in T\cup \lbrace i\rbrace \setminus \lbrace A\rbrace }\sqrt{r_k\lambda _k}-\sqrt{\sum _{k\in T\cup \lbrace i\rbrace \setminus \lbrace A\rbrace }r_k\lambda _k}\right) \end{aligned}$$

    Note that

    $$\begin{aligned}&v(T\cup \lbrace i\rbrace )-v(T)-\left( v(S\cup \lbrace i\rbrace )-v(S)\right) \\&\quad =2\left( \sqrt{r_i\lambda _i+r_j\lambda _j}-\sqrt{r_j\lambda _j}\right) \\&\qquad -2\left( \sqrt{r_i\lambda _i+\sum _{k\in T\setminus \lbrace A\rbrace }r_k\lambda _k}-\sqrt{\sum _{k\in T\setminus \lbrace A\rbrace }r_k\lambda _k}\right) \end{aligned}$$

    Since $r_i,\lambda _i,r_j,\lambda _j>0$ for $i,j\in N$, let $x=\sum _{k\in T\setminus \lbrace A\rbrace }r_k\lambda _k$, $y=r_j\lambda _j$ and $a=r_i\lambda _i$. Since $\lbrace j\rbrace =S\setminus \lbrace A\rbrace \subset T\setminus \lbrace A\rbrace$, therefore $x>y>0$. According to Lemma A.2: $\sqrt{r_i\lambda _i+r_j\lambda _j}-\sqrt{r_j\lambda _j}>\sqrt{r_i\lambda _i+\sum _{k\in T\setminus \lbrace A\rbrace }r_k\lambda _k}-\sqrt{\sum _{k\in T\setminus \lbrace A\rbrace }r_k\lambda _k}\Rightarrow v(T\cup \lbrace i\rbrace )-v(T)>v(S\cup \lbrace i\rbrace )-v(S)$.

  • Subcase 4.4:$\varvec{S\setminus \lbrace A\rbrace \subset T\setminus \lbrace A\rbrace \subseteq N\setminus \lbrace i\rbrace }$ where $\varvec{|S\setminus \lbrace A\rbrace |\ge 2}$. According to (15)

    $$\begin{aligned} v(S)&=2\left( \sum _{j\in S\setminus \lbrace A\rbrace }\sqrt{r_j\lambda _j}-\sqrt{\sum _{j\in S\setminus \lbrace A\rbrace }r_j\lambda _j}\right) \\ v(S\cup \lbrace i\rbrace )&=2\left( \sum _{j\in S\cup \lbrace i\rbrace \setminus \lbrace A\rbrace }\sqrt{r_j\lambda _j}-\sqrt{\sum _{j\in S\cup \lbrace i\rbrace \setminus \lbrace A\rbrace }r_j\lambda _j}\right) \\ v(T)&=2\left( \sum _{j\in T\setminus \lbrace A\rbrace }\sqrt{r_j\lambda _j}-\sqrt{\sum _{j\in T\setminus \lbrace A\rbrace }r_j\lambda _j}\right) \\ v(T\cup \lbrace i\rbrace )&=2\left( \sum _{j\in T\cup \lbrace i\rbrace \setminus \lbrace A\rbrace }\sqrt{r_j\lambda _j}-\sqrt{\sum _{j\in T\cup \lbrace i\rbrace \setminus \lbrace A\rbrace }r_j\lambda _j}\right) \end{aligned}$$

    Note that

    $$\begin{aligned}&v(T\cup \lbrace i\rbrace )-v(T)-\left( v(S\cup \lbrace i\rbrace )-v(S)\right) \\&\quad =2\left( \sqrt{r_i\lambda _i+\sum _{j\in S\setminus \lbrace A\rbrace }r_j\lambda _j}-\sqrt{\sum _{j\in S\setminus \lbrace A\rbrace }r_j\lambda _j}\right) \\&\qquad -2\left( \sqrt{r_i\lambda _i+\sum _{j\in T\setminus \lbrace A\rbrace }r_j\lambda _j}-\sqrt{\sum _{j\in T\setminus \lbrace A\rbrace }r_j\lambda _j}\right) \end{aligned}$$

    Since $r_i,\lambda _i,r_j,\lambda _j>0$ for $i,j\in N$, let $x=\sum _{j\in T\setminus \lbrace A\rbrace }r_j\lambda _j$, $y=\sum _{j\in S\setminus \lbrace A\rbrace }r_j\lambda _j$ and $a=r_i\lambda _i$. Since $S\setminus \lbrace A\rbrace \subset T\setminus \lbrace A\rbrace$, therefore $x>y>0$. According to Lemma A.2

    $$\begin{aligned}&\sqrt{r_i\lambda _i+\sum _{j\in S\setminus \lbrace A\rbrace }r_j\lambda _j}-\sqrt{\sum _{j\in S\setminus \lbrace A\rbrace }r_j\lambda _j}\\&\quad>\sqrt{r_i\lambda _i+\sum _{j\in T\setminus \lbrace A\rbrace }r_j\lambda _j}-\sqrt{\sum _{j\in T\setminus \lbrace A\rbrace }r_j\lambda _j}\\&\quad \Rightarrow v(T\cup \lbrace i\rbrace )-v(T)>v(S\cup \lbrace i\rbrace )-v(S) \end{aligned}$$

To summarize, $(N_A,v)$ is convex. $\square$