2. Model

There are $n \geq 2$ space agencies indexed by $[n] = \{1, \ldots, n\}$ . Each agency i has a project of workload $Q_i > 0$ and possesses resources $R_i \geq 0$ . Agencies may execute parts of projects for one another. For each ordered pair (i,j), the scalar $a_{ij} \in [0,1]$ denotes the fraction of i’s project that i delegates to j, with the convention $a_{ii} \in [0,1]$ the fraction retained by i and $\sum_{j \in [n] \setminus i} a_{ij} \leq 1$ . Delegations define a directed network with arc $i \to j$ when $a_{ij} > 0$ .

The cost of executing project work is modeled by a family of functions $f=\{\,f_{ij}:[0,\infty)\to\mathbb{R}_{\ge 0}\ :\ i,j\in[n]\,\}$ , where, for a given pair (i,j), the value $f_{ij}(q)$ is the operational cost borne by agency j to execute an amount $q\ge 0$ of workload belonging to project i. We assume that, for each (i,j), the function $f_{ij}$ is twice continuously differentiable, satisfies $f_{ij}(0)=0$ , $f'_{ij}(q)\ge 0$ and $f''_{ij}(q)\ge 0$ for all $q\ge 0$ , meaning that f is non-decreasing and has convex marginal cost.

To induce acceptance of delegated work, agency i designates nonnegative incentives $\theta_{ij}\ge 0$ for each potential partner j, under the resource feasibility $\sum_{j=1}^n \theta_{ij}\le R_i$ . If agency i fails to complete its project–either alone or with the accepted collaborations–its performance deteriorates; this deterioration is summarized by a parameter $\beta\in[0,1]$ that scales down the effective incentives it can deliver ex post. The network is enhanced as follows: the labels of each node are the resources and the workload that the corresponding agency has, and the labels of the edges are the incentives that an agency offers to a specific agency to delegate a portion of its project, see Figure 1.

Figure 1.

An interagency network.

The incentives actually transferred from i to j are defined piecewise by

\begin{equation*}x_{ij} \;=\; \begin{cases}\theta_{ij}, & \text{if project } \textit{i} \text{ is completed,}\\[6pt]\displaystyle \beta\!\left(R_i+\sum_{k=1}^n x_{kj}\right)\!\frac{\theta_{ij}}{\sum_{k=1}^n \theta_{ik}}, & \text{if project } \textit{i} \text{ is not completed,}\end{cases}\end{equation*}

with the convention that the second branch applies only when $\sum_{k} \theta_{ik}>0$ . Given $\theta=(\theta_{ij})$ , the above defines $x=(x_{ij})$ as the solution of a well-posed linear system in the non-completion case. Noteworthy the $x_{ij}$ s are not necessarily unique. However, we focus on those that point-wise maximize all corresponding incentives. Such maximal incentives exist uniquely, and can be computed in polynomial time, see Rogers and Veraart (Reference Rogers and Veraart2013).

A pure strategy for agency i is a pair $s_i=(O_i,D_i)$ , where $O_i\subseteq\{j\in[n]:a_{ji}>0\}$ lists the agencies from which i accepts delegated work, and $D_i\subseteq\{j\in[n]:a_{ij}>0\}$ lists the agencies to which i delegates work; the strategy profile is $\mathbf{s}=(s_1,\dots,s_n)$ . Acceptance is binary, so for fixed fractions $a_{ij}$ , either the offer $i\leftarrow j$ is accepted ( $j\in O_i$ ) or rejected ( $j\notin O_i$ ). In case where an agency picks only one decision the it has a pure strategy. Otherwise, it it randomizes over its pure strategies, in has a mixed strategy. Here, $s_{-i}$ represents the pure strategy profile for all agencies except i.

For a given profile $\mathbf{s}$ , the amount of i’s project executed by itself equals $\left(1-\sum_{j\in D_i} a_{ij}\right)Q_i$ , which yields own-execution cost $f_{ii}\!\left(\left(1-\sum_{j\in D_i} a_{ij}\right)Q_i\right)$ . The total work of project i executed by partner $j\in D_i$ equals $a_{ij}Q_i$ and is costed by $f_{ij}(a_{ij}Q_i)$ on j’s side; incentives $\theta_{ij}$ (or $x_{ij}$ ) are funded by i. Symmetrically, when i accepts from $j\in O_i$ a fraction $a_{ji}$ of j’s project, it executes $a_{ji}Q_j$ and bears cost $f_{ji}(a_{ji}Q_j)$ while receiving incentives $\theta_{ji}$ (or $x_{ji}$ ) from j.

The utility of agency i under $\mathbf{s}$ is

\begin{equation*}u_i(\mathbf{s}) \;=\; R_i\;-\; f_{ii}\!\left(\left(1-\!\!\sum_{j\in D_i}\! a_{ij}\right)Q_i\right)\;-\; \sum_{j\in D_i}\theta_{ij}\;+\; \sum_{j\in O_i}\theta_{ji}\;-\; \sum_{j\in O_i} f_{ji}(a_{ji}Q_j).\end{equation*}

We consider Nash equilibrium as our strategic solution concept for liquidation games. A Nash equilibrium is a strategy profile in which no agency has an incentive to unilaterally deviate, and therefore improve its utility. If such a profile is pure, we call it a pure-strategy Nash equilibrium. Formally, a pure-strategy Nash equilibrium is a profile $\mathbf{s}^\ast$ such that, for every $i\in[n]$ , $u_i(s_i^\ast,\mathbf{s}_{-i}^\ast)\;\ge\;u_i(s_i,\mathbf{s}_{-i}^\ast)$ for all admissible $s_i$ .

The individual rationality condition for accepting a specific offer $j\to i$ requires nonnegative net gain, $\theta_{ji}\;-\;f_{ji}(a_{ji}Q_j)\;\ge\;0$ , and, in a relaxed aggregate form, unconditional acceptance of all incoming offers by i is justified whenever $\sum_{j\in O_i}\theta_{ji}\;-\;\sum_{j\in O_i}f_{ji}(a_{ji}Q_j)\;\ge\;0$ .

Finally, writing $\Theta_i^{-}(\mathbf{s})=\sum_{j\in D_i}x_{ij}$ and $\Theta_i^{+}(\mathbf{s})=\sum_{j\in O_i}x_{ji}$ for the total incentives paid and received by i, one has the resource feasibility $\Theta_i^{-}(\mathbf{s})\;\le\; R_i$ in the completion case and $\Theta_i^{-}(\mathbf{s})\;\le\;\beta\Big(R_i+\sum_{k}x_{ki}\Big)$ in the non-completion case, both implied by the definition of x.

Illustrative example. Consider the interaction between two agencies $a_1$ , $a_2$ where $R_1 = 2$ , $R_2 = 1$ , $Q_1 = 1$ , $Q_2 = 3/2$ , and $\theta_{12} = \theta_{21} = b_{12} = b_{21} = 1/2$ . Furthermore, $\beta = 1$ and we also have $f_{ii}(x) = xQ_i + \varepsilon$ , with $\varepsilon > 0$ , and $f_{ij}(x) = xQ_i$ . The network is depicted in Figure 2.

Figure 2.

Interagency network in the illustrative example.

Clearly, agency $a_2$ can not provide incentives, so it can only accept or abstain from the incentives. On the other hand, if $a_1$ decides to not provide incentives, that is $\mathbf{s} = (\emptyset, \emptyset)$ , then the utilities of agents become $u_{1}(\mathbf{s}) = R_1 - f_{11}(1) = 1 - \varepsilon, \quad u_{2}(\mathbf{s}) = R_2 - f_{22}(1) = -1/2 - \varepsilon$ . Else, if $a_1$ decides to provide incentives, that is $\mathbf{s} = (\{a_2\}, \emptyset)$ , then the utilities of agencies become $u_{1}(\mathbf{s}) = R_1 - f_{11}(1 - b_{12}) - \theta_{12} = 1, \quad u_{2}(\mathbf{s}) = R_2 - f_{22}(1) + \theta_{12} - f_{12}(b_{12}) = -1/2$ . Since $\varepsilon > 0$ , agency $a_1$ improves its utility if it provides incentives to agency $a_2$ , and $a_2$ has incentives to accept them.