On Risk Evaluation and Control of Distributed Multi-Agent Systems

The widely accepted axiomatic framework for coherent measures of risk was proposed in [2] and further analyzed in [8], [14], [20], [29, 30], [25] and many others works. It is worth noting that another axiomatic approach was initiated in [18] and this line of thinking was developed to an entire framework in [27]. For a detailed exposition, we refer to [31] and the references therein. Let $\mathcal{L}_{p}(\Omega,\mathcal{F},P)$ be the space of real-valued random variables, defined on the probability space $(\Omega,\mathcal{F},P)$, that have finite $p$-th moments, $p\in[1,\infty)$, and are indistinguishable on events with zero probability. We shall assume that the random variables represent random costs or losses. A lower semi-continuous functional $\varrho:\mathcal{L}_{p}(\Omega,\mathcal{F},P)\to\mathbb{R}\cup\{+\infty\}$ is a coherent risk measure if it is convex, positively homogeneous, monotonic with respect to the a.s. comparison of random variables, and satisfies the the following translation property

$$\varrho[X+a]=\varrho[X]+a\text{ for all }X\in\mathcal{L}_{p}(\Omega,\mathcal{F},P),\;a\in\mathbb{R}.$$

If $\varrho[\cdot]$ is monotonicity, convex, and satisfies the translation property, then it is called a convex risk measure. Some examples of coherent measures of risk include Average Value-at-Risk (also called Conditional Value-at-Risk) and mean-semideviations measure, which are defined as follows. The Average Value-at-Risk at level $\alpha$ for a random variable $Z$ is defined as

$$\operatorname{AVaR}_{\alpha}[Z]=\inf_{\eta\in\mathbb{R}}\Big{\{}\eta+\frac{1}{\alpha}\mathbb{E}\big{[}(Z-\eta)_{+}\big{]}\Big{\}}$$

It is a special case of the higher-order measures of risk:

$$\varrho[Z]=\min_{t\in\mathbb{R}}\bigg{\{}t+\frac{1}{\alpha}\big{\|}\big{(}Z-t\big{)}_{+}\big{\|}_{p}\bigg{\}},\quad\alpha\in(0,1),$$

where $\|\cdot\|_{p}$ refers to the norm in $\mathcal{L}_{p}(\Omega,\mathcal{F},P)$. The mean semi-deviation of order $p$ is given by

$$\varrho[Z]=\mathbb{E}[Z]+\varkappa\big{\|}\big{(}Z-\mathbb{E}[Z]\big{)}_{+}\big{\|}_{p},\quad\varkappa\in[0,1].$$

The space $\mathcal{L}_{p}(\Omega,\mathcal{F},P)$ equipped with its norm topology is paired with the space $\mathcal{L}_{q}(\Omega,\mathcal{F},P)$ equipped with the weak^∗ topology where $\frac{1}{p}+\frac{1}{q}=1$. For any $Z\in\mathcal{Z}$ and $\xi\in\mathcal{Z}^{*}$, we use the bilinear form:

$$\langle\xi,Z\rangle=\int_{\Omega}\xi(\omega)Z(\omega)dP(\omega).$$

The following result is known as a dual representation of coherent measures of risk. A proper lower semicontinuous coherent risk measure $\varrho$ has a dual representation

$$\varrho[Z]=\sup_{\xi\in\mathcal{A}_{\varrho}}\langle\xi,Z\rangle,\quad Z\in\mathcal{Z},$$

(1)

where $\mathcal{A}_{\varrho}\subset\{\xi\in\mathcal{Z}^{*}~{}|~{}\xi\geq 0,~{}\int_{\Omega}\xi(\omega)P(d\omega)=1\}$ is the convex-analysis subdifferential $\partial\varrho[0]$.

Risk measures have also been defined by specifying a set of desired values for the random quantity in question; this set is called an acceptance set. Denoting the acceptance set by $\mathcal{K}\subset\mathbb{R}$, the risk of a random outcome $Z$ is defined as:

$$\varrho_{\mathcal{K}}[X]=\inf\{m\in\mathbb{R}~{}|~{}X-m\in\mathcal{K}\}.$$

(2)

In finance, this notion of risk is interpreted as the minimum amount of capital that needs to be invested to make the final position acceptable. It is easy to verify that $\varrho[\cdot]$ in (2) is a coherent measure if and only if $\mathcal{K}$ is a convex cone (cf. [13]).

As the risk is not additive, when we deal with distributed complex systems, we need to address the question of risk evaluation for the entire system. This risk is usually called systemic in financial literature and the proposed measures for its evaluation are termed systemic risk measures.

Assume that the system consists of $m$ agents. One approach to evaluating the risk of a system is to use an aggregation function, $\Lambda:\mathbb{R}^{m}\to\mathbb{R}$, and univariate risk measures. Let $X\in\mathcal{L}_{p}(\Omega,\mathcal{F},P;\mathbb{R}^{m})$ be an $m$-dimensional random vector comprising the costs incurred by the system, where each component $X_{i}$ corresponds to the costs of one agent. The first approach to systemic risk is to choose a univariate risk measure $\varrho_{0}$ and apply it to the aggregated cost $\Lambda(X).$ If we prefer to use an acceptance set $\mathcal{K}$ as in (2), the systemic risk can be defined as:

$\displaystyle\varrho[X]=\varrho_{0}[\Lambda(X)]=\inf\big{\{}z\in\mathbb{R}~{}|~{}\Lambda(X)-z\in\mathcal{K}\big{\}}.$

(3)

In ([7] this point of view is analyzed in finite probability spaces and it is shown that any monotonic, convex, positively homogeneous function provides a risk evaluation as in (3) as long as it is consistent with the preferences represented in the definition of $\mathcal{K}$. The point of view presented in definition (3) is further extended in [11], where the authors analyzed convex risk measures defined on a general measurable space and proposed examples of aggregation functions suitable for a financial system. In both studies, the structural decomposition of the systemic risk measure (3) is established when the aggregation function $\Lambda$ satisfies properties similar to the axioms postulated for risk measures. In [4], the authors considered a particular case of an aggregation function, proposing an evaluation method for the risk associated with the cumulative externalities or costs endured by financial institutions. Note that these evaluation methods rely on a choice of one aggregation function suitable for a specific problem.
The translation property for constant vectors is introduced in [5] for convex risk measures defined for bounded random vectors. This property differs from the one we propose here. The authors analyzed the maximal risk over a class of aggregation functions rather than using one specific function. We refer to [28] for an overview of the risk measures constructed this way. A similar approach is taken in [10], where law-invariant risk measures for bounded random vectors are investigated for the purpose of obtaining a Kusuoka representation. The axioms proposed in [5, 10] are closest to ours and we provide more detailed discussion in section 3.

Another approach to risk evaluation of complex systems consists of evaluation of the risk of individual agents first and aggregation of the obtained values next. This method is used, for example, in [3] and [12]. Using the notion of acceptance sets the systemic risk measure is defined in [3] in the following way:

$\displaystyle\varrho[X]$

$\displaystyle=\varrho_{0}[\Lambda(X)]=\inf\Big{\{}\sum_{i=1}^{m}z_{i}~{}|~{}z\in\mathbb{R}^{m},~{}\Lambda(X-z)\in\mathcal{K}\subset\mathbb{R}^{m}\Big{\}}.$

The proposed measures of risk in section 3 also accommodate this point of view. A further extension in [3] replaces the constant vector $z\in\mathbb{R}^{m}$ by a random vector $Y\in\mathcal{C}$, where $\mathcal{C}$ is a given set of admissible allocations. This formulation of the risk measure allows to decide scenario-dependent allocations, where the total amount $\sum_{i=1}^{m}z_{i}$ can be determined ahead of time while individual allocations $z_{i}$ may be decided in the future when uncertainty is revealed. In [12] a set-valued counterpart of this approach is proposed by defining the systemic risk measure as the set of all vectors that make the outcome acceptable. Once the set of all acceptable allocations is constructed, one can derive a scalar-valued efficient allocation rule by minimizing the weighted sum of components of the vectors in the set. Set-valued risk measures were proposed in [17], see also [1, 16] for duality theory including the dual representation for certain set-valued risk measures. In fast majority of literature, the systemic risk depends on the choice of the aggregation function $\Lambda$ and how well it captures the interdependence between the components. To capture the dependence, an approach based on copula theory was put forward in [24]. It is assumed that independent operation does not carry systemic risk and, hence, the local risk can be optimized by each agent independently. The systemic risk measures are then constructed based on the copulas of the distributions.

Another line of work includes methods that use some multivariate counterpart of the univariate risk measures. The main notion here is the Multivariate Value-at-Risk ($\operatorname{MVaR}$) for random vectors, which is identified with the set of $p$-efficient points. Let $F_{X}(\cdot)$ be the right-continuous distribution function of a random vector $X$ with realizations in $\mathbb{R}^{m}$. A $p$-efficient points for $X$ is a point $v\in\mathbb{R}^{m}$ such that $F_{X}(v)\geq p$ and there is no point $z$ that satisfies $F_{X}(z)\geq p$ with $z\leq v$ componentwise. This notion plays a key role in optimization problems with chance constraints (see e.g. [31]). Multivariate Value-at-Risk satisfies the properties of translation equivariance, positive homogeneity and monotonicity. This notion is used to define Average Value-at-Risk for multivariate distributions ($\operatorname{MAVaR}$) in [19, 23, 26]. Let ${Z}_{p}$ be the set of all points, each of which is component-wise larger than some $p$-efficient point:

$$Z_{p}=\bigcup_{s\in\operatorname{MVaR}_{p}(X)}(s+\mathbb{R}^{m}_{+}).$$

In [19], Lee and Prekopa define the $\operatorname{MAVaR}$ of a random vector $X$ at level $p$ as

$$\operatorname{MAVaR}_{p}(X)=\mathbb{E}(\Lambda(X)~{}|~{}X\in D_{p}\},$$

(4)

where $\Lambda$ is assumed integrable with respect to $F_{X}$, i.e., $\mathbb{E}(\Lambda(X))$ is finite. It is shown in [19] that $\operatorname{MAVaR}$ is translation equivariant, positive homogeneous and subadditive only when all of the components of the random vector are independent.

While the definition of $\operatorname{MAVaR}$ above is scalar-valued, in [22] the authors define a Multivariate Average Value-at-Risk ($\operatorname{MAVaR}$) using the notion of $p$-efficient points as $\operatorname{MVaR}_{p}(X)$ and the extremal representation of the Average Value-at-Risk. First for given probability $p\in(0,1)$, we consider the vectors

$$\operatorname{MAVaR}_{p}(X;v)=v+\frac{1}{p}\mathbb{E}[(X-v)_{+}],$$

where $[(X-v)_{+}]_{i}=\max(0,X_{i}-v_{i})$, $i=1,\dots,m$. Then, the following vector-optimization problem is solved:

$$\operatorname{VMAVaR}_{p}(X)=\min\{\operatorname{MAVaR}_{p}(X;v):v\in\operatorname{MVaR}_{p}(X)\}.$$

The vector-valued Multivariate Average Value-at-Risk is monotonic, positively homogeneous, translation equivariant, but is not subadditive. Note that in both $\operatorname{MVaR}$ and $\operatorname{MAVaR}$, one needs to use a scalarization function to obtain a scalar value for the risk.

We shall compare our proposal to the aforementioned risk measures in section 4.

We shall consider further the following property.

• Normalization a coherent measure of risk $\varrho:\mathcal{Z}\to\mathbb{R}\cup\{+\infty\}$ is normalized iff $\varrho[\mathbb{I}]=1$.

In the paper [5], the authors have adopted the following translation axiom:

We also observe that a particular implication of Theorem 2 is that risk measures are linear on constant vectors.

In [10], the authors have analyzed law-invariant risk measures for bounded random vectors. They have introduced a set of axioms that are closest to ours: their axioms include our axioms together with the two normalization properties $\varrho[\mathbb{I}]=1$ and $\varrho[0]=0$. We do not need these normalization properties to establish the dual representation for general random vectors with finite $p$-moments, $p\geq 1$; we derive that the risk of the deterministic zero vector is zero from the dual representation. The property of strong coherence of risk measures, introduced in that paper implies in particular that $\varrho[a+b]=\varrho[a]+\varrho[b],$ which appears to be a strong assumption.

Using the dual representation of the coherent risk measure $\varrho$ for scalar-valued random variables, we obtain the following:

$\displaystyle\varrho[c^{\top}X]$

$\displaystyle=\sup_{\xi\in\mathcal{A}_{\varrho}}\int_{\Omega}\xi(\omega)c^{\top}X(\omega)P(d\omega)=\sup_{\zeta\in\tilde{\mathcal{A}}}\int_{\Omega}\zeta(\omega)X(\omega)P(d\omega)\quad\text{with }\tilde{\mathcal{A}}=\{\xi c:\xi\in\mathcal{A}_{\varrho}\}$

(14)

Additionally, a measurable selection $\nu_{X}(\omega)\in\operatorname{\arg\max}_{c\in S}c^{\top}X(\omega)$ exists by the Kuratowski-Ryll-Nadjevski theorem; we shall use the notation $\nu_{X}\in S$ for any such selection.

	$\displaystyle\varrho[\max_{c\in S}c^{\top}X]$	$\displaystyle=\sup_{\xi\in\mathcal{A}_{\varrho}}\int_{\Omega}\xi(\omega)\max_{c\in S}c^{\top}X(\omega)P(d\omega)=\sup_{\xi\in\mathcal{A}_{\varrho}}\int_{\Omega}\xi(\omega)\nu_{X}^{\top}X(\omega)P(d\omega)$
		$\displaystyle=\sup_{\zeta\in\tilde{\mathcal{A}}^{\prime}}\int_{\Omega}\zeta(\omega)X(\omega)P(d\omega)\quad\text{with }\tilde{\mathcal{A}}^{\prime}=\{\xi\nu_{X}:\xi\in\mathcal{A}_{\varrho}\}$

Notice that the representations just derived have the form of the dual representation in (6), however we have not established that $\tilde{\mathcal{A}}$ coincides with the domain of its conjugate function.

We observe the following properties of the aggregation by a single linear scalarization.

The penultimate relation implies the second claim of the theorem.

We observe that $c^{\top}X$ and $c^{\top}Y$ have the same distribution for any vector $c\in\mathbb{R}^{m}$. Hence, for any real number $r$, the following relations hold:

$\displaystyle P(X_{S}\leq r)$

$\displaystyle=P(c^{\top}X\leq r,\;\forall c\in S)=P(c^{\top}Y\leq r,\;\forall c\in S)=P(Y_{S}\leq r),$

which shows the equality of the distribution functions.

The second aggregation method that falls within the scope of our axiomatic framework is that of nonlinear scalarization. This class of risk measures cannot be obtained within the framework of aggregations by non-linear functions, and does not fit the axiomatic approaches in [7] or in [5]. Furthermore, we shall see that this method of evaluating systemic risk allows to maintain fairness between the system’s participants.

We define $\Omega_{m}=\{1,\dots,m\}$ and consider a probability space $(\Omega_{m},\mathcal{F}_{c},c)$, where $c\in S^{+}_{m}$ and $\mathcal{F}_{c}$ contains all subsets of $\Omega_{m}$. We view $c$ as a probability mass function of the space $\Omega_{m}$. Given an $m$-dimensional random vector $X\in\mathcal{Z}=\mathcal{L}_{p}(\Omega,\mathcal{F},P;\mathbb{R}^{m})$ and a collection of $m$ univariate measures of risk $\varrho_{i}:(\Omega,\mathcal{F},P)\to\mathbb{R}$, $i=1,\dots,m,$ we define the random variable $X_{R}$ on the space $\Omega_{m}$ as follows:

$$X_{R}(i)=\varrho_{i}[X_{i}],\quad i=1,\dots m.$$

(15)

Choosing a scalar measure of risk $\varrho_{0}:(\Omega_{m},\mathcal{F}_{c},c)\to\mathbb{R}$, the measure of systemic risk $\varrho_{s}:\mathcal{L}_{p}(\Omega_{m},\mathcal{F}_{c},c)\to\mathbb{R}$ is defined as follows:

$$\varrho_{s}[X]=\varrho_{0}[X_{R}].$$

(16)

This is a nonlinear aggregation of the individual risks $\varrho[X_{i}]$, hence this approach falls within the category of methods that evaluate the risk of each component first and then aggregate their values. The measure $\varrho_{s}[X]$ satisfies the axioms postulated for systemic risk measures.