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\begin{document}

\title{Adversarial Risk Analysis: applications to basic counterterrorism models}

\author{Jesus Rios\inst{1} and David Rios Insua\inst{2}}

\institute{Manchester Business School, UK \and Royal Academy of Sciences, Spain}

\maketitle

\begin{abstract}

Interest in counterterrorism modelling has increased recently. A common theme in the approaches adopted
is the need to develop methods to analyse decisions when there are intelligent opponents
 ready to increase our risks. Most of the approaches have a clear game theoretic flavour,
 although there have been some decision analytic based approaches. We have recently introduced a framework
 for adversarial risk analysis, aimed at dealing with problems with intelligent opponents and uncertain outcomes.
  In this paper, we shall explore how such framework may cope with two of the standard counterterrorism
  model formulations: sequential defend-attack and simultaneous defend-attack moves.
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

Recent high-profile terrorist attacks have demanded significant
  investments in protective responses. This has stirred a great deal of interest in modelling issues
  to deal with counterterrorism decisions. Good accounts and introductions to this field may be
  seen in Parnell et al. (2008) and Bier and Azaiez (2008). Besides some reliability analysis studies based on tools
  such as fault trees, much of this literature has a distinct game theoretic flavour.
Two examples include Zhuang and Bier (2007), who compute best responses and Nash
equilibria as a basis for allocating resources against terrorism when the defender and attacker have
different multiattribute utility functions, in situations of both simultaneous and sequential play; and Brown et
al.\ (2006), who present max-min, min-max and min-max-min optimization models for
defender-attacker, attacker-defender and defender-attacker-defender problems.
   Following Raiffa (2002), we remain skeptical about the relevance of such concepts
  in counterterrorism modelling, based on common knowledge assumptions that entail that
  parts have too much information about their counterparts, in a field in which secrecy tends to be
  an advantage.

The other mainstream literature in the field has a decision analytic flavour.
Among others, we mention
Von Winterfeldt and O'Sullivan (2006), who use decision trees to evaluate Man-Portable Air Defense Systems
countermeasures; and Pinker (2007), who applies influence diagrams to assess the deployment of various short-term
countermeasures.  Their recurrent (critical and criticised) problem is the need to assess the probabilities of the actions of the
others, which is a key issue of the Bayesian approach to games, see Kadane and Larkey (1982) or
Raiffa (2002).
Banks and Anderson (2006) provide a simple numerical comparison to both approaches
to game theory within a smallpox threat problem. This tension between game theoretic and decision analytic approaches to decision making problems with adversaries is not exclusive of counterterrorism models but appears in other
business and industrial areas, see e.g. van Bingsbergen and Marx (2007) or Rothkopf (2007).

In Rios Insua et al. (2009), we have introduced Adversarial Risk Analysis (ARA) a framework to cope with risk
analysis situations in which we have one or more opponents ready to increase our risks. We applied it in
 simple auction
contexts. ARA has a Bayesian game theoretic flavour. We use principled procedures which employ the game
theoretical structure and other information available to assess probabilities on the opponents's actions. In
this paper, we explore the application of such framework to simplified versions of two standard
models used in counterterrorism contexts, see Zhuang and Bier (2007):
sequential defend-attack and simultaneous defend-attack models. We aim at supporting
the Defender in choosing her best defense against the attacker. We use
coupled decision trees to illustrate our discussion. Our emphasis is on how we may coherently
assess the probabilities of various attacks by the attacker. We first study the sequential model. We then consider the simultaneous one. We end up with
some discussion on possible extensions and actual applications of our streamlined models. We shall assume all throughout the paper that both the defender and the attacker only have two actions available.


%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Attack-Defend ModelsBrown et al.\ (2006) and
%Brown, Carlyle and Wood (2007)}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Defend-Attack Sequential Model} \label{Defend-Attack-Seq}

We start by considering a simple sequential Defend-Attack situation, in which the Defender
starts by choosing a defense from a discrete set ${\cal D}=\{d_1,d_2\}$ and, then, the Attacker,
 having observed the defense, chooses
an attack within the discrete set ${\cal A}=\{a_1,a_2\}$. The problem has a clear sequential game theoretic
structure, as in Stackelberg games, see Aliprantis et al (2000).
 The only uncertainty deemed relevant is a binary outcome $S\in \{0,1 \}$  representing the success or
failure of the Attack. For both players, the consequences depend on the success of this attack.

%Fig.~\ref{D-Amodel} shows a coupled influence diagram
%and a coupled decision tree representing this situation. Recall, that arcs into
%a utility node represent functional dependence. Thus, the utility functions over the consequences for the Defender
%and the Attacker are, respectively, $u_D(d,S)$ and $u_A(a,S)$. The influence diagram shows explicitly that the
%uncertainty associated with the success of an attack $S$ is probabilistically dependent on the actions of
%both the Attacker and the Defender: $S \mid d,a$. The arc in the influence diagram from the the Defender's decision %node to the Attacker's reflects that the Defender's choice is observed by the Attacker. The tree clearly shows the
%sequential nature.

%\begin{figure}[h]
%    \caption{The Defend-Attack sequential decision game}
%    \begin{center}
%        $
%        \begin{array}{c@{\hspace{25pt}}c}
%        \epsfxsize=2in \epsffile{GRAPHICS/S_TAdv3.eps}   &   \epsfxsize=2in \epsffile{GRAPHICS/TAdv4c.eps} \\
%        \mbox{(a) Influence diagram } & \mbox{(b) Decision tree}\\
%5        \end{array}
%        $
%    \end{center}
%    \label{D-Amodel}
%"\hspace{· pt}" defines the distance between the figures,
%"\epsfxsize= · in" their size
%\end{figure}

 Fig.~\ref{D-Amodel} shows a coupled decision tree representing this situation. The utility functions over the consequences for the Defender
 and the Attacker are, respectively, $u_D(a, d,S)$ and $u_A(a,d, S)$, depending on both actions and the eventual success $S$ of the attack, which is probabilistically dependent on the actions of
 both the Attacker and the Defender: $S \mid d,a$. The sequence of nodes reflects that the Defender's choice is observed by the Attacker.

\begin{figure}[h]
    \caption{The Defend-Attack sequential decision game}
    \begin{center}
        $
          \epsfxsize=2in \epsffile{GRAPHICS/TAdv4c.eps}
                $
    \end{center}
    \label{D-Amodel}
%"\hspace{· pt}" defines the distance between the figures,
%"\epsfxsize= · in" their size
\end{figure}

The standard game theoretic approach computes Nash equilibria under strong common knowledge
assumptions: in this case, the Defender would know the beliefs and preferences of the Attacker, modeled, respectively, in $(p_A, u_A)$, which is not realistic. Note that the attacker does not need to know the Defender's probabilities and utilities $(p_D, u_D)$ as he observes the Defender's decision. We now weaken such common knowledge assumptions in this sequential decision game. Suppose, thus, that
the Defender does not
know $(p_A, u_A)$. The Defender's decision tree denotes uncertainty about the Attacker's attack by replacing
\fbox{A} with $\;$ \tcircledI{A} $\;$.
\begin{figure}[h]
    \caption{The decision problem as seen by the Defender}
    \begin{center}
        $
                \epsfxsize=1.75in \epsffile{GRAPHICS/TAdvDT5D.eps}
            $
    \end{center}
    \label{Adv5}
%"\hspace{· pt}" defines the distance between the figures,
%"\epsfxsize= · in" their size
\end{figure}
Note that, in order to solve this game, the Defender has already assessed $p_D(S|d,a)$ and $u_D(a, d,S)$, but
she also needs $p_D(A=a|d)$, which is her assessment of the probability that the Attacker will choose attack
$a$ after observing that the Defender has chosen defense $d$. This assessment requires the Defender to
analyze the problem from the Attacker's perspective, as follows.

\begin{figure}[h]
    \caption{The Defender's analysis of the Attacker's problem}
    \begin{center}
        $
                   \epsfxsize=1.75in \epsffile{GRAPHICS/TAdvDT5A.eps}
                $
    \end{center}
    \label{Adv5A}
%"\hspace{· pt}" defines the distance between the figures,
%"\epsfxsize= · in" their size
\end{figure}


%This is a key issue, and was one of the central criticisms in the
% Bayesian approach to ARA (Harsanyi, 1982).


%Consider the Defender's problem.
%She could make initial guesses of $\pi_A(c | a, d, S)$, $p(S=0|a,d)$
%and $u_A(c)$, then use these in the situation shown in
%Fig.~\ref{Adv5A} to solve the Attacker's version of the problem.
%the Defender obtains an estimate of the Attacker's best response to her decision,
%under her assumption about his utilities:
%$\hat{a}^*(d) = \mbox{argmax}_{a \in {\cal A}} \hat{\psi}_A(a,d)$,
%where $\hat{\psi}_A(a,d)$ is her estimate of the Attacker's expected
%utility.
%Then the Defender could use the strategy in Pate-Cornell and Guikema (2002)
%to set $\hat{p}_D(a|d) \propto \hat{\psi}_A(a,d)$.
%
% But this approach does not explicitly consider the Attackeras a utility
% maximizer.
%A better approach notes

For that, the Defender must place herself in the Attacker's shoes, and consider his decision problem.
 Fig.~\ref{Adv5A}
represents the Attacker's problem, as seen by the Defender. We assume that the Defender
 will analyze the Attacker's problem considering that he is an expected utility maximizer.
 Thus, she will use all the information and judgment that she can
about the Attacker's utilities and probabilities. Therefore, to find $p_D(A|d)$, she should first estimate
the Attacker's utility function and his probabilities about success $S$, conditional on $(d,a)$, and,
consequently, compute the required probability. However, instead of using point estimates for $p_A$ and $u_A$
to find the Attacker's optimal decision $a^*(d)$, the Defender's uncertainty about the Attacker's decision
should derive from her uncertainty about the Attacker's $(p_A, u_A)$, which we describe through a
distribution $F$. This will induce a distribution on the Attacker's expected utility
$\psi_A(a,d)$. Thus, assuming the Attacker is rational, the Defender's predictive distribution on the
Attacker's attack choice given her defense choice $d$ is
$$
p_D(A=a|d) = \Pb_F[ a = \mbox{argmax}_{x \in {\cal A}} \Psi_A(d,x)], \quad \forall a \in {\cal A},
$$
where
$$
\Psi_A(d,a) = P_A(S=0 \mid d,a) \; U_A(a,d, S=0)  +  P_A(S=1 \mid d,a)  \; U_A(a,d,S=1)
$$ for $(P_A, U_A) \sim F$.

She may use Monte Carlo simulation to approximate $p_D(A|d)$ by drawing $n$ samples $\left\{\left( p_A^i,
u_A^i \right)\right\}_{i=1}^n $ from $F$, which produce $ \{\psi_A^i\}_{i=1}^n \sim \Psi_A $, and
approximating $p_D(A=a|d)$ by
\begin{equation*} %\label{eq:MC}
\hat {p} _D(A=a|d) = \frac{\# \{ a =  \mbox{argmax}_{x \in {\cal A}} \;
  \psi_A^i(d,x) \} }{ n } .
\end{equation*}
Once the Defender has completed these assessments, she can solve her problem. Her expected utilities at node $\;$
\tcircledI{S} $\;$ in Figure~\ref{Adv5} for each $(d,a) \in {\cal D} \times {\cal A}$ are:
\begin{eqnarray*}
\psi_D(d,a) = p_D(S=0|d,a) \; u_D(a, d, S=0) \; + \;
              p_D(S=1|d,a) \; u_D(a, d, S=1) .
\end{eqnarray*}
Working up the tree in Fig.~\ref{Adv5}, her estimated expected utilities at node $\;$ \tcircledI{A} $\;$ for
each $d \in {\cal D}$ are:
$$
\hat \psi_D(d) = \psi_D(d,a_1) \; \hat p_D(a_1 | d) + \psi_D(d,a_2)\; \hat p_D(a_2 | d).
$$
Finally, her approximate optimal decision is $d^* = \mbox{argmax}_{d \in {\cal D}} \; \hat \psi_D(d)$.

We summarize the previous discussion with the following procedure to find a recommendation for the Defender in the
Defend-Attack sequential decision model.
\begin{enumerate}
    \item Assess $(p_D, u_D)$ from the Defender
    \item Assess $F=(P_A, U_A)$, describing the Defender's uncertainty about $(p_A, u_A)$
    \item For each $d$, simulate to approximate $p_D(A|d)$ as follows:
    \begin{enumerate}
        \item Generate  $(p^i_A,u^i_A)\sim F$, $i=1,\ldots,n$
        \item Solve  $a_i^*(d) =\mbox{argmax}_{a\in {\cal A}} \; \psi_A^i(d,a)$
        \item Approximate $ \hat p_D(A=a|d) = {\# \{ a = a_i^*(d) \} } / {n} $
    \end{enumerate}
   \item Solve the Defender's problem $$ d^*=\mbox{argmax}_{d \in {\cal D}} \; \psi_D(d,a_1)\; \hat p_D(a_1|d) \;+\;  \psi_D(d,a_2)\;
\hat p_D(a_2|d)$$
\end{enumerate}




%\subsection{The Bayesian Solution}
%The previous discussion suggests a general Bayesian solution to the fully general ARA problem, which we now
% summarise.
%Figure~\ref{Adv6}, to be compared with 7(a) and 9(a), shows the influence
%diagram when the Attackerdoes not observe the Defender's decision but the Attacker%has imperfect information, expressed in the distribution $G$, on how
%the Defender would behave.
%This is reflected in the chance node labeled ?? which
%incorporates the Attacker's intelligence concerning the Defender's decision.
%the Attackerwould use the preceding analysis to find the Bayesian solution
%that determines his decision.

%\begin{figure}[htb]
%\caption{ Influence Diagram from the Attacker perspective of a simultaneous decision game with imperfect
%information.}
%    \begin{center}
%         \epsfig{file=Adv6.eps, width=8cm}
%    \end{center}
%    \label{Adv6}
% \end{figure}
%
%More broadly,

%Fig.~\ref{ourProposal} extends Fig.~6(a) and summarizes the general ARA problem. The figure consists of two
%parallel influence diagrams with interlinking nodes that represent partial information. It describes the
%scenario in which both the Attackerand the Defender must make their decisions without knowledge of the other's decision,
%nor of the other's utilities and probabilities. But both the Attackerand the Defender have probabilistic expressions of
%their uncertainty that can be used to derive their best play. These distributions are represented by the
%chance nodes $\;$ \tcircledI{A} $\;$ and $\;$ \tcircledI{D} $\;$. The $\;$ \tcircledI{A} $\;$ chance node is
%used by the Defender and summarizes her beliefs about the Attacker's situation. The $\;$ \tcircledI{D} $\;$ chance node
%is used by the Attackerand summarizes his beliefs about the Defender's circumstance.

%\begin{figure}[htb]
%\caption{Influence Diagram from both perspectives with probabilistic information about the opponents decision.}
%    \begin{center}
%         \epsfig{file=BN_ourProposal.eps, width=0.4\textwidth}
%    \end{center}
%    \label{ourProposal}
%\end{figure}
%As our objective is to support the Defender, we describe algorithmically how to support her,

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Defender-Attacker Simultaneous Action Model}

We now present a simultaneous-move case in which the Defender  and the Attacker must make their defense and attack decisions, without knowing
the action chosen by each other. Again, each of them has just two options to choose from: ${\cal D}=\{d_1,d_2\}$ for
the Defender and ${\cal A}=\{a_1,a_2\}$ for the Attacker. The only uncertainty is $S$: the success or
failure of the Attack. This problem can be represented as a joint decision tree as in Fig.~\ref{Simultaneous}.

\begin{figure}[h]
    \caption{The two players simultaneous decision game}
    \begin{center}
        $
                  \epsfxsize=1.75in \epsffile{GRAPHICS/TAdv2_Attacker_Defender.eps}
        $
    \end{center}
    \label{Simultaneous}
%"\hspace{· pt}" defines the distance between the figures,
%"\epsfxsize= · in" their size
\end{figure}

The standard game-theoretic approach computes Nash equilibria under the common knowledge assumption
that both the Defender's and Attacker's preferences and beliefs are available. This is unrealistic in
the counterterrorism context. When these are unknown, another standard game theoretic approach
computes Bayes-Nash equilibria under the
common knowledge assumption about priors over types of defenders and attackers, as in Harsanyi (1967). But again,
such assumption is unrealistic in the counterterrorism setting.
We now weaken this common prior knowledge assumption and
solve the Defender-Attacker simultaneous action game for the Defender. In this model, the assessment of
probabilities on the adversary's actions tend to be more elaborated than in the Defend-Attack sequential game
from Section~\ref{Defend-Attack-Seq}, as we shall see.

\begin{figure}[h]
    \caption{The Defender's decision analysis}
    \begin{center}
        $
                  \epsfxsize=2in \epsffile{GRAPHICS/Simultaneous_DT_D.eps}
                $
    \end{center}
    \label{Simultaneous-D}
\end{figure}

The Defender has to choose a defence action $d \in {\cal D}$, whose consequences depend on the
success of the attack chosen simultaneously by the Attacker, as reflected in Fig.~\ref{Simultaneous-D}. By standard
decision theory, see French and Rios Insua (2000), the defender should maximize her expected utility. Therefore, the decision problem she should
solve is
\begin{equation*}%\label{Simultaneous:d*}
    d^* = \mbox{argmax}_{d \in {\cal D}} \,
    \sum_{a \in {\cal A}}
    \left[ \sum_{s \in \{0,1\}} u_D (a, d, s) \; p_D (S=s \mid d,a) \; \right]
    \pi_D ( A = a ),
\end{equation*}

\noindent where the Defender's utility function $u_D(a, d,s)$ and her personal probabilistic assessment $p_D$ over $S$
conditional on $(d,a)$ are known to her. As the Defender does not know the Attacker's decision at node $A$,
she expresses her uncertainty through a probability distribution $\pi_D ( A = a )$, over all $a \in {\cal A}$.

\begin{figure}[h]
    \caption{The Attacker's decision analysis as seen by the Defender}
    \begin{center}
        $
                    \epsfxsize=2in \epsffile{GRAPHICS/Simultaneous_DT_A.eps}
        $
    \end{center}
    \label{Simultaneous-A}
\end{figure}

The main difficulty for the Defender is assessing the probability distribution
$\pi_D (A)$, which represents her uncertainty about what attack will the Attacker choose.
 To do so, she can think of
the Attacker as an expected utility maximizer who tries to solve a decision problem as the one shown in
Fig.~\ref{Simultaneous-A}, by finding the attack $a \in {\cal A}$ that provides him maximum
expected utility:
\begin{equation*}
    a^* = \mbox{argmax}_{a \in {\cal A}} \,
    \sum_{d \in {\cal D}}
    \left[ \sum_{s \in \{0,1\}} u_A (a,d, s) \; p_A (S=s \mid d,a) \; \right]
    \pi_A ( D = d ) .
\end{equation*}

\noindent However, the Defender cannot solve the Attacker's decision problem, and, thus, find his maximum expected utility attack $a^*$,
since in general she will not know the Attacker's true probabilities and utility function $(u_A,p_A,\pi_A)$. But,
she may model all information available to her about $(u_A, p_A, \pi_A)$ through a probability
distribution $(U_A,P_A, \Pi_A)\sim F$. % representing her probabilistic knowledge about $(u_A,p_A,\pi_A)$. Thus,
Thus, the Defender's assessment of $\pi_D(A)$ is reduced to the computation of the following probability distribution
\begin{equation} \label{Simultaneous:A}
    A \mid D  \sim \mbox{argmax}_{a \in {\cal A}} \,
    \sum_{d \in {\cal D}}
    \left[ \sum_{s \in \{0,1\}} U_A (a, d, s) \; P_A (S=s \mid d,a) \; \right]
    \Pi_A ( D = d ) .
\end{equation}


\noindent Note that $\Pi_A$ represents the Defender's assessment of the probabilistic model $\pi_A$ used by
the Attacker to predict what defense the Defender will choose, whereas $D$ represents the Attacker's beliefs
about what choice the Defender will make. We are, thus, identifying two sources of uncertainty:
\begin{enumerate}
\item  one
stemming from the Attacker's uncertainty about the Defender's choice ($D$), and
\item  one coming from the
Defender's uncertainty about how the Attacker assesses $D$ through $\Pi_A$.
\end{enumerate}
 Again, the Defender may
presume that the Attacker thinks she is an expected utility maximizer trying to solve a decision problem analogous
to that in Fig.~\ref{Simultaneous-D}. Therefore, in order for the Defender to solve Eq.~\eqref{Simultaneous:A},
she will elicit $(U_A , P_A, \Pi_A) \sim F$ from her viewpoint, and assess $D$ through the analysis of her
decision problem as thought by the Attacker, reducing the assessment of $ D $ to the computation of the following
distribution
\begin{equation*}
    D \mid A^1  \sim  \mbox{argmax}_{d \in {\cal D}} \,
    \left[ \sum_{a \in {\cal A}}
    \sum_{s \in \{0,1\}} U_D (a, d, s) \; P_D (S=s \mid d,a) \; \right]
    \Pi_{D} ( A^1 = a ) ,
\end{equation*}
where the Defender needs to elicit $(U_D , P_D, \Pi_D) \sim G$ representing her assessment on the Attacker's
estimation of her utility function $u_D(a,d,s)$ and her probability $p_D$ over $S \mid d,a$, when she
analyzes how the Attacker thinks about her decision problem, as well as the Defender's confidence in her
assessment model leading to $A^1$, which represents the Attacker's decision when he is modeled by the
Defender within her second level of recursive modeling. This may require again further recursive thinking
from the Defender, which will lead to the following recursive computations, starting with $i=0$.

\begin{itemize}
  \item[] Repeat
        \begin{eqnarray*}\nonumber
            A^i \mid {D^i} &\sim& \mbox{argmax}_{a \in {\cal A}} \,
            \sum_{d \in {\cal D}}
            \left[ \sum_{s \in \{0,1\}} U_A^{i} (a, d, s) \; P_A^{i} (S=s \mid d,a) \; \right]
            \Pi_{A^i} ( D^i = d ) \\ \nonumber
                && \text{where } (U_A^{i} , P_A^{i} , \Pi_{A^i}) \sim F^{i} \\ \nonumber
%
            D^i \mid {A^{i+1}} &\sim& \mbox{argmax}_{d \in {\cal D}} \,
            \sum_{a \in {\cal A}}
            \left[ \sum_{s \in \{0,1\}} U_D^{i} (a,d, s) \; P_D^{i} (S=s \mid d,a) \; \right]
            \Pi_{D^i} ( A^{i+1} = a ) \\ \nonumber
                && \text{where } (U_D^{i} , P_D^{i} , \Pi_{D^i}) \sim G^{i} \nonumber
        \end{eqnarray*}
        $i=i+1$
\end{itemize}

To simplify the discussion, we have assumed that the decision models recursively used to assess $A^i$ and $
D^i$, from $i=0$ and so forth, are a reflection of each other and have the same structure as those in
Figs.~\ref{Simultaneous-A} and ~\ref{Simultaneous-D}, respectively. Moreover, the choice sets for the Defender
and the Attacker are the same in all models in the recursive hierarchy of analysis: ${\cal D} $ and ${\cal A}$,
respectively.
This hierarchy of nested models would stop at the level in which the Defender lacks the kind of information
necessary to assess the distribution $F^{i}$ (or $G^{i}$). At this point, the Defender will holistically
assign an unconditional probability distribution over $A^i$ (or $ D^i$), without going deeper in the
hierarchy. Of course, if she feels that she still has no information available to do so, she can assign a
noninformative or reference probability distribution, see French and Rios Insua (2000).
% Luce & Raiffa (1957) to see elicitation procedure to do this (Chapter 13)



\section{Discussion}

We have provided here an account of how Bayesian decision analysis can support a Defender against an
intelligent adversary (the Attacker), in contrast with the standard game-theoretic approach, in two stylised problems. In our framework,
the Defender assesses the probabilities of the adversarial actions before computing her maximum expected utility defense
strategy. We assume that the Attacker is an expected utility maximizer and
the Defender's uncertainty about the Attacker' decision stems from her uncertainty about his decision
analysis, specifically about his probabilities and utilities. The Bayesian approach weakens the unrealistic common
knowledge assumptions necessary in the game-theoretic approach.

Of course, real problems in counterterrorism are extremely complex. For example, they involve hundreds of
possible decisions, and there are large uncertainties associated with the goals and resources of the
terrorists. For this reason, we have focused on illustrating with two simple examples a general methodology
based on decision analysis principles to support decision making against an intelligent Attacker, making
parallel comparisons with the standard solutions proposed by the game-theoretic approach. More complex
dynamic interactions, say as in defend-attack-defend models or general coupled defend and attack influence
diagrams, require more complex analysis but the methodology would stay essentially the same. Extensions of
the methodology to the case in which there are more than one Attacker, or an uncertain number of Attackers,
and more than one defender also need to be explored. This last one might require negotiations about risk
sharing, as in Rios and Rios Insua (2010).


\subsection*{Acknowledgements} Research supported by grants from the Spanish Ministry of Science and Innovation, the
E-DEMOCRACIA-CM program and the COST Algorithmic Decision Theory Action. We are grateful to discussions by the referees.

%%%%%%%%%%%%%%%%%%%%%%%

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\end{description}



\end{document}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Defend-Attack-Defend Model}

We now consider a situation of sequential moves with two players (a Defender and an Attacker) in which the
Defender makes first a initial defense investment in ${\cal D}_1$, then the Attacker observes her investment
and chooses an attack in ${\cal A}$; and, finally, the Defender chooses an action from ${\cal D}_2$ after the
attack, to mitigate or respond to it using the resources deployed from her investment at her first move. The
only uncertainty is the outcome $S$ of the Attack, which depends probabilistically on $(d_1,a) \in {\cal D}_1
\times {\cal A}$. The consequences for the Defender and the Attacker depend respectively on $(d_1,s,d_2)$ and
$(a,s,d_2)$. Fig.~\ref{D-A-D model} shows a coupled influence diagram and a decision tree representing this
situation, where nodes $ D_1 $ and $ D_2 $ correspond to the Defender's first and second decisions,
respectively.


\begin{figure}[h]
    \caption{The Defend-Attack-Defend decision game}
    \begin{center}
        $
        \begin{array}{c@{\hspace{30pt}}c}
            \epsfxsize=2.2in \epsffile{GRAPHICS/DAD_ID.eps}   &
            \epsfxsize=3in \epsffile{GRAPHICS/DAD_DT.eps}    \\
            \mbox{(a) Influence diagram} & \mbox{(b)  Decision tree }\\
        \end{array}
        $
    \end{center}
    \label{D-A-D model}
\end{figure}


We now solve the problem for the Defender treating the Attacker's behaviour at node $A$ as uncertain and
modeling her uncertainty about it as a random variable. This is reflected in the influence diagram and
decision tree from Fig.~\ref{D-A-D Defender}, where the Attacker's decision node has been converted into a
chance node, by replacing \fbox{A} with $\;$ \tcircledI{A} $\;$.

\begin{figure}[h]
    \caption{the Defender's decision problem}
    \begin{center}
        $
        \begin{array}{c@{\hspace{30pt}}c}
            \epsfxsize=2.2in \epsffile{GRAPHICS/DAD_ID_Defender.eps}   &
            \epsfxsize=3in \epsffile{GRAPHICS/DAD_DT_Defender.eps}    \\
            \mbox{(a) Influence diagram} & \mbox{(b)  Decision tree }\\
        \end{array}
        $
    \end{center}
    \label{D-A-D Defender}
\end{figure}

To solve her decision problem, the Defender not only needs to know her $u_D(d_1,s,d_2)$ and $p_D(S \mid
d_1,s)$ but also her assessment of $p_D(A \mid d_1)$ for all $d_1 \in {\cal D}_1$. Thus, working backwards
the tree in Fig.~\ref{D-A-D Defender}, the Defender can compute  at node $D_2$ her action of maximum utility
$d_2^*(d_1,s)$ for each $(d_1,s) \in {\cal D}_1 \times S$, and, later,
%
at node $S$ her expected utility $ \psi_D (d_1,a)$ for each $(d_1,a) \in {\cal D}_1 \times A$.
%
At this point, she can use her probabilistic assessment about what the Attacker will do, $p_D(A \mid d_1)$,
to compute her expected utility at node $A$ for each $d_1 \in {\cal D}_1$,
$$ \psi_D (d_1) = \int \psi_A (d_1,a) \; p_D( a \mid d_1) \; \mathrm{d}a  , $$
%
to finally find her decision of maximum expected utility at node $D_1$:
$$d_1^* = \mbox{argmax}_{d_1 \in {\cal D}_1}  \psi_D (d_1).$$

Based on this approach, the Defender's best strategy for the Defend-Attack-Defend problem is to play first
$d_1^* \in {\cal D}_1$ at node $D_1$, and later $d_2^*(d_1^*,s) \in {\cal D}_2$ at node $D_2$ after observing
$s \in S$. We note in order for the Defender to be able to solve her decision problem she needs to elicit
$p_D(A \mid d_1)$, her predictive distribution about what attack the Attacker will choose at node $A$ after
observing her defense choice $d_1 \in {\cal D}_1$.


The key issue for the Defender to be able to solve her decision problem is the assessment of her predictive
distributions about which attack the Attacker will choose for each of her first defense moves $d_1 \in {\cal
D}_1$, represented by the $p_D(A \mid d_1)$. She can use a standard statistical elicitation method if
historical data on the Attacker’s behavior in prior similar situations is available. A judgemental assessment
of $p_D(A \mid d_1)$ can also incorporate expert opinion. However, we propose to model the Defender's
uncertainty about the Attacker’s decision assuming he is a expected utility maximizer and that the Defender’s
uncertainty stems from her uncertainty about the Attacker’s probabilities and utilities. Thus, the assessment
of $p_D(A \mid d_1)$ is reduced to the analysis of the Attacker’s decision problem as seen by the Defender,
shown in Fig.~\ref{D-A-D Attacker}, and the assessment of his probabilities and utilities from the Defender's
perspective, using as sources of information all available data as well as expert opinions. In addition, an
advantage of structuring the Attacker's problem is that we can isolate different uncertainty parts as well as
value parts of his problem and accommodate different expertise by bringing in different experts to provide
information about the different parts of the Attacker's problem. Should this kind of information not be
available, we will use a noninformative (or reference) distribution to describe $p_D(A \mid d_1)$.

\begin{figure}[h]
    \caption{the Defender's view of the Attacker's decision problem}
    \begin{center}
        $
        \begin{array}{c@{\hspace{15 pt}}c}
        \epsfxsize=2in \epsffile{GRAPHICS/DAD_ID_Attacker.eps}   &
        \epsfxsize=2.75in \epsffile{GRAPHICS/DAD_DT_Attacker.eps}   \\
        \mbox{(a) Influence diagram} & \mbox{(b) Decision tree}\\
         \end{array}
        $
    \end{center}
    \label{D-A-D Attacker}
\end{figure}


Thus, to find $p_D(A \mid d_1)$, the Defender needs to assess $u_A(a,s,d_2)$, $p_A(S \mid d_1, a)$ as well as
$p_A(D_2 \mid d_1, s)$. If the Defender would have access to the Attacker's true probabilities and utilities,
she would be able to know the Attacker's attack of maximum expected utility $a^* (d_1)$ for each $d_1 \in
{\cal D}_1$, and therefore $p_A(A = a^* (d_1) \mid d_1) = 1$. However, she does not know these true
quantities, but she can acknowledge her uncertainty about them through a probability distribution $F=\left(
U_A(a,s,d_2) , P_A(S \mid d_1, a) , P_A(D_2 \mid d_1, s) \right)$ and solve the Attacker's decision problem
using backward induction over the decision tree in Fig.~\ref{D-A-D Attacker} as follows.

\begin{itemize}
  \item At chance node, compute $D_2$:
        $ (d_1, a, s) \rightarrow \Psi_A (d_1, a, s) = \int U_A(a,s,d_2) \; P_A(D_2=d_2 \mid d_1, s) \; \mathrm{d} d_2 $

  \item At chance node $S$:
        $ (d_1, a) \rightarrow \Psi_A (d_1, a) = \int \Psi_A (d_1, a, s)  \; P_A(S=s \mid d_1, a) \; \mathrm{d} s $

  \item At decision node $A$:
        $ d_1 \rightarrow A^* (d_1) = \mbox{argmax}_{a \in {\cal A}} \Psi_A (d_1, a) $
\end{itemize}
\noindent obtaining the Defender's predictive distribution of the Attacker's attack after he observes the
Defender's first move $d_1 \in {\cal D}_1$:
\begin{itemize}
%  \item $p_D(A = a \mid d_1) = \Pr ( A^* (d_1) = a )$ if $A \mid d_1$ is discrete, or
  \item $ \int_0^a p_D(A = x \mid d_1) \mathrm{d}x= \Pr ( A^* (d_1) \leq a )$, as we assume that $A \mid d_1$ is absolutely continuous.
\end{itemize}

\noindent Note that this distribution can be computed using Monte-Carlo simulation methods by drawing $n$
samples
$$\{ \left( u^i_A(a,s,d_2) , p^i_A(S \mid d_1, a) , p^i_A(D_2 \mid d_1, s) \right) \}_{i=1}^n \sim F , $$
and computing $\{ a^*_i(d_1) \}_{i=1}^n $ by
\begin{itemize}
  \item At chance node $D_2$:
        $ (d_1, a, s) \rightarrow \psi_A^i (d_1, a, s) = \int u^i_A(a,s,d_2) \; p^i_A(D_2=d_2 \mid d_1, s) \; \mathrm{d} d_2 $

  \item At chance node $S$:
        $ (d_1, a) \rightarrow \psi_A^i (d_1, a) = \int \psi_A^i (d_1, a, s)  \; p^i_A(S=s \mid d_1, a) \; \mathrm{d} s $

  \item At decision node $A$:
        $ d_1 \rightarrow a_i^* (d_1) = \mbox{argmax}_{a \in {\cal A}} \psi_A^i (d_1, a) $
\end{itemize}
\noindent to obtain the following approximations:
\begin{itemize}
  \item $p_D(A = a \mid d_1) \approx {\# \{ a_i^*(d) = a \} } / {n} $ if $A \mid d_1$ is discrete, or
  \item $ \int_0^a p_D(A = x \mid d_1) \mathrm{d}x \approx {\# \{ a_i^*(d) \leq a \} } / {n}$ if $A \mid d_1$ is
  absolutely
continuous.
\end{itemize}

Hence the assessment of $p_D(A \mid d_1)$ is straightforward from the Defender's elicitation of $F$. However,
$P_A(D_2 \mid d_1, s)$ in $F$ could be problematic, as the Defender may want to exploit some information
available to her about how the Attacker is analyzing her problem. Of course, if there is no information that
the Defender can use to do this, she will put a noninformative, or reference, distribution over $P_A(D_2 \mid
d_1, s)$. The ARA analysis will continue until no more information can be used by the Defender to analyze
recursively the problem concerning the thinking-about-what-the-other-is-thinking-about. Thus, this recursive
analysis will stop in some point, maybe after using some simplification leading to an heuristic distribution
for modeling a player's thinking at some step of the recursive analysis as illustrated in Rios at al (2009).
