# Feasible Action-Space Reduction as a Metric of Causal Responsibility in
Multi-Agent Spatial Interactions

Ashwin George Corresponding Author. Email<EMAIL_ADDRESS>Luciano
Cavalcante Siebert  David Abbink  Arkady Zgonnikov  Delft University of
Technology

###### Abstract

Modelling causal responsibility in multi-agent spatial interactions is crucial
for safety and efficiency of interactions of humans with autonomous agents.
However, current formal metrics and models of responsibility either lack
grounding in ethical and philosophical concepts of responsibility, or cannot
be applied to spatial interactions. In this work we propose a metric of causal
responsibility which is tailored to multi-agent spatial interactions, for
instance interactions in traffic. In such interactions, a given agent can, by
reducing another agent’s feasible action space, influence the latter.
Therefore, we propose feasible action space reduction (FeAR) as a metric of
causal responsibility among agents. Specifically, we look at ex-post causal
responsibility for simultaneous actions. We propose the use of Moves de
Rigueur (MdR) — a consistent set of prescribed actions for agents — to model
the effect of norms on responsibility allocation. We apply the metric in a
grid world simulation for spatial interactions and show how the actions,
contexts, and norms affect the causal responsibility ascribed to agents.
Finally, we demonstrate the application of this metric in complex multi-agent
interactions. We argue that the FeAR metric is a step towards an
interdisciplinary framework for quantifying responsibility that is needed to
ensure safety and meaningful human control in human-AI systems.

##  1 Introduction

The rapid development of AI methods have sparked debates on the ethics of AI
[3, 5, 14, 26]. Ethical principles like responsibility, trust, and fairness
need to be operationalised for making AI systems sensitive to human values
[13]. In particular, understanding responsibility of agents is crucial for
designing autonomous systems and distributing rewards and punishments during
their operation in an appropriate way. For example, consider an automated
vehicle (AV) that brakes abruptly to prevent hitting a child that walked on to
the road. Who would be responsible if the AV ends up in a rear-end collision?
How should the AV be designed to prevent situations when no agent is actively
responsible for preventing negative outcomes? [27] Understanding the
responsibility of agents is therefore crucial for the users, designers and
regulatory bodies of such an automation [2, 24, 26].

On the practical side, researchers working on design and engineering of
automated driving systems have been trying to incorporate the notion of
responsibility in motion planning frameworks for such systems. For example,
the Responsibility-Sensitive Safety (RSS) approach formalises an
interpretation of "Duty of Care" from tort law and promises verifiable safety
when all agents are following RSS [29, 30]. This approach provides the "proper
responses" for all the agents and those deviating from the "proper response"
would be considered responsible if something goes wrong. However, for mixed
traffic (in which both AVs and human-driven vehicles share the road), not all
agents might know the "proper response" or be capable of executing it. Hence,
this approach is not ideal for mixed-traffic with humans having variable
behaviours. Another approach is to include an extra term in the cost function
of trajectory planning algorithms of AVs to account for responsibility. This
term can incorporate an interpretation of responsibility based on traffic
rules and physical constraints [15], or be learned from driving data [7].
However, the simplified notions of "responsibility" captured by these
approaches are not grounded in the established definitions of responsibility
from ethics and philosophy.

On the more fundamental side, metrics and models of responsibility have been
developed that aim to quantify different flavours of responsibility, e.g.,
moral responsibility, legal liability, blameworthiness, praiseworthiness, role
responsibility, and causal responsibility [34]. Of these, causal
responsibility — the causal importance of an agent’s action for the
implementation of an event [12] — is a prerequisite for other forms of
responsibility, including moral responsibility and blame [33, 34], and
therefore received most attention in terms of existing metrics and models.
Models of causal responsibility traditionally involve counterfactual and
probabilistic reasoning about an agent’s action being pivotal for an event,
[1, 6, 12, 32]. Collective causal responsibility of groups of agents in an
interaction has also been modelled by exploring how groups of agents by
concerted actions can cause or prevent an event [35, 36, 37]. Specifically in
the context of human-AI interactions, comparative human causal responsibility
has been modelled using information theory to quantify the influence of human
agents on an outcome [9, 11, 10]. These more fundamental metrics and models
are suitable for relatively abstract scenarios where agents can choose not to
act at a given instant. However, in spatial interactions like traffic, all the
agents are physically embedded in the scene and they cannot choose not to act
— not moving or not stopping would also have an impact on the safety of the
interaction. This makes it difficult to apply the existing work on quantifying
causal responsibility to spatial interactions of embodied agents.

Contribution.  To bridge the gap between the existing practical approaches to
quantifying responsibility in traffic and the theoretically-grounded metrics
of responsibility, here we propose a metric for causal responsibility designed
specifically for spatial interactions. We assume perfect information
conditions and simultaneous actions to propose a metric for ex-post causal
responsibility for one instant of interaction. The metric is based on the
principle that an agent restricting the feasible action space of another agent
should be held responsible for the state and safety of the latter. To make the
metric sensitive to norms, we propose the use of Moves de Rigueur — a
consistent set of default actions that are prescribed for all the agents.

Outline.  In Section 2 we describe the nature of spatial interactions and how
we model these in a grid world simulation. Section 3 introduces the Feasible
Action-Space Reduction (FeAR) metric and the notion of Move de Rigueur (MdR).
The sensitivity of the FeAR metric to actions and norms, and a potential
application of the metric to multi-agent interactions are demonstrated with
case studies in Section 4. These are followed by the discussion and
conclusions (Section 5).

##  2 Grid World for Spatial Interaction

(a) Real world interaction

(b) Grid world representation

Figure 1: The grid-world representation of a real-world interaction at an
intersection. The yellow car is slowing down at the intersection because the
blue car is aggressively entering the intersection. This in turn affects the
red car which has to slow down so as not to hit the yellow car. The red,
yellow and blue cars are represented by agents $i$, $j$ and $k$ respectively
in the grid world.

To define the metric, we need a model environment that captures the essence of
spatial interactions. In spatial interactions the assertive moves of one agent
can restrict the action space of another agent and the latter may be forced to
move out of the way. Consider two cars approaching an intersection (yellow and
blue cars in Fig. 1(a)). If the blue car accelerates to cross first, it
usually forces the yellow car to decelerate. On the hand, if the blue vehicle
decelerates, it gives the yellow car more freedom to decide whether to
accelerate or decelerate. Thus when the blue car accelerates, it is acting
assertively because it has more influence on the state of the yellow car than
when it decelerates. Now consider a third car (red) which is following the
yellow car. If the yellow car slows down, the red car is also forced to slow
down. Thus the assertiveness of the blue car can have an impact on the red car
too. To model these interactions, it is necessary to consider the speed of the
cars and the collisions between them in multi-agent settings.

We use grid world simulations to model multi-agent spatial interactions like
the one described above. For example, the real-world scenario depicted in Fig.
1(a) would be represented by the grid world instance shown in Fig. 1(b). We
only consider the position and speed of the agents — we assume that agents
move with constant velocity in each instance. The actions chosen by the agents
would correspond to their speed. An action would always be in one direction,
and the speed would correspond to the number of cells an agent would move
through during an action. For our cases, we assume that all agents are
identical and have the same choice of actions.

The maximum speed corresponds to actions that move through four cells. The
actions are represented using the first letter of the direction followed by
the number of cells it moves through - S0 for stay, R1 for one cell to the
right, L1 for one cell to the left, U1 for one cell up, D1 for one cell down,
etc. So each agent has 17 possible actions - S0, R1, R2, R3, R4, L1, L2, L3,
L4, U1, U2, U3, U4, D1, D2, D3 and D4. For each instant, collision checks are
done to see if any agents are colliding as they move. When a collision is
detected, the agents that collide would stop at the last safe cell they
occupied. Agents would be embedded in a map with valid locations. Actions of
agents that try to push them outside the valid locations would also be treated
as causing collisions. Details of the collision-check and step-update
algorithms can be found in the supplementary materials111Detailed explanation
of the functions and algorithms can be found in the supplementary materials:
https://osf.io/5s9m2..

##  3 The FeAR Metric

###  3.1 Preliminaries

We consider a grid world with a state-space $\StateSpace$ where,
$\aState\in\StateSpace$ represents the map of the grid world and the locations
of agents. $\Agents$ represents the set of $k$ agents $\\{1,..,k\\}$.
$\ActionSpace{i}$ is the action space of agent $i$. The joint action space of
the all the agents is $\JointActionSpace=\times_{i=1}^{k}\ActionSpace{i}$. The
action chosen by agent $i$ is $\action{i}$ and the joint action of all the
agents is $\jointAction=\times_{i=1}^{k}\action{i}$. Each agent in prescribed
a Move de Rigueur (MdR) $\mdr{i}(\aState)$ for a given state $\aState$. The
joint MdR for all the agents is
$\JointMdR(\aState)=\times_{i=1}^{k}\mdr{i}(\aState)$. We only consider one
state per case, so for brevity from here on, we will refer to
$\mdr{i}(\aState)$ as $\mdr{i}$ and $\JointMdR(\aState)$ as $\JointMdR$. We
assume that the states and actions are finite and discrete. We also assume
that each agent chooses one action per instance.

For a given state $\aState$ and joint action $\jointAction$, the function
$\validCount{\aState}{\jointAction}{j}$ gives the count of feasible moves
available for agent $j$††footnotemark: . An intervention of replacing
$\action{i}$ (the move of agent $i$) with another action $a^{\prime}_{i}$ is
denoted as $\jointAction_{i}\leftarrow a^{\prime}_{i}$. Thus, the intervention
of replacing $\action{i}$ with $\mdr{i}$ (the MdR of agent $i$) would then be
$\jointAction_{i}\leftarrow\mdr{i}$. $\nijMdR$ gives the number of feasible
actions of agent $j$ when agent $j$ would have chosen the MdR $\mdr{i}$.
$\niiMdR$ gives the number of feasible moves available to agent $i$ when all
other agents $\neg i$, choose their MdRs $\mdr{\neg i}$. To map the value of
the metric to the range $[-1,1]$, the function $Z()$ was used.

$Z(x)=\begin{cases}1,&\text{if }x\geq 1\\\ x,&\text{if }-1<x<1\\\ -1,&\text{if
}x\leq-1\\\ \end{cases}$ (1)

###  3.2 Feasible Action-Space Reduction

This work proposes a metric based on the idea that one agent taking actions
that restrict the feasible actions of another agent must be held responsible
(at least to some extent) for the state of the latter agent. Intuitively, in
spatial interactions, an agent’s action that leaves another agent with zero
feasible moves, makes the latter unsafe in all possible contingencies, and
thus should bear total responsibility for the state of the second. Also, we
intuit that when the actions of the former do not change the feasible action
space of the latter, the former has no influence on the state of the latter.
In the intermediate case, a greater reduction in the feasible action space of
the latter should reflect in greater influence of the former on the latter. In
this paper, when the action of an agent $i$ reduces the feasible action space
of an agent $j$, we say that $i$ is assertive towards $j$. Similarly, when the
action of agent $i$ increases the feasible action space of $j$, we say that
$i$ is courteous towards $j$. Based on these intuitions, we propose a metric
based on feasible action space reduction to capture the influence of one agent
on another.

Responsibility ascribed to agents is highly dependent on contexts and norms
[8, 20, 34]. Agents in particular situations might experience a reduction in
feasible actions that are not caused by the actions of other agents. For
example, an agent in a dead-end might have a very small feasible action space.
Therefore to capture the effect of contexts, we formulate the metric as the
ratio of feasible action space reduction in comparison to a nominal case. To
define the nominal case, we utilise the idea of Moves de Rigueur (MdR) — a
consistent set of default actions that are prescribed to agents. Thus the
Feasible Action-space Reduction (FeAR) metric is defined based on the ratio by
which the action of an agent $i$ reduces the feasible action space of another
agent $j$ ($\nij$) in relation to the feasible action space of the latter when
$i$ would have chosen the MdR ($\nijMdR$) as:

$\FeAR_{i,j}=\begin{cases}\clip{\frac{\nijMdR-\nij}{\nijMdR+\epsilon}},&\text{if
}i\neq j,\\\ \\\ \clip{\frac{\nii}{\niiMdR+\epsilon}},&\text{if }i=j.\\\
\end{cases}$ (2)

$\epsilon$ is added to the denominator to ensure that the metric can still be
defined when $\nijMdR$ or $\niiMdR$ are zero ($0<\epsilon\ll 1$).

Consider the FeAR ($\FeAR^{\prime}_{i,j}$) for an alternative action
$a^{\prime}_{i}$ instead of $\action{i}$ for agent $i$. If
$\FeAR^{\prime}_{i,j}>\FeAR_{i,j}$, then $a^{\prime}_{i}$ is more assertive
towards $j$ than action $\action{i}$. If $\FeAR^{\prime}_{i,j}<\FeAR_{i,j}$,
then $a^{\prime}_{i}$ is more courteous towards $j$ than action $\action{i}$.
When $\FeAR_{i,j}=1$, it means that $i$’s action reduces the feasible action
space of $j$ to an empty set and thus $i$ should be ascribed complete
responsibility for the state of $j$.

###  3.3 Move de Rigueur

The Move de Rigueur $\mdr{i}$ is used to provide a baseline condition with
which an agent’s action is compared. In counterfactual analysis for causal
responsibility, the pivotality of an action for an outcome is evaluated based
on whether the outcome would have happened if the agent had not chosen that
action [17, 12]. In a spatial interactions, all agents have to choose an
action — even if it is not to move or not to stop. We propose MdR as a means
of relaxation of the counterfactual condition of not choosing an action.

In our definition, Moves de Rigueur are a consistent set of default actions
that are prescribed to all the agents in an interaction. The prescribed MdRs
have to be consistent in the sense that if all agents follow their MdRs, there
should not be any collisions. The FeAR analysis is dependent on the assumption
that all agents know their MdRs. We consider MdR as a heteronomous mandate
[16] which, if followed by the agents, must yield zero responsibility ascribed
to these agents. Consequently, as per the definition Eq. 2, $\FeAR_{i,j}(i\neq
j)$ when $i$ chooses $\mdr{i}$ is zero.

$\FeAR_{i,i}$ indicates how much of the feasible action space of $i$ when all
the other agents $\neg i$ had chosen their MdRs remains under the chosen joint
action $\jointAction$. So, it is better to think of $\FeAR_{i,i}$ as the
Feasible Action-space Remaining for agent $i$. When $\FeAR_{i,i}=1$, agent $i$
has at least as many feasible actions as it would have had when all the others
choose their MdR, i.e, $i$ has as much freedom as the MdRs can afford.

The greater the FeAR caused by $i$ on $j$, the greater the influence of $i$ on
$j$. $\FeAR_{i,j}>0$ indicates that, agent $i$ choosing the current action,
offers fewer actions to $j$, than when $i$ would have chosen the MdR i.e.,
$i$’s action is more assertive than the MdR. Similarly X.

##  4 Case Studies

In this section we use a series of simple case studies222The simulations were
run on CPU (Intel i7) in a Dell Latitude 7320 laptop with 16 GB of RAM. The
source code for reproducing all case studies can be found at
https://github.com/DAI-Lab-HERALD/FeAR. to demonstrate how the FeAR value is
dependent on 1) the action chosen by the agent, 2) on the Moves de Rigueur
(MdR) and 3) on the actions of other agents. The last case study illustrates
the use of the metric in multi-agent scenarios with large number of possible
interaction outcomes.

###  4.1 Two agents on a lane: the effect of agents’ actions on FeAR

Grid world instance - state and action selection

For actor’s MdR $\mdr{i}$

For actor’s move $\action{i}$

FeAR values

(a)

(b)

(c)

Figure 2: Case Study 1 explores the effect of an agent’s action on the FeAR
values for Mdr $\JointMdR=\text{S0-S0}$. The actions selected by the agents
for each instance are depicted in the first column. The second and third
columns show the feasible actions available to the agents when under the other
agents’ MdR $\mdr{i}$ — $\nijMdR$ — or their chosen action $\action{i}$ —
$\nij$. The move S0 is depicted in the center with a grey rectangle around it
with other left and right moves on either side. The feasible moves are shown
in green. The final column states the FeAR values that are calculated based on
the number of feasible actions. $\FeAR_{i,j}$ for $i\neq j$, represents the
Feasible Action-Space Reduction of $j$ by the action of $j$. $\FeAR_{i,i}$
represent Feasible Action-Space Remaining for agent $i$ left by the actions of
all other agents. Since the definition of $FeAR_{i,i}$ is slightly different,
they are shown in the plots with an extra white rectangle around them. For
$i\neq j$, if $\nij>\nijMdR$, then $\FeAR_{i,j}<0$ and vice versa.

The first case study demonstrates how the actions chosen by the agents affect
the FeAR values. The location of agents 1 and 2 are asymmetric in the sense
that 1 is more constrained than 2. In Fig. 2 we consider three instances where
(a) both agents move one step away from each other, (b) both agents move one
step to the right, and (c) both agents move a step closer to each other. In
(a) and (b) there are no collisions while in (c) both agents collide.

The MdR for the calculation is taken to be $\JointMdR=\text{S0-S0}$. When
agent 1 chooses the MdR (S0), agent 2 has $\nijMdR=6$ feasible moves (L1, S0,
R1, R2, R3 and R4). When 2 chooses the MdR (S0), 1 has $\nijMdR=4$ feasible
moves (L2, L1, S0 and R1).

In Instance (a) 1 chooses L1 which leaves 2 with $\nij=7$ feasible moves (L2,
L1, S0, R1, R2, R3 and R4). So $\FeAR_{1,2}=\frac{6-7}{6}=-0.17$ 333Z() only
makes an impact when the magnitude of the ratios exceed 1. The $\epsilon$ in
the denominator is only relevant in cases where the denominator is zero. For
brevity, these are excluded in the explanations when unnecessary.. And when 2
chooses R1, it increases the feasible actions to $\nij=5$ (L2, L1, S0, R1 and
R2). Thus, $\FeAR_{2,1}=\frac{4-5}{4}=-0.25$. So both agents have negative
FeAR and are courteous to each other. Since 1 has more feasible moves
($\nij=5$) than when the other agent chooses the MdR ($\nijMdR=4$),
$\FeAR_{1,1}=Z(5/4)=1.0$. Similarly for 2, $\FeAR_{2,2}=Z(7/6)=1.0$. Thus, in
this case, since both agents have more feasible moves than afforded by other
agents choosing MdR, they have the maximum possible FeAR values.

In instance (b) 1 chooses R1 which leaves 2 with $\nij=5$ feasible moves (S0,
R1, R2, R3 and R4) , and 2 chooses R1 which leaves 1 with $\nij=5$ feasible
moves (L2, L1, S0, R1 and R2). Thus,
$\FeAR_{1,2}=\frac{6-5}{6}=\frac{1}{6}=0.17$ is positive and 1 is being
assertive to 2, while, $\FeAR_{2,1}=\frac{4-5}{4}=-0.25$ is negative and 2 is
being courteous to 1. This causes a reduction in $\FeAR_{2,2}=5/6=0.83$ while
$\FeAR_{1,1}=Z(5/4)=1$ remains at the maximum possible value.

In instance (c) 1 chooses R1 which leaves 2 with $\nij=5$ feasible moves (S0,
R1, R2, R3 and R4), and 2 chooses L1 which leaves 1 with $\nij=3$ feasible
moves (L2, L1 and S0). Thus, $\FeAR_{1,2}=\frac{6-5}{6}=0.17$ and
$\FeAR_{2,1}=\frac{4-3}{4}=0.25$ are both positive which means that both
agents are being assertive to each other. This leads to reductions in
$\FeAR_{1,1}=5/6=0.83$ and $\FeAR_{2,2}=3/4=0.75$.

The above examples demonstrate how the actions of agents, determine the FeAR
values. In the given locations and $\JointMdR=\text{S0-S0}$, the actions L1 of
1 and R1 of 2 are considered courteous from the calculated FeAR values, while
the actions R1 of 1 and L1 of 2 are considered assertive from the FeAR values.
$\FeAR_{1,2}$ is dependant on the move of 1 and $\FeAR_{2,1}$ is dependant on
the move of 2. On the other hand, $\FeAR_{1,1}$ is dependant on the move of 2
and $\FeAR_{2,2}$ is dependant on the move of 1. So, $\FeAR_{i,j}$ where
$i\neq j$ capture the assertiveness/courteousness of $i$ towards $j$ , while
$\FeAR_{i,i}$ captures the complement of the collective
assertiveness/courteousness of all other agents on $i$ . Thus, the chosen
moves in combination with context determine the FeAR values and how agents are
considered courteous or assertive.

The examples also showcase how the causal responsibility ascribed to agents
can be asymmetric. In (a), both agents are courteous. In (c) both agents are
assertive. In (b) one agent is assertive while the other is courteous. It is
also worth noting that even in the case where both agents are found to be
courteous (a), there is a small asymmetry in the FeAR values
($\FeAR_{1,2}=-0.17$ and $\FeAR_{2,1}=-0.25$). This stems from the asymmetry
in the context — 1 is more restricted than 2 and has fewer feasible moves when
considering the MdRs. Thus, the causal influence of agents can be asymmetric
and are highly dependent on the context.

###  4.2 Two agents on a lane: the effect of agents’ Move de Rigueur on FeAR

Grid world instance - state and action selection

$\JointMdR$: S0-S0

$\JointMdR$: R1-R1

$\JointMdR$: R2-R2

(a)

(b)

(c)

Figure 3: Case Study 2 explores the effect of MdR on the FeAR metric. We
consider the same instances as in Case Study 1 (Fig. 2). Here three different
MdRs are considered — $\JointMdR=\text{S0-S0}$, $\JointMdR=\text{R1-R1}$ and
$\JointMdR=\text{R2-R2}$. $\FeAR_{i,j}$ for $i\neq j$, represents the Feasible
Action-Space Reduction of $j$ by the action of $j$. $\FeAR_{i,i}$ represent
Feasible Action-Space Remaining for agent $i$ left by the actions of all other
agents. For $i\neq j$, if $\nij>\nijMdR$, then $\FeAR_{i,j}<0$ and vice versa.
Thus changes in $\nijMdR$ affect the FeAR values.

In the first case study, we considered only one MdR, $\JointMdR=\text{S0-S0}$.
In this case study, we explore how changing the MdRs can affect the FeAR
values, again for the same instances considered before.
$\JointMdR=\text{R1-R1}$ considers the norm where both agents are supposed to
move one step to the right — like on a lane where cars are not supposed to
stop in the middle of the lane. $\JointMdR=\text{R2-R2}$ considers a norm
where the agents are supposed to keep moving — now at a higher speed than when
$\JointMdR=\text{R1-R1}$. For ease of reference, we are repeating the FeAR
values for $\JointMdR=\text{S0-S0}$ here too (See Fig. 3). For instances (a),
(b) and (c) the feasible actions available to agents 1 and 2 for each case
remain the same as they were for the previous case study.

For $\JointMdR=\text{S0-S0}$, 1 has $\nijMdR=4$ feasible moves (L2, L1, S0 and
R1) when 2 chooses the MdR, and 2 has $\nijMdR=6$ feasible moves (L1, S0, R1,
R2, R3 and R4) when 1 chooses the MdR. For $\JointMdR=\text{R1-R1}$, 1 has
$\nijMdR=5$ feasible moves (L2, L1, S0, R1 and R2) when 2 chooses the MdR
(R1), and 2 has $\nijMdR=5$ feasible moves (S0, R1, R2, R3 and R4) when 1
chooses the MdR (R1). For $\JointMdR=\text{R2-R2}$, 1 has $\nijMdR=6$ feasible
moves (L2, L1, S0, R1, R2 and R3) when 2 chooses the MdR (R2), and 2 has
$\nijMdR=4$ feasible moves (R1, R2, R3 and R4) when 1 chooses the MdR (R2).

The move L1 of 1 (instance (a)) offers more feasible actions to 2 than all the
MdRs considered (S0, R1 and R2). Thus, when 1 chooses L1, $\FeAR_{1,2}$ for
all the MdRs are negative and indicate that 1 is being courteous. It can also
be noted that the move L1 of 1 is considered more courteous under
$\JointMdR=\text{R2-R2}$ than under $\JointMdR=\text{S0-S0}$ (and
$\JointMdR=\text{R1-R1}$ being an intermediate condition).

Similarly, the move L1 of 2 (instance c)) offers fewer feasible actions to 1
than all the MdRs considered (S0, R1 and R2). Thus when 2 chooses L1,
$\FeAR_{2,1}$ for all the MdRs are positive and indicate that 2 is being
assertive. Also, as we go from $\JointMdR=\text{S0-S0}$ to
$\JointMdR=\text{R2-R2}$, the 2 choosing L1 is considered increasingly
assertive ($\FeAR_{2,1}$ for $\JointMdR=\text{S0-S0}$ $<$ $\FeAR_{2,1}$ for
$\JointMdR=\text{R1=R1}$ $<$$\FeAR_{2,1}$ for $\JointMdR=\text{R2-R2}$).

As we go from $\JointMdR=\text{S0-S0}$ to $\JointMdR=\text{R2-R2}$, the move
R1 of 1 (instances (b and c)) goes from being considered assertive
($\FeAR_{1,2}>0$) to being considered courteous($\FeAR_{1,2}<0$), while, the
move R1 of 2 (instances(a and(b)) goes from being considered courteous
($\FeAR_{2,1}<0$) to being considered assertive ($\FeAR_{1,2}>0$). These
examples illustrate how the complex interplay of actions, contexts and norms
(represented by the MdRs) can be represented by the FeAR metric.

###  4.3 Three agents at an Intersection: the effect of other agents’ actions
on FeAR

This case study illustrates the use of the FeAR metric in interactions of
multiple agents. We consider interactions of three agents at an intersection
(Fig. 4). Agents 1 and 2 are approaching the intersection from the left and
agent 3 is approaching it from above. In instance (a), there are no collisions
and agent 3 safely passes the intersection. If agent 1 decides to move faster
(R4), it causes agent 1 to rear-end agent 2 (instance (b)). This rear-ending
can be prevented if agent 2 moves forward (R1) as seen in instance (c). But,
if agent 2 moves forward faster (R2), it causes a T-bone collision with agent
3 (instance (d)). This T-bone collision could have been prevented if agent 3
had chosen to stay (S0) as in instance (e). Thus, we explore a series of
interactions where agent 2 is caught between agents 1 and 3.

As we move from instances (a) to (b), the action of agent 1 becomes more
restrictive to agent 2 and $\FeAR_{1,2}$ increases from $0.3$ to $0.7$. This
value of $\FeAR_{1,2}$ remains unchanged for instances (b), (c) and (d) —
irrespective of whether agent 2 collides with agent 1 or 3. In instance (e),
when agent 3 chooses (S0), it increases the feasible action space of 2. This
leads to $\FeAR_{1,2}$ decreasing to $0.4$ even though the action of agent 1
does not change from instances (b), (c) and (d). Thus, $\FeAR_{1,2}$ is
dependent on the actions of both agents 1 and 3. Similarly, we can see the
influence of agent 1’s action on $\FeAR_{3,2}$ as we go through instances (e),
(d), (c), (b) and (a) ($\FeAR_{3,2}=0.0$ for (e), $\FeAR_{3,2}=0.6$ for (d),
(c) and (b), and $\FeAR_{3,2}=0.4$ for (a)).

Now, let us consider the effect of the actions of agent 2 on agents 1 and 3.
In instances (a) and (b), agent 2 is choosing S0 which is the MdR. Hence,
$\FeAR_{2,1}=0$ and $\FeAR_{2,3}=0$. When agent 2 chooses R1 in instance (c),
it increases the feasible action space of 1 and thus $\FeAR_{2,1}$ decreases
to $-0.2$. Since choosing R1 does not put agent 2 in the path of agent 3, the
feasible action space of agent 3 remains unchanged and $\FeAR_{2,3}=0$. But,
when agent 2 chooses R2 (instances (d) and (e)), it moves into the path of
agent 3 restricting the feasible action space of agent 3. Thus, $\FeAR_{2,3}$
increases to $0.2$. Thus in instances (d) and (e), agent 2 is acting
assertively towards agent 3 ($\FeAR_{2,3}>0$) and courteously towards agent 1
($\FeAR_{2,1}$)

In our analysis, we only considered one MdR $\JointMdR=\text{S0-S0-S0}$ which
gives equal priority at all. In real traffic, certain lanes are given priority
over others at intersections. These changes in priorities could be captured
using MdRs like $\JointMdR=\text{S0-S0-D2}$ (which prioritises the vertical
lane) or $\JointMdR=\text{R2-R2-S0}$ (which prioritises the horizontal lane).
Analyses of these scenarios can be found in the supplementary materials.

Overall, this cases study illustrates how the complexities of multi-agent
interactions are captured in the FeAR metric. We also show how agents caught
in the middle can be assertive to some agents while being courteous to other
agents.

Grid world instance

$\JointMdR$: S0-S0-S0

(a) No collisions

(b) Collision between 1 and 2

(c) No collisions

(d) Collision between 2 and 3

(e) No collisions

Figure 4: Case Study 3 explores the FeAR metric in the context of multi-agent
interactions. We consider the interactions between three agents at an
intersection. Agent 2 is caught between Agents 1 and 3. Instance (c) is the
central condition where no collisions happen. But, if agent 2 slightly changes
its move, it can collide with agent 1 (instance (b)) or with agent 3 (instance
(d)). The collisions in instances (b) and (d) could also be prevented by the
actions of agent 1 (instance (a)) or agent 3 (instance (d)) respectively.
Thus, in these interactions $\FeAR_{1,2}$ and $\FeAR_{3,2}$ are dependent on
the actions chosen by third parties too.

###  4.4 Four agents on a lane: illustrating complex interactions

Instances with extreme values of
$\left(\sum_{i}^{k}\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{k}(\FeAR_{i,j})^{2}\right)$ (a) Instance with smallest sum
of non-diagonal FeAR values

(b) Instance with biggest sum of non-diagonal FeAR values

Figure 5: Case Study 4 illustrates the application of the metric in scenarios
with potentially complex interactions of multiple agents. These two instances
were selected from 10,000 instances based on the FeAR values. In Fig. 5(a) —
the case with minimum sum of squares of FeAR — all agents choose MdR (or the
action equivalent to MdR due to hitting a border). In Fig. 5(b) — the case
with maximum sum of square of FeAR — agents are acting assertively to the
agents ahead of them, while being courteous to the agents behind them. All of
them deviate from the MdR and together they avoid collisions.

To highlight the insights provided by the FeAR metric in more complex
scenarios, we analysed the interactions of 4 agents on a single lane. To do
this, we randomly generated 10,000 instances that all involved 4 agents on a
lane 10 units long. We randomized the locations of the agents as well as their
action choices (S0, R1, R2, R3 and R4). This set of actions was selected to
mimic how agents would behave on a one-way lane. For each agent, the location
on the lane and the action were chosen at random with equal likelihood.

We expected these randomly generated instances to vary substantially in
complexity of the interaction. Here we investigated two extreme cases: an
instance in which the agents acted most independently from each other, and an
instance with a high level of interconnectedness between the agents. To do so,
among the 10,000 generated instances we selected the ones with the minimum and
maximum sum of squares of the $\FeAR_{i,j}$ for $i\neq j$ (Fig. 5(a) and Fig.
5(b), respectively).

In the minimum sum-of-squares case Fig. 5(a), agents 1,2 and 3 choose S0;
agent 4’s move to the right hits the boundary. Thus all agents would
effectively be doing S0. Which means that effectively all agents are following
the MdR and hence FeAR values are zero. It must be noted that this is not the
only case where the sum of squares of FeAR is zero (the others are not shown
here).

In the maximum sum-of-squares case Fig. 5(b), there are no collisions, but all
the agents are deviating from the MdR (S0). If agents 2, 3 or 4 choose the
MdR, they would cause a collision for all the agents behind them. Thus their
current action increases the feasible action space of all agents behind them —
indicated by the negative FeAR values below the diagonal. At the same time,
since agents 1,2 and 3 choose R4, they severely restrict the feasible action
of the agent in front of them, compared to when they choose the MdR.

These two cases demonstrate that in multi-agent interactions, the sum of non-
diagonal FeAR values captures the intuitions about the
independence/interconnectedness of the agents’ actions. In a similar vein,
large positive (non-diagonal) FeAR values across multiple agents would flag
cases where agents are being very assertive. Conversely, instances where
agents are being very courteous would be marked by consistently negative
values of FeAR (see the supplementary materials for details).

Overall, this case study illustrates how the FeAR metric can be analysed in
more complex scenarios in which the number of possible combinations of agent
positions and actions is very large (more than 3 million in this case). To
this end, comparisons between different outcomes of the interaction (e.g.
different actions of the agents for a given set of positions) can be
facilitated by aggregate FeAR values across all the agents.

##  5 Discussion and Conclusion

Our Feasible Action-space Reduction metric can be used as an indicator for
causal responsibility in spatial interactions where each agent is physically
present in the scene and, therefore, has to take at least one action. The FeAR
metric captures the effects of actions of agents, norms (represented as Moves
de Rigueur (MdR)), and the physical layout of the grid world. The case studies
also illustrate how the responsibility ascription among agents can be
asymmetric ($\FeAR_{i,j}\neq\FeAR_{j,i}$). We also illustrate how agents can
be assertive to some agents ($\FeAR_{i,j}>0$), while being courteous to others
($\FeAR_{i,j}<0$). A positive FeAR value does not necessarily mean that a
collision happens, but if a collision does happen, it indicates the share of
responsibility of a given agent. For example, with three agents interacting at
an intersection Fig. 4, when two agents (1 and 3) collectively constrain an
agent (2), irrespective of whether it collides with 1 or 3, both agents share
some responsibility for the state of 2. In real-world traffic interactions,
when a car swerves to avoid hitting another car but ends up hitting a third
car, it can be unclear who is responsible for the interaction. The FeAR metric
along with a suitably chosen set of MdRs would be able to help guide the
debate on the distribution of responsibility in such controversial cases.

We used Moves de Rigueur to represent norms. MdRs are actions that are
expected of or prescribed to agents in an interaction. Ideally, theses actions
should be aligned with human behaviour and cognitive capacity. For example,
consider the case where humans are used/trained to stop at an intersection. In
this case, prescribing an MdR to keep driving through an intersection would be
inappropriate. Similarly, automated vehicles require fallback actions and
minimal risk manoeuvres to ensure safety in case of emergencies. Well-defined
MdRs aligned with human behaviour could provide good fallback actions for
automated-vehicle algorithms.

We proposed Feasible Action-Space Reduction as a metric for causal
responsibility among agents in a spatial interaction. While other models of
responsibility [1, 6, 9, 12, 32, 35] ascribe responsibility for events, in
spatial interactions, the FeAR metric assigns causal responsibility for the
state and safety of agents. Causal responsibility has traditionally been
evaluated based on probabilistic reasoning over sequences of events. Our
formulation does not consider the probabilities of actions — we consider that
all actions of an agent are equally likely. Our metric only considers an
instant of the interaction and does not consider the history of events like
other approaches. Because of the different nature of these metrics, no
straightforward comparison can be made between FeAR and other models of causal
responsibility. Developing methods to meaningfully compare disparate ways of
quantifying responsibility is an important avenue for future work.

Existing literature on spatial navigation has several theories and models that
try to address various aspects of how pedestrians and drivers engage in
spatial interactions [18, 19, 21, 25, 28, 31]. Such models, typically
incorporate several assumptions to explain and predict human behaviour. In
contrast, the goal of this paper is not to describe human behaviour but to
develop a model-agnostic framework for quantifying causal responsibility.
However, we do believe that such models would be crucial for anticipating the
behaviour of other agents when using FeAR for estimating responsibility of
future actions (ex-ante responsibility).

Methods like Responsibility Sensitive Safety [29] prescribe "proper responses"
for all agents and ascribe greater responsibility to agents that deviate from
the "proper response". The MdR approach offers more freedom to the agents —
agents do not have to follow the MdR strictly and additionally have the
possibility of being more courteous than the MdR. Furthermore, designing
consistent MdRs can help humans and automation to have mutually compatible
representations, which can help reduce the responsibility gaps in human-AI
interactions [5, 27].

Limitations and future work. FeAR does not look at past interactions, states
or beliefs of the agents. To get a more complete picture of causal
responsibility as well as capture moral/role/capacity responsibility [34],
more factors should be considered. The current implementation of FeAR only
works for discrete and finite action and state spaces. In real-life
interactions, an agent might have an uncountable number of actions to choose
from. Our goal was to provide the first steps and the foundation towards
measuring causal responsibility in spatial settings, and we felt that adding
more complexity would have made the problem intractable. By making these
assumptions, we were able to derive a metric that will then serve as a basis
for investigating more complex settings, including continuous time and action
space. Also, the current formulation of the metric scales exponentially with
the number of agents and thus is not practical for real-time applications. A
possible improvement point for future works would be to consider the
calculated FeAR values as ground truth to train a supervised learning method
to approximate the FeAR in a given instance.

Defining MdRs for all possible scenarios is a non-trivial task; especially if
done manually. Establishing good norms for driving (like, ‘always maintain
safe time headway’) would limit occurrence of scenarios where defining MdRs is
challenging. Nevertheless, there might be situations where defining MdRs ahead
of time might not be feasible, due to unknown circumstances or unpredictable
behaviour of agents. Methods like Norm Synthesis [22, 23] or Responsibility-
Sensitive Safety [29, 30] could be useful to automate the process of defining
conflict-free MdRs. Making the MdR method applicable in practice is a
challenging issue that future work should address.

We opted to present the metric in an intuitive way and use case studies to
evaluate such intuitions and how it could be applied for understanding causal
responsibility in multi-agent spatial interactions. In future works,
especially when dealing with continuous domains, we believe that a formal
evaluation might be necessary to achieve necessary insights on the robustness
of the metrics and its impact on safety

Conclusions. We proposed Feasible Action-space Reduction (FeAR) as a metric
for quantifying the causal responsibility a given agent’s actions have on the
state and safety of other agents in an interaction. To represent norms, we
introduced Moves de Rigueur (MdR) — consistent sets of default actions that
are prescribed to each agent in an interaction. We used case studies to
illustrate how the metric captures the effects of agents’ actions, norms and
multi-agent interactions. We also demonstrated an application of the metric to
flag instances with high levels of interconnectedness/independence among the
actions chosen by agents.

Ensuring safety and implementing meaningful human control in automated driving
requires a common interdisciplinary framework that engages conversations
between road users, vehicle manufacturers, traffic designers, regulatory
agencies and policy makers [4, 24]. The FeAR metric and the concept of MdR is
a step towards this interdisciplinary framework.

This work is supported by the TU Delft AI Labs programme.

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Lund jet plane for Higgs tagging

Charanjit K. Khosa

Dipartimento di Fisica, Università di Genova and INFN, Sezione di Genova, Via
Dodecaneso 33, 16146, Italy

<EMAIL_ADDRESS>
50th International Symposium on Multiparticle Dynamics
(ISMD2021)
12-16 July 2021 10.21468/SciPostPhysProc.?

## Abstract

We study the boosted Higgs tagging using the Lund jet plane. The convolutional
neural network is used for the Lund images data set to classify hadronically
decaying Higgs from the QCD background. We consider $H\to b\bar{b}$ and $H\to
gg$ decay for moderate and high Higgs transverse momentum and compare the
performance with the cut based approach using the jet color ring observable.
The approach using Lund plane images provides good tagging efficiency for all
the cases.

##  1 Introduction

Improved Higgs tagging will be beneficial for new physics searches as well as
for precise standard model measurements at the Large Hadron Collider (LHC).
Recent developments for Higgs tagging explored the potential of Machine
Learning (ML) techniques along with novel observable construction, see e.g.
[1, 2]. Idea of using ML for jet tagging using low-level raw information (jet
images) has been suggested a couple of years ago. We consider Lund jet images
and Convolutional Neural Networks (CNN)111CNNs provide very efficient image
classification for different types of input data in High Energy Physics see
e.g. [6, 7] for the signal and background classification for hadronically
decaying boosted Higgs boson[3]. The Lund jet plane is a theory inspired jet
representation which is used for jet tagging recently[4, 5]222For experimental
measurement of the Lund jet plane density see [8, 9].. It uses the information
of the radiation pattern inside the jet and is a log-log
$(\ln\frac{1}{\Delta},\ln\frac{k_{t}}{\text{GeV}})$ plane of the angle of the
emission and the transverse momentum w.r.t the jet. We consider the primary
Lund jet plane. We assess the performance of this approach for both $H\to
b\bar{b}$ and $H\to gg$ using two $p_{T}$ benchmarks (250 GeV and 550 GeV). We
compare the classification accuracy with the standard cut based approach where
one would consider observables. We consider a single observable named jet
color ring, which by construction is an optimal observable for (two-prong)
color singlet tagging[10]. It is defined as

$\mathcal{O}=\frac{\Delta_{ka}^{2}+\Delta_{kb}^{2}}{\Delta_{ab}^{2}}$ (1)

where a and b are the primary (leading or tagged according to the decay
products properties) subject, k is the remaining leading subject and $\Delta$
is the separation in the rapidity-azimuth plane.

In our analysis, signal process we consider is $pp$ $\rightarrow$ $ZH$ where
$Z\rightarrow\mu^{+}\mu^{-}$ and $H\rightarrow$ $b\bar{b}$ or $gg$, and
background processes are $Zbb$ and $Zjj$ for the $H\to b\bar{b}$ and $H\to gg$
analyses, respectively. We simulate events with generation level
$p_{T}^{\mu\mu}$ cut of 200 and 500 GeV using Madgraph 2.7.2 [11] at
$\sqrt{s}=14$ TeV. Pseudo-rapidity cuts for leptons ($|\eta_{l}|<$2.5) and
jets ($|\eta_{j}|<$5.0) are also imposed at the generation level. Pythia 8.303
[12] (with default Monash 2013 tune [13]) is used for the parton-shower and
the hadronization.

We consider the particles excluding muons from Z decay and form radius R=1
jets using the anti-$k_{t}$ algorithm [14] implementation in Fastjet 3.3.3
[15]. We consider the leading jet with a $p_{T}$ cut of 250 GeV (or 550 GeV)
and jet mass $110$ GeV $<m_{J}<$ 140 GeV for the analysis. We further need to
identify the subjects inside the jet for the jet color ring. For that, we
cluster the charged particles with $p_{T}$ >500 MeV using anti-$k_{t}$
algorithm (radius R=0.2) and form sub-jets. For sub-jets with $p_{T}>$ 5 GeV,
we check if the $\Delta<0.8$ with the leading large radius jet. For the $H\to
b\bar{b}$ analysis, we further identify these subjets as b-jets if $\Delta$
separation with the large radius jet is less than 0.2. Analysis efficiency is
different for the signal and background processes as well as for different
$p_{T}$ benchmarks. A large sample of events is generated such that after
selections we get at-least 100K events for all the data sets. We re-cluster
the constituents of the large radius jet using C/A algorithm, and consider the
declustering history of this jet to get the primary Lund jet plane. We
consider 25 by 25 pixels for the Lund jet images. For the CNN training, the
data set of 200 K (equal proportions) signal and background events is divided
into three sets: training set, validation set and test set in 60:20:20 ratios.

Figure 1: The averaged primary Lund jet plane images and the jet color ring
distributions for the $p_{T}>$ 250 GeV and 550 GeV benchmarks of $H\to
b\bar{b}$ analysis. Figures are taken from Ref[3] under the Creative Commons
Attribution 4.0 International license.

##  2 H $\to$ $b\bar{b}$ Analysis

Averaged Lund jet plane images and jet color ring distributions of the signal
and background are shown in Fig. 1 for both the $p_{T}$ benchmark points
(BPs). The main difference in the Lund plane images is in the high $k_{T}$ and
large $\Delta$ region between the two $p_{T}$ benchmarks. Last column in Fig.
1 has the jet color ring signal and background distributions. Signal jet color
ring distributions are peaked towards the lower values and background
distributions are comparatively flat. The CNN architecture is presented in
Table 1. These benchmarks are labelled as BP1 ($p_{T}>$ 250 GeV) and BP2
($p_{T}>$ 550 GeV). We compare the Receiver Operating Characteristic (ROC)
curves AUC333AUC is the area under the curve. Note that we use
$\mathcal{A}$=1-AUC as a performance measure. in Fig. 2. We can see CNN
classification accuracy for the Lund jet images is slightly better than that
of using a cut on the jet color ring (which by construction is an optimal
observable to tag a color singlet).

| BP1 | BP2 | BP3 | BP4
---|---|---|---|---
$N_{1}$ Conv2D | 16 | 16 | 16 | 16
$N_{2}$ Conv2D | 16 | 16 | 16 | 16
Dropouts | 0.25 | 0.05 | 0.20 | 0.20
$N_{3}$ Conv2D | 32 | 16 | 16 | 32
$N_{4}$ Conv2D | 32 | 16 | 16 | 32
Dropouts | - | 0.05 | 0.30 | 0.30
Flat Layer | 800 | 800 | 800 | 800
Epochs | 15 | 15 | 20 | 20
Batch Size | 1000 | 1000 | 800 | 700
Table 1: CNN architectures used for different data sets. Here $N_{i},i=$1,..4
represents the number of filters in the $i^{th}$ convolutional layer.

Figure 2: CNN (used for the Lund jet plane images) and color ring ROC curves
for $H\to b\bar{b}$ analysis [3].

##  3 H $\to$ gg Analysis

Lund jet plane averaged images and jet color ring distributions are given in
Fig. 3. We can easily spot the difference between the signal and background
Lund jet images however, jet color ring distributions are almost overlapping.
Signal distributions are the same as that of $H\to b\bar{b}$ analysis but the
background distributions are very different. In this case several possible
color configurations are contributing to the background, contrary to the
earlier case where the background was mainly due to $g\to b\bar{b}$ splitting.
As shown in Fig. 4, in this case color ring offers no distinction but CNN can
efficiently distinguish the signal and the background. CNN architecture for
250 GeV and 550 GeV benchmark is given by BP3 and BP4 columns in Table 1.

Figure 3: The averaged primary Lund jet plane images and the jet color ring
distributions for the $p_{T}>$ 250 GeV and $p_{T}>$ 550 GeV benchmarks of
$H\to gg$ analysis. Figures are taken from Ref[3].

Figure 4: CNN (used for the Lund jet plane images) and color ring ROC curves
for the $H\to gg$ analysis [3].

##  4 Conclusions

We consider the Lund jet plane for Higgs tagging. Using CNN for the signal-
background classification, we compare the performance of this set-up with the
jet color ring observable. We found the Lund plane and CNN combination
provides equally good tagging efficiency for the $H\to b\bar{b}$ and $H\to gg$
analysis. For the first case, the jet color ring also provides similar tagging
performance, but it does not offer any separation for the $H\to gg$ decay.
This study further motivates the combined analysis of color sensitive
observables and Lund jet plane for the optimal Higgs tagging.

## Acknowledgements

I thank Simone Marzani for the collaboration on the work presented here. This
work is supported by Università di Genova under the curiosity-driven grant
“Using jets to challenge the Standard Model of particle physics” and by the
Italian Ministry of Research (MUR) under grant PRIN 20172LNEEZ.

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