Master’s Thesis 要旨

A Microscopic Pedestrian Simulation Model with Theory of Mind
(心の理論を考慮した微視的歩行者シミュレーションモデル)
Dept. of Mathematical Informatics 48096201
Supervisor

Ryo Adachi (安達 涼)
Professor Kazuyuki Aihara (合原 一幸教授)

1 Introduction
Modeling and simulating pedestrian behavior have contributed
to design congested public spaces and improve the efficacy ofpedestrian flows for the last few decades. Recently, a paper intended to capture the game theoretic aspects among pedestrians
which are conspicuous in collision avoidance behavior and in its
negotiation process ([1]). Although their works were elaborateand had been well implemented for practical applications, collision avoidance behavior is simplified too much with classicalgame theory and there must be some fundamental and important
perspectives that are missing by such over-simplification.
In this paper, we propose a microscopic pedestrian simulation
model by adopting a universal modeling method in theoreticaldecision science. Also, we probe into pedestrians’ mental attribution to infer the behavior of other pedestrians, which can beregarded to be Theory of Mind ([2]). Furthermore, we present
how depth of inference levels affects the efficacy of a pedestrianflow as a whole with computer simulations.

2 The model
Our modeling procedure is composed of three parts. We first
stipulate available transition of pedestrians and then define their
movements as a sequence of decision making tasks. Then, we
incorporate the idea of Theory of Mind to our model.
2.1 A pedestrian modelA pedestrian must be presented with some alternatives to choose
from so that they can evaluate the value of utility for each of
their options. Therefore, we discretize pedestrians’ applicabledirection and speed at a certain step and give a finite numberof choices. The choice set of a pedestrian denoted by Ωi (τ ) is
composed of two sets; one is the choice set of direction denotedby Ωθi (τ ) and it is given asΩθi (τ ) = {θi (τ )| θi (τ ) = θi (τ − 1)2m − n+φ, m = 0, 1, · · · , n},n
and the other is the choice set of speed denoted by Ωvi and it isstipulated askΩvi = {vi | vi = vid , k = 0, 1, · · · , l}.
lThen, the choice set Ωi (τ ) is given asΩi (τ ) = Ωθi (τ ) × Ωvi .
Above configuration is summarized in Figure 1.
2.2 Modeling pedestrian behavior with decision theory
We assume that pedestrian behavior can be divided into two factors and we name them goal-directed (GD) behavior and riskoriented (RO) behavior. GD behavior reflect the fact that pedestrians basically attempt to minimize the time that is required toreach their goals and RO behavior reflect the fact that pedestrians hate physical contacts with other pedestrians or walls. To

Fig. 1. Choice set of pedestrian i. The fan-shape is consistent
with our experience as pedestrians.

represent such behavior, we define the utility of a state si (τ + T )for pedestrian i who is in the state si (τ ) in a simple form as
U (si , s−i ) = αT vi (τ ) cos(θi (τ ) − θid (τ ))∑f (|si (τ + T ) − sj (τ + T )|)+β
j:sj (τ )∈Ai (τ )+γ∑

f (dist(si (τ + T ), wall)),wallwhere s−i denotes the set of positions of pedestrians other thanpedestrian i and the state si (τ + T ) is completely stipulated bythe choice set Ωi (τ ). Also, Ai (τ ) represents the set of pedestrians who are in the eye sight of pedestrian i at step τ and theanticipatory period which is denoted by T represents how distantfuture events a pedestrian focus on to decide his action. Furthermore, we specify the function f as f (x) = −e−ax so that we canignore the interaction between a pedestrian and other pedestrians
or walls when the value of x that corresponds to their distance islarge enough.
Also, utilities must determine the transition probability and it
is obtained by adopting the softmax function aspi (si (τ + T )|si (τ )) =exp(λE(Ui (si (τ + T ))))∑.
exp(λE(Ui (s′i (τ + T ))))s′i (τ +T )Here, λ is the inverse-temperature parameter and it controls how
likely a pedestrian deviates from the rational choice for unexplained reasons. Note that we employ the expected utility forstipulating the transition probability.
2.3 Theory of Mind
The concept of Theory of Mind is considered in pedestrian behavior such that, “In a certain environment, a pedestrian makes
belief about other pedestrians’ future movements and decides hisaction based on the belief.”. We assume types in pedestrian behavior and denote them as L(k) where k represents the k-th order
strategic thinking. We assume that a pedestrian with type L(1)does not represent the behavior of other pedestrians, thus does
not anticipate the action of others. In contrast, a pedestrian withA Microscopic Pedestrian Simulation Model with Theory of Mind(心の理論を考慮した微視的歩行者シミュレーションモデル) (Ryo Adachi (安達 涼))(p(L(1)), p(L(2)), p(L(3))) = (0.25, 0.00, 0.75) in Figure 5. In
Figure 5, we see dynamical lane or cluster formation in pedestrian behavior.
[3 Simulation results
In this section, we present simulation results which are intended
to verify how the distribution of types in pedestrians affect theefficacy of the flows by evaluating average speed into the desired direction and frequency of collisions. The simulations inthis section consider interactions among about forty pedestrianssimultaneously. Also, we consider three types of order.

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is beneficial for pedestrians if the point is plotted in thebottom-right of the figure and we see that it is achieved
with some mixtures of order types.

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3.1 Bi-directional flow
We present simulation results for a bi-directional flow, in whichsome pedestrians are walking from left to right and others arewalking from right to left along a road whose length is 10 [m]and width is 5 [m]. We present the results for some distributions which are denoted by (p(L(1)), p(L(2)), p(L(3))) in Figure 2 and show trajectories that corresponds to each distributionin Figure 3.
IUHTXHQF\RIFROOLVLRQVtype L(k) (k ≥ 2) is assumed to evaluate and predict other pedestrians’ actions to determine the behavior of his own. In practical,
we assume that a pedestrian with type L(1) always chooses hisdesired direction and speed regardless of the existence of any interference. Also, for a pedestrian whose order type is equal toor higher than 2, his formed beliefs are completely reflected in
his utility calculation. The utility of a pedestrian with type L(k)is calculated assuming that all other pedestrians in his eye sighthave type L(k − 1), which is one level lower than his own.

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; PFig. 3. Trajectories of pedestrian movements in a bidirectional flow. A pedestrian is added at [0, 0] or[10, 0] in every two successive steps by turns. Also,
their goal point is set to be [10, 0] or [0, 0], respectively. Each figure from above left to right corresponds
to each distribution in Figure 2 in order.

3.2 Flow at an intersection
We show the simulation result for the flows which are observed in an intersection, where two roads each of which is
10 [m] long and 5 [m] wide are crossing orthogonally. We
first show the results in Figure 4 and then we plot the position of pedestrians at some particular time steps for the case of4Conclusion

We proposed a microscopic pedestrian simulation model with auniversal method in decision theory and incorporated the ideaof Theory of Mind by introducing types of order in strategic
thinking levels that are reflected in their behavior. Also, we applied our model to large scale pedestrian flows to test how thedistribution of types of order affects the efficacy of pedestrianflows as a whole. Although our modeling procedure is one of
the simplest ones than those of previous models, we could focus on collision avoidance behavior and its mental attribution,
which have never been shed much light in past researches. Also,
our results support the idea that high order strategies induce cooperative behavior since they contributed to the lane formation
among pedestrians.

Bibliography[1] M. Asano, T. Iryo and M. Kuwahara. Microscopic pedestrian simulation model combined with a tactical model for route choice behavior. Transportation Research Part C, 18(6), pp. 842–855, 2010.
[2] D. Premack and G. Woodruff. Does the chimpanzee have theory ofmind?. Behavioral Brain Science, 1(4), pp. 515–526, 1978.
[3] R. Adachi and K. Aihara. Theory of mind in a microscopic pedestrian simulation model. Proc. of Int’l Conf. on Artificial Life andRobotics, 2011, in press.