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by breaking its symmetry via the turn taking mecha-
nism itself. This suggests that such a quantization is
not a good model for the real world, it hides the main
problem of the scenario from the start. A related mo-
deling issue around time relates to Zeno’s Paradox. In
the models presented here, time ticks are discrete and
of equal length. It might be argued that two Zeno-like
players could choose to define each of their ticks to
have half the duration of the previous one, and thus
create an infinite number of ticks which would be cer-
tain to eventually lead to an asymmetric yield and
avoidance of collision. Formal mathematical analy-
sis of this claim could form future work, though in
practice, any human or machine compute system has
some finite limit on its computation speed.
Extended models should handle speed more re-
alistically. Rather than just two discrete speeds, a
continuum of speeds should be available, including
stopping at a complete halt. Continuous speeds may
require sampling approximations to compute over,
while complete halts allow for potentially infinite
time games which require further consideration to
model. Human drivers when faced with, for example,
a busy motorway merge, may gradually slow down
towards a halt at the end of the slip-road, while dri-
vers in their path may do the same. Perhaps under a
continuous speed model this behavior can be shown
to converge safely as everyone slows down towards a
halt and reduces both the probability and penalty of
collisions. Nevertheless, the underlying logic must
still hold - that there must be a credible threat of
a non-zero probability of causing some collision, in
order that the other party cannot take advantage of
the AV every time. Future models should add furt-
her realistic details to the framework. Real drivers
do not know each other’s utility functions and must
infer them in an information game during the inte-
raction. This could include giving and reading signals
about utility such as the model, age, colour and clean-
liness of their cars, their lateral positions on the road,
their facial expressions and hand gestures as well as
more formal car signaling via light flashing and horn
usage. Real drivers may not have Markovian time de-
lay utilities and more detailed models should allow for
time dimensioned value functions v(y,x,t) rather than
than simpler v(y, x) used here. Traffic regulations and
conventions such as legally binding and non-binding
signs and lights, and the cost of public humiliation or
legal action for being seen or recorded breaking them
should be added to modify utilities. Such models sug-
gest new signaling conventions for autonomous vehi-
cles, such as use of V2V radio communications and
virtual currencies to aid negotiations; or the use of
small AV-mounted water-pistols to induce small ne-
gative utilities in assertive pedestrians in place of ac-
tually hitting them.
ACKNOWLEDGEMENTS
This project has received funding from the European
Union’s Horizon 2020 Research and Innovation pro-
gramme under grant agreement InterACT No 723395.
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