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ontology classifies the accident likelihood in discrete terms like "none" (value 0), "negligible"
(value 1) and "significant" (value 2), then the resulting matrix
A
could look like
A
(
s
,
l
)(
f
,
l
)(
s
,
r
)(
f
,
r
)(
s
,
s
)(
f
,
s
)
(
)
s
,
f
1
2
0
0
0
0
(
l
,
m
)
0
0
1
2
1
2
(
o
,
h
)
0
0
0
0
1
2
(
)=
Solving this game for its equilibrium value gives
v
A
0.5 with equilibrium strategies
x
∗
=(
and
y
∗
=(
. Observe that this indicates that - aiming at
maximal safety - we should never be advised to turn over, and can take either of the remaining
choices with equal probability. Following this rule, we end up having a less than negligible
chance (the value is 0.5 and as such strictly less than the negligible-value 1) of having an
accident.
This process can be repeated for different scenarios, but crucially hinges on the ontology to be
correct
and perform reasoning
efficiently
.
Each query to the system yields a different matrix-game with strategy sets
PS
1
,
PS
2
, with
its own unique Nash-equilibrium solution (which can as well be pre-computed). The actual
recommendation provided to the user is a random selection from
PS
1
, where the particular
choice is drawn from the equilibrium profile. This solution, among the valid alternatives,
is presented as the recommendation, along with possible alternatives that are not explicitly
recommended but possible.
This method is indeed computationally feasible, as a large set of possible games along
with their corresponding Nash-equilibria can be
pre-computed
and stored for later usage
in a
game-database
(indeed, all we need is the strategy set
PS
1
and the corresponding
Nash-equilibrium; the game matrix itself can be discarded). If the input parameters are
discrete, then the set of queries is finite and hence the number of such games remains tractable.
This is even more substantiated by the fact that a human operator will most likely not enter
more than a few parameters as well as these will not be entered at arbitrary precision. As
humans tend to reason in fuzzy terms, any natural mapping of these to parameters of a query
answering system will consist of a small number of inputs to specify within small ranges to
get an answer.
The ontology is then used to select the particular game at hand and provide the best
behavior under uncertain behavior of others. The overall workflow is depicted in figure 3.
The only block to be further explained in this picture is the
sampler
, drawing the concrete
recommendation from the set of possible ones according to the Nash-equilibrium. This is
a trivial task, as it amounts to sampling from a discrete probability distribution. We refer
the interested reader to Gibbons (1992) for a formal justification, as we will restrict our
presentation to giving the sampling algorithm: assume that a discrete distribution is given
by
1/2, 1/2, 0
)
1/2, 0, 1/2, 0, 0, 0
)
(
)
.
1. Generate a uniform random number
x
within the interval
p
1
,...,
p
n
.
2. Find and return (as the result) the smallest integer
k
such that
x
[
0, 1
]
i
1
p
i
.
So the overall process of reasoning with assurance can be described in a sequence of simple
steps, presupposing a logically consistent ontology:
1. Unless the current query is found in the set of pre-computed ones,
2. generate the set of candidate recommendations,
≤
∑
=
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