Information Technology Reference
In-Depth Information
Table 6.1
A payoff matrix for
a 'two person zero sum' game
Action (frequency)
Expected
hypothesis
Expected NOT
hypothesis
−
−
Exp1 (f
h
) Ask for Head
I
h
* (0)
(1
I
h
)*(
1)
Exp2 (f
t
) Ask for Tail
I
t
* (0)
(1
−
I
t
)*(
−
1)
Table 6.2
An example of a
payoff matrix
Action (frequency)
Expected
hypothesis
Expected NOT
hypothesis
Exp1 (f
h
) Ask for Head
0.41 * (0)
0.59 * (
−
1)
Exp2 (f
t
) Ask for Tail
0.40 * (0)
0.60 * (
−
1)
We iterate this mixed strategy in each cycle of the belief-revision process. Inclu-
sion of this strategy makes the application of game theory to science and to learning
more appropriate than a single-step decision procedure would be. If f
h
is the fre-
quency of applying the strategy 'ask-for-head' and f
t
is the frequency of 'ask for
tail', then we can calculate for each actor the gain (maximum-security or minimum
possible loss) between the agent's two options, as:
f
h
=
(1
−
I
t
)
/
[(1
−
I
h
)
+
(1
−
I
t
)]
=
0
.
59
/
[0
.
59
+
0
.
60]
=
0
.
496
f
t
=
−
1
f
h
=
0
.
504
So for optimum results the frequency of 'Ask for a Head' should be
0.496
and 'Ask
for a Tail' should be
0.504
. Having only a slight bias towards asking for head reflects
the strong prior belief that the coin is good; the coin is fair. This course of action
would be more rational than asking for a tail every time, given that the agent believes
that the coin is good.
The above equations are extended to deal with decisions about which experiment
to perform or which agent to consult. Agents evaluate a distribution of choices in
order to select those most likely to maximize the gain (confidence in the overall view
of the world). A consultee is chosen by extending the calculations to consider the
entropy of every pair of agents', including the agent making the decision, average
confidence in each hypothesis. Self-consultation then becomes an option. As with
evaluating experiments, the payoff should invoke a 'maximum security' level from
the entropy pairings. So an agent's best bet is to decide which other agent it is to
be influenced by in order to gain a greater overall confidence. The mechanism of
a 'conversation', where agents infer each other's internal views of music, has been
investigated by Addis and Billinge (
2004
) and described in Chap. 13.
