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log E max
E ( n )
log( R max )
log E ( n )
E min
log( R max )
η ( n )= 1
2
0 . 3+0 . 2
+0 . 3
0 . 2
Where E max is the connectivity of agents in the fully connected group and E min
is the connectivity of agents in the weakly connected ring, and n is the number of
connections of the final agent. If we require this average to be the standard ZIP average
of 0.3 this equation can be solved to give:
n = E max E min
Here, we employ a fully-connected clique of fifty-one agents and a ring of fifty-one
agents, giving E max =50and E min =2, requiring n =10connections in total, or
five connections to each population.
The results of 40000 repetitions are shown in figure 2(right). As can be seen, the
modified ZIP agent using an adaptive learning rate converges faster and to a signifi-
cantly lower asymptotic value than the standard ZIP agent (t-test, p< 0 . 0001). Hence,
the adaptive learning rate rule has a positive effect on convergence beyond simply in-
creasing the learning rate.
5
Discussion
In this paper we wished to explore the effect on convergence of market structure in
terms of trader connectivity. The result obtained in the first part of this paper clearly
show that the more connected an agent is, the faster it is able to converge, and the closer
it is able to get to the optimum price. In the short term the more well-connected traders
receive more information, and so are able to adapt faster. In the longer term, this greater
volume of information means that they have a better overall picture of the market, and
so may evaluate the optimum price more accurately.
As a consequence, the source of a shout has an impact on the quality of the in-
formation obtained. This was demonstrated by factoring information quality into the
Widrow-Hoff adaptation rule via an adaptive learning rate. Our results show that agents
who employ this strategy value the commodity more accurately in the majority of cases.
The fact that a small number of well-connected agents do worse by adopting this
adaptive learning rate rule demonstrates that it is not of universal utility. The reason for
this may be related to who these individuals are connected to. In the market structures
studied here, for the vast majority of the time, highly connected agents will receive their
information from less well-connected individuals. Since such information is judged (by
the learning rule) to be of relatively low quality, less attention is paid to it. This may be a
good decision in the long run. However, when the market opens this proves to be detri-
mental. At the beginning of a market every agent has a random valuation. Therefore,
it would pay to attend to any information, even if it originates from poorly connected
individuals. Whereas the most poorly-connected individuals do just this, the most well-
connected agents tend to trust their initial valuation to a greater extent. This results
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