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In order to reason out of the presented results, assume there are many agents
but only one of them has the best strategy. Even if this strategy suggests a
correct prediction for itself the prediction of the whole system can be potentially
incorrect, provided there is suciently many agents with bad strategies. Such
situation is less probable when the number of strategies per agent is increased
and concurrently the number of agents is decreased. This is impossible if there
is only one agent because then the best strategy is always used. Similarly, if
the constraint on NS is shifted towards larger values and number of agents
is preserved, then the probability, that more and more agents have a correct
strategy becomes large. Hence, in the limit NS
, all agents have the correct
strategy and the eciency of the group is the same as the eciency of a single
agent equipped with the aggregated pool of strategies.
Considering Ψ as a function of the S value, the larger value of S , the better
results, as there is a larger probability that better strategy is drawn by an agent.
However, if S is above some threshold, the pool of strategies is oversampled
and many agent's strategies are the same. Since there is no particular gain if
agent has more than one best strategy at his disposal, the success rate does not
increase. Given this, one can wonder if a random draw is the most ecient way
to generate strategies. Indeed it is not. Only the fully probed strategy space
assures that, if the best strategy exists, then it is used in the game, provided
N = 1. The fringe benefit is that there is also no redundancy between strategies,
what reduces the computational costs. Hence, the best MG predictor is based
on single agent equipped with all pairwise different strategies, i.e. all strategies
covering Full Strategy Space (FSS). However, in cases when m is large, it can
be hardly possible to generate so huge set of strategies. Therefore, for higher m
it seems reasonable to use all strategies from Reduced Strategy Space (RSS).
The RSS consists of only 2 m +1 strategies which are pairwise uncorrelated or
anticorrelated, i.e. the normalized Hamming distance between them is equal
to 0 . 5 or 1 [6]. The RSS apparently reduces the complexity and still assures
good quality of prediction as FSS is regularly probed. These statements are
visualized in Fig. 2 where marks 'x' and '*' represent results for single agent and
all strategies related to RSS (16 strategies) and FSS (256 strategies) respectively.
Despite of a numerous differences between both pools the results are close, which
indicates a superior power of RSS usage.
The presented above methodology differs significantly from approaches pre-
sented by other researchers [11,8,7]. In those papers the whole bunch of individ-
uals is used and the strategy distribution is randomly initiated. Although those
methods work either, the approach presented here is optimal.
→∞
5.3 Nonstationary Signals and
λ
-GCMG Predictor
The agents' behavior is determined by the strategies' utilities which they have
at their disposal. Generally, the strategies that collect more frequently a posi-
tive payoff are characterized by a monotonously rising utilities. The utilities are
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