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F(x)
G(x)
x
Fig. 6.
Functions involved in ERE
10
.
0
have been chosen for the parameters
α
and
γ
, respectively. It is worth noting that
the values for
e
,
r
,
α
,ad
γ
have been chosen so to guarantee stability of the differ-
ence equations involved in the learning process (i.e., equations 5 and 9). In order to
understand effectiveness of the proposed Enhanced Roth and Erev algorithm and the
interrelation between learning convergence and economic results, we firstly studied the
behavior and the convergence of the learning in the power exchange model. We have as-
sumed the initial (i.e., at
t
=0
) propensities
S
j,t
(
a
j
)
in equation 5 to be uniformly dis-
tributed among the possible strategies in the strategy space. Furthermore, as discussed
in Section 3.3, the strategy space is related to the mark-up variables. In all computa-
tional experiments discussed hereafter we have considered a uniformly spaced grid for
μ
i
in the range [0.8, 2.3] with step 0.05. This results in a set of 31 possible strategies
for each of the N = 175 generators. Stated this simulation context, the evolution of the
strategy probabilities pointed out three groups of agents:
-
those whose bids are lower than clearing prices and are always accepted by the mar-
ket. We denote them as price-takers agents and are characterized by a convergence
of the strategy probabilities;
-
those whose bids are higher than clearing prices and are always rejected by the mar-
ket. We denote them as out-of-the-market agents and are generally characterized by
randomly chosen strategies, as they do not participate to the market price formation
and accordingly receive always negative payoffs;
-
those whose bids are able to set the Locational Marginal Price. We denote them
as price-maker agents and are characterized by the faster convergence time in the
learning process.
Figure 7 shows an example of reference convergence time-path. For the sake of repre-
sentativeness, the strategy characterized by the largest final probability (i.e., the action
most willing to be played ) of three reference Gencos is considered and their probabili-
ties plotted as function of the simulation iterations. Figure 7 points out that both price-
taker and price-maker are characterized by a learning process that select the preferred
action strategy (i.e., the one whose probability converge to 1). Conversely, it is worth
noting that only some of the out-of-the-market agents are characterized by a conver-
gence of the strategy probabilities. Indeed, those agents whose bids are slightly higher
than the LMP tend to converge even if their bids are always rejected by the market. This
can be interpreted as a result of an almost complete exploration process of their strategy
spaces that allows them to conclude that the strategies played by the near competitors
