The El Farol Bar Problem for next generation systems

24 CHAPTER 3. ANALYSIS AND EXTENSION OF THE STOCHASTIC ALGORITHM
table 3.1 and in the first two plots of figure 3.2.
N /M 60/100 600/1000
iterations 2 · 10 8 2 · 10 8
µ 0.01 0.001
average 59.072334 598.99924
std 0.259638 0.052933
Table 3.1: Behaviour of the ‘tax’ algorithm with ctax = 6.
This algorithm could be more fair for the agents if those who do not attend
the bar were encouraged to attend it. By definition of the algorithm this cannot
be done through the parameter µ, since there is no change for the probabilities
pi of those who choose or perhaps are forced not to attend.
A rather aggressive way to change this, is to multiply these probabilities with
a number close to 1. This way in every iteration the attendance probability for
all those that do not attend will slightly increase and finally they will ‘decide’ to
attend the bar. That leads to the following equations for the partial algorithm
with taxing:
⎧
0,
⎪⎨
where µppi(k) − µtax(N(k) − N )xi(k) < 0
pi(k + 1) = 1,
where µppi(k) − µtax(N(k) − N )xi(k) > 1
⎪⎩
pi(k) − µppi(k) − µtax(N(k) − N )xi(k), otherwise
(3.2.1)
µp and µtax are defined as following:
1, xi(k) = 1
µp =
(3.2.2)
cp, xi(k) = 0
where i = 1 . . . M and c1 a constant which affects the degree pi is changing for
those agents that do not attend (xi(k) = 0)
µ, N(k) ≤ N
µtax =
(3.2.3)
ctaxµ, N(k) > N
where c2 > 1 is a constant that defines the degree pi is changing for selfish agents
who insist to attend even if the bar was crowded the previous time.
The behaviour of this algorithm is summarised in the 3rd and 4th plots of
figure 3.2. Simulations showed that in a reasonable taxed system (ctax ≥ 3),
large values of mp result in an underutilised bar (Table:3.2. An interpretation of
this is that, although people are encouraged to attend, taxation prevents them