The El Farol Bar Problem for next generation systems

3.2. TAXING/PAYOFFS ALGORITHMS 27
In the following a set of equations that describe the stationary state behaviour
of the probabilities will be derived pi(k).
The number of agents attending at time k is N(k) = M i=1 xi(k) and xi(k) are
independent Bernoulli trials given by:
1, pi(k)
xi(k) =
(3.2.5)
0, 1 − pi(k)
In the stationary state we have:
pi(k + 1) = pi(k) ∀i ∈ N
Taking the expectation values results to:
E(pi(k + 1)) = E(pi(k)) ⇒ E(pi(k + 1)) − E(pi(k)) = 0 ⇒
E(µtax(N(k) − N )) = 0
N(k) is the number of agents attending at time k is equal to M i=1 xi(k), where
xi(k) are independent Bernoulli trials.
The expectation value is calculated using the following formula:
E(XY ) =
fx,y(x, y)
where X,Y random variables and fx,y(x, y) joined mass function. In this case
Y = µtax and X = N(k) − N . By definition, Y = µtax is a function of N(k). So,
Y = g(X) and fx,y(x, y) = fx(x, g(x)) = fx(x). Hence:
E(µtax(N(k)) · (N(k) − N )) =
(N(k) − N ) · µtax(N(k)) · fN(k)(N(k)) ⇒
cµ
N(k)>N
N(k)−N
(N(k) − N ) · fN(k)(N(k)) + µ
cµ
N(k)>N
µ
N(k)≤N
x
N(k)≤N
N(k) · fN(k)(N(k)) − cµN
(N(k) − N ) · fN(k)(N(k)) ⇒
N(k)>N
N(k) · fN(k)(N(k)) − µN
N(k)≤N
fN(k)(N(k))+
fN(k)(N(k)) (3.2.6)
Kolmogorov-Smirnov test indicated that z = N(k) might follow Poisson distribution
f(z) = λz
z! e−λ , with λ = E(z). Hence:
E(f(z)) =
zf(z) =
zf(z) +
zf(z) = λ ⇒
z
z≤N
z>N