Ex 13.2: time t* at which the entropy of the natural flow is maximal, t* = logit(1/A) - logit(p0)  (one rewarded action among A)
      A p0=0.1      p0=0.01     p0=0.001   
      5       0.811       3.209       5.520
     20      -0.747       1.651       3.962
    100      -2.398       0.000       2.312
  scan for A = 20, p0 = 0.01: maximum at t = 1.651, H = 2.9957 (log 20 = 2.9957)

Ex 13.3: natural flow keeps the ratio between two correct actions fixed; vanilla flow does not
  natural: t=  0.0: pi_a/pi_b = 5.0000   t= 20.0: pi_a/pi_b = 5.0000   t= 40.0: pi_a/pi_b = 5.0000   t= 60.0: pi_a/pi_b = 5.0000
  vanilla: t=  0.0: pi_a/pi_b = 5.0000   t= 20.0: pi_a/pi_b = 44.4030   t= 40.0: pi_a/pi_b = 424.7559   t= 60.0: pi_a/pi_b = 798.3492

Ex 13.6: the optimum of E r + beta H is pi ~ exp(r / beta);  P(action 0) in the bandit of Listing 13.4
  beta = 0.02 : P(a0) = 1.0000, P(decoy) = 0.0000, expected reward 1.0000
  beta = 0.1  : P(a0) = 0.9969, P(decoy) = 0.0009, expected reward 0.9972
  beta = 0.5  : P(a0) = 0.1292, P(decoy) = 0.0318, expected reward 0.1387
  beta = 2.0  : P(a0) = 0.0324, P(decoy) = 0.0229, expected reward 0.0393
