21.1  N, E[u*], true reward at E over samples
  N=    1  E[u]=  0.01  E[true]= -0.88
  N=    4  E[u]=  3.08  E[true]=  1.68
  N=   16  E[u]=  5.30  E[true]=  2.23
  N=   64  E[u]=  7.03  E[true]=  1.91
  N=  256  E[u]=  8.48  E[true]=  1.15
  N= 1024  E[u]=  9.77  E[true]=  0.11
  N= 4096  E[u]= 10.90  E[true]= -1.07
  peak of u-0.1u^2 at u=5, value 2.5
21.2  LP value vs dual  min_lam [ max_a (r_a - lam c_a) + lam D ]
  seed 0: LP -0.1501  dual -0.1500 at lambda= 2.790
  seed 1: LP  0.3972  dual  0.3972 at lambda= 4.611
  seed 2: LP  0.1552  dual  0.1552 at lambda= 3.229
  seed 3: LP  0.6908  dual  0.6908 at lambda= 3.761
  seed 4: LP  0.3325  dual  0.3326 at lambda= 2.744
21.3  largest action probability and entropy of the last iterate; support of the LP optimum
  seed 0: max prob 0.9679  entropy 0.1820  LP support 2  cost 0.147  lambda 5.05
  seed 1: max prob 0.9996  entropy 0.0044  LP support 2  cost 0.410  lambda 45.51
  seed 2: max prob 0.9838  entropy 0.0963  LP support 2  cost 0.085  lambda 11.33
  seed 3: max prob 0.9991  entropy 0.0100  LP support 2  cost 0.245  lambda 0.00
  seed 4: max prob 0.9978  entropy 0.0207  LP support 2  cost 0.267  lambda 0.00
21.4  q=0.8: success prob and informative-group probability (G=8)
  H=  1  p=0.80000  P(group informative)=0.83223  p(1-p)=0.16000
  H=  5  p=0.32768  P(group informative)=0.95812  p(1-p)=0.22031
  H= 10  p=0.10737  P(group informative)=0.59695  p(1-p)=0.09584
  H= 20  p=0.01153  P(group informative)=0.08860  p(1-p)=0.01140
  H= 40  p=0.00013  P(group informative)=0.00106  p(1-p)=0.00013
21.5  Gaussian: E[max over N of (m_i - k s)] where true value v_i, noise sd s: shrinkage factor
  k=0: selected true value 2.539  (k does not matter for constant s)
  k=1: selected true value 2.516  (k does not matter for constant s)
  k=2: selected true value 2.531  (k does not matter for constant s)
  analytic: regression coefficient 1/(1+0.25)=0.8, selected value 0.8*E[max]
  E[max of 256 of N(0,1.25)] = 3.153  times 0.8
21.6  cost of the averaged policy over time, seed 0
  T=  1000  avg cost 0.606  avg reward 0.394
  T=  5000  avg cost 0.246  avg reward -0.382
  T= 20000  avg cost 0.285  avg reward -0.221
