Table 2 Detailed probability distributions for \(X_{20},X_{21},X_{22},X_{23},X_{24},X_{25},X_{100}\).

From: Optimal strategy of the simultaneous dice game Pig for multiplayers: when reinforcement learning meets game theory

Prob.

0

k

k+1

k+2

k+3

k+4

k+5

\(X_{20}\)

0.6245

0.0997

0.0950

0.0742

0.0542

0.0352

0.0172

\(X_{21}\)

0.6412

0.0950

0.0908

0.0708

0.0518

0.0338

0.0166

\(X_{22}\)

0.6570

0.0908

0.0866

0.0676

0.0496

0.0325

0.0158

\(X_{23}\)

0.6721

0.0866

0.0828

0.0648

0.0476

0.0310

0.0151

\(X_{24}\)

0.6866

0.0828

0.0792

0.0620

0.0454

0.0296

0.0144

\(X_{25}\)

0.7004

0.0792

0.0758

0.0592

0.0434

0.0282

0.0138

\(X_{100}\)

0.9898

0.0027

0.0026

0.0020

0.0015

0.0010

0.0005

  1. The range \(\{0,k,k+1,k+2,k+3,k+4,k+5\}\) is not the same for all variables. Instead, it depends on the random variable \(X_k\). For instance, the range of \(X_{20}\) is \(\{0,20,21,22,23,24,25\}\) while the range of \(X_{25}\) is \(\{0,25,26,27,28,29,30\}\).
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