After much research and troubleshooting finally some progress in the analysis of RPS (Rock Paper Scissors) game model.

Why the RPS game is of so much interest to me? Because RPS represents one of the simplest models (two players with three probabilistic states each) in fact RPS game
is used by researches in game theory, statistics and chaos theory to study phenomena in non-linear dynamics.

In the game theory  the simplest goal is to improve payout matrix (as its in the lottery which is a type of probabilistic game, albeit very extreme case of one!)

One thing RPS game has in common with lottery is its random probability distribution between game iterations, of course RPS probability states are only 3 where else in a typical lottery game let say mega lottery, there are 56 different  probability states between each iteration, nevertheless its still of value to study simpler models
(possible reduction of large dimensional probabilistic state game (such as lottery) to simpler model is unknown at this stage, I don't even know if at all possible!)

With RPS game, 3 different trials were run, each iterating 4 times across 500 game runs and each run was divided to one of two bets:

1. blind bets, using RPS game learning algorithm that at best yields > 0.333% improvement over statistical medium

2. using Local Lyapunov Threshold to consistently (and selectively) bet on Local Lyapunov Threshold bands

Both 1 and 2 game outcome were collected and are displayed below (note that consistent improvement has been made of 2 over 1, yielding some credibility to the fact that using Local Lyapunov Threshold can in fact improve game's outcome matrix.

This in fact is really surprising to me, to recall RPS game probabilistic states are chosen from the random numbers! So in short, Lyapunov strategy compensates for the random!

There are many things not done as yet such as introduction of:

1. adaptive threshold (to inject statistical inference to Local Lyapunov Threshold)

2. entropy filtering

Below is output from rps_game.cpp, improved trials (using LLE threshold) were marked witth "GAIN" label, there were 12 trials run resulting in 10 improved (GAIN) payouts with LLE over 2 that did not use LLE Threshold.

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Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1143 Losses: 898 Ties: 459 Ratio Win/Total: 0.4572

Strategy with   (+) Local Lyapunov Threshold (min: -0.1, max: 0.1)
Total Game Iterations: 2525 Super Agent Played: 586 times.   Wins: 274   Losses: 209 Ties: 103 Ratio Win/Total: 0.467577       [GAIN]

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1136 Losses: 930 Ties: 434 Ratio Win/Total: 0.4544

Strategy with   (+) Local Lyapunov Threshold (min: -0.5, max: 0.5)
Total Game Iterations: 2525 Super Agent Played: 1688 times. Wins: 813   Losses: 590 Ties: 285 Ratio Win/Total: 0.481635       [GAIN]

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1194 Losses: 915 Ties: 391 Ratio Win/Total: 0.4776

Strategy with   (+) Local Lyapunov Threshold (min: -1, max: 0)
Total Game Iterations: 2525 Super Agent Played: 537 times.   Wins: 259   Losses: 199 Ties: 79   Ratio Win/Total: 0.482309       [GAIN]

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1120 Losses: 927 Ties: 453 Ratio Win/Total: 0.448

Strategy with   (+) Local Lyapunov Threshold (min: 0, max: 1)
Total Game Iterations: 2525 Super Agent Played: 1654 times. Wins: 753   Losses: 610 Ties: 291 Ratio Win/Total: 0.45526         [GAIN]

Press any key to continue . . .

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1177 Losses: 943 Ties: 380 Ratio Win/Total: 0.4708

Strategy with   (+) Local Lyapunov Threshold (min: -0.1, max: 0.1)
Total Game Iterations: 2525 Super Agent Played: 617 times.   Wins: 301   Losses: 224 Ties: 92   Ratio Win/Total: 0.487844      [GAIN]

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1162 Losses: 933 Ties: 405 Ratio Win/Total: 0.4648

Strategy with   (+) Local Lyapunov Threshold (min: -0.5, max: 0.5)
Total Game Iterations: 2525 Super Agent Played: 1693 times. Wins: 828   Losses: 598 Ties: 267 Ratio Win/Total: 0.489073      [GAIN]

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1110 Losses: 936 Ties: 454 Ratio Win/Total: 0.444

Strategy with   (+) Local Lyapunov Threshold (min: -1, max: 0)
Total Game Iterations: 2525 Super Agent Played: 596 times.   Wins: 304   Losses: 195 Ties: 97   Ratio Win/Total: 0.510067      [GAIN]

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1217 Losses: 869 Ties: 414 Ratio Win/Total: 0.4868

Strategy with   (+) Local Lyapunov Threshold (min: 0, max: 1)
Total Game Iterations: 2525 Super Agent Played: 1568 times. Wins: 785   Losses: 535 Ties: 248 Ratio Win/Total: 0.500638      [GAIN]

Press any key to continue . . .

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1113 Losses: 940 Ties: 447 Ratio Win/Total: 0.4452

Strategy with   (+) Local Lyapunov Threshold (min: -0.1, max: 0.1)
Total Game Iterations: 2525 Super Agent Played: 568 times.   Wins: 270   Losses: 189 Ties: 109 Ratio Win/Total: 0.475352      [GAIN]

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1189 Losses: 898 Ties: 413 Ratio Win/Total: 0.4756

Strategy with   (+) Local Lyapunov Threshold (min: -0.5, max: 0.5)
Total Game Iterations: 2525 Super Agent Played: 1738 times. Wins: 810   Losses: 626 Ties: 302 Ratio Win/Total: 0.466053      [LOSS]

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1247 Losses: 883 Ties: 370 Ratio Win/Total: 0.4988

Strategy with   (+) Local Lyapunov Threshold (min: -1, max: 0)
Total Game Iterations: 2525 Super Agent Played: 568 times.   Wins: 272   Losses: 202 Ties: 94   Ratio Win/Total: 0.478873      [LOSS]

Strategy without(-)
Total Game Iterations: 2525 Super Agent Played: 2500 times. Wins: 1159 Losses: 904 Ties: 437 Ratio Win/Total: 0.4636

Strategy with   (+) Local Lyapunov Threshold (min: 0, max: 1)
Total Game Iterations: 2525 Super Agent Played: 1497 times. Wins: 709   Losses: 531 Ties: 257 Ratio Win/Total: 0.473614      [GAIN]

Press any key to continue . . .