Here are some random Weighted Markov Chains with, 20, outgoing states, 3, absorbing states , degrees from, 1, and , 4 Minimum weight, 0, max. weight, 20 followed by their statistical analysis: prob. of lending in the abrobing st\ ates, list of expected gains, and list of expected number of rounds conditioned on it finishing before, 50, rounds. Starting at state 1. ------------------------------------- If the Weighted Markov Chain is [{[14, 1, 4]}, {[5, 1/3, 0], [11, 1/3, 0], [12, 1/3, 18]}, {[2, 1/4, 8], [4, 1/4, 8], [9, 1/4, 10], [11, 1/4, 19]}, {[6, 1/3, 13], [9, 1/3, 12], [11, 1/3, 11]}, {[9, 1/4, 8], [11, 1/4, 17], [16, 1/4, 20], [23, 1/4, 4]}, {[1, 1/4, 16], [3, 1/4, 17], [18, 1/4, 8], [19, 1/4, 4]}, {[2, 1, 11]}, {[3, 1/4, 7], [17, 1/4, 1], [19, 1/4, 0], [20, 1/4, 11]}, {[2, 1/3, 8], [7, 1/3, 13], [14, 1/3, 12]}, {[5, 1, 14]}, {[3, 1/2, 16], [16, 1/2, 8]}, {[4, 1/2, 20], [23, 1/2, 13]}, {[8, 1/4, 16], [10, 1/4, 19], [15, 1/4, 14], [16, 1/4, 4]}, {[2, 1/2, 16], [17, 1/2, 20]}, {[21, 1, 9]}, {[2, 1/3, 20], [4, 1/3, 4], [6, 1/3, 4]}, {[5, 1, 2]}, {[14, 1/4, 0], [17, 1/4, 3], [21, 1/4, 13], [22, 1/4, 17]}, {[3, 1/3, 1], [9, 1/3, 17], [14, 1/3, 3]}, {[7, 1/4, 16], [10, 1/4, 17], [12, 1/4, 20], [20, 1/4, 9]}] Then the statistical information is [[0.056440403537009972313, 0.056440403537009972313, 0.88711919292598005537], [197.45164409001219052, 201.45164409001219052, 156.40334160356906497], [19.056427645080541973, 19.056427645080541973, 15.088149960624654238], 0.056911404044582004813] ------------------------------------- If the Weighted Markov Chain is [{[1, 1/2, 1], [23, 1/2, 2]}, {[6, 1/4, 12], [15, 1/4, 20], [21, 1/4, 0], [22, 1/4, 19]}, {[8, 1, 10]}, {[3, 1/4, 9], [6, 1/4, 18], [21, 1/4, 3], [22, 1/4, 6]}, {[6, 1/2, 6], [22, 1/2, 16]}, {[7, 1/2, 0], [12, 1/2, 1]}, {[15, 1, 16]}, {[3, 1/4, 19], [9, 1/4, 15], [17, 1/4, 8], [18, 1/4, 7]}, {[16, 1/3, 15], [17, 1/3, 2], [20, 1/3, 18]}, {[1, 1, 14]}, {[11, 1, 17]}, {[2, 1/2, 17], [8, 1/2, 0]}, {[11, 1/4, 18], [15, 1/4, 12], [20, 1/4, 20], [23, 1/4, 19]}, {[9, 1/2, 18], [11, 1/2, 9]}, {[2, 1/4, 16], [11, 1/4, 7], [15, 1/4, 15], [18, 1/4, 18]}, {[3, 1/2, 7], [17, 1/2, 7]}, {[19, 1, 13]}, {[4, 1/4, 14], [7, 1/4, 6], [9, 1/4, 10], [16, 1/4, 4]}, {[15, 1, 13]}, {[5, 1/2, 16], [10, 1/2, 20]}] Then the statistical information is [[0., 0., 1.], [0., 0., 2.9999999999999555911], [0., 0., 1.9999999999999555911], -15 0.88817841970012523234 10 ] ------------------------------------- If the Weighted Markov Chain is [{[2, 1/3, 6], [4, 1/3, 16], [20, 1/3, 6]}, {[3, 1/4, 4], [8, 1/4, 3], [13, 1/4, 4], [23, 1/4, 4]}, {[2, 1/3, 1], [17, 1/3, 11], [20, 1/3, 10]}, {[3, 1/2, 9], [14, 1/2, 1]}, {[9, 1/2, 6], [10, 1/2, 20]}, {[19, 1, 15]}, {[2, 1/2, 1], [16, 1/2, 19]}, {[17, 1, 12]}, {[1, 1/2, 16], [23, 1/2, 2]}, {[9, 1, 10]}, {[2, 1/2, 10], [22, 1/2, 5]}, {[1, 1/4, 15], [3, 1/4, 13], [5, 1/4, 13], [18, 1/4, 11]}, {[4, 1/3, 1], [13, 1/3, 8], [22, 1/3, 4]}, {[20, 1, 16]}, {[5, 1/3, 11], [8, 1/3, 12], [15, 1/3, 20]}, {[11, 1/4, 6], [12, 1/4, 6], [20, 1/4, 16], [22, 1/4, 7]}, {[7, 1, 2]}, {[2, 1/4, 7], [3, 1/4, 13], [19, 1/4, 9], [22, 1/4, 17]}, {[4, 1/4, 2], [12, 1/4, 11], [20, 1/4, 5], [21, 1/4, 3]}, {[22, 1, 19]}] Then the statistical information is [[0.00055697503798836453662, 0.83925912033330800150, 0.16018390462870363397], [88.622812288290200604, 44.633573845302178022, 27.646015014172150163], [11.264638905300190459, 4.4993238687561348600, 4.7125054333542263748], -5 0.26907632436692574071 10 ] ------------------------------ If the Weighted Markov Chain is [{[6, 1/3, 2], [16, 1/3, 14], [17, 1/3, 8]}, {[10, 1/2, 20], [22, 1/2, 6]}, {[8, 1/2, 1], [22, 1/2, 10]}, {[12, 1/2, 4], [16, 1/2, 20]}, {[7, 1, 19]}, {[2, 1/3, 7], [16, 1/3, 19], [20, 1/3, 5]}, {[1, 1/4, 3], [2, 1/4, 9], [14, 1/4, 12], [17, 1/4, 3]}, {[8, 1/3, 16], [13, 1/3, 19], [21, 1/3, 16]}, {[4, 1/2, 18], [16, 1/2, 15]}, {[9, 1/4, 8], [14, 1/4, 15], [20, 1/4, 9], [23, 1/4, 18]}, {[4, 1/4, 2], [7, 1/4, 9], [15, 1/4, 0], [17, 1/4, 13]}, {[3, 1, 19]}, {[3, 1, 13]}, {[9, 1/2, 11], [17, 1/2, 17]}, {[6, 1/4, 6], [15, 1/4, 4], [17, 1/4, 7], [21, 1/4, 12]}, {[20, 1, 9]}, {[3, 1/2, 3], [17, 1/2, 1]}, {[3, 1/4, 8], [13, 1/4, 8], [20, 1/4, 17], [23, 1/4, 4]}, {[1, 1, 11]}, {[14, 1/2, 16], [17, 1/2, 4]}] Then the statistical information is [[0.31018513421013004781, 0.67592597265480976315, 0.013888893135060189045], [80.451710325275861781, 61.305097208319038948, 47.], [9.1247400393437561190, 7.2446325859404943742, 4.], -6 0.30572424014389235553 10 ] ------------------------------------- If the Weighted Markov Chain is [{[3, 1/4, 10], [15, 1/4, 4], [19, 1/4, 17], [20, 1/4, 3]}, {[1, 1/4, 16], [18, 1/4, 12], [19, 1/4, 4], [20, 1/4, 2]}, {[14, 1, 17]}, {[9, 1/3, 10], [12, 1/3, 10], [15, 1/3, 0]}, {[1, 1/3, 20], [9, 1/3, 13], [13, 1/3, 16]}, {[6, 1/3, 18], [15, 1/3, 14], [21, 1/3, 20]}, {[2, 1/4, 20], [5, 1/4, 13], [16, 1/4, 5], [22, 1/4, 18]}, {[5, 1/4, 13], [7, 1/4, 18], [8, 1/4, 5], [15, 1/4, 4]}, {[2, 1/4, 20], [9, 1/4, 7], [15, 1/4, 15], [21, 1/4, 2]}, {[5, 1/3, 9], [9, 1/3, 9], [14, 1/3, 5]}, {[18, 1/2, 7], [19, 1/2, 17]}, {[16, 1, 12]}, {[6, 1, 7]}, {[4, 1/3, 1], [12, 1/3, 2], [20, 1/3, 18]}, {[20, 1, 12]}, {[8, 1/4, 10], [14, 1/4, 13], [17, 1/4, 17], [23, 1/4, 14]}, {[20, 1/2, 1], [21, 1/2, 15]}, {[1, 1/4, 12], [10, 1/4, 5], [16, 1/4, 9], [20, 1/4, 12]}, {[8, 1/3, 1], [13, 1/3, 7], [20, 1/3, 1]}, {[1, 1/2, 14], [20, 1/2, 0]}] Then the statistical information is [[0.65538586102654067699, 0.077481608392769109967, 0.26713253058069021305], [200.63785997808117647, 201.65103101778185122, 195.41100913435703593], [21.366693024807650685, 21.196603410051182871, 21.417149207174513949], 0.25418358266755353687] ------------------------------------- If the Weighted Markov Chain is [{[8, 1/2, 11], [21, 1/2, 14]}, {[3, 1/3, 0], [4, 1/3, 7], [15, 1/3, 20]}, {[16, 1/2, 7], [18, 1/2, 11]}, {[7, 1/4, 6], [11, 1/4, 19], [12, 1/4, 11], [13, 1/4, 17]}, {[2, 1, 6]}, {[5, 1/4, 17], [7, 1/4, 5], [8, 1/4, 15], [19, 1/4, 16]}, {[11, 1, 2]}, {[18, 1, 19]}, {[1, 1/3, 11], [15, 1/3, 3], [19, 1/3, 3]}, {[7, 1/4, 16], [9, 1/4, 12], [14, 1/4, 12], [15, 1/4, 13]}, {[3, 1/3, 9], [4, 1/3, 14], [11, 1/3, 15]}, {[15, 1, 9]}, {[8, 1, 9]}, {[4, 1/3, 18], [11, 1/3, 12], [17, 1/3, 5]}, {[22, 1, 1]}, {[10, 1/3, 5], [21, 1/3, 6], [22, 1/3, 1]}, {[12, 1, 7]}, {[2, 1/2, 1], [6, 1/2, 1]}, {[5, 1/4, 18], [10, 1/4, 3], [17, 1/4, 12], [19, 1/4, 12]}, {[13, 1, 12]}] Then the statistical information is [[0.56542690608109641675, 0.43457309391890358325], [25.008764483224837421, 107.28865089751747252], [2.3316315951679591967, 11.613770991451481465], 0.0010297911629219591140] If the Weighted Markov Chain is [{[13, 1/4, 5], [14, 1/4, 17], [18, 1/4, 19], [19, 1/4, 15]}, {[9, 1/2, 14], [10, 1/2, 17]}, {[5, 1/4, 16], [6, 1/4, 20], [7, 1/4, 0], [15, 1/4, 13]}, {[6, 1/4, 9], [15, 1/4, 15], [17, 1/4, 6], [22, 1/4, 3]}, {[6, 1/3, 16], [11, 1/3, 1], [13, 1/3, 6]}, {[21, 1, 19]}, {[2, 1/2, 5], [18, 1/2, 5]}, {[1, 1/2, 3], [20, 1/2, 8]}, {[15, 1, 13]}, {[1, 1/2, 5], [15, 1/2, 0]}, {[7, 1/3, 3], [15, 1/3, 0], [20, 1/3, 10]}, {[2, 1, 4]}, {[6, 1/3, 12], [12, 1/3, 0], [19, 1/3, 10]}, {[10, 1, 19]}, {[21, 1, 18]}, {[7, 1/2, 2], [9, 1/2, 13]}, {[3, 1/3, 0], [5, 1/3, 13], [23, 1/3, 11]}, {[4, 1/2, 9], [16, 1/2, 0]}, {[4, 1/4, 2], [8, 1/4, 10], [9, 1/4, 7], [20, 1/4, 20]}, {[7, 1/4, 9], [8, 1/4, 10], [9, 1/4, 4], [12, 1/4, 9]}] Then the statistical information is [[0.89607516413360423554, 0.077943627824782126054, 0.025981208041613638408], [70.818383818806101149, 41.992781483342598852, 55.992762030928708318], [6.2174575728649031775, 4.7691374511786129243, 5.7691353040943039142], -6 0.20179992705337189551 10 ] ------------------------------------- If the Weighted Markov Chain is [{[11, 1/2, 14], [20, 1/2, 19]}, {[7, 1/4, 1], [11, 1/4, 1], [14, 1/4, 10], [18, 1/4, 2]}, {[12, 1/4, 11], [18, 1/4, 3], [19, 1/4, 18], [22, 1/4, 10]}, {[10, 1, 14]}, {[5, 1/4, 17], [6, 1/4, 13], [8, 1/4, 9], [16, 1/4, 4]}, {[22, 1, 20]}, {[7, 1/2, 11], [23, 1/2, 5]}, {[16, 1/2, 16], [20, 1/2, 3]}, {[9, 1/3, 4], [15, 1/3, 5], [18, 1/3, 4]}, {[2, 1/3, 11], [19, 1/3, 14], [20, 1/3, 16]}, {[3, 1/3, 15], [12, 1/3, 7], [14, 1/3, 11]}, {[14, 1/3, 0], [15, 1/3, 17], [19, 1/3, 10]}, {[13, 1/4, 9], [18, 1/4, 20], [19, 1/4, 18], [22, 1/4, 14]}, {[4, 1/3, 3], [14, 1/3, 1], [21, 1/3, 13]}, {[20, 1, 0]}, {[8, 1/2, 4], [13, 1/2, 18]}, {[11, 1/2, 4], [16, 1/2, 16]}, {[16, 1/2, 18], [17, 1/2, 2]}, {[8, 1/4, 2], [9, 1/4, 19], [13, 1/4, 1], [19, 1/4, 6]}, {[15, 1, 14]}] Then the statistical information is [[0.53794206953599777631, 0.41723130999645402328, 0.044826620467548200408], [46.797545858047399351, 88.554903686071168583, 78.778077752597610512], [4.8257867222737494529, 7.7617608672686861045, 9.8239774852082273528], 0.74818865224680325738] ------------------------------------- If the Weighted Markov Chain is [{[4, 1/4, 19], [5, 1/4, 12], [8, 1/4, 16], [23, 1/4, 2]}, {[20, 1, 1]}, {[11, 1/2, 6], [14, 1/2, 8]}, {[5, 1/2, 15], [8, 1/2, 19]}, {[6, 1, 7]}, {[9, 1/2, 3], [11, 1/2, 0]}, {[3, 1/2, 19], [22, 1/2, 10]}, {[3, 1/3, 6], [5, 1/3, 3], [7, 1/3, 1]}, {[5, 1/2, 19], [13, 1/2, 20]}, {[9, 1/4, 3], [10, 1/4, 10], [14, 1/4, 15], [15, 1/4, 1]}, {[1, 1/2, 9], [10, 1/2, 12]}, {[10, 1/3, 16], [13, 1/3, 8], [16, 1/3, 3]}, {[1, 1/4, 16], [5, 1/4, 6], [15, 1/4, 11], [21, 1/4, 18]}, {[14, 1, 3]}, {[5, 1/3, 18], [17, 1/3, 20], [23, 1/3, 12]}, {[13, 1/2, 16], [23, 1/2, 5]}, {[20, 1/2, 2], [23, 1/2, 8]}, {[1, 1, 6]}, {[5, 1/3, 14], [18, 1/3, 5], [20, 1/3, 14]}, {[1, 1/4, 0], [4, 1/4, 6], [9, 1/4, 5], [23, 1/4, 16]}] Then the statistical information is [[0.14588144010873099364, 0.13495998673639850478, 0.71915857315487050158], [135.18661442775631168, 71.515596631784768589, 62.850438118622557999], [12.765250962384303454, 7.3365200286397635299, 7.2221226576113687317], 0.28620436304772337497] ------------------------------------- If the Weighted Markov Chain is [{[1, 1/4, 17], [6, 1/4, 5], [7, 1/4, 9], [21, 1/4, 0]}, {[2, 1/4, 18], [7, 1/4, 17], [15, 1/4, 16], [19, 1/4, 3]}, {[6, 1, 7]}, {[19, 1, 2]}, {[11, 1/3, 5], [13, 1/3, 19], [23, 1/3, 13]}, {[7, 1/3, 15], [10, 1/3, 15], [14, 1/3, 0]}, {[12, 1/3, 8], [15, 1/3, 0], [17, 1/3, 3]}, {[1, 1/3, 3], [10, 1/3, 13], [11, 1/3, 12]}, {[10, 1/3, 2], [19, 1/3, 17], [23, 1/3, 14]}, {[9, 1/2, 10], [22, 1/2, 1]}, {[6, 1/4, 10], [9, 1/4, 5], [11, 1/4, 6], [16, 1/4, 4]}, {[4, 1/2, 20], [17, 1/2, 10]}, {[18, 1, 13]}, {[15, 1/3, 16], [16, 1/3, 14], [23, 1/3, 2]}, {[9, 1/3, 1], [19, 1/3, 0], [21, 1/3, 12]}, {[3, 1/4, 12], [6, 1/4, 14], [14, 1/4, 7], [17, 1/4, 4]}, {[1, 1/2, 16], [12, 1/2, 4]}, {[9, 1/2, 18], [16, 1/2, 3]}, {[19, 1/2, 18], [21, 1/2, 5]}, {[2, 1/2, 18], [6, 1/2, 17]}] Then the statistical information is [[0.79063829030560721364, 0.099574471782444277448, 0.10978723791194850892], [41.939659514145230082, 39.373695206954867590, 41.102773627718080210], [4.8215534931594099272, 5.2163497065998036139, 5.3833807062951332930], -6 0.13611364206891517216 10 ]