Almost optimal

[Quuxplusone]

Here's an approach that can get 16.67 cents per game! I just let this script run for a long time in a terminal with a small font. Unfortunately it still eventually stops making progress. You probably don't want to merge this as-is, but it should give you some good inspiration.

## Winning strategy
- With probability 0.0023%: Guess [62, 17, 9, 83], then right-straight binary search avoiding [87, 26, 1, 98].
- With probability 0.0146%: Guess [57, 23], then left-straight binary search avoiding [30, 26].
- With probability 0.0291%: Guess [52, 22, 73], then right-straight binary search avoiding [26, 1, 87].
- With probability 0.0384%: Guess [62, 17, 9, 83], then left-straight binary search avoiding [87, 26, 1, 98].
- With probability 0.0517%: Guess [47, 17, 3, 31, 71, 56], then left-straight binary search avoiding [27, 14, 2, 26, 43, 30].
- With probability 0.0612%: Guess [60, 45, 82], then right-straight binary search avoiding [26, 1, 98].
- With probability 0.0890%: Guess [50, 30, 69], then right-straight binary search avoiding [27, 13, 43].
- With probability 0.1141%: Guess [35, 11], then left-straight binary search avoiding [85, 27].
- With probability 0.1261%: Guess [46, 30], then left-straight binary search avoiding [98, 13].
- With probability 0.1328%: Guess [49, 30, 70], then left-straight binary search avoiding [65, 34, 97].
- With probability 0.1368%: Guess [44], then left-leaning binary search.
- With probability 0.1388%: Guess [52, 32, 17, 70], then left-straight binary search avoiding [42, 26, 23, 97].
- With probability 0.1473%: Guess [46, 17, 13, 30, 62, 52], then right-straight binary search avoiding [26, 2, 1, 3, 95, 87].
- With probability 0.1530%: Guess [39, 18], then left-straight binary search avoiding [87, 1].
- With probability 0.1604%: Guess [53, 26, 10, 41, 77, 65], then left-leaning binary search avoiding [74, 3, 2, 12, 97, 87].
- With probability 0.1876%: Guess [33, 13, 67], then left-straight binary search avoiding [26, 2, 97].
- With probability 0.1942%: Guess [49, 30, 70], then right-straight binary search avoiding [65, 34, 97].
- With probability 0.2036%: Guess [57, 22], then left-straight binary search avoiding [66, 44].
- With probability 0.2090%: Guess [35, 18, 68], then right-straight binary search avoiding [2, 1, 87].
- With probability 0.2221%: Guess [36, 13], then left-straight binary search avoiding [72, 14].
- With probability 0.2296%: Guess [55, 23, 78], then right-leaning binary search avoiding [2, 1, 98].
- With probability 0.2486%: Guess [55, 35], then left-straight binary search avoiding [21, 1].
- With probability 0.2669%: Guess [62, 36], then left-straight binary search avoiding [95, 3].
- With probability 0.2864%: Guess [50, 30, 15, 74, 65], then left-straight binary search avoiding [13, 2, 1, 98, 25].
- With probability 0.2913%: Guess [58, 29, 85], then outward-leaning binary search avoiding [14, 12, 61].
- With probability 0.3121%: Guess [45, 31], then outward-leaning binary search avoiding [43, 26].
- With probability 0.3232%: Guess [45, 22, 12, 83], then right-straight binary search avoiding [44, 14, 2, 72].
- With probability 0.3483%: Guess [53, 25, 77], then left-leaning binary search avoiding [34, 2, 97].
- With probability 0.3821%: Guess [48, 20, 12, 68], then left-straight binary search avoiding [93, 92, 26, 97].
- With probability 0.4211%: Guess [47, 22], then right-straight binary search avoiding [97, 34].
- With probability 0.4227%: Guess [51, 29], then right-straight binary search avoiding [87, 12].
- With probability 0.4429%: Guess [48, 23], then right-straight binary search avoiding [13, 3].
- With probability 0.4511%: Guess [58, 29], then right-straight binary search avoiding [95, 26].
- With probability 0.4720%: Guess [54, 24, 84], then outward-leaning binary search avoiding [11, 1, 35].
- With probability 0.5005%: Guess [45, 21], then outward-leaning binary search avoiding [14, 12].
- With probability 0.5172%: Guess [58, 28, 79], then left-straight binary search avoiding [92, 87, 93].
- With probability 0.5443%: Guess [56, 28, 17, 42, 82, 67], then left-straight binary search avoiding [71, 43, 23, 68, 91, 85].
- With probability 0.5478%: Guess [53, 25, 77], then outward-leaning binary search avoiding [34, 2, 97].
- With probability 0.5997%: Guess [53, 25, 77], then right-straight binary search avoiding [34, 2, 97].
- With probability 0.6070%: Guess [61, 27, 13, 45, 77, 69], then left-straight binary search avoiding [93, 21, 20, 92, 98, 94].
- With probability 0.6077%: Guess [62, 38, 17], then right-straight binary search avoiding [3, 2, 87].
- With probability 0.7039%: Guess [64, 29], then left-straight binary search avoiding [98, 68].
- With probability 0.7113%: Guess [53, 25], then right-straight binary search avoiding [68, 31].
- With probability 0.7150%: Guess [62], then left-straight binary search.
- With probability 0.7288%: Guess [47, 29, 66], then left-straight binary search avoiding [35, 24, 97].
- With probability 0.7819%: Guess [49, 23], then right-straight binary search avoiding [68, 1].
- With probability 0.7890%: Guess [44, 17], then right-straight binary search avoiding [77, 22].
- With probability 0.8510%: Guess [45, 22, 9, 75], then right-straight binary search avoiding [48, 31, 21, 93].
- With probability 0.9024%: Guess [46, 16], then outward-leaning binary search avoiding [72, 14].
- With probability 0.9590%: Guess [50, 29, 69], then right-straight binary search avoiding [32, 28, 90].
- With probability 0.9711%: Guess [43], then left-straight binary search.
- With probability 1.0090%: Guess [57, 19, 78], then left-straight binary search avoiding [52, 11, 93].
- With probability 1.1893%: Guess [48, 23], then left-straight binary search avoiding [13, 3].
- With probability 1.3410%: Guess [37], then right-straight binary search.
- With probability 1.3625%: Guess [53, 25], then left-straight binary search avoiding [71, 65].
- With probability 1.3887%: Guess [36], then left-straight binary search.
- With probability 1.4602%: Guess [60, 30], then left-straight binary search avoiding [98, 1].
- With probability 1.5104%: Guess [46, 22], then left-straight binary search avoiding [72, 14].
- With probability 1.5187%: Guess [45, 22], then left-straight binary search avoiding [70, 14].
- With probability 1.6163%: Guess [53], then right-straight binary search.
- With probability 1.6390%: Guess [47], then right-straight binary search.
- With probability 1.7078%: Guess [48, 24, 15, 39, 75, 63], then left-straight binary search avoiding [72, 34, 26, 54, 92, 85].
- With probability 1.7287%: Guess [45, 13], then right-leaning binary search avoiding [90, 14].
- With probability 1.7706%: Guess [51, 25], then outward-leaning binary search avoiding [69, 2].
- With probability 1.8651%: Guess [48, 24], then right-straight binary search avoiding [93, 22].
- With probability 1.9731%: Guess [49, 33, 81], then left-straight binary search avoiding [35, 13, 97].
- With probability 2.0546%: Guess [39, 19], then right-straight binary search avoiding [68, 1].
- With probability 2.0719%: Guess [51, 23, 13, 35, 80, 65], then right-straight binary search avoiding [54, 7, 2, 8, 92, 70].
- With probability 2.0972%: Guess [59, 28, 14, 46, 82, 73], then right-straight binary search avoiding [72, 6, 2, 8, 93, 86].
- With probability 2.1215%: Guess [55, 23, 78], then left-leaning binary search avoiding [2, 1, 98].
- With probability 2.2720%: Guess [49, 19, 78], then left-straight binary search avoiding [39, 36, 86].
- With probability 2.4479%: Guess [44, 29], then outward-leaning binary search avoiding [31, 13].
- With probability 2.4973%: Guess [62, 30], then left-straight binary search avoiding [78, 71].
- With probability 2.6554%: Guess [51], then left-straight binary search.
- With probability 2.8926%: Guess [54], then right-straight binary search.
- With probability 3.2921%: Guess [38, 15], then outward-leaning binary search avoiding [68, 31].
- With probability 3.9052%: Guess [42], then right-straight binary search.
- With probability 4.2010%: Guess [47], then right-leaning binary search.
- With probability 4.2797%: Guess [43], then outward-leaning binary search.
- With probability 4.4162%: Guess [56, 24], then outward-leaning binary search avoiding [68, 31].
- With probability 4.5947%: Guess [50], then outward-leaning binary search.
- With probability 5.1925%: Guess [49, 25, 73], then outward-leaning binary search avoiding [36, 14, 86].
- With probability 6.5795%: Guess [52], then left-leaning binary search.
## Average wins for each number
0: $0.1667 (stdev $1.0449) 1: $0.1788 (stdev $0.9867) 2: $0.1725 (stdev $1.0396) 3: $0.1740 (stdev $1.0790) 4: $0.1667 (stdev $1.1125) 5: $0.1667 (stdev $1.0203) 6: $0.1667 (stdev $1.0572) 7: $0.1667 (stdev $1.1796) 8: $0.1677 (stdev $1.0544) 9: $0.1667 (stdev $1.1916)
10: $0.1667 (stdev $1.0918) 11: $0.1667 (stdev $1.1744) 12: $0.1717 (stdev $1.1145) 13: $0.1667 (stdev $1.1489) 14: $0.1713 (stdev $1.1871) 15: $0.1667 (stdev $1.1484) 16: $0.1667 (stdev $1.0916) 17: $0.1667 (stdev $1.1222) 18: $0.1667 (stdev $1.1915) 19: $0.1667 (stdev $1.1353)
20: $0.1667 (stdev $1.1616) 21: $0.1667 (stdev $1.1723) 22: $0.1667 (stdev $1.1801) 23: $0.1667 (stdev $1.2117) 24: $0.1667 (stdev $1.2139) 25: $0.1667 (stdev $1.2835) 26: $0.1814 (stdev $1.0497) 27: $0.1667 (stdev $1.1813) 28: $0.1667 (stdev $1.1172) 29: $0.1687 (stdev $1.2339)
30: $0.1667 (stdev $1.1628) 31: $0.1667 (stdev $1.1432) 32: $0.1667 (stdev $1.0837) 33: $0.1667 (stdev $1.1448) 34: $0.1667 (stdev $1.1319) 35: $0.1667 (stdev $1.1513) 36: $0.1667 (stdev $1.1965) 37: $0.1667 (stdev $1.1248) 38: $0.1667 (stdev $1.2149) 39: $0.1667 (stdev $1.2230)
40: $0.1667 (stdev $1.0888) 41: $0.1667 (stdev $1.0973) 42: $0.1667 (stdev $1.1743) 43: $0.1741 (stdev $1.2363) 44: $0.1667 (stdev $1.1602) 45: $0.1667 (stdev $1.2485) 46: $0.1667 (stdev $1.1904) 47: $0.1667 (stdev $1.2744) 48: $0.1667 (stdev $1.2270) 49: $0.1667 (stdev $1.3464)
50: $0.1667 (stdev $1.2355) 51: $0.1667 (stdev $1.2827) 52: $0.1667 (stdev $1.2753) 53: $0.1682 (stdev $1.2203) 54: $0.1667 (stdev $1.2038) 55: $0.1667 (stdev $1.1516) 56: $0.1667 (stdev $1.2294) 57: $0.1667 (stdev $1.1274) 58: $0.1667 (stdev $1.1228) 59: $0.1667 (stdev $1.1476)
60: $0.1667 (stdev $1.2085) 61: $0.1667 (stdev $1.1115) 62: $0.1667 (stdev $1.2234) 63: $0.1667 (stdev $1.1254) 64: $0.1667 (stdev $1.1052) 65: $0.1667 (stdev $1.1838) 66: $0.1667 (stdev $1.1298) 67: $0.1667 (stdev $1.0436) 68: $0.1667 (stdev $1.2399) 69: $0.1667 (stdev $1.1546)
70: $0.1667 (stdev $1.1699) 71: $0.1667 (stdev $1.1499) 72: $0.1752 (stdev $1.1579) 73: $0.1667 (stdev $1.1998) 74: $0.1667 (stdev $1.1534) 75: $0.1667 (stdev $1.2469) 76: $0.1667 (stdev $1.1237) 77: $0.1667 (stdev $1.2401) 78: $0.1667 (stdev $1.1939) 79: $0.1667 (stdev $1.1330)
80: $0.1667 (stdev $1.1037) 81: $0.1667 (stdev $1.1919) 82: $0.1667 (stdev $1.1883) 83: $0.1667 (stdev $1.1122) 84: $0.1668 (stdev $1.1563) 85: $0.1674 (stdev $1.1180) 86: $0.1677 (stdev $1.1158) 87: $0.1746 (stdev $1.1531) 88: $0.1667 (stdev $1.1297) 89: $0.1667 (stdev $1.1769)
90: $0.1667 (stdev $1.1276) 91: $0.1667 (stdev $1.0596) 92: $0.1667 (stdev $1.1130) 93: $0.1671 (stdev $1.1053) 94: $0.1667 (stdev $1.0470) 95: $0.1667 (stdev $1.0994) 96: $0.1667 (stdev $1.0544) 97: $0.1667 (stdev $1.0046) 98: $0.1761 (stdev $1.0410) 99: $0.1667 (stdev $1.0236)
Avg win if Ballmer chooses randomly: $0.16760422893884921
Win if Ballmer chooses adversarially: $0.16671211189426988

[gukoff]

Thanks, I'll get to reviewing this soon!