Overview

This document presents the first report for “Bigger Fish” by Underdog Games. We include a game description as per the digital implementation in our software, a brief overview of AI players used and data they create, and several metrics offering game information.

Key Takeaways

  1. Is there a first-player advantage? Only very slight, and most pronounced for 2 players. However, a trend remains with more players that those earlier in turn order are more likely to win.
  1. How common are comebacks, where players behind in score can catch up and win the game? Quite common with 2 players, but less so with 3+ players in the game. It can still happen (although rarely) that players up to 9 points behind at some point in the game end up winning.
  1. Do players get stuck with un-scorable buckets? Yes, but rarely. At least one player usually can score at least one of their buckets, but this does decrease in games with more players to 88%. The likelihood of not being able to score a player’s buckets (immediately, or given positive game progression) does increase with the number of players. The highest percentage of games where one (and only one of the five) bucket becomes impossible to score happens for 5-player games (below 7% of the games played).
  1. How much should we push the luck? We see winning players take a lot less Fish actions per game than losing players, and the ratio of Fish - Stop fishing actions suggests that losing players push their luck too far. We also see an increase in this ratio with player count, suggesting bigger risks are needed to win in games with more players.
  1. How often do games end due to a failed score attempt? The likelihood of this happening increases from just 14% in 2-player games, to 51% in 5-player games! With more opponents in the game, the chances of gaining points from opponent failed score attempts increase, and so does the probability of this winning scenario.

Game Overview


Bigger Fish is a push-your-luck card game in which 2-5 players attempt to be the first to reach 10 points (15 in 2-player games).

A central deck of cards contains 20 cards of each of 5 colors. A public pool may contain 0-20 cards in each colored deck, as cards from the deck get flipped and added to the pool through the ‘Fish’ action. Each player has their own collection of 5 decks (one of each color), that may contain 0-20 cards. The card on top in either the public pool or player buckets is the one comparisons are drawn to.

The game ends early if the deck of cards runs out. In that case, the player with the most points wins. In case of a draw, the tie break is the higher number of cards in the players’ buckets.

Actions

  • Fish: flip card from deck; if a higher card of that color exists in the pool, player busts. Otherwise, the card joins the pool and the player can keep fishing or decide to stop.
    • Fish (continue)
      • (if bust) Opponent Fish: each opponent takes any deck in the pool and adds it to their bucket.
    • Stop fishing: player collects all decks in the pool and adds them on top of their buckets.
  • Score: flip card from deck; if the player doesn’t have a bucket of that color, or the card on top of their bucket is bigger, they fail. Otherwise, they score 1 point for each card of that color in their bucket, plus the one flipped. All those cards are discarded.
    • (if fail) Opponent Score: all opponents score any card on top of a bucket for 1 point (and discard it).

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Game parameters

  • Number of colors (5)
  • Number of cards per color (20)
  • Number of points to win (10, or 15 for 2-player games)
  • Number of cards flipped per Fish action (1)
  • Number of decks taken by opponents per bust Fish actions (1)
  • Number of cards flipped per Score action (1)
  • Number of points won per card in successful Score actions (1)
  • Number of cards scored by opponents in failed Score actions (1)

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Data

AI Players

The players used to generate the data are all based on a tree-search algorithm (Monte Carlo Tree Search). At each decision point in the game, this algorithm creates a tree of statistics describing possible outcomes of various actions tried, up to a certain depth into the game (often the end for short games). It then picks the action most likely to lead to good outcomes.

In “Bigger Fish”, outcomes are considered better if the player gets closer to winning the game, and opponents do not. A bias is given to actions that have a high probability of winning or losing the game, based on the current state of the pool and player buckets.

Data extraction

This report shows results of a total of 15,466 games played for 2, 3, 4 and 5-player configurations, using 589 AI player configurations. The final ranking of all players for all configurations is given below, sorted by the Glicko-2 rating system. The table shows the win rate across all games, number of games played, glicko-2 rating and average score in all games (overall, on wins and on losses).

2-Player games (data)

There were a total of 5,034 2-player games recorded, played by 124 variations of the AI player with different skill levels.

AI Player Plays Win % Glicko-2 Rating (SD) Avg. Score Avg. Score on win Avg. Score on loss
Agent [71] 112 69.64% 1728 (98) 14.88 17.44 9.00 ± 3.83
Agent [30] 140 68.57% 1722 (91) 13.76 16.01 8.86 ± 4.08
Agent [105] 72 69.44% 1721 (103) 14.58 16.88 9.36 ± 4.80
Agent [36] 142 69.01% 1693 (95) 14.04 16.41 8.77 ± 4.41
Agent [75] 113 61.95% 1660 (93) 12.96 16.24 7.60 ± 4.00
Agent [10] 184 63.04% 1658 (86) 13.60 16.09 9.35 ± 4.03
Agent [57] 137 61.31% 1658 (92) 13.81 16.74 9.17 ± 3.62
Agent [40] 138 65.22% 1650 (95) 13.91 16.50 9.04 ± 3.37
Agent [24] 173 65.90% 1639 (92) 13.83 16.18 9.29 ± 3.83
Agent [119] 48 72.92% 1635 (102) 14.21 16.14 9.00 ± 3.76
Agent [87] 95 55.79% 1634 (94) 12.55 16.17 7.98 ± 4.21
Agent [38] 131 56.49% 1619 (92) 13.40 16.18 9.81 ± 2.97
Agent [53] 136 58.09% 1617 (89) 13.02 16.30 8.47 ± 3.79
Agent [89] 103 50.49% 1610 (90) 12.01 15.94 8.00 ± 4.10
Agent [69] 107 56.07% 1609 (91) 12.82 16.48 8.15 ± 3.90
Agent [106] 71 52.11% 1605 (97) 12.46 16.14 8.47 ± 4.14
Agent [2] 162 54.94% 1596 (85) 12.23 15.93 7.71 ± 4.06
Agent [60] 125 52.00% 1593 (88) 13.14 17.06 8.90 ± 4.03
Agent [16] 160 55.00% 1589 (87) 13.52 16.78 9.53 ± 3.62
Agent [3] 161 51.55% 1588 (86) 11.99 16.12 7.59 ± 4.45
Agent [113] 45 48.89% 1588 (115) 11.11 15.86 6.57 ± 4.09
Agent [114] 42 54.76% 1586 (114) 12.74 16.83 7.79 ± 3.90
Agent [6] 170 51.76% 1573 (88) 12.74 16.57 8.63 ± 3.67
Agent [79] 94 53.19% 1571 (93) 11.77 16.18 6.75 ± 3.26
Agent [46] 127 58.27% 1566 (89) 13.01 16.35 8.34 ± 3.59
Agent [94] 88 51.14% 1564 (91) 12.08 16.20 7.77 ± 4.40
Agent [61] 108 51.85% 1561 (88) 11.93 15.93 7.62 ± 4.07
Agent [118] 50 52.00% 1557 (102) 12.88 16.81 8.62 ± 3.67
Agent [95] 87 49.43% 1557 (91) 12.92 17.16 8.77 ± 3.23
Agent [25] 159 52.83% 1556 (86) 12.69 16.31 8.64 ± 3.34
Agent [31] 139 52.52% 1555 (90) 11.89 16.18 7.15 ± 4.30
Agent [63] 112 50.89% 1554 (86) 12.54 16.35 8.58 ± 3.40
Agent [108] 42 45.24% 1553 (104) 12.38 16.26 9.17 ± 3.71
Agent [43] 127 44.88% 1552 (86) 10.80 16.05 6.51 ± 3.97
Agent [64] 92 46.74% 1547 (88) 12.50 16.53 8.96 ± 3.67
Agent [0] 147 53.74% 1544 (89) 12.16 15.90 7.81 ± 3.98
Agent [103] 45 48.89% 1542 (97) 12.24 16.55 8.13 ± 3.67
Agent [18] 171 50.88% 1541 (85) 12.39 16.54 8.08 ± 3.51
Agent [99] 82 45.12% 1530 (88) 11.85 16.65 7.91 ± 3.38
Agent [19] 131 53.44% 1527 (90) 11.87 16.63 6.41 ± 4.38
Agent [11] 159 50.94% 1526 (86) 12.54 16.36 8.58 ± 3.41
Agent [120] 44 45.45% 1525 (107) 11.30 15.85 7.50 ± 4.49
Agent [54] 144 52.08% 1522 (83) 12.45 16.53 8.01 ± 4.32
Agent [23] 157 45.22% 1522 (84) 12.04 16.18 8.62 ± 3.36
Agent [47] 142 44.37% 1520 (82) 11.87 16.70 8.01 ± 3.91
Agent [96] 80 45.00% 1519 (92) 11.85 16.83 7.77 ± 3.37
Agent [26] 158 47.47% 1515 (86) 11.68 15.92 7.84 ± 3.88
Agent [90] 88 46.59% 1510 (88) 12.00 16.10 8.43 ± 3.80
Agent [77] 95 44.21% 1509 (90) 11.60 17.45 6.96 ± 3.64
Agent [29] 149 53.69% 1509 (88) 11.92 15.97 7.22 ± 3.99
Agent [5] 167 44.91% 1509 (84) 12.22 16.68 8.59 ± 3.45
Agent [7] 174 43.68% 1504 (87) 11.84 16.39 8.32 ± 3.64
Agent [41] 147 51.70% 1503 (81) 13.07 16.22 9.69 ± 3.25
Agent [83] 92 41.30% 1503 (94) 11.77 16.66 8.33 ± 3.76
Agent [73] 97 48.45% 1501 (91) 12.30 16.60 8.26 ± 3.75
Agent [88] 73 56.16% 1500 (87) 12.89 16.44 8.34 ± 4.41
Agent [37] 114 44.74% 1498 (89) 12.49 16.78 9.02 ± 3.63
Agent [123] 35 37.14% 1497 (124) 11.11 16.15 8.14 ± 3.01
Agent [39] 129 46.51% 1496 (92) 12.35 16.35 8.87 ± 3.76
Agent [115] 56 51.79% 1493 (99) 13.27 16.21 10.11 ± 3.68
Agent [86] 95 49.47% 1492 (86) 12.93 16.13 9.79 ± 3.47
Agent [4] 148 41.89% 1492 (86) 12.53 16.15 9.93 ± 3.25
Agent [81] 91 47.25% 1489 (87) 11.32 16.00 7.12 ± 4.55
Agent [100] 73 43.84% 1488 (89) 11.47 16.38 7.63 ± 3.57
Agent [80] 94 42.55% 1488 (92) 11.63 16.48 8.04 ± 3.77
Agent [42] 146 43.84% 1487 (88) 12.16 16.48 8.78 ± 3.62
Agent [22] 126 37.30% 1485 (86) 11.48 16.49 8.49 ± 4.04
Agent [107] 60 48.33% 1480 (93) 11.68 16.41 7.26 ± 3.82
Agent [55] 128 46.09% 1478 (85) 11.99 16.51 8.13 ± 3.22
Agent [66] 87 44.83% 1478 (86) 11.90 15.87 8.67 ± 4.07
Agent [28] 153 43.14% 1476 (85) 12.14 16.02 9.21 ± 3.04
Agent [12] 151 41.06% 1475 (87) 11.25 16.29 7.73 ± 3.70
Agent [97] 70 45.71% 1473 (90) 12.30 16.28 8.95 ± 3.35
Agent [32] 129 50.39% 1470 (91) 12.73 16.71 8.69 ± 3.85
Agent [121] 12 33.33% 1470 (127) 10.50 16.75 7.38 ± 3.81
Agent [15] 153 45.75% 1469 (89) 11.85 16.23 8.16 ± 3.92
Agent [82] 71 42.25% 1466 (88) 12.10 17.33 8.27 ± 3.93
Agent [91] 88 43.18% 1466 (92) 12.05 16.89 8.36 ± 3.65
Agent [111] 30 53.33% 1464 (107) 11.67 15.69 7.07 ± 4.14
Agent [92] 48 54.17% 1460 (94) 12.44 16.50 7.64 ± 2.68
Agent [51] 100 46.00% 1458 (85) 11.45 15.91 7.65 ± 3.71
Agent [117] 37 40.54% 1450 (105) 10.73 15.93 7.18 ± 3.36
Agent [101] 54 42.59% 1450 (89) 11.35 15.65 8.16 ± 3.43
Agent [49] 155 41.94% 1449 (82) 11.48 16.14 8.11 ± 3.76
Agent [84] 70 45.71% 1449 (92) 12.61 17.12 8.82 ± 3.02
Agent [17] 159 42.77% 1448 (88) 11.62 16.76 7.78 ± 3.38
Agent [44] 95 45.26% 1447 (86) 10.99 16.14 6.73 ± 4.14
Agent [74] 91 48.35% 1446 (85) 13.07 17.36 9.04 ± 3.13
Agent [93] 78 34.62% 1445 (95) 10.58 16.19 7.61 ± 4.15
Agent [52] 106 34.91% 1439 (89) 10.59 15.89 7.75 ± 3.88
Agent [62] 85 37.65% 1432 (87) 11.52 16.88 8.28 ± 4.24
Agent [33] 111 40.54% 1429 (85) 11.21 16.20 7.80 ± 3.55
Agent [56] 106 28.30% 1417 (92) 10.03 16.20 7.59 ± 3.52

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3-Player games (data)

There were a total of 4,014 3-player games recorded, played by 134 variations of the AI player with different skill levels.

AI Player Plays Win % Glicko-2 Rating (SD) Avg. Score Avg. Score on win Avg. Score on loss
Agent [130] 20 60.00% 1699 (98) 7.60 11.00 2.50 ± 1.93
Agent [117] 60 55.00% 1696 (75) 8.22 10.88 4.96 ± 2.62
Agent [126] 41 36.59% 1696 (81) 7.41 10.47 5.65 ± 2.90
Agent [57] 148 50.68% 1682 (66) 8.36 10.71 5.95 ± 2.49
Agent [20] 183 51.37% 1679 (65) 8.18 10.73 5.48 ± 2.35
Agent [84] 103 46.60% 1672 (67) 7.92 10.75 5.45 ± 2.36
Agent [118] 48 45.83% 1670 (84) 7.27 10.64 4.42 ± 3.02
Agent [92] 118 47.46% 1660 (63) 7.45 10.95 4.29 ± 2.89
Agent [38] 161 45.96% 1659 (64) 7.99 10.80 5.61 ± 2.57
Agent [96] 119 42.86% 1638 (65) 7.63 10.67 5.35 ± 2.74
Agent [119] 66 40.91% 1637 (65) 7.61 10.44 5.64 ± 2.64
Agent [24] 201 49.25% 1628 (65) 7.76 10.58 5.02 ± 2.34
Agent [112] 61 37.70% 1626 (71) 6.98 11.04 4.53 ± 2.73
Agent [55] 146 41.78% 1624 (65) 7.69 10.79 5.47 ± 2.38
Agent [74] 106 39.62% 1623 (65) 7.59 11.14 5.27 ± 2.41
Agent [41] 186 42.47% 1620 (64) 7.62 10.76 5.31 ± 2.43
Agent [95] 115 42.61% 1609 (66) 7.47 10.96 4.88 ± 2.44
Agent [31] 201 37.81% 1606 (67) 7.39 10.83 5.30 ± 2.56
Agent [11] 183 40.98% 1604 (61) 6.80 10.71 4.09 ± 2.57
Agent [90] 91 34.07% 1600 (69) 7.27 10.87 5.42 ± 2.48
Agent [54] 124 42.74% 1599 (67) 7.43 10.58 5.07 ± 2.73
Agent [132] 20 25.00% 1598 (91) 6.45 10.40 5.13 ± 2.72
Agent [93] 84 27.38% 1595 (66) 6.95 10.70 5.54 ± 2.59
Agent [123] 37 27.03% 1593 (67) 7.08 11.30 5.52 ± 2.83
Agent [43] 155 41.94% 1588 (67) 7.94 10.54 6.07 ± 2.27
Agent [97] 102 43.14% 1584 (64) 6.87 10.73 3.95 ± 2.70
Agent [6] 181 39.78% 1572 (61) 6.92 10.88 4.30 ± 2.43
Agent [70] 119 43.70% 1570 (70) 7.02 10.81 4.07 ± 2.65
Agent [23] 181 43.09% 1565 (69) 7.83 10.74 5.63 ± 2.46
Agent [56] 146 28.08% 1561 (66) 7.12 10.71 5.72 ± 2.50
Agent [64] 134 42.54% 1558 (64) 7.05 10.95 4.17 ± 2.86
Agent [51] 132 35.61% 1558 (64) 7.25 11.19 5.07 ± 2.41
Agent [14] 186 26.34% 1557 (63) 6.80 10.65 5.42 ± 2.45
Agent [27] 162 35.80% 1557 (63) 7.26 10.97 5.19 ± 2.49
Agent [4] 153 33.99% 1557 (63) 7.07 10.58 5.27 ± 2.52
Agent [76] 89 31.46% 1555 (69) 6.92 11.04 5.03 ± 2.74
Agent [81] 70 35.71% 1553 (63) 7.03 10.48 5.11 ± 2.77
Agent [82] 94 28.72% 1552 (67) 6.96 10.93 5.36 ± 2.29
Agent [127] 18 22.22% 1549 (82) 6.33 10.25 5.21 ± 2.42
Agent [32] 185 37.84% 1548 (66) 7.62 10.73 5.72 ± 2.41
Agent [9] 193 36.79% 1548 (63) 7.40 10.83 5.40 ± 2.74
Agent [30] 166 33.13% 1541 (67) 7.22 11.18 5.26 ± 2.29
Agent [1] 184 31.52% 1541 (65) 6.90 11.14 4.95 ± 2.51
Agent [46] 133 38.35% 1540 (66) 6.73 11.00 4.07 ± 2.40
Agent [121] 37 13.51% 1540 (81) 5.46 10.80 4.62 ± 2.89
Agent [106] 100 30.00% 1537 (67) 6.16 10.73 4.20 ± 2.70
Agent [94] 75 28.00% 1537 (65) 6.52 11.57 4.56 ± 2.44
Agent [42] 163 29.45% 1536 (61) 7.01 11.21 5.25 ± 2.57
Agent [7] 173 33.53% 1534 (66) 6.49 10.86 4.28 ± 2.64
Agent [101] 23 21.74% 1532 (72) 6.04 11.00 4.67 ± 2.45
Agent [99] 40 22.50% 1532 (69) 6.62 11.78 5.13 ± 2.68
Agent [58] 130 31.54% 1530 (67) 7.05 10.88 5.28 ± 2.38
Agent [12] 184 31.52% 1528 (64) 6.87 10.79 5.06 ± 2.43
Agent [36] 168 35.71% 1528 (64) 7.14 10.63 5.19 ± 2.53
Agent [62] 91 31.87% 1526 (67) 6.60 10.62 4.73 ± 2.86
Agent [105] 59 23.73% 1522 (67) 6.75 10.71 5.51 ± 2.26
Agent [13] 210 32.38% 1521 (61) 7.21 10.91 5.44 ± 2.53
Agent [122] 43 25.58% 1521 (77) 6.44 11.09 4.84 ± 2.63
Agent [60] 152 34.21% 1521 (63) 6.91 11.33 4.62 ± 2.36
Agent [65] 112 28.57% 1520 (64) 6.93 11.25 5.20 ± 2.39
Agent [25] 182 28.57% 1517 (63) 6.53 11.02 4.73 ± 2.54
Agent [52] 118 31.36% 1517 (65) 6.90 10.97 5.04 ± 2.38
Agent [85] 90 25.56% 1517 (69) 6.48 10.91 4.96 ± 2.27
Agent [34] 187 31.55% 1516 (64) 6.70 10.68 4.87 ± 2.51
Agent [87] 66 27.27% 1514 (66) 6.68 11.28 4.96 ± 2.36
Agent [5] 180 29.44% 1513 (65) 6.82 10.92 5.10 ± 2.33
Agent [28] 168 34.52% 1513 (64) 7.10 11.21 4.93 ± 2.50
Agent [83] 104 25.96% 1511 (63) 6.20 11.00 4.52 ± 2.61
Agent [66] 133 19.55% 1510 (64) 5.62 10.46 4.44 ± 2.43
Agent [67] 123 29.27% 1510 (66) 6.53 10.64 4.83 ± 2.37
Agent [108] 57 28.07% 1510 (66) 6.07 10.69 4.27 ± 2.38
Agent [3] 179 30.17% 1508 (65) 6.63 10.69 4.87 ± 2.79
Agent [98] 42 26.19% 1507 (64) 6.93 11.36 5.35 ± 3.01
Agent [78] 95 26.32% 1506 (62) 6.19 10.96 4.49 ± 2.39
Agent [63] 116 31.03% 1503 (64) 6.66 11.28 4.58 ± 2.39
Agent [107] 18 33.33% 1502 (70) 6.11 10.67 3.83 ± 2.25
Agent [59] 151 29.14% 1500 (66) 6.97 11.16 5.24 ± 2.23
Agent [71] 76 27.63% 1499 (63) 6.72 10.71 5.20 ± 2.29
Agent [21] 167 32.93% 1495 (64) 6.47 10.55 4.47 ± 2.62
Agent [40] 140 33.57% 1495 (69) 6.97 11.00 4.94 ± 2.47
Agent [45] 135 26.67% 1492 (62) 6.36 10.89 4.71 ± 2.28
Agent [49] 102 20.59% 1488 (64) 6.32 10.67 5.20 ± 2.38
Agent [33] 170 29.41% 1488 (63) 7.17 10.98 5.58 ± 2.32
Agent [17] 166 25.90% 1486 (67) 6.72 11.67 4.99 ± 2.70
Agent [69] 106 28.30% 1483 (65) 6.47 11.03 4.67 ± 2.29
Agent [114] 13 38.46% 1483 (68) 7.15 10.60 5.00 ± 2.83
Agent [50] 160 28.75% 1481 (61) 6.48 11.02 4.65 ± 2.56
Agent [15] 148 15.54% 1480 (68) 5.40 10.83 4.40 ± 2.23
Agent [72] 121 29.75% 1478 (62) 6.52 10.83 4.69 ± 2.51
Agent [77] 55 36.36% 1478 (66) 6.40 10.90 3.83 ± 2.06
Agent [53] 104 29.81% 1476 (67) 7.03 10.94 5.37 ± 2.52
Agent [35] 179 26.82% 1475 (63) 7.01 11.00 5.55 ± 2.55
Agent [37] 187 24.06% 1475 (61) 6.05 11.00 4.49 ± 2.44
Agent [0] 178 23.03% 1472 (67) 5.61 10.76 4.07 ± 2.12
Agent [44] 133 21.05% 1471 (65) 5.65 10.39 4.39 ± 2.40
Agent [103] 50 20.00% 1471 (68) 5.66 11.00 4.33 ± 2.29
Agent [133] 1 0.00% 1469 (84) 5.00 N/A 5.00 ± NA
Agent [29] 155 30.32% 1469 (64) 6.91 10.70 5.26 ± 2.34
Agent [22] 157 22.93% 1466 (67) 6.46 10.61 5.22 ± 2.42
Agent [16] 165 21.82% 1464 (65) 6.50 10.67 5.33 ± 2.40

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4-Player games (data)

There were a total of 4,023 4-player games recorded, played by 181 variations of the AI player with different skill levels.

AI Player Plays Win % Glicko-2 Rating (SD) Avg. Score Avg. Score on win Avg. Score on loss
Agent [172] 44 34.09% 1694 (65) 7.48 10.47 5.93 ± 2.52
Agent [90] 124 48.39% 1683 (56) 7.86 10.27 5.61 ± 2.11
Agent [165] 37 27.03% 1660 (60) 7.11 10.20 5.96 ± 2.26
Agent [69] 172 34.30% 1657 (52) 7.44 10.58 5.81 ± 2.41
Agent [169] 34 38.24% 1643 (63) 7.06 10.92 4.67 ± 2.78
Agent [115] 98 32.65% 1639 (59) 6.96 10.28 5.35 ± 2.38
Agent [152] 47 34.04% 1626 (62) 7.13 10.12 5.58 ± 2.33
Agent [164] 28 28.57% 1622 (65) 6.82 11.00 5.15 ± 2.16
Agent [153] 55 29.09% 1618 (56) 6.85 10.62 5.31 ± 2.56
Agent [22] 226 35.40% 1618 (50) 7.31 10.45 5.59 ± 2.45
Agent [36] 170 39.41% 1618 (56) 7.56 10.63 5.57 ± 2.57
Agent [30] 187 33.16% 1609 (57) 6.93 10.69 5.06 ± 2.58
Agent [32] 199 35.18% 1608 (54) 7.22 10.53 5.42 ± 2.46
Agent [4] 196 40.31% 1607 (60) 7.11 10.54 4.79 ± 2.42
Agent [92] 122 30.33% 1606 (52) 7.14 10.51 5.67 ± 2.47
Agent [125] 82 32.93% 1601 (55) 6.71 10.70 4.75 ± 2.32
Agent [97] 120 30.83% 1598 (50) 7.37 10.43 6.00 ± 2.15
Agent [11] 221 23.53% 1598 (54) 6.68 10.42 5.53 ± 2.36
Agent [55] 192 31.77% 1595 (55) 6.72 10.52 4.95 ± 2.52
Agent [103] 141 27.66% 1593 (52) 7.01 10.97 5.49 ± 2.32
Agent [91] 118 34.75% 1592 (55) 6.84 10.63 4.82 ± 2.23
Agent [126] 116 25.86% 1591 (55) 7.10 10.60 5.88 ± 2.18
Agent [16] 178 19.66% 1587 (54) 6.33 10.80 5.24 ± 2.34
Agent [56] 168 26.79% 1585 (56) 7.25 10.60 6.02 ± 2.16
Agent [170] 37 27.03% 1585 (69) 6.86 11.20 5.26 ± 2.60
Agent [70] 163 36.81% 1584 (55) 7.12 10.68 5.04 ± 2.54
Agent [124] 101 31.68% 1583 (55) 7.28 10.72 5.68 ± 2.15
Agent [23] 219 38.81% 1583 (57) 7.45 10.49 5.51 ± 2.46
Agent [12] 205 18.05% 1583 (55) 6.51 10.78 5.57 ± 2.45
Agent [5] 208 28.37% 1582 (58) 6.34 10.37 4.74 ± 2.43
Agent [145] 83 27.71% 1582 (54) 6.45 10.91 4.73 ± 2.33
Agent [120] 89 29.21% 1581 (52) 6.87 10.38 5.41 ± 2.15
Agent [163] 40 20.00% 1580 (61) 6.53 10.00 5.66 ± 2.21
Agent [113] 89 32.58% 1578 (58) 7.03 11.21 5.02 ± 2.76
Agent [59] 165 38.18% 1577 (55) 7.42 10.60 5.45 ± 2.29
Agent [9] 224 32.14% 1577 (55) 7.08 10.46 5.47 ± 2.38
Agent [151] 48 25.00% 1572 (59) 6.48 10.75 5.06 ± 2.22
Agent [140] 75 25.33% 1572 (57) 6.53 10.53 5.18 ± 2.48
Agent [68] 169 33.73% 1572 (54) 6.82 10.63 4.88 ± 2.47
Agent [18] 208 15.87% 1569 (53) 6.13 10.94 5.22 ± 2.40
Agent [15] 221 19.91% 1566 (54) 6.80 10.32 5.92 ± 2.39
Agent [29] 207 24.64% 1566 (53) 6.34 10.73 4.91 ± 2.29
Agent [26] 171 30.99% 1565 (58) 6.77 10.58 5.05 ± 2.21
Agent [57] 189 29.10% 1565 (49) 5.90 10.71 3.93 ± 2.13
Agent [180] 25 20.00% 1565 (61) 5.84 10.60 4.65 ± 2.66
Agent [127] 80 27.50% 1565 (59) 7.16 10.45 5.91 ± 2.03
Agent [175] 18 22.22% 1563 (65) 6.11 10.25 4.93 ± 2.73
Agent [87] 130 31.54% 1561 (59) 6.95 10.34 5.39 ± 2.63
Agent [128] 62 25.81% 1561 (54) 7.35 10.25 6.35 ± 2.36
Agent [47] 147 18.37% 1560 (54) 6.54 10.96 5.55 ± 2.26
Agent [37] 193 21.76% 1559 (58) 6.42 11.07 5.13 ± 2.36
Agent [13] 211 27.49% 1557 (52) 6.77 10.76 5.25 ± 2.47
Agent [160] 20 15.00% 1556 (65) 6.35 10.00 5.71 ± 1.99
Agent [109] 101 31.68% 1555 (54) 7.00 10.47 5.39 ± 2.38
Agent [79] 106 24.53% 1555 (54) 6.70 10.35 5.51 ± 2.44
Agent [63] 101 26.73% 1555 (56) 6.97 10.96 5.51 ± 2.18
Agent [75] 144 32.64% 1553 (55) 6.51 10.43 4.62 ± 2.43
Agent [89] 139 33.09% 1550 (58) 7.24 10.41 5.68 ± 2.21
Agent [14] 193 33.68% 1549 (56) 6.60 10.48 4.62 ± 2.50
Agent [53] 154 20.78% 1549 (52) 6.51 10.94 5.35 ± 2.39
Agent [58] 147 25.17% 1549 (52) 6.80 10.73 5.47 ± 2.47
Agent [40] 144 23.61% 1548 (53) 6.49 10.62 5.21 ± 2.27
Agent [8] 229 24.45% 1540 (52) 6.57 10.62 5.25 ± 2.55
Agent [82] 88 22.73% 1538 (52) 6.34 10.70 5.06 ± 2.09
Agent [39] 164 29.88% 1536 (50) 6.43 10.45 4.71 ± 2.45
Agent [100] 99 14.14% 1536 (52) 6.13 11.07 5.32 ± 2.21
Agent [88] 95 21.05% 1536 (53) 6.84 10.65 5.83 ± 2.20
Agent [31] 169 22.49% 1535 (54) 6.30 10.87 4.98 ± 2.36
Agent [162] 27 14.81% 1533 (62) 6.11 10.50 5.35 ± 2.01
Agent [98] 84 21.43% 1532 (55) 5.77 10.67 4.44 ± 2.11
Agent [95] 109 20.18% 1531 (54) 6.34 10.77 5.22 ± 2.39
Agent [136] 87 16.09% 1531 (54) 6.47 10.86 5.63 ± 2.31
Agent [45] 149 25.50% 1531 (53) 6.16 10.82 4.57 ± 2.50
Agent [117] 79 20.25% 1531 (51) 6.32 10.94 5.14 ± 2.19
Agent [10] 168 17.86% 1529 (54) 6.08 10.63 5.09 ± 2.34
Agent [60] 153 26.80% 1527 (52) 6.48 10.51 5.01 ± 2.32
Agent [96] 110 22.73% 1526 (56) 6.22 10.84 4.86 ± 2.36
Agent [43] 161 32.30% 1525 (55) 7.38 10.69 5.80 ± 2.21
Agent [114] 77 15.58% 1520 (54) 6.18 10.58 5.37 ± 2.15
Agent [166] 17 23.53% 1517 (59) 7.12 10.75 6.00 ± 2.45
Agent [142] 69 24.64% 1516 (52) 7.30 10.71 6.19 ± 2.16
Agent [155] 22 27.27% 1516 (62) 6.55 10.67 5.00 ± 2.37
Agent [48] 146 17.81% 1514 (56) 5.99 10.27 5.06 ± 2.29
Agent [73] 106 16.04% 1513 (52) 6.18 10.76 5.30 ± 2.04
Agent [171] 38 21.05% 1512 (61) 7.03 11.25 5.90 ± 2.23
Agent [147] 48 25.00% 1511 (57) 6.29 10.00 5.06 ± 2.00
Agent [71] 123 21.14% 1511 (56) 5.91 10.92 4.57 ± 2.33
Agent [76] 74 14.86% 1510 (52) 6.31 11.64 5.38 ± 2.20
Agent [110] 82 23.17% 1509 (49) 6.39 10.68 5.10 ± 2.09
Agent [52] 130 20.00% 1509 (54) 6.04 10.88 4.83 ± 2.32
Agent [7] 147 21.77% 1508 (52) 5.95 10.66 4.63 ± 2.24
Agent [86] 83 20.48% 1507 (53) 6.40 10.76 5.27 ± 2.22
Agent [54] 159 16.35% 1507 (52) 6.20 10.77 5.31 ± 2.28
Agent [49] 156 21.15% 1506 (55) 6.31 10.39 5.22 ± 2.31
Agent [27] 163 22.09% 1503 (55) 6.02 10.75 4.69 ± 2.09
Agent [81] 107 23.36% 1503 (63) 6.64 10.60 5.44 ± 2.35
Agent [77] 112 14.29% 1502 (52) 6.32 10.88 5.56 ± 2.16
Agent [33] 152 16.45% 1501 (56) 6.18 10.56 5.31 ± 2.20
Agent [83] 86 19.77% 1499 (54) 6.45 11.24 5.28 ± 2.27
Agent [38] 180 26.11% 1499 (52) 6.35 10.62 4.84 ± 2.52
Agent [72] 160 25.62% 1499 (53) 6.60 10.83 5.14 ± 2.30
Agent [131] 86 27.91% 1498 (52) 6.64 10.58 5.11 ± 2.31
Agent [123] 85 20.00% 1498 (53) 6.13 10.47 5.04 ± 2.27
Agent [107] 80 26.25% 1496 (54) 7.04 10.67 5.75 ± 1.99
Agent [6] 239 17.99% 1495 (59) 5.62 10.72 4.51 ± 2.29
Agent [67] 90 22.22% 1495 (53) 6.17 10.65 4.89 ± 2.18
Agent [138] 75 16.00% 1494 (54) 6.20 10.42 5.40 ± 2.45
Agent [94] 111 20.72% 1493 (53) 6.28 10.39 5.20 ± 2.08
Agent [112] 65 15.38% 1493 (55) 6.14 10.30 5.38 ± 2.13
Agent [46] 137 18.98% 1490 (53) 6.18 10.38 5.20 ± 2.17
Agent [174] 9 22.22% 1490 (65) 6.89 10.50 5.86 ± 2.12
Agent [116] 87 10.34% 1490 (53) 6.11 11.00 5.55 ± 2.25
Agent [34] 170 18.24% 1490 (54) 6.26 10.42 5.34 ± 2.40
Agent [129] 68 11.76% 1489 (55) 5.96 10.00 5.42 ± 2.10
Agent [20] 161 17.39% 1489 (53) 5.97 11.11 4.89 ± 2.35
Agent [122] 82 19.51% 1486 (54) 6.12 10.69 5.02 ± 2.23
Agent [35] 147 18.37% 1484 (56) 6.01 10.85 4.92 ± 2.23
Agent [119] 52 9.62% 1484 (53) 5.98 10.40 5.51 ± 2.41
Agent [28] 216 30.09% 1484 (56) 6.76 10.65 5.09 ± 2.51
Agent [65] 139 12.95% 1483 (55) 5.71 10.50 4.99 ± 2.40
Agent [133] 85 16.47% 1483 (53) 6.29 10.36 5.49 ± 2.27
Agent [111] 39 23.08% 1482 (54) 6.28 11.56 4.70 ± 2.44
Agent [99] 79 11.39% 1480 (51) 5.72 10.44 5.11 ± 2.26
Agent [62] 181 18.23% 1480 (56) 5.95 10.79 4.87 ± 2.37
Agent [141] 39 12.82% 1479 (51) 5.72 10.40 5.03 ± 2.04
Agent [74] 104 19.23% 1471 (53) 6.11 10.70 5.01 ± 2.35
Agent [21] 198 16.67% 1468 (54) 5.84 10.70 4.87 ± 2.42
Agent [2] 194 28.35% 1467 (56) 6.08 10.89 4.17 ± 2.53
Agent [146] 43 20.93% 1464 (55) 6.40 10.44 5.32 ± 2.48
Agent [137] 38 18.42% 1462 (53) 5.74 10.29 4.71 ± 2.44
Agent [148] 22 13.64% 1461 (55) 5.18 10.33 4.37 ± 1.89
Agent [3] 189 17.99% 1458 (52) 6.05 10.74 5.02 ± 2.38
Agent [178] 6 16.67% 1457 (66) 6.33 10.00 5.60 ± 1.52
Agent [0] 167 16.77% 1457 (56) 5.56 10.50 4.57 ± 2.11
Agent [50] 100 19.00% 1456 (55) 6.31 10.84 5.25 ± 2.20

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5-Player games (data)

There were a total of 2,395 5-player games recorded, played by 150 variations of the AI player with different skill levels.

AI Player Plays Win % Glicko-2 Rating (SD) Avg. Score Avg. Score on win Avg. Score on loss
Agent [24] 202 32.67% 1638 (47) 7.26 10.44 5.71 ± 2.20
Agent [29] 190 33.68% 1636 (49) 7.36 10.31 5.87 ± 2.20
Agent [111] 83 28.92% 1633 (45) 7.30 10.54 5.98 ± 2.17
Agent [139] 62 24.19% 1626 (47) 6.92 10.40 5.81 ± 2.48
Agent [131] 44 20.45% 1619 (56) 6.39 10.67 5.29 ± 2.18
Agent [99] 73 32.88% 1617 (49) 7.49 10.71 5.92 ± 2.25
Agent [68] 134 35.07% 1608 (55) 6.99 10.21 5.25 ± 2.44
Agent [137] 56 17.86% 1605 (51) 6.45 10.60 5.54 ± 2.41
Agent [132] 44 13.64% 1600 (51) 6.07 10.17 5.42 ± 2.32
Agent [37] 148 39.86% 1599 (54) 7.53 10.32 5.69 ± 2.24
Agent [144] 59 10.17% 1593 (50) 5.47 9.83 4.98 ± 2.25
Agent [123] 54 22.22% 1591 (50) 6.15 10.25 4.98 ± 2.24
Agent [6] 189 31.22% 1590 (49) 6.86 10.59 5.17 ± 2.41
Agent [35] 138 33.33% 1581 (48) 7.09 10.37 5.46 ± 2.28
Agent [115] 61 19.67% 1581 (49) 6.26 10.25 5.29 ± 2.37
Agent [79] 136 12.50% 1578 (49) 5.82 10.35 5.17 ± 2.18
Agent [11] 158 24.05% 1575 (49) 6.49 10.47 5.23 ± 2.49
Agent [117] 84 34.52% 1573 (52) 7.27 10.21 5.73 ± 2.27
Agent [146] 53 22.64% 1573 (53) 6.43 10.67 5.20 ± 2.25
Agent [88] 129 21.71% 1571 (47) 6.22 10.96 4.91 ± 2.52
Agent [134] 69 18.84% 1571 (49) 6.10 10.23 5.14 ± 2.14
Agent [17] 199 35.18% 1571 (48) 7.21 10.43 5.46 ± 2.05
Agent [61] 114 22.81% 1569 (53) 6.29 10.27 5.11 ± 2.19
Agent [34] 159 25.79% 1569 (48) 7.14 10.27 6.06 ± 2.06
Agent [116] 90 28.89% 1568 (48) 6.53 10.31 5.00 ± 2.22
Agent [119] 85 29.41% 1567 (45) 6.62 10.44 5.03 ± 2.22
Agent [81] 156 37.18% 1565 (56) 7.44 10.48 5.63 ± 2.42
Agent [69] 122 24.59% 1561 (47) 6.70 10.57 5.45 ± 2.22
Agent [118] 83 10.84% 1561 (47) 5.82 10.11 5.30 ± 2.35
Agent [70] 108 21.30% 1558 (49) 6.61 10.43 5.58 ± 2.35
Agent [130] 43 23.26% 1552 (50) 6.35 10.30 5.15 ± 2.40
Agent [59] 123 21.95% 1552 (47) 6.74 10.59 5.66 ± 2.28
Agent [53] 124 20.97% 1550 (47) 6.73 10.35 5.78 ± 2.05
Agent [84] 112 22.32% 1550 (50) 6.40 10.40 5.25 ± 2.61
Agent [0] 183 7.65% 1549 (49) 5.95 10.50 5.57 ± 2.13
Agent [94] 111 19.82% 1544 (45) 6.35 11.05 5.19 ± 2.34
Agent [135] 62 27.42% 1544 (47) 6.16 10.88 4.38 ± 2.58
Agent [112] 47 14.89% 1542 (49) 5.85 10.00 5.12 ± 2.31
Agent [16] 152 17.11% 1541 (49) 6.35 10.50 5.49 ± 2.06
Agent [54] 153 16.34% 1540 (49) 6.54 11.00 5.66 ± 1.77
Agent [27] 132 10.61% 1540 (49) 6.29 10.21 5.82 ± 2.22
Agent [82] 103 16.50% 1539 (48) 6.05 10.59 5.15 ± 2.35
Agent [142] 36 19.44% 1539 (51) 6.44 10.43 5.48 ± 1.92
Agent [104] 78 28.21% 1538 (45) 6.81 10.23 5.46 ± 2.38
Agent [143] 25 20.00% 1537 (57) 4.80 11.20 3.20 ± 1.58
Agent [3] 150 16.67% 1531 (48) 5.99 10.04 5.18 ± 2.27
Agent [66] 120 19.17% 1529 (49) 6.45 10.35 5.53 ± 2.36
Agent [89] 119 24.37% 1527 (55) 6.61 10.41 5.39 ± 2.34
Agent [56] 146 26.03% 1526 (48) 6.41 10.61 4.94 ± 2.40
Agent [101] 84 22.62% 1524 (44) 6.79 10.58 5.68 ± 2.02
Agent [114] 63 20.63% 1523 (44) 5.87 10.23 4.74 ± 2.28
Agent [7] 183 22.40% 1523 (55) 6.17 10.63 4.88 ± 1.98
Agent [40] 146 22.60% 1521 (49) 5.78 10.61 4.37 ± 2.24
Agent [67] 122 14.75% 1521 (43) 6.02 10.17 5.31 ± 2.19
Agent [86] 127 22.05% 1521 (48) 6.57 10.46 5.47 ± 2.25
Agent [12] 189 23.28% 1521 (51) 6.43 10.39 5.23 ± 2.15
Agent [38] 131 22.14% 1520 (48) 6.36 10.52 5.18 ± 2.21
Agent [9] 150 15.33% 1520 (51) 6.13 10.61 5.32 ± 2.24
Agent [128] 60 21.67% 1519 (48) 6.33 10.54 5.17 ± 2.48
Agent [14] 194 32.47% 1519 (54) 7.42 10.52 5.92 ± 2.13
Agent [13] 164 16.46% 1519 (49) 6.01 10.52 5.12 ± 2.29
Agent [57] 107 20.56% 1518 (49) 6.50 10.05 5.59 ± 2.03
Agent [45] 108 18.52% 1518 (49) 6.60 10.20 5.78 ± 2.22
Agent [19] 156 21.15% 1518 (47) 6.60 10.39 5.59 ± 2.33
Agent [76] 68 16.18% 1517 (48) 6.18 11.09 5.23 ± 2.27
Agent [124] 19 15.79% 1517 (54) 6.32 11.33 5.38 ± 2.47
Agent [110] 78 10.26% 1516 (47) 5.38 10.38 4.81 ± 2.27
Agent [107] 99 19.19% 1516 (47) 6.52 10.47 5.58 ± 1.99
Agent [106] 72 25.00% 1516 (49) 6.57 10.72 5.19 ± 2.46
Agent [18] 115 8.70% 1515 (48) 6.07 10.30 5.67 ± 2.10
Agent [122] 25 20.00% 1511 (43) 6.76 10.60 5.80 ± 2.48
Agent [71] 92 9.78% 1509 (47) 6.15 10.11 5.72 ± 2.11
Agent [48] 132 15.91% 1509 (52) 6.33 10.57 5.53 ± 1.95
Agent [72] 92 25.00% 1508 (47) 6.03 10.43 4.57 ± 2.29
Agent [4] 144 12.50% 1507 (46) 5.92 10.50 5.26 ± 2.22
Agent [75] 56 8.93% 1507 (46) 5.75 10.40 5.29 ± 2.15
Agent [55] 119 20.17% 1505 (46) 6.63 11.00 5.53 ± 2.35
Agent [113] 53 9.43% 1502 (51) 5.89 11.00 5.35 ± 2.46
Agent [36] 149 12.08% 1501 (46) 6.15 10.11 5.61 ± 2.17
Agent [97] 86 12.79% 1500 (47) 5.83 10.36 5.16 ± 2.06
Agent [103] 84 17.86% 1497 (47) 6.62 10.93 5.68 ± 2.25
Agent [87] 67 20.90% 1497 (50) 6.09 10.50 4.92 ± 2.39
Agent [33] 148 16.22% 1496 (62) 6.53 10.42 5.78 ± 2.21
Agent [1] 189 16.40% 1495 (48) 6.29 10.68 5.42 ± 2.06
Agent [49] 138 3.62% 1494 (42) 5.12 10.80 4.91 ± 2.20
Agent [5] 192 14.58% 1490 (59) 5.88 10.46 5.09 ± 2.20
Agent [102] 43 13.95% 1489 (47) 5.58 10.17 4.84 ± 2.29
Agent [20] 149 20.13% 1486 (52) 6.49 10.40 5.50 ± 2.20
Agent [83] 53 15.09% 1485 (49) 6.25 10.00 5.58 ± 1.71
Agent [140] 9 11.11% 1484 (56) 4.78 10.00 4.12 ± 1.96
Agent [50] 112 16.96% 1484 (48) 6.21 10.74 5.29 ± 2.29
Agent [105] 55 16.36% 1483 (48) 6.22 10.56 5.37 ± 2.42
Agent [42] 117 11.97% 1482 (51) 6.02 10.57 5.40 ± 2.41
Agent [52] 117 14.53% 1482 (47) 5.73 10.71 4.88 ± 2.20
Agent [26] 121 11.57% 1480 (51) 6.26 10.57 5.69 ± 2.09
Agent [41] 117 19.66% 1479 (50) 6.27 10.91 5.14 ± 2.57
Agent [39] 123 11.38% 1479 (48) 5.88 10.57 5.28 ± 2.28
Agent [92] 55 9.09% 1477 (46) 5.87 10.20 5.44 ± 2.34
Agent [47] 109 12.84% 1475 (52) 5.85 10.21 5.21 ± 2.27
Agent [78] 114 14.91% 1475 (50) 6.13 11.18 5.25 ± 2.25
Agent [64] 74 14.86% 1474 (51) 5.82 10.55 5.00 ± 2.46
Agent [31] 100 14.00% 1473 (46) 6.43 10.43 5.78 ± 2.28
Agent [62] 77 15.58% 1472 (52) 6.13 10.50 5.32 ± 2.03
Agent [80] 93 16.13% 1472 (48) 6.22 10.40 5.41 ± 2.16
Agent [28] 160 18.75% 1472 (55) 6.39 10.13 5.52 ± 2.29
Agent [126] 9 22.22% 1471 (48) 7.11 10.50 6.14 ± 1.21
Agent [15] 151 15.23% 1470 (51) 5.95 10.87 5.07 ± 2.22
Agent [95] 57 7.02% 1470 (46) 5.39 10.75 4.98 ± 2.29
Agent [93] 76 9.21% 1469 (48) 5.55 10.43 5.06 ± 1.85
Agent [73] 94 14.89% 1469 (46) 5.70 10.29 4.90 ± 2.38
Agent [2] 199 12.56% 1466 (48) 6.00 10.44 5.36 ± 2.11
Agent [44] 89 13.48% 1463 (49) 6.30 11.08 5.56 ± 2.50

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Win rates

This section shows winning rates statistics. We include the number of games, percentage of victories and average of scores obtained (overall, on wins and on losses). The analysis is presented for all players as a whole (across all games) and for each player count.

2P

When summarising the data of all players across all games and divide it by play order, we see a very small first player advantage for win rates, although the difference is not significant.

Play Order Plays Win % Avg. Score Avg. Score on win Avg. Score on loss
1 5034 52.76% 12.61 ± 4.90 16.39 ± 1.73 8.39 ± 3.72
2 5034 47.24% 12.05 ± 5.10 16.37 ± 1.66 8.17 ± 3.88

3P

For 3-player games, the same slight first player advantage signal is present.

Play Order Plays Win % Avg. Score Avg. Score on win Avg. Score on loss
1 4014 35.45% 7.13 ± 3.49 10.84 ± 1.26 5.09 ± 2.51
2 4014 32.76% 6.89 ± 3.53 10.85 ± 1.34 4.96 ± 2.50
3 4014 31.79% 6.80 ± 3.57 10.88 ± 1.32 4.90 ± 2.54

4P

Although this is less impactful with 4 or more players.

Play Order Plays Win % Avg. Score Avg. Score on win Avg. Score on loss
1 4023 27.17% 6.76 ± 3.19 10.63 ± 1.17 5.31 ± 2.40
2 4023 25.21% 6.56 ± 3.14 10.57 ± 1.05 5.21 ± 2.36
3 4023 25.01% 6.56 ± 3.18 10.67 ± 1.20 5.19 ± 2.35
4 4023 22.62% 6.31 ± 3.17 10.65 ± 1.18 5.04 ± 2.33

5P

Play Order Plays Win % Avg. Score Avg. Score on win Avg. Score on loss
1 2395 22.25% 6.57 ± 2.93 10.46 ± 1.00 5.46 ± 2.27
2 2395 21.34% 6.48 ± 2.94 10.52 ± 1.03 5.38 ± 2.25
3 2395 20.75% 6.39 ± 2.91 10.45 ± 0.97 5.33 ± 2.23
4 2395 18.29% 6.24 ± 2.91 10.54 ± 1.18 5.28 ± 2.24
5 2395 17.33% 6.20 ± 2.83 10.40 ± 0.90 5.32 ± 2.25

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Scores

This section focuses on game scores, both during the game, and at the end of the game.For clarity, a turn is understood as all the actions taken since a player has to choose between Fish and Score (where applicable) until just before the next player has to choose between Fish and Score (where applicable). This is, a turn of player A will include the original action selection, plus any subsequent actions taken (by themselves and any other player) until the next player has a Fish or Score selection.

Score Differences

The plots below show the frequency of score differences between the first player and the second (left), or the first player and the last player (right), at any point during the game, calculated across all games played (with different number of players taking part in the game on each row. The horizontal axis shows the distinct score differences, while the vertical axis shows how many times such a score difference was observed at least once in a game. The vertical dashed line shows the median value of all data. We observe very similar profiles across all player counts.

The plots below show the score difference for all games between the second player and the winner (left), or the last player and the winner (right). Negative values represent the winner being behind in score, while positive values mean the winner being ahead in score. The vertical dashed line shows the median value of this score difference. A majority of score differences are ‘0’, which is the case in the beginning of all games. However, the winner does generally have a lead in score throughout the game. We observe the spread of score differences to reduce slightly as we increase the number of players. We do notice several negative values representing comebacks from a score deficit, and we explore this more in the ‘Comebacks’ section.

Score Progression

The following figures aggregate the absolute value of the score difference per game turn. The horizontal axis shows the turn of the game, while the vertical axis shows the average and standard deviation of the absolute score difference. Note that these plots do not show changes of game leader (see Lead Change section next). This analysis shows that, on average, with a 95% confidence, the difference in scores between two players (either the first and second, left; or first and last, right) does not rise above 4 points before turn 10 (i.e. both players playing 5 turns each). With more than 2 players, the score progression profiles are very similar and with reduced averages, as players do require less points to end the game. We do see a consistent trend of lower differences to the second player than the last, suggesting players in the lead to be quite close in score throughout the game. The standard deviation reduces as well as we increase the number of players, suggesting tighter battles for the win.

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Lead Change

This section provides an analysis on the number of times a lead change happened. We define a lead change as a flip in which player is ahead in points during the game. For a flip to happen, the score difference must change from positive (player A winning) to negative (player A losing), excluding cases where the difference is 0 (players tied in score).

We analyze lead changes for all games played. The table below shows that in 30.37% of all 2-player games played there was no lead change. Effectively, this means that in 30.37% of all 2-player games, a player that got ahead in points remained ahead and ended up winning (note this may include cases in which score difference dropped to 0, but a flip of score was never completed). For more than 2 players, we can check the lead change with respect to the second player (how many times the player that was in second place became first), or with respect to the last player (how many times the player in last place became first).

On average, the mean lead changes per game with respect to the second player are close to 1 in all player counts, with a maximum of 4 (or 5 in 2-player games). However, it is interesting that even in 5-player games, the player order changed dramatically up to 3 times, such that the last player took the lead (although this happens in less and less games as the number of players increases).

Changes wrt Number of Players Games with no changes (%) Min P25 Mean Median P75 Max
Second Player 2-Player Games 30.37% 0 0 1.10 ± 0.96 1 2 5
Second Player 3-Player Games 28.90% 0 0 1.01 ± 0.84 1 1 4
Second Player 4-Player Games 26.00% 0 0 1.03 ± 0.80 1 1 4
Second Player 5-Player Games 27.99% 0 0 0.98 ± 0.79 1 1 4
Last Player 2-Player Games 30.37% 0 0 1.10 ± 0.96 1 2 5
Last Player 3-Player Games 52.22% 0 0 0.64 ± 0.76 0 1 4
Last Player 4-Player Games 59.88% 0 0 0.51 ± 0.69 0 1 4
Last Player 5-Player Games 67.50% 0 0 0.39 ± 0.62 0 1 3

This data is more clearly visualised in the figures below, which show the distribution of the number of score changes across all games. The horizontal axis shows the number of lead changes, and the vertical axis indicates how many of these changes took place across all games. The dashed vertical line shows the median of the data (1 change).

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Comebacks

The following plots show the percentages of games where a player, being behind a specific number of points compared to the leader at the time, was able to turn the tables to win the game.

In the histograms below, each bar corresponds to the maximum score deficit a player had during a game and showing:

  • The number of games played where a player was behind the leader in score by that amount.
  • The percentage of times a player with such disadvantage in score was able to win the game in the end.

This section analyzes comebacks in all games played. The horizontal axis shows the maximum score deficit for a player during a game, while the vertical axis shows the percentage of times when the player with this score deficit was able to recover and win the game.

For example, there were 757 2-player games where one player was no more than 5 points behind the opponent. In 46% of these games, the player that was behind was able to win in the end (not being able to recover in 54% of the games played where this difference was present).

These results show that changes in score are relatively frequent, but the likelihood of comebacks does decrease with higher player counts. Not falling behind in points against the opponent for more than 4 points still leaves a high chance of winning (over 40% comebacks in all cases, and over 70% for 2-player games), and being behind 1 point is so common in 2-player games that it does not determine who wins (as it happens in over 90% of the games).

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Actions

This section offers an statistics on the different actions executed in the game. We run analysis based on five different actions: the two main actions (Fish and Score) and the three second-tier actions: Stop Fishing (available after taking the Fish action), Opponent Scored (when a Score action the opponent took fails) and Opponent Fished (when a Fish action the opponent took busts).

Overview

First, we show the distribution and percentage of actions for all games played (as a percentage of all actions taken in all games). The line of play Fish - Stop Fishing is clearly dominant in 2 and 3 player counts, with a combined percentage of approximately 65% of the actions ever taken. The action Score is taken less than half of the times of the Fish action (about 47% versus 18% in 2-player games), with an even bigger difference as we increase the number of players in a game (only 8% of actions are Score for 5-player games). In terms of actions failed, we observe failing to Score a much more common event (10% for 2-players, increasing steadily to 18.5% for 5-players to become the second most common event after fishing) than failing to Fish (below 5% for 2-player games, increasing to 10.4% in 4- and 5-player games). The increase of failed score attempts with more players suggests higher competition and more scoring opportunities with more opponents attempting to get ahead.

Below we show the percentage of actions taken in each tick of a game (a tick refers to any action decision made by a player). Our data shows a predominance of the Fish action in all player counts (and the corresponding Stop Fishing) during the first stages of the game, stabilizing below 50% mid-way through the game. Conversely, Score reaches a 25% predominance in the mid game after a steady increase for 2-player games, but Opponent Scored takes this space instead with 4 and 5 players, supporting the narrative above.

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By Winner

The following table shows statistics about actions played across all games We differentiate statistics in the columns for winning and losing players. One thing to note is the difference in Fish actions played across all player counts, which tends to be higher in the losing players (and increasingly higher with more players in the game). An average of 10 Fish actions per game appears to be sufficient. With higher player counts, this may be reduced due to the high occurrence of opponent failed scoring observed, which adds more cards in the pool for other players to take later, with lower fishing risks.

We do observe, that the stop fishing action, while seeing no difference with 2 players in the game, when the number of players increases, is picked a lot more by losing players than winning players.

Action Type Players Winner (Median) Winner (Mean) Non-Winner (Median) Non-Winner (Mean)
Fish 2-Player Games 14.00 14.27 ± 5.67 18.00 18.36 ± 6.83
Opponent Fished 2-Player Games 2.00 2.37 ± 2.03 1.00 1.05 ± 1.21
Opponent Scored 2-Player Games 4.00 3.70 ± 2.11 3.00 3.54 ± 1.93
Score 2-Player Games 7.00 7.19 ± 2.19 5.00 5.37 ± 2.74
Stop fishing 2-Player Games 7.00 6.79 ± 2.33 7.00 6.90 ± 2.49
Fish 3-Player Games 5.00 5.38 ± 2.38 13.00 13.51 ± 4.30
Opponent Fished 3-Player Games 1.50 1.42 ± 0.99 1.50 1.93 ± 1.27
Opponent Scored 3-Player Games 1.50 1.75 ± 0.96 3.00 3.07 ± 1.55
Score 3-Player Games 2.00 1.86 ± 0.69 2.50 2.47 ± 1.14
Stop fishing 3-Player Games 2.00 2.20 ± 0.86 4.50 4.45 ± 1.55
Fish 4-Player Games 3.00 3.17 ± 1.39 11.33 11.94 ± 3.30
Opponent Fished 4-Player Games 1.00 0.99 ± 0.65 2.00 2.17 ± 1.29
Opponent Scored 4-Player Games 1.33 1.42 ± 0.67 3.67 3.74 ± 1.60
Score 4-Player Games 1.00 1.04 ± 0.41 2.00 1.98 ± 0.78
Stop fishing 4-Player Games 1.33 1.22 ± 0.49 3.67 3.76 ± 1.19
Fish 5-Player Games 2.25 2.43 ± 1.06 11.00 11.27 ± 2.82
Opponent Fished 5-Player Games 0.75 0.75 ± 0.46 2.25 2.26 ± 1.18
Opponent Scored 5-Player Games 1.25 1.21 ± 0.49 4.25 4.14 ± 1.57
Score 5-Player Games 0.50 0.65 ± 0.29 1.75 1.69 ± 0.56
Stop fishing 5-Player Games 0.75 0.89 ± 0.36 3.50 3.62 ± 1.08

The overall ratio of Fish / Stop fishing actions is very similar for all cases. This does show a trend of increasing with the player count, and also being higher for losing players, giving a small signal of losing players pushing their luck too far, while more luck pushing is also required with more players.

Action Type Players Winner (Median) Winner (Mean) Non-Winner (Median) Non-Winner (Mean)
Fish Ratio 2-Player Games 2.00 2.17 ± 0.76 2.43 3.10 ± 2.66
Fish Ratio 3-Player Games 2.33 2.61 ± 1.26 2.91 3.42 ± 2.00
Fish Ratio 4-Player Games 2.50 2.81 ± 1.43 3.11 3.48 ± 1.68
Fish Ratio 5-Player Games 2.67 2.98 ± 1.56 3.09 3.33 ± 1.19

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Points by Action

This section breaks down the source of points for all players and analyses how many points players can earn from the different actions available in the game.

Average Points per Score Action

The tables below show a summary of the amount of points won by a player as a result of a Score action (if successful ones are considered only; or all attempted score actions, including failed score attempts that result in 0 points). We further isolate the Score actions which lead to ending the game. The average points obtained for winning the game is higher than the overall average in all cases, indicating that game-winning moves use larger buckets accumulated throughout the game.

Players Category Average points (Scored) Average points (Attempted)
2-Player games Avg. Points Score action (Overall) 3.55 ± 1.62 1.58 ± 2.07
2-Player games Avg. Points Score action (Game End) 4.47 ± 1.96 4.47 ± 1.96
3-Player games Avg. Points Score action (Overall) 3.25 ± 1.41 1.49 ± 1.88
3-Player games Avg. Points Score action (Game End) 3.98 ± 1.71 3.98 ± 1.71
4-Player games Avg. Points Score action (Overall) 3.14 ± 1.38 1.35 ± 1.80
4-Player games Avg. Points Score action (Game End) 4.00 ± 1.74 4.00 ± 1.74
5-Player games Avg. Points Score action (Overall) 3.05 ± 1.26 1.25 ± 1.70
5-Player games Avg. Points Score action (Game End) 3.84 ± 1.60 3.82 ± 1.61
Progression of average score per action and turn

We plot below the progression of points obtained throughout the game from Score actions, by player turn, and by successful score actions only (left) and overall score attempts, including unsuccessful ones (right). We observe the average number of points growing steadily throughout the game, arriving to an average over 4 points per successful score action towards the end for all player counts.

There are also a few edge cases in the beginning of the game where the first player fails to fish after building up a single color stack, and the second player takes it from the pool into their buckets and successfully scores it on their turn; this is shown by the odd point averages in the beginning in the plots of successful score attempts only.

Count of actions that finish games

The next table analyses how many times games end from a player’s own successful scoring attempt (48% in 5-player games up to 85% in 2-player games), or instead from an opponent’s failed scoring attempt (52% in 5-player games down to 15% in 2-player games). With more opponents in the game (as the player count increases), the chances of gaining points from opponent failed score attempts increase, and so does the probability of this winning scenario.

Players Action Count Percentage
2-Player games Opponent Scored 732 14.54%
2-Player games Score 4302 85.46%
3-Player games Opponent Scored 1244 30.99%
3-Player games Score 2770 69.01%
4-Player games Opponent Scored 1745 43.42%
4-Player games Score 2274 56.58%
5-Player games Opponent Scored 1230 51.72%
5-Player games Score 1148 48.28%

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Duration

We study the duration of the games, as measured in turns.

The table below shows the number of turns a game lasts, calculated over all games played. The mean total turns are lowest with 3 players in a game and highest with 5 players. However, the number of turns per player reduces from 14 on average for 2-player games, to just over 6 on average for 5-player games. While 2-player games can end the quickest with a minimum of 7 total player turns, 5-player games may take longest with a maximum of 56 total player turns.

Players min P25 median mean sd P75 max IQR
2-Player games 7 25 28 29 5 32 54 7
3-Player games 9 21 25 25 5 28 52 7
4-Player games 12 24 28 28 5 31 54 7
5-Player games 12 28 32 32 6 36 56 8

The plots below show the frequency of game duration in turns over all games played. The vertical dashed line shows the median on 30 turns. The large majority of games fall between 20 and 42 turns.

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Bucket scorability analysis

This section shows a bucket scorability analysis, both as a percentage of games where a particular circumstance takes place, and as a percentage of decisions where a player can try to Score buckets.

Percentage of Games

This section indicates the percentages of games played where a metric is recorded for 0 to 5 suits. The situations analyzed are:

  • At some point during the game, a bucket can’t be scored because it’s empty.
  • At some point during the game, it is possible to score the top card of a bucket.
  • At some point during the game, the top card in 0-5 buckets can’t be scored.
  • At some point during the game, 0-5 buckets became impossible to score because there are no more cards of the given suit in the draw deck.
  • At some point during the game, 0-5 buckets became impossible to score despite having more cards of the given suit in the draw deck.

We show separately for all situations:

  • % of games, at least 1 player: this indicates the percentage of games for which the metric affects at least one of the players.
  • % of games, all players: this indicates the percentage of games for which the metric affects all players.

Empty buckets

This table shows percentages of games where it happened that a bucket could not be scored because of being empty. The column “None” refers to the case where none of the buckets are empty (i.e. there’s at least one card on the 5 buckets), which happens in 93% of the 2-player games, but slightly less with more players in the game. “1” means that only one of the buckets is empty, while “5” means that all buckets are empty (i.e. start of the game, hence happening in all games).

Percentage of games/action-decisions in which, at least once, players had empty buckets for N suits
Players Empty buckets: None 1 2 3 4 5
2-Player games % of games, at least 1 player 93.008% 99.781% 99.762% 96.861% 88.339% 100.000%
2-Player games % of games, all players 56.297% 91.299% 90.306% 68.574% 42.729% 100.000%
3-Player games % of games, at least 1 player 88.440% 99.377% 99.900% 98.555% 92.950% 100.000%
3-Player games % of games, all players 19.108% 64.574% 74.016% 49.626% 27.603% 100.000%
4-Player games % of games, at least 1 player 86.279% 99.727% 99.975% 99.727% 97.415% 100.000%
4-Player games % of games, all players 5.767% 43.251% 68.307% 46.557% 23.167% 100.000%
5-Player games % of games, at least 1 player 87.724% 99.958% 100.000% 99.916% 99.207% 100.000%
5-Player games % of games, all players 1.837% 24.885% 58.330% 42.756% 25.344% 100.000%

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Can score

This table shows percentages of games where, at least once, it was possible to score the top card of a bucket. In other words, games where for a different number of buckets there was at least one card of the relevant suit in the draw deck bigger than the current top card of the bucket.

The column “None” indicates situations in which none of the existing buckets (which are not empty) can be scored, which happens in all games. Column “1” indicates that, in 90% of the 2-player games, there was once when only 1 bucket could be scored for one of the players. Also in over 90% of the games (“5”) there was a moment when at least one of the players was able to successfully score all 5 buckets (note that this does not mean that the action would necessarily succeed: this specifically means that there is a card on the draw deck bigger than the top card for each of the 5 buckets). Statistics remain similar for all player counts when the metric is checked for at least one player, but reduces significantly if the condition must be satisfied by all players in the game: only 0.9% of games see a situation where all 5 players in a game could score all of their 5 buckets.

Percentage of games/action-decisions in which, at least once, players could score the top card
Players Scorable buckets: None 1 2 3 4 5
2-Player games % of games, at least 1 player 100.000% 90.048% 97.894% 99.901% 99.722% 90.505%
2-Player games % of games, all players 100.000% 46.524% 73.858% 92.849% 90.147% 48.252%
3-Player games % of games, at least 1 player 100.000% 94.544% 99.153% 100.000% 99.178% 85.077%
3-Player games % of games, all players 100.000% 30.618% 55.032% 76.756% 60.837% 14.325%
4-Player games % of games, at least 1 player 100.000% 98.459% 99.751% 100.000% 99.553% 82.028%
4-Player games % of games, all players 100.000% 26.547% 52.150% 69.625% 38.678% 3.878%
5-Player games % of games, at least 1 player 100.000% 99.541% 99.958% 100.000% 99.791% 83.132%
5-Player games % of games, all players 100.000% 28.351% 49.937% 59.332% 20.376% 0.877%

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Can’t score now

This table shows percentages of games where, at least once, it was not possible to score a bucket immediately, but it could be possible if the card on top was removed. In other words, games where there were cards in the draw deck that are bigger than other cards down the bucket or in the pool of that suit, for none to 5 buckets.

It happens in all games that at least once the game is in a state when this situation doesn’t happen for all players (“None”). It does happen often, in 78% of the 2-player games (and increasing to 95% for 5-player games), that there is at least once when 1 bucket can’t be scored immediately for one player. It is very rare that this situation happens simultaneously for 3 buckets or more for any player (and 2 buckets or more for all players).

Percentage of games in which, at least once, players could not score a bucket without removing the top card first
Players Can’t score immediately in… None 1 bucket 2 buckets 3 buckets 4 buckets 5 buckets
2-Player games % of games, at least 1 player 100.000% 78.526% 11.800% 0.675% 0.040% 0.000%
2-Player games % of games, all players 100.000% 24.454% 0.278% 0.000% 0.000% 0.000%
3-Player games % of games, at least 1 player 100.000% 83.308% 13.503% 0.947% 0.050% 0.000%
3-Player games % of games, all players 100.000% 6.104% 0.000% 0.000% 0.000% 0.000%
4-Player games % of games, at least 1 player 100.000% 90.753% 16.083% 0.969% 0.075% 0.000%
4-Player games % of games, all players 100.000% 2.163% 0.000% 0.000% 0.000% 0.000%
5-Player games % of games, at least 1 player 100.000% 95.198% 21.253% 1.879% 0.167% 0.000%
5-Player games % of games, all players 100.000% 2.004% 0.000% 0.000% 0.000% 0.000%

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Can’t score (no cards)

This table shows percentages of games where, at least once, it became impossible to score a number of buckets for the rest of the game: there are no more cards of the given suit in the draw deck.

It never happened in any game that all buckets were never able to be scored (“None”) because running out of cards of a given suit. It happens very rarely that 1-5 buckets become impossible to score during the game due to drawing all cards of a given suit from the deck, for one or all players.

P Percentage of games in which, at least once, it was impossible for players to score a bucket (no more cards of that suit in the draw deck)
Players Impossible to Score… None 1 bucket 2 buckets 3 buckets 4 buckets 5 buckets
2-Player games % of games, at least 1 player 100.000% 0.040% 0.000% 0.000% 0.000% 0.000%
2-Player games % of games, all players 100.000% 0.020% 0.000% 0.000% 0.000% 0.000%
3-Player games % of games, at least 1 player 100.000% 0.050% 0.000% 0.000% 0.000% 0.000%
3-Player games % of games, all players 100.000% 0.000% 0.000% 0.000% 0.000% 0.000%
4-Player games % of games, at least 1 player 100.000% 0.373% 0.124% 0.075% 0.025% 0.000%
4-Player games % of games, all players 100.000% 0.050% 0.000% 0.000% 0.000% 0.000%
5-Player games % of games, at least 1 player 100.000% 2.839% 0.835% 0.418% 0.209% 0.000%
5-Player games % of games, all players 100.000% 0.167% 0.000% 0.000% 0.042% 0.000%

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Can’t score (cards in deck)

This table shows percentages of games where, at least once, it became impossible to score a number of buckets for the rest of the game, despite still having cards of the given suit in the draw deck. This takes place when the minimum card of a given suit in both the bucket and the pool deck of this suit is greater than all cards that suit in the draw deck, and the cards in the draw deck of this suit are in strictly decreasing order of value.

It never happened in any game that all buckets where never able to be scored (“None”) because of the order of the cards in the deck. It happens very rarely that 1 bucket become impossible to score because of this (although the probability of this occurrence increases with more players in the game), and it only happens with 4 or more players that more than 1 bucket became simultaneously impossible to score during the game.

Percentage of games in which, at least once, it was impossible for players to score a bucket (with cards of that suit still in the draw deck)
Players Impossible to score in… None 1 bucket 2 buckets 3 buckets 4 buckets 5 buckets
2-Player games % of games, at least 1 player 100.000% 0.040% 0.000% 0.000% 0.000% 0.000%
2-Player games % of games, all players 100.000% 0.000% 0.000% 0.000% 0.000% 0.000%
3-Player games % of games, at least 1 player 100.000% 0.174% 0.000% 0.000% 0.000% 0.000%
3-Player games % of games, all players 100.000% 0.000% 0.000% 0.000% 0.000% 0.000%
4-Player games % of games, at least 1 player 100.000% 0.671% 0.050% 0.000% 0.000% 0.000%
4-Player games % of games, all players 100.000% 0.000% 0.000% 0.000% 0.000% 0.000%
5-Player games % of games, at least 1 player 100.000% 4.718% 0.752% 0.209% 0.042% 0.000%
5-Player games % of games, all players 100.000% 0.125% 0.000% 0.000% 0.000% 0.000%

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Percentage of Action-Decisions

This section indicates the scorability of buckets when Score actions can be played, across all games. We analyze:

  • Percentage of decisions when a bucket can’t be scored because it’s empty.
  • Percentage of decisions when it is possible to score the top card of a bucket.
  • Percentage of decisions when the top card in 0-5 buckets can’t be scored.
  • Percentage of decisions when 0-5 buckets became impossible to score because there are no more cards of the given suit in the draw deck.
  • Percentage of decisions when 0-5 buckets became impossible to score despite having more cards of the given suit in the draw deck.

All these circumstances are analyzed with respect to the player that is making the decision of scoring or not.

Empty buckets

This table shows, for every time a player has a chance to Score, how often 0 to N of their buckets are empty. The column “None” shows that, in 17% of the times, there is a card in every bucket for 2-player games (though this reduces with higher player counts). The number of buckets that are simulteneously empty decrease from 27% (1 bucket) to 9% (all buckets) in 2-player games, with slight variations with higher player counts.

Percentage of games/action-decisions in which, at least once, players had empty buckets for N suits
Players Empty buckets: None 1 2 3 4 5
2-Player games % of action-decisions 17.172% 27.366% 23.836% 14.805% 8.173% 8.648%
3-Player games % of action-decisions 13.260% 24.266% 24.261% 16.719% 9.921% 11.574%
4-Player games % of action-decisions 9.909% 21.798% 25.353% 19.258% 11.666% 12.016%
5-Player games % of action-decisions 8.769% 19.345% 24.340% 20.960% 13.406% 13.179%

Can score

This table shows percentages of decisions for a player when it was possible to score the top card of a bucket. In other words, how often for a different number of buckets there was at least one card of the relevant suit in the draw deck bigger than the current top card of the bucket.

Percentage of games/action-decisions in which, at least once, players could score the top card
Players Scorable buckets: None 1 2 3 4 5
2-Player games % of action-decisions 8.989% 8.937% 16.491% 25.650% 26.404% 13.528%
3-Player games % of action-decisions 12.058% 10.973% 18.353% 25.153% 22.938% 10.526%
4-Player games % of action-decisions 12.582% 12.904% 20.572% 25.820% 20.397% 7.725%
5-Player games % of action-decisions 13.815% 14.729% 22.415% 24.534% 17.722% 6.785%

Can’t score now

This table shows percentages of decisions for a player when it was not possible to score a bucket immediately, but it could be possible if the card on top was removed. In other words, moments where there were cards in the draw deck that are bigger than other cards down the bucket or in the pool of that suit, for none to 5 buckets. For over 88% of the time, the game is not in a situation where this happens. Otherwise, this happens to 1 bucket in 11% of the times when a Score action can be played, for all player counts, with less than 1% for more than one bucket.

Percentage of games/action-decisions in which, at least once, players could not score a bucket without removing the top card first
Players Can’t score immediately in… None 1 bucket 2 buckets 3 buckets 4 buckets 5 buckets
2-Player games % of action-decisions 88.292% 10.959% 0.720% 0.028% 0.001% 0.000%
3-Player games % of action-decisions 88.763% 10.533% 0.663% 0.040% 0.001% 0.000%
4-Player games % of action-decisions 89.375% 10.037% 0.566% 0.021% 0.001% 0.000%
5-Player games % of action-decisions 89.288% 10.026% 0.655% 0.030% 0.001% 0.000%

Can’t score (no cards)

This table shows percentages of decisions for a player when it was impossible to score a number of buckets at that instance and for the rest of the game. This is the particular case when there are no more cards of the given suit in the draw deck.

The data is clear in that close to 100% of the time at least 1 bucket is scorable (“None” column). In very few cases, one bucket was impossible to be scored due to no cards of its color remaining in the draw deck. It was never the case that this happened to more than 1 bucket during the 2-player games played, although the situation does arise (albeit rarely) with higher player counts.

Percentage of decisions in which, at least once, it was impossible for players to score a bucket (no more cards of that suit in the draw deck)
Players Impossible to Score… None 1 bucket 2 buckets 3 buckets 4 buckets 5 buckets
2-Player games % of action-decisions 100.000% 0.000% 0.000% 0.000% 0.000% 0.000%
3-Player games % of action-decisions 99.999% 0.001% 0.000% 0.000% 0.000% 0.000%
4-Player games % of action-decisions 99.993% 0.005% 0.001% 0.001% 0.000% 0.000%
5-Player games % of action-decisions 99.946% 0.044% 0.008% 0.002% 0.001% 0.000%

Can’t score (cards in deck)

This table shows percentages of decisions for a player when it was impossible to score a number of buckets at that instance and for the rest of the game, despite still having cards of the given suit in the draw deck. This takes place when the minimum card of a given suit in both the bucket and the pool deck of this suit is greater than all cards that suit in the draw deck, and the cards in the draw deck of this suit are in strictly decreasing order of value.

Similar to the previous case, the data shows that in the majority of the moments of the game at least one bucket is scorable (“None” column), with 1 bucket becoming impossible to score in less than 1% of the turns in which the current player could select the Score action; this probability increases with the player count.

Percentage of decisions in which, at least once, it was impossible for players to score a bucket (with cards of that suit still in the draw deck)
Players Impossible to score in… None 1 bucket 2 buckets 3 buckets 4 buckets 5 buckets
2-Player games % of action-decisions 100.000% 0.000% 0.000% 0.000% 0.000% 0.000%
3-Player games % of action-decisions 99.997% 0.003% 0.000% 0.000% 0.000% 0.000%
4-Player games % of action-decisions 99.988% 0.012% 0.000% 0.000% 0.000% 0.000%
5-Player games % of action-decisions 99.910% 0.084% 0.005% 0.001% 0.000% 0.000%

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End Game

This section presents a variety of end-game statistics, summarising 2 main game features:

  • Game score
  • Number of cards left in the draw deck at the end

Score Difference to Winner

First, we examine the ending scores for the players, and the difference between the winner and the second player, or bewteen the winner and the last player (which will be the same values for 2-player games). The overall score difference at the end of the game, across all games played, is above 8 for 2-player games, but decreases to 3 points for 5-player games (and close to 7 points if we consider the difference to the last player), with a minimum of 1 (a very close game) and a maximum of 24 (a huge advantage for one player). We do see 2 ties in score appearing in 4-player games and 4 others in 5-player games (minimum difference is 0).

Players Reference Games Min Mean Max Ties
2-Player games To Second 5034 1 8.11 ± 4.28 24 0
3-Player games To Second 4014 1 4.73 ± 2.68 17 0
4-Player games To Second 4023 0 3.80 ± 2.39 15 2
5-Player games To Second 2394 0 3.28 ± 2.12 14 4
2-Player games To Last 5034 1 8.11 ± 4.28 24 0
3-Player games To Last 4014 1 7.02 ± 2.65 18 0
4-Player games To Last 4023 1 7.00 ± 2.34 16 0
5-Player games To Last 2394 1 6.85 ± 2.12 16 0

The figures below show the same data as the distribution of all point differences observed across all games, with the specific point difference observed on the horizontal axis, and the number of times that point difference occurred on the vertical axis. The dashed line indicates the median of the data. We show the difference from the winner to the second player on the left, and to the last player on the right, with a different player count per row.

The majority of the 2-player games lie between 0 and 15 point difference, with 4.35% of the games showing more than 15 points of difference. The distribution reduces and skews more towards 0 as we increase the number of players.

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Number of cards in draw deck

We analyse the number of cards remaining in the draw deck at the end of the game. This is a reflection of game duration, as games with less cards remaining will last longer. One of the end-game conditions is the draw deck being depleted, therefore it is interesting to observe how close games get to this point.

The first table describes statistics across all games. The right half of the table offers a detailed break-down into several limits (with column header format: lower_bound - upper_bound). As such, we can see that 0.02% of 2-player games end due to the draw deck running out of cards, with this statistic increasing to 0.88% for 5-player games. Only 0.52% of games end with more than 75 cards in the draw deck for 3 players, and this never occurs with 4 or more players in the game.

Number of cards in the draw deck at the end of the game
Player Games Min Mean Median Max = 0 0 - 5 6 - 10 11 - 25 26 - 50 51 - 75 > 75
2-Player games 5034 0 55 56 82 0.02% 0.00% 0.04% 0.42% 26.76% 72.61% 0.16%
3-Player games 4014 3 54 55 81 0.00% 0.02% 0.10% 1.27% 32.14% 65.94% 0.52%
4-Player games 4023 0 46 47 73 0.10% 0.15% 0.47% 4.65% 59.43% 35.20% 0.00%
5-Player games 2395 0 36 37 70 0.88% 1.04% 2.34% 15.87% 69.14% 10.73% 0.00%

These statistics are further exemplified in the following distribution plots, showing the number of cards remaining in the deck at the end of the game on the horizontal axis, and the count of times each specific number occurs in a game on the vertical axis.

The median is shown with the dashed blue line, and the data sees many games ending with many cards in the draw deck, while there are less examples of longer games with 2 or 3 players (at the left end of the distribution). The distribution moves significantly to the left for 4 and 5 player games, indicating more cards are used throughout the game.

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