WAR: Is the Card Game Really a Game?

123456Hello! My name is Quentin Heise. I am a rising senior, a mathematics major, and a data science minor. This summer, I am collaborating with Professor Johnson (Physics) to continue the work Kelvin Cupay (class of 22) started two years ago. We are analyzing the card game WAR! WAR is a game that many young kids play, which involves two players with half a deck of cards each. The objective is to gain all of your opponent’s cards. On a skirmish, both players reveal the top card of their deck. The highest card wins (after 10, the ranks in ascending order are jack, queen king, and ace)! The winner of the skirmish takes both cards and puts them on the bottom of their deck. Finally, if the skirmish involves cards of equal rank, there is a special event called a WAR that begins. Players remove three cards from the top of their decks, and another skirmish occurs. The WAR does not conclude until one player wins a skirmish. Each subsequent skirmish after the first changes the war to a double war, a triple war, etc.! Once one player obtains all of their opponent’s cards, the match concludes, and the player is declared the match victor.

123456I mentioned above that WAR is a “game.” However, one can argue that WAR is not a game at all. Consider that the order in which the winner of a skirmish or WAR returns cards to the bottom of their deck is fixed (ex: ascending order). There is no randomness, and players do not have control over the order of their decks. Thus, the match victor of the “game” is determined as soon as one shuffles and deals the initial deck!

123456Before I continue, there are three vital definitions.

  1. Win percentage is the percent of the time that a half-deck (the players’ initial decks) with specified characteristics will win.
  2. Deck Weight is where each value (two through ace) is assigned an integer from [-6,6]. Ideally, the higher the deck weight, the higher the win percentage.
  3. The initial advantage is the difference between the number of times both players win cards from their opponent over the first 52 cards.
    • For example, if player one wins ten skirmishes and player two wins 16, the players would have initial advantages of -6 and six, respectively. Unfortunately, there is ambiguity about how to take WARS into account. For example, if a player wins a triple war, they win 13 cards from their opponent (five from the first skirmish, four from both the second and third). Consider if the other player wins (a maximum of) 13 skirmishes to complete the first 52 cards. If we count the triple war as gaining the cards only once, the first player will have an initial advantage of -12. However, that player gains twice as many cards as their opponent (26 vs. 13). Furthermore, counting wars only once will reduce the initial advantage of the player that wins them. (One way I am attempting to improve the calculation of initial advantage is to exclude the matches with wars in the first round).

123456Last summer, Kelvin used Python to replicate two things from a 2006 paper by Jacob Haqqmisra [1]. These were how the win percentage correlated with deck weight and initial advantage. The correlations are nearly perfect, at 0.994 and 0.999, respectively! One might wonder why I am continuing the research this summer. I do so because many questions were previously or are currently unanswered!

123456The deck weight scale has several significant flaws. First, it does not account for when a player has cards at opposite ends of the scale. For example, if a deck has four twos and four aces, these eight cards would yield a deck weight of zero. However, the four aces are significantly better than the four twos are worse. Over 300,000 matches, the win percentage of decks with four twos and four aces is 83%! Yet, the deck weight of zero would suggest a win percentage of only 50%. I am currently experimenting with different deck weight assignments, which are all nonnegative.

123456Second, the deck weight is only usable immediately after dealing cards and before the first skirmish. This restricted use occurs by definition; one calculates deck weight over the first 26 cards only. If a player wins cards from their opponent, the earned cards are more likely to have negative values than not. Indeed, if one measures the deck weight after every skirmish, the deck weights of both players trend toward zero (the deck weight of the 52-card deck is zero).

123456Finally, I can assign deck weights to cards exponentially instead of linearly. For example, have the jack be worth 10, the queen be worth 12, the king be worth 15, and the ace be worth 20. These values would be consistent with work I did earlier this summer, where I observed that a half deck with all four aces has a win percentage of 83%. However, a half deck with all four kings only has a win percentage of 56%.

123456Another question I answered earlier this summer is how introducing randomness changes things. Previously, after winning a skirmish or war, the winner would return the cards to the bottom of their deck in a set order. Instead, I experimented with returning the cards in a random order. Interestingly, the win percentage stays the same. Things that change (increase) include the mean and average median number of games in a match and match length. I saw some that lasted over 2,000 skirmishes or WARS combined! That is a long time!

123456The final significant thing I observed is that results converge at around 60,000 matches. Comparing the results from 60,000 matches and several million matches, the differences in results are insignificant. For comprehensiveness, the results include the win percentage, the mean and standard deviation of the number of games in a match, average median and standard deviation of the number of games in a match, shortest and longest match (by the number of skirmishes and WARS combined), average starting deck weight, and average initial advantage and its standard deviation.

Raw Data

123456The left-most column is the configuration of cards in an initial 26-card deck. Either the deck is (randomly) configured to be a certain deck weight, or specific cards are in the deck. (Note that for the random configuration, there was no requirement for the half-decks). For example, “Four [9-A]s” means that the initial 26-card deck includes four nines, four tens, four jacks, four queens, four kings, and four aces. The other two cards are picked randomly from the rest of the deck. Every deck configuration was simulated in WAR games 300,000 times, except for the random deck configuration, which was simulated five million times. (Note that Professor Johnson had simulated over 100 million games last summer, but it takes a LONG time to process all that data and is borderline impossible to be thorough with limited ram and time.)

1234556Finally, the left-most column is sorted by win percentage, with the highest being at the top.

aaaaaaaaaaaaaaaaaaaaaaa
Win %
Mean # of Games in a Matcha
STD of left
Median # of Games in a matcha
STD of left
a
Shortest Match
a
Longest Match
Avg Starting Deck Weighta
Average Initial Advantage
a
STD of left
Four [9-A]s99.928.1513.08282.9718110978.0024.062.40
Four [10, J, Q, K, A]s98.837.9643.75317.4110161465.0020.443.44
Four [J, Q, K, A]s97.152.0365.65368.906144052.0016.654.50
Four [2, 3, Q, K, A]s97.065.7166.42508.9013146213.004.454.10
Four [Q, K, A]s94.970.9985.284516.315154038.9712.685.49
Four [2, 3, K, A]s92.2102.75100.356626.691015680.030.505.14
Four [2, 3, 4, K, A]s92.0110.15102.427226.69181689-13.00-3.684.13
Four [K, A]s91.099.24105.136131.138162826.038.596.28
Deck Weight 50+87.7105.56119.315638.5510122652.3216.474.53
Four Aces83.2144.69126.4710068.205223612.954.346.94
Four [2, A]s83.0149.77125.7710668.20102136-0.010.396.50
Deck Weight 4582.3127.59127.987966.7210117645.0014.234.97
Deck Weight 4078.4138.32130.799075.6110104140.0012.785.19
Deck Weight 3577.0151.44138.5310184.518121735.0011.085.53
Deck Weight 3073.5161.07139.3811694.896117330.009.455.66
Deck Weight 2569.5169.80139.0912799.3310116325.007.945.75
Three Aces67.9182.36140.0314199.33617896.492.197.16
Deck Weight 2065.6176.39138.7713699.335124820.006.315.91
Deck Weight 1561.9186.70145.48146105.266147615.004.896.10
Deck Weight 1058.6187.42140.68148103.786148610.003.226.08
Four Kings55.8186.06142.65146106.756200810.823.557.07
Deck Weight 554.4192.67143.70152103.78818555.001.636.19
Deck Weight 050.8192.19139.83152103.781013890.000.036.21
Two Aces50.5198.53143.12160106.75518320.010.007.24
Random50.5186.36141.61146103.78422770.000.007.48
Four Twos46.3187.02141.61146103.7861873-13.02-3.947.00
Four [2, 3]s41.0184.20141.19144103.7861761-26.01-7.946.39
Four [2, 3, 4]s34.9175.57140.73134103.7861858-39.01-11.975.62
Four [2, 3, 4, 5]s27.4158.33139.3311597.8581844-52.02-16.004.68
Four [2-6]s18.5127.60131.137566.7291718-65.01-20.003.58
Four [2-7]s7.271.8297.783411.86181681-78.00-24.012.34
Four [2-8]s + Two Eights0.025.451.38260.001826-84.00-25.842.01

Correlations

The correlations (R) are categorized by strength, which is determined using J. D. Evan’s scale in his 1996 book, Straightforward Statistics for the Behavioral Sciences [2]:

  • Very Weak:120.2 > |R|
  • Weak:1234560.2 |R| < 0.4
  • Moderate:120.4 |R| < 0.6
  • Strong:123450.6 |R| < 0.8
  • Very Strong:10.8 < |R|
a
Win %
Mean # of Games in a Matcha
STD of left
Median # of Games in a Matcha
STD of left
Shortest MatchLongest MatchAvg Starting Deck WeightAvg HM Advantage
Mean # of Games in a Match-0.28
(Weak)
STD of Above-0.20
(Weak)
0.93 (Very_Strong)
Median # of Games in a Match-0.29
(Weak)
0.98 (Very_Strong)0.84 (Very_Strong)
STD of Above-0.34
(Weak)
0.98 (Very_Strong)0.88 (Very_Strong)0.97 (Very_Strong)
Shortest Match-0.06
(Very_Weak)
-0.64
(Strong)
-0.65
(Strong)
-0.64
(Strong)
-0.66
(Strong)
Longest Match0.04
(Very_Weak)
0.45 (Moderate)0.51
(Moderate)
0.42 (Moderate)0.36
(Weak)
-0.48 (Moderate)
Avg_Starting Deck_Weight0.87 (Very_Strong)-0.14
(Very_Weak)
-0.11
(Very_Weak)
-0.13
(Very_Weak)
-0.13
(Very_Weak)
-0.20
(Weak)
-0.09
(Very_Weak)
Avg HM Advantage0.88 (Very_Strong)-0.15
(Very_Weak)
-0.11
(Very_Weak)
-0.14
(Very_Weak)
-0.13
(Very_Weak)
-0.20
(Weak)
-0.09
(Very_Weak)
0.9999 (Very_Strong)
STD of Above0.10
(Very_Weak)
0.82 (Very_Strong)0.77
(Strong)
0.82 (Very_Strong)0.76
(Strong)
-0.81 (Very_Strong)0.58 (Moderate)0.16
(Very_Weak)
0.16
(Very_Weak)

123456But wait, there is more! A fun game (it IS a game!) that I play with Professor Johnson and other members of my lab (and other labs occasionally) is Hacky Sack. The goal is to keep a beanbag off the ground without using one’s arms or hands. A successful “Hack” is when each person in the circle touches the beanbag at least once. As you can imagine, this is exponentially harder as more people join!

123456I am proud to say that I am the only one that has successfully “elevened” Professor Johnson this summer! “Elevening” involves hitting the beanbag between a person’s legs while their feet are flat on the ground. It is considered shameful for that person because it typically occurs when they are not paying attention! (Or they do not have proficient defenses, which is an unwritten expectation).

References

[1]1234J. Haqqmisra, “Predictability in the Game of War,” The Science Creative Quarterly, October 5, 2006.

123456 https://www.scq.ubc.ca/predictability-in-the-game-of-war/

[2]1234J. D. Evans, “Straightforward Statistics for the Behavioral Sciences,“ Brooks/Cole Pub. Co., 1996.

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