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**Victory in the Card Game ‘War’ and Initial Hand Strength**

Roger W. Johnson South Dakota School of Mines & Technology

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**Rules of ‘War’ Deck, face down, divided evenly among the two players**

In unison, each player reveals top card of their stack – this is a "battle" – player with higher card takes both cards (placed at bottom of their stack)

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Rules of ‘War’ If equal battle cards, then “War". A face-down card followed by new battle card. If battle cards equal again then repeat until one player's battle card is higher. Game ends when one player, the winner, has all of the cards.

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**Rules of ‘War’ Pathological Cases: Running out of cards during a War**

Draws Infinite length games Simulation: Running out of cards during a War considered a loss Cards won placed in random order at bottom of deck

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**An Initial Measure Average Initial Rank Notes:**

Card ranks: 2,3,. . ., 10, J, Q, K, A assigned ranks 2,3,. . ., 10, 11, 12, 13, 14 One measure: Average Initial Rank Notes: Average rank over entire deck = 8 Player’s initial average rank between about 4.8 & 11.2 4.77 ~ 124/26 <= rank <= 292/26 ~ 11.23

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A Second Measure where is the rank of the ith card in A’s deck, is the rank of the jth card in B’s deck and

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**Equivalence of Measures**

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**Simulation involved 100,000 games with one down card**

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**Fitted Linear Models Winning Proportion for Player A**

4.77 ~ 124/26 <= rank <= 292/26 ~ 11.23

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**Thank you for your attention!**

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**Generalization of Equivalence**

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