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

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Victory in the Card Game ‘War’ and Initial Hand Strength
Roger W. Johnson South Dakota School of Mines & Technology

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)

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.

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

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

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

Equivalence of Measures

Simulation involved 100,000 games with one down card

Fitted Linear Models Winning Proportion for Player A
4.77 ~ 124/26 <= rank <= 292/26 ~ 11.23

Thank you for your attention!

Generalization of Equivalence

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