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

1. Victory in the Card Game ‘War’ and Initial Hand Strength Roger W. Johnson roger.johnson@sdsmt.edu South Dakota School of Mines & Technology

2. 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)

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

4. 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

5. An Initial Measure • 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

6. 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

7. Equivalence of Measures

9. Fitted Linear Models Winning Proportion for Player A

10. Thank you for your attention! roger.johnson@sdsmt.edu

12. Generalization of Equivalence