Strategic War (with cards)

Written by: Stephen Hsu

Primary Source: Information Processing

War is a simple card game played by children. The most common version does not require decisions, so it’s totally deterministic (outcome is determined) once the card order in each deck is fixed. Nevertheless it can be entertaining to watch/play: there are enough fluctuations to engage observers, mainly due to the treatment of ties. The question of how to determine the winner from the two deck orderings (without actually playing the entire game, which can take a long time) was one of the first aspects of computability / predictive modeling / chaotic behavior I thought about as a kid. This direction leads to things like classification of cellular automata and the halting problem.

My children came home with a version designed to teach multiplication — each “hand” is two cards, rather than the usual single card, and the winner of the “battle” is the one with the higher product value of the two cards (face cards are removed).  I thought this was still too boring: no strategy (my kids understood this right away, along with the meaning of deterministic; this puts them ahead of some philosophers), so I came up with a variant that has been quite fun to play.

Split the deck into red and black halves, removing face cards. Each hand (battle) is played with two cards, but they are chosen by each player. One card is placed faced down simultaneously by each player, and the second cards played are chosen after the first cards have been revealed (flipped over). Winner of most hands is the victor.

This game (“strategic war”) is simple to learn, but complex enough that it involves bluffing, calculation, and card counting (keeping track of which cards have been played). A speed version, with, say, 10 seconds allowed per card choice, goes very fast.

Has anyone seen heard of this game before? It’s a bit like repeated two card poker (heads up), drawing from a fixed deck. Note the overall strength of hands for each player (combined multiplicative value of all cards) is fixed and equal. Playing strong hands early means weaker hands later in the game. The goal is to win each hand by as small a margin as possible.

Are there strategies which dominate random play (= select first card at random, second card from range not exceeding highest card required to guarantee a win)?

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Stephen Hsu
Stephen Hsu is vice president for Research and Graduate Studies at Michigan State University. He also serves as scientific adviser to BGI (formerly Beijing Genomics Institute) and as a member of its Cognitive Genomics Lab. Hsu’s primary work has been in applications of quantum field theory, particularly to problems in quantum chromodynamics, dark energy, black holes, entropy bounds, and particle physics beyond the standard model. He has also made contributions to genomics and bioinformatics, the theory of modern finance, and in encryption and information security. Founder of two Silicon Valley companies—SafeWeb, a pioneer in SSL VPN (Secure Sockets Layer Virtual Private Networks) appliances, which was acquired by Symantec in 2003, and Robot Genius Inc., which developed anti-malware technologies—Hsu has given invited research seminars and colloquia at leading research universities and laboratories around the world.
Stephen Hsu

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