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Wednesday, June 04, 2014

Strategic War (with cards)

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 face 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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