## 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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#### 5 comments:

Marc Cohen said...

Reminds me a little of Open Face Chinese Poker http://en.wikipedia.org/wiki/Open-face_Chinese_poker

Christopher Chang said...

A fellow math major and I used to joke a bit about "The Art of War", a hypothetical book on War strategy... i.e., knowing which card to put on top when one wins a "battle". (The Wikipedia article on War correctly notes this nondeterministic element of the game.)

steve hsu said...

We play another hand -- this adds excitement and can end the game quickly :-)

ben_g said...

Egyptian Ratscrew is my favorite. It has the simplicity of War but with a competition on speed/reflexes (rather than strategy) http://en.wikipedia.org/wiki/Egyptian_Ratscrew

Javier said...

There are dominated choices for the second card. For example, if you have 1 vs 9, you shouldn't play 3-8. Also as the deck gets depleted, it's possible to guarantee yourself wins on the remaining rounds by playing certain combinations.

This reminds me of the game "12 days", where the deck consists of 12 twelves, 11 elevens, ..., 1 one. On each round (day), the players bid and the lowest card wins the day. You get points for winning each day, increasing in value from 1 to 12 points. But at the end of the game whoever kept the most twelves gets 12 pts, and so on, giving value to the "bad" cards. You can only keep a total of 12 cards in your hand at any time so very interesting decisions arise.

Another game that's a huge hit with kids and adults alike is King of Tokyo, a dice game with similarities to Yahtzee but featuring fighting monsters and power cards.

Links to Wil Wheaton's Tabletop episodes:
12 days http://www.youtube.com/watch?v=Hp3IwPbZYSE
King of Tokyo http://www.youtube.com/watch?v=BGaCjM2hal4