Statistical Analysis of War

Introduction

For a while I was playing the card game War with my son a lot. I mean, a lot. It's a pretty low-brainpower game, so I had time to think. I thought about War, the statistics of the game, and possible strategies to win. Following in the footsteps of the esteemed Kellog and Smith, I decided to run some stats and see what I could discover.

Rules

I used what I consider fairly standard US rules for the game:

Two players only.
Three cards down in a war. Some rules call for one card down, others call for n cards down, where n is the value of the warring cards.
Jokers trump all cards. Some rules call for the appearance of a joker to cause an immediate war.
Taken cards are put into a pile. That pile is recycled into your hand when your hand runs out of cards.

Questions

How are the length of the game and the number of wars affected by the size of the deck?
Are there any simple strategies that can affect the outcome of the game?
Is there a correlation between minimum take profit and winning the game? Take profit is the difference between the winning card and the losing card in a round. My theory was that taking cards using just-slightly-better cards would help win the game because it saves your better cards for better uses.

Execution

I wrote this Python script to collect statistics. The script runs 1000 reps of the game in a few different configurations and gathers the following information:

Average number of rounds per game.
Average percentage of rounds that resulted in a war.
Number of games won by each player.
Number of games that resulted in a stalemate, i.e. an infinitely-looping series of rounds.
Average take profit advantage of the winning player. This is the difference between the winning player's take profit and the losing player's.

The configurations run by the script are as follows:

Deck sizes ranging from 6 to 18 cards per suit, with the number of jokers ranging from 0 to 4.
All combinations of the following four strategies using a standard-sized deck (14 cards per suit plus 2 jokers):

Shuffle Take: randomly arrange the cards won in each round before returning them to your pile. This is pretty close to the default behavior in a real game.
Sort Take: order the cards won in each round, highest-to-lowest, before returning them to your pile. My son favors this strategy for some reason.
Shuffle Hand: randomly arrange the cards in your pile when recycling it and before the start of the game. This is the strategy I ususally use against my son. This strategy is also employed at the start of the game for consistency.
Sort Hand: order the careds in your pile, highest-to-lowest, when recycling it and before the start of the game.

Results

The script produced this output, which I imported into Excel. As you might have expected, the length of the game is non-linearly proportional to the size of the deck, and the number of wars in a game is inversely proportional:

As for the strategies, the table below summarizes the results.

Balance of Wins, Number of Rounds,
and Percent of Rounds in War

Player 1 Strategy

Player 2 Strategy

Sort Hand

Shuffle Hand

Sort Take

Shuffle Take

281

3.4%

370

5.2%

349

5.1%

369

5.2%

Sort Take

243*

3.4%

360

5.1%

240

5.6%

Shuffle Hand

287

3.9%

372

5.1%

Sort Hand

6.0%

Notes:

* This combination is the only one that produces stalemates.

A few things can be gleaned from this table:

Any form of sorting is a very good strategy. "Sort Hand" won easily over all other strategies, and "Sort Take" did much better than average against everything but "Sort Hand". So, it turns out my son was right.
Any form of sorting produces shorter games, "Sort Hand" more so than "Sort Take".
"Sort Hand" produces far fewer wars than other strategies, unless it's up against itself, in which case you get a very short, very combative game.

Finally, my theory on the take profit did not bear out. In all cases, the winner's take profit was larger than the loser's, showing that the winner did not consistently use slightly-better cards when taking. I think this is because of two things:

Once you start to win a game, the need for using slightly-better cards diminishes because you have a surplus of good cards.
The surplus of good cards produces very high take profits, wiping out any small profits from early in the game.

Perhaps modifying the methodology to calculate take profits based only on the first run through the deck would produce better results.

Chez Zeus:Statistical Anaylysis of War