Software developer; math enthusiast; he/him; MTG casual; https://github.com/birchmd https://www.typedriven.ca/portfolios/
![A graph of the histogram for frequencies of water sort puzzles with specific numbers of optimal moves to solve. The puzzle setup in this case is containers that are 5 segments high (N=5), with 5 distinct colours (C=5) and 2 empty containers (K=2) at the start/end. The most common number of moves is 19 and the bulk of the distribution is in the range [18, 20]. The distribution is bell-like, but skewed towards smaller values. It is fitted with a skew normal distribution curve that has parameters ξ=20.4, ω=2.38, and α=-1.76.](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:tmjbce3bsvzvp5bi7cmyticr/bafkreihbyxsbungsgnrey3yit7gczhntcfzpa7dsf5mlmaapnkxflwmcm4)
![A graph of the histogram for frequencies of water sort puzzles with specific numbers of optimal moves to solve. The puzzle setup in this case is containers that are 6 segments high (N=6), with 4 distinct colours (C=4) and 2 empty containers (K=2) at the start/end. The most common number of moves is 18 and the bulk of the distribution is in the range [17, 19]. The distribution is bell-like, but skewed towards smaller values. It is fitted with a skew normal distribution curve that has parameters ξ=19.3, ω=2.56, and α=-1.57.](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:tmjbce3bsvzvp5bi7cmyticr/bafkreidjzyhpl22ezwb752ecq6b5wjvpnmnvlcdlajaz4vjbbkxzf4hrk4)
![A graph of the histogram for frequencies of water sort puzzles with specific numbers of optimal moves to solve. The puzzle setup in this case is containers that are 5 segments high (N=5), with 4 distinct colours (C=4) and 1 empty container (K=1) at the start/end. The most common number of moves is 15 and the bulk of the distribution is in the range [14, 16]. The distribution is bell-like, but skewed towards smaller values. It is fitted with a skew normal distribution curve that has parameters ξ=16.6, ω=2.50, and α=-1.49.](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:tmjbce3bsvzvp5bi7cmyticr/bafkreibz3hmrro5k7bq7rfbawfnsaxcqpiihqnb5pml7cqvz7ersdqunfu)
![A graph of the histogram for frequencies of water sort puzzles with specific numbers of optimal moves to solve. The puzzle setup in this case is containers that are 5 segments high (N=5), with 4 distinct colours (C=4) and 2 empty containers (K=2) at the start/end. The most common number of moves is 15 and the bulk of the distribution is in the range [14, 16]. The distribution is bell-like, but skewed towards smaller values. It is fitted with a skew normal distribution curve that has parameters ξ=16.1, ω=2.38, and α=-1.77.](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:tmjbce3bsvzvp5bi7cmyticr/bafkreifjuqrshmcachk7ltsddhpme2ypq6teyrsca6xjljwwsr5gxs32ue)
![This figure shows the result of simulating 20 million games of War. To describe what is being plotted we first need to know the mathematical model I used for the analysis.
Let's assign each card in the deck a value 2 through 14. Then the outcome of a round of the game is determined solely by comparing these values. And we can measure how well a player is doing by considering the total value of cards in their deck. A value of 416 means that player has all the cards. We also track the the number of cards each player has. A player with all the cards as 52.
Without loss of generality, we can focus on player 1 since if player 1 has a deck value V containing N cards then player 2 has a deck value of 416 - V and 52 - N cards, so dynamics of player 2 are easily deduced from the dynamics of player 1.
To make our lives easier, we rescale the variables: x = V/416 and y = N/52. Note x and y are both in the range [0, 1].
We'll derive equations for how x and y change in the next figure, but for now we have enough context to say what is plotted in this figure. It is the (x, y) state space for the games. Each colored line is the average over many games with the same (V, N) initial conditions. All lines start at y=0.5 since each player begins with half the deck. They can start at different points in the x direction because the cards are randomly assigned to the players (so player 1 may get more or less high-value cards).
All the trajectories move diagonally until they approach the line y = x, then they all follow this line to the center point (0.5,0.5). Thus we see the games evolving to this steady state where they will stay forever, as described in the main post.
Now let's see if we can explain this numerical result with our mathematical model.](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:tmjbce3bsvzvp5bi7cmyticr/bafkreiawvugakbtqexxvw3fribja466mxhs6xscqwz7onsgs2rejhziuva)
Software developer; math enthusiast; he/him; MTG casual; https://github.com/birchmd https://www.typedriven.ca/portfolios/