Snakes And Ladders

The Julia code snakes_and_ladders.jl analyzes the single-player game of snakes and ladders as a discrete-time Markov chain with a single absorbing state, and calculates

1. the expected number of turns till the game finishes,
2. the probability that a player ever moves to a square i on the board, given that she started the game in square j (i and j can take on any integer value from 0 to 100),
3. the least number of moves (or, turns) to complete the game,
4. the most likely path (or, sequence of board moves) to complete the game, and
5. the probability that a player completes the game in exactly k moves, where k can equal 1, or 2, or 3, and so on.

We assume that the game begins from a initial position (and not the first square on the board) that we call square 0. A player rolls a fair die before each move, and game is said to end when the player lands on the last square (or square 100).

In keeping with tradition, a player on one of the squares 94 to 99 must roll a exact number to land on the last square, or remain in the same position if she overshoots 100.

Choice Of Board

1. Data for the Milton Bradley version, is saved in numberofsquares.txt, snakes.txt, and ladders.txt under Data.
2. To play this highly punitive version, change the input files to snakes_punitive.txt and ladders_punitive.txt.

Example

Running snakes_and_ladders.jl for the punitive board will generate the following output.

Mean number of moves to complete the game = 54.4735849242718.

The game can be completed in a minimum of 6.0 moves.
One such set of moves is [0, 6, 26, 32, 38, 77, 100].


The most likely path from square 0 to square 100 is [0, 6, 26, 32, 38, 77, 100].
This path requires 6 moves and is achieved with probility 2.1433470507544607e-5.

The random variable representing the number of moves to complete the game (starting from square 0) follows a discrete phase-type distribution, whose probability mass function and density are saved under Plots.

Code Credits

@kalyaninagaraj

Resources

Lecture notes, videos, and book chapters:

1. Introduction to Probability Models, 11th Ed., Sheldon M. Ross, Section 4.6
2. Introduction to Probability, 2nd Ed., Charles M. Grinstead and J. Laurie Snell, Chapter 11 [pdf]
3. Lecture on Expected Time to Absorption by John Tsitsiklis, RES.6-012 Introduction to Probability. Spring 2018. Massachusetts Institute of Technology: MIT OpenCourseWare [video]
4. Lecture notes for Professor Bo Friis Nielsen's course on Stochastic Processes, DTU Compute, Denmark, Course 02407, Fall 2020 [pdf] [course homepage]