A model built on “where am I now?”
In many board games, what happens next depends only on where your token stands and what you roll. It does not matter how you got there. A token on square 12 faces the same prospects whether it arrived by a long cautious route or a lucky jump.
A Markov chain is a model with exactly this feature. It has a set of states (here, squares or situations) and a list of probabilities for moving from each state to each other state in one step. The defining assumption is that the next step depends only on the current state, not on the history. Real games sometimes break this assumption, but when they do you can usually repair the model by making the state richer, for example “square 12, having already rolled doubles this turn.”
A small worked example
Imagine a four-square track numbered 0 to 3, with 3 as the finish. On each turn you flip a fair coin: heads moves you one square, tails moves you two. Overshooting the finish still counts as finishing. This is an invented illustration, not any real game.
The transition probabilities from each square are:
| From | To 1 | To 2 | To 3 |
|---|---|---|---|
| Square 0 | 0.5 | 0.5 | 0 |
| Square 1 | 0 | 0.5 | 0.5 |
| Square 2 | 0 | 0 | 1 |
| Square 3 | finished, stays at 3 |
Every row sums to 1, as it must: from any square the token has to go somewhere. A state like square 3, which you cannot leave, is called absorbing.
Getting average game length from the table
You can compute the expected number of turns to finish by working backwards. Let E(s) be the expected turns remaining from square s.
- E(3) = 0, since the game is over.
- E(2) = 1, because any move from square 2 reaches or passes the finish.
- E(1) = 1 + 0.5 × E(2) + 0.5 × E(3) = 1 + 0.5 = 1.5.
- E(0) = 1 + 0.5 × E(1) + 0.5 × E(2) = 1 + 0.75 + 0.5 = 2.25.
So the average game on this track lasts 2.25 turns. A quick check: the shortest game is 2 turns (two tails), and the longest is 3 turns (heads, heads, then anything from square 2). The average has to sit between those, and 2.25 does. This is the same idea as finding an expected duration from per-turn probabilities, extended to situations where the chance of finishing changes as the token advances.
Adding snakes, ladders and special squares
Special squares fit naturally. If a ladder on square 4 lifts the token to square 9, then “landing on 4” is not a state the token ever rests on. In the transition table, the probability of moving to square 4 is simply added to the probability of moving to square 9. A snake works the same way in the opposite direction. A “lose a turn” square can be handled by adding an extra state such as “waiting on square 7” that leads back to square 7 with probability 1.
This is one reason the model is popular with game designers. You can change a rule, update the table, recompute the average length and see the effect without playing a single game.
Dice rolls as transition probabilities
With real dice, the probability of each move comes from the dice themselves. Two dice make the move distribution lumpy: the sums from two dice are not equally likely, and the pattern behind two-dice sums means a move of seven is the most common. A single die gives six equal moves. Whichever you use, the row for each square lists the chance of each destination, and the roll odds of reaching a particular square from where you stand are exactly the landing odds that other articles discuss in detail.
For a board with a fixed number of squares, you can set up the table once and get answers to several questions:
- The average number of turns to finish from the start.
- The chance of finishing within a given number of turns.
- The chance of landing on a particular penalty square at some point during the game.
Looping boards and long-run frequencies
Some boards are loops with no finish line. There the interesting question changes from “how long until it ends?” to “how often is each square visited over a long game?” For many loops, the visit frequencies settle into a stable pattern called the stationary distribution, regardless of where the token started. This is the reasoning behind questions such as which squares players land on most in a famous property trading game. The idea is more advanced than the finite track above, and the actual numbers depend on the exact rules, so it is best to compute them from the specific board rather than guess.
Checking the model by simulation
The tidy calculations above are exact only for the model as written. A Monte Carlo simulation provides an independent check: play the track thousands of times and average the number of turns. For the four-square track the simulated average should settle near 2.25. If a simulation of a more elaborate board disagrees with the table, one of them has a rule wrong, and tracking down which is a good way to catch mistakes.
Limits of the Markov view
- Hidden memory. If a rule depends on earlier events, such as a card drawn three turns ago, the state must include that information or the model will be wrong.
- Player decisions. The basic chain has no choices. When players pick between moves, you need to specify how they choose before the model gives answers.
- Large boards. The table grows with the number of states. Software handles this easily, but hand calculation does not scale.
- Averages hide spread. Like any average, the 2.25 above says nothing about how much individual games vary.
Despite these limits, a Markov chain is a clear way to describe progress in a game in a few lines of probabilities. If you can list the states and the chances of moving between them, you can often answer questions about length, risk and frequency with simple arithmetic.