Markov Chains in Football Analytics Explained

A Markov chain is a statistical model that represents football as movement between defined states. Those states might describe pitch zones, action types or possession outcomes. A transition matrix records the estimated probability of moving from each current state to every possible next state.

In football analytics, Markov chains can help describe possession sequences, compare playing styles and estimate the probability that an attack reaches a shot or ends with the ball being lost. Their main simplifying assumption is that the next state depends on the current state rather than the complete history of the move. That makes the method transparent and useful, but it can also omit important information about timing, tactics and earlier actions.

What Is a Markov Chain in Football Analytics?

A discrete-time Markov chain represents a process as a sequence of states. If the ball is currently in state i, the model assigns a probability to every state j that could follow.

The one-step transition probability is:

P(i,j) = probability that the next state is j, given that the current state is i

The defining Markov assumption is that the current state contains all the information needed to model the next transition. In a first-order model, knowing that possession is currently in the attacking third should be enough to estimate the next state. The model does not separately remember whether the ball arrived through a counterattack, a long passing sequence or a defensive mistake unless that information is included in the state definition.

This is why state design is central. A model can only use information that its states preserve.

The Main Inputs of a Football Markov Model

States

States are the situations between which the process moves. A simple possession model might use:

• Defensive-third possession.
• Middle-third possession.
• Attacking-third possession.
• Shot.
• Possession lost.

A more detailed model could divide the pitch into dozens of zones and add action type, pressure, score, time, player identity or tactical phase. Extra detail may improve the football representation, but it also increases the amount of data required.

Transition counts

The analyst counts how often each state is followed by another state. If middle-third possession is observed 1,000 times and the next state is the attacking third on 400 occasions, the raw estimated transition probability is 0.40.

In general:

Estimated P(i,j) = observed transitions from i to j / all observed departures from i

Analysts may apply smoothing or shrinkage when some states contain little data. Otherwise, a transition that was merely unobserved in a small sample can be assigned an unrealistic probability of zero.

The transition matrix

The probabilities are arranged in a matrix. Every row represents the current state, and every column represents the next state. Each row should sum to 1 because the next transition must go somewhere within the model.

The initial state and modelling horizon

The model also needs a starting state or an initial probability distribution. The analyst must then decide whether the question concerns one action, a fixed number of actions or the eventual outcome of the possession.

Absorbing states

An absorbing state is a terminal outcome for the process being modelled. If a possession model ends whenever a shot is taken or the ball is lost, those outcomes can be made absorbing. This does not mean that football stops after a shot. It means that the analytical possession sequence has reached its defined endpoint.

A 2014 Markovian possession study used a compact system involving recovery, possession, ball loss and shots. Later football research has used more granular pitch and action states to study complete possession sequences.

A Simple Possession Transition Matrix

Consider the following illustrative model. The probabilities are invented educational inputs, not observed facts from a real team or competition.

The defensive-state row says that the next recorded state remains defensive with probability 0.20, moves to the middle third with probability 0.55, jumps directly to the attacking third with probability 0.05 or ends in a loss with probability 0.20.

The shot and loss rows contain a probability of 1 for remaining in the same state because they are terminal outcomes within this possession model.

How to Calculate a Sequence Probability

Suppose an analyst wants the model probability of one exact route:

1. Defensive third to middle third.
2. Middle third to attacking third.
3. Attacking third to shot.

Multiply the relevant transition probabilities:

0.55 × 0.40 × 0.35 = 0.077

The model assigns this exact three-transition route a probability of 7.7%.

This is not the overall probability that a possession beginning in the defensive third will eventually produce a shot. Other routes can also reach the shot state. The ball could remain in the middle third for another action, enter the attacking third more than once or move backwards before advancing again.

Matrix multiplication accounts for all eligible routes. The entries of Pn, the transition matrix raised to the power n, give the probability of moving between states over exactly n transitions. Absorbing-chain calculations can also estimate eventual shot or loss probabilities and the expected number of state visits before the possession ends.

Why Sensitivity Matters

The 7.7% result depends completely on the model inputs. If the attacking-third-to-shot probability falls from 0.35 to 0.25, the exact route probability becomes:

0.55 × 0.40 × 0.25 = 0.055

That is 5.5%. If the final transition rises to 0.45, the route probability becomes 9.9%.

A modest change to one row of the matrix can therefore change downstream sequence probabilities materially. Analysts should examine whether conclusions survive plausible changes to state definitions, samples, smoothing choices and contextual controls.

How Markov Chains Are Used in Football

Possession outcomes

A model can estimate how frequently possession starting in a particular state is expected to reach a shot, goal, stoppage or turnover. It can also estimate likely possession length and the states most frequently visited before an outcome.

Playing-style comparisons

Transition matrices can reveal differences in how teams progress the ball. One team may move rapidly from defensive to attacking states, while another may have high probabilities of remaining in the same zone or returning backwards.

These differences describe patterns, not automatic quality. Direct progression can create danger but also increase turnover risk. Longer circulation may represent control, difficulty breaking a block or both.

Possession and action value

Markov transitions can support possession-value models that estimate how actions change the future probability of scoring or conceding.

Expected Threat (xT) is a related location-based approach. A typical xT model studies shooting, movement and turnover probabilities from pitch zones, then estimates the future scoring potential associated with possession in each zone.

A Markov chain is the transition mechanism. Possession value is the analytical quantity an organisation may choose to estimate from those transitions. The two concepts overlap, but they are not synonyms.

Tactical and match-state analysis

Separate matrices can be estimated for different tactical conditions. An analyst might compare transitions while level, leading and trailing, or before and after a substitution.

This can expose game-state effects, but the samples can become small quickly. Apparent tactical differences may instead reflect opponent strength, venue, player availability or random variation.

Markov Chains vs Monte Carlo Simulation

Markov chains and Monte Carlo simulation are related but different.

• A Markov chain specifies states and the probabilities of moving between them.
• Monte Carlo simulation repeatedly samples possible paths or outcomes from supplied probability rules.

An analyst can use Monte Carlo simulation to generate thousands of possession paths from a Markov transition matrix. The chain defines the process; the simulation approximates the distribution of what that process produces.

Neither method creates reliable inputs by itself. A precise simulation built from weak transition estimates remains a weak model.

Limitations and Common Errors

Assuming the current state contains enough information

A first-order chain may treat two possessions in the same pitch zone as equivalent even when one follows a fast counterattack and the other involves slow circulation against a settled defence. If the recent path matters, the state may need to include action history, speed, pressure or tactical phase.

Ignoring time between actions

A standard discrete chain focuses on transition order rather than how long each state lasts. A 2026 semi-Markov football study modelled both state transitions and action duration, reporting better performance than a memoryless timing baseline on its evaluated datasets. This does not make every standard Markov model invalid, but it shows that timing can matter when the analytical target has a fixed time horizon.

Treating the matrix as permanently stable

A time-homogeneous model assumes that transition probabilities remain constant throughout the period being analysed. Football behaviour changes with score, fatigue, substitutions, red cards, opponent pressure and tactical instructions. Analysts may need separate matrices, time-varying probabilities or richer states.

Using states that are too broad or too narrow

Broad states are stable and easy to interpret but can hide important differences. Fine states provide greater detail but create sparsity. The correct resolution depends on the data volume and decision being supported.

The same principle underlies sample-size reliability: adding categories does not create more information and may leave each estimate supported by too few observations.

Allowing data definitions to drive hidden errors

Results depend on how possessions, carries, duels, interceptions, rebounds and stoppages are defined. Two providers can create different state sequences from the same match. Analysts should document boundaries and test whether conclusions survive reasonable alternative definitions.

Confusing description with causation

A transition matrix can show that a team frequently moves from wide attacking states to shots. It does not prove that using wide areas causes better outcomes. Team strength, opponents, game state and player quality may explain both the transition pattern and the results.

Evaluating only the training data

A matrix estimated from historical transitions will reproduce parts of those data by construction. The important question is whether it remains useful on later matches, other opponents and the exact outcome the analyst wants to predict.

GoalIQAI Analytical Interpretation

Markov chains are most useful as transparent maps of a football process. They force the analyst to define what a state means, identify which transitions matter and show how local probabilities combine into longer sequences.

They should not be treated as automatic forecasting machines. A strong application should:

1. Define possession and state boundaries before examining the desired result.
2. Separate observed transition counts from estimated or smoothed probabilities.
3. Use chronological training and testing periods.
4. Compare the model with simpler benchmarks.
5. Test different state resolutions and contextual variables.
6. Report sensitivity rather than relying on one matrix.
7. Check whether the output improves the real football decision being made.

The model is valuable when its simplification makes the process clearer without removing the information necessary for the question. If earlier actions, elapsed time or tactical context materially change the next transition, a higher-order, time-varying or semi-Markov model may be more appropriate.

Key Takeaways

• A Markov chain represents football as movement between defined states.
• A transition matrix records the estimated probability of every next state given the current state.
• Sequence probabilities are calculated by multiplying transitions along a route, while matrix powers account for all routes over multiple steps.
• Shots, goals, turnovers or stoppages can be treated as absorbing outcomes when they end the process being studied.
• Markov chains can support possession analysis, playing-style comparison and possession-value modelling.
• The Markov assumption can be too restrictive when action history, timing or tactical context matters.
• State design, data definitions, sample size and out-of-sample validation are as important as the matrix calculation.
• Markov chains define transition processes; Monte Carlo simulation can sample paths from those processes but is not the same method.

Related Guides

• Building a simple football model
• Professional model validation
• Signal from noise in football data

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