A context-aware framework to evaluate passing decisions in soccer

Robert Bajons | Joint work with Tobias Harringer

Vienna University of Economics and Business

May 5, 2026

Motivation

Messi pass
Germany pass

Problem overview

Analyze decision-making: Evaluate pass relative to available options

Two-step process:

Problem overview

Analyze decision-making: Evaluate pass relative to available options

Two-step process:

Value any hypothetical pass based on:

Pitch control: Who controls a location if the ball gets there?
Pitch value: How valuable is having possession at a location?

Combined: Expected pass value (xPV) of any hypothetical pass

Problem overview

Analyze decision-making: Evaluate pass relative to available options

Two-step process:

Identify how realistic a pass is based on:

Pass likelihood: Where does the ball usually end up in a situation?
Pass feasibility: What was reasonable and physically possible?

Combined: Likelihood of realistic options (behavioral policy) at moment of pass

Problem overview

Analyze decision-making: Evaluate pass relative to available options

Two-step process:

Identify how realistic a pass is based on:

Pass likelihood: Where does the ball usually end up in a situation?

Pass feasibility: What was reasonable and physically possible?

Combined: Likelihood of realistic options (behavioral policy) at moment of pass

Problem overview

Analyze decision-making: Evaluate pass relative to available options

Two-step process:

Identify how realistic a pass is based on:

Pass likelihood: Where does the ball usually end up in a situation?
Pass feasibility: What was reasonable and physically possible?

Combined: Likelihood of realistic options (behavioral policy) at moment of pass

Problem overview

Analyze decision-making: Evaluate pass relative to available options

Two-step process:

Identify how realistic a pass is based on:

Pass likelihood: Where does the ball usually end up in a situation?
Pass feasibility: What was reasonable and physically possible?

Combined: Likelihood of realistic options (behavioral policy) at moment of pass

Problem overview

Analyze decision-making: Evaluate pass relative to available options

Two-step process:

Value any hypothetical pass based on:

Pitch control: Who controls a location if the ball gets there?
Pitch value: How valuable is having possession at a location?

Combined: Expected pass value (xPV) of any hypothetical pass

Identify how realistic a pass is based on:

Pass likelihood: Where does the ball usually end up in a situation?
Pass feasibility: What was reasonable and physically possible?

Combined: Likelihood of realistic options (behavioral policy) at moment of pass

\(\rightarrow\) Decision-making evaluation: Compare value (xPV) of actual pass to average value of realistic and likely options

Mathematical formulation

Decision-theoretic framework:
- State \(S\): Current game state \(\rightarrow\) situational context (positions / velocities of players and ball, etc.)
- Pass action \((x, y) \in \mathcal{A}(S)\): Potential pass defined by end location
- Pass value function \(\text{xPV}(x, y \mid S)\): Our model xPV model \(\rightarrow\) expected goal threat generated by pass to \((x,y)\) in situation \(S\)
- Behavioral policy \(\pi(x, y \mid S)\): Density over realistic pass destinations given \(S\)
- Context-aware situational value \(V(S) = \mathbb{E}_{(x,y) \sim \pi(\cdot \mid S)}[\text{xPV}(x, y \mid S)]\): Average expected goal threat across all realistic passes in \(S\)

Decision Score

Evaluate observed pass \((x^*, y^*)\) in observed situation \(S^*\) relative to context-aware situation baseline:

\[DS(S^*, x^*, y^*) = \underbrace{\text{xPV}(x^*, y^* \mid S^*)}_{\text{Value of pass }(x^*,y^*)} - \underbrace{V(S^*)}_{\text{context- aware situational value}}\]

\(DS > 0\): Pass outperforms behavioral average \(\rightarrow\) good decision
\(DS < 0\): Pass underperforms behavioral average \(\rightarrow\) poor decision

Interpretation: \(DS = 0.02\) \(\rightarrow\) pass created 2 percentage points more expected goal threat than the average realistic pass in the same situation

Behavioral Policy

Quality of \(DS\) depends heavily on quality of baseline \(V(S)\):
- Poor \(\pi\) \(\rightarrow\) arbitrary baseline \(\rightarrow\) uninterpretable advantage
- \(\pi\) must reflect domain-reasonable average behavior in situation \(S\)

Estimating \(\pi\) in continuous action space \(\mathcal{A}(S) \subset \mathbb{R}^2\) is non-trivial:
- Observed passes are biased — only attempted passes recorded, infeasible regions never sampled
- Execution noise (deflections, mis-hits) contaminates direct density estimation
- Naive density estimate \(\neq\) distribution over deliberate pass intentions

Decompose into two interpretable components:
\[\pi(x, y \mid S) \propto \underbrace{p(x, y \mid S)}_{\text{pass likelihood}} \cdot \underbrace{f(x, y \mid S)}_{\text{pass feasibility}}\]
- \(p(x, y \mid S)\): Where does the ball tend to end up given \(S\)?
- \(f(x, y \mid S)\): Was a pass to \((x, y)\) structurally sensible given \(S\)?

Behavioral Policy: Visualization

Pass likelihood

Goal: Estimate \(p(x, y \mid S)\) — density over pass end locations given situation \(S\)

Requirements for a good model:
- Flexible: Pass destinations are multimodal — players pass to teammates, not uniformly
- Conditional: Density changes with situation \(S\) (pressure, teammate positions, etc.)
- Proper density: Must integrate to 1 over \(\mathcal{A}(S)\) for interpretable \(\pi\)

\(p\) models where the ball ends up — this is not necessarily where player intended to pass
- Trained on all passes (including unsuccessful passes): Captures execution noise, deflections, forced passes
- Features are situation-only: passer position/velocity, teammate/defender positions, pressure, match state

Pass likelihood: MDN

Mixture density network (MDN): Model \(p(x, y \mid S)\) as a Gaussian mixture with situation-dependent parameters:

\[p(x, y \mid S) = \sum_{k=1}^K \alpha_k(S)\, \mathcal{N}\!\left((x, y) \mid \mu_k(S), \Sigma_k(S)\right)\]

Parameters \(\alpha_k(S), \mu_k(S), \Sigma_k(S)\), \(k = 1,\dots,K\), estimated by neural network taking \(S\) as input:
- Trained by maximum likelihood: \(\max \sum_i \log p(x_i, y_i \mid S_i)\)
- \(K\) components — tuning parameter controlling model flexibility and multimodality

Advantages of MDNs:
- Gaussian mixtures are universal approximators of densities
- Yields closed-form density — directly usable in \(\pi\) without numerical integration
- Mixture components capture distinct passing tendencies in a given situation

Pass feasibility

Goal: Estimate \(f(x, y \mid S)\) — is pass to \((x, y)\) structurally sensible given \(S\)?

\(f\) is conceptually distinct from \(p\):
- Would a rational player deliberately target \((x, y)\)?
- Trained only on successful passes
- \(f\) is a score, not a proper density (does not need to integrate to 1)
- Interplay between \(p\) and \(f\) \(\rightarrow\) positive (near-zero) \(p\), near-zero (positive) \(f\) possible/sensible

\(f\) uses location-interaction features — computed at each candidate \((x, y)\):
- Features encode physics and geometry of passing to each location
- Defender cone features: Defenders in passing lane toward \((x, y)\)
- Teammate reachability: Time-to-intercept for teammates/defenders at \((x, y)\)
- Pass geometry: Distance, angle, angular mismatch with passer orientation

Challenge: No negative labels — only attempted passes observed, infeasible regions never sampled

Pass feasibility: NIPP

Model pass destinations as a spatial inhomogeneous Poisson process (IPP) on pitch:
\[N(B) \mid S \sim \text{Poisson}\!\left(\int_B \lambda(x,y \mid S)\, d(x,y)\right) \quad \forall\, B \subseteq \mathcal{A}\]
- \(N(B)\): Random number of pass destinations falling in region \(B \subseteq \mathcal{A}\)
- Governed by inhomogenous intensity function \(\lambda(x,y \mid S) > 0\) — varies across pitch and situation

Intensity function is the key part:
- \(\lambda(x,y \mid S) > 0\) is a score, not a density — perfect for \(f\) \(\rightarrow\) Set \(f(x, y \mid S) := \lambda(x, y \mid S)\) in policy decomposition
- Directly yields feasibility surface over full pitch \(\mathcal{A}\)
- Parameterized by neural network taking \((x,y)\) \(\rightarrow\) Neural IPP (NIPP)
- Intuitively:
  - High \(\lambda\): Passes frequently directed here given \(S\) \(\rightarrow\) high feasibility
  - Low \(\lambda\): Passes rarely directed here given \(S\) \(\rightarrow\) low feasibility

Pass feasibility: Estimation

Estimate by maximizing log-likelihood for observed pass destination \((x_i, y_i)\) under situation \(S_i\):

\[\ell = \sum_{i=1}^N \overbrace{\log \lambda(x_i, y_i \mid S_i)}^{\text{encourages } \lambda \text{ to be large where passes are observed}} - \underbrace{\int_{\mathcal{A}} \lambda(x, y \mid S_i)\, d(x,y)}_{\text{penalizes total intensity — no explicit negatives needed}}\]

The integral term is the key insight:
- Discourages \(\lambda\) from spreading mass arbitrarily across the pitch
- Acts as natural regulariser — replaces the need for negative sampling
- Infeasible regions accumulate no pass events \(\rightarrow\) integral penalty dominates \(\rightarrow\) low \(\lambda\)
- For estimation, integral approximated by Riemann sum over discrete grid: \[\int_{\mathcal{A}} \lambda\, d(x,y) \approx \Delta A \sum_j \lambda(x_j, y_j \mid S_k)\]

Decision score estimation

Recall:

\[DS(S, x, y) = \operatorname{xPV}(x,y \mid S) - V(S)\] \[V(S) = \mathbb{E}_{(x,y) \sim \pi(x,y\mid S)}[\operatorname{xPV}(x,y\mid S)] = \int_{\mathcal{A(S)}} \operatorname{xPV}(x,y|S) \pi(x,y|S) d(x,y)\]

Decision score:
- Step 1: Obtain xPV for pass and xPV surface for given situation
- Step 2: Obtain behavioral policy surface for given situation
- Step 3: Numerically evaluate above integral

Results: Messi pass example

Pitch control surface

Pitch value surface

xPV surface

xPV of actual pass: 0.0395

Results: Messi pass example

Pitch control surface

Pitch value surface

xPV surface

xPV of actual pass: 0.0395

Pass likelihood surface

Pass feasibility surface

Behavioral policy surface

Value of the Situation: 0.0193

Results: Messi pass example

Pitch control surface

Pitch value surface

xPV surface

xPV of actual pass: 0.0395

Pass likelihood surface

Pass feasibility surface

Behavioral policy surface

Value of the Situation: 0.0193

Decision Score: 0.0202

Good Decision

Results: Germany pass example

Pitch control surface

Pitch value surface

xPV surface

xPV of actual pass: 0.0211

Results: Germany pass example

Pitch control surface

Pitch value surface

xPV surface

xPV of actual pass: 0.0211

Pass likelihood surface

Pass feasibility surface

Behavioral policy surface

Value of the Situation: 0.0291

Results: Germany pass example

Pitch control surface

Pitch value surface

xPV surface

xPV of actual pass: 0.0211

Pass likelihood surface

Pass feasibility surface

Behavioral policy surface

Value of the Situation: 0.0291

Decision Score: -0.008

Poor Decision

Results: Player analyses

Broadcast tracking data from the 2022 FIFA World Cup

Results: Player analyses

Broadcast tracking data from the 2022 FIFA World Cup
Carefully filter passes:
- Checks for sensible passes (broadcast tracking is a difficult problem)
- Only passes played by foot (no headers)
- Total amount of passes used: 34195

Results: Player analyses

Broadcast tracking data from the 2022 FIFA World Cup
Carefully filter passes:
- Checks for sensible passes (broadcast tracking is a difficult problem)
- Only passes played by foot (no headers)
- Total amount of passes used: 34195
DS centered around zero but strongly heavy-tailed \(\rightarrow\) evaluating players requires care:
- Extreme outliers near goals (high xPV values) \(\rightarrow\) average unreliable
- Median, trimmed mean favors low-frequency passers
- Passing is high-volume action, interested in players that consistently make good decision \(\rightarrow\) winsorized sum

Results: Player analysis

Results: Decision making under pressure

Conclusion

Evaluating pass decision-making requires counterfactual reasoning \(\rightarrow\) comparing chosen pass to realistic alternatives

We propose two step framework:
- Estimate pass value surface \(\rightarrow\) xPV model
- Estimate realistic pass distribution for given situation \(\rightarrow\) behavioral policy:
  - Pass likelihood model: Where does the ball tend to end up given \(S\)?
  - Pass feasibility model: Which passes are structurally sensible given \(S\)?

Decision score measures pass quality relative to situation:
- Interpretable as increase in expected goal threat above behavioral average
- Enables context-aware player decision quality analyses

Framework is modular — components useful beyond decision scoring:
- Player style profiling, space creation analysis, box presence

Thank you for your attention!

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Appendix

MDN: Details

Neural network architecture:
- 2 hidden layers with 256 hidden nodes, ReLu activation function, dropout
- Mixture weights \(\alpha_k(S)\) sum up to 1 \(\rightarrow\) softmax activation
- Mean vector for each component \(\mu_k(S)\): linear activation function
- Covariance matrices \(\Sigma_k(S)\): positive semidefinite contraints
- \(K = 8\)
Features for MDN:
- Player information:
  - Ordered by distance to passer
  - Position, distance and velocities relative to passer
  - Goalkeeper information separately
- Information on pass: One-touch, foot/head
- Location and duration of previous action
- angular mismatch between body orientation of passer and previous action location

NIPP: Details

Neural network architecture:
- 3 hidden layers with 256 hidden nodes, ReLu activation function, layer normalization, dropout
- Positivity constraint on \(\lambda\) output
- Approximation of integral on grid of 1x1 grid cells
Features for NIPP:
- Reachability and interceptability features:
  - Times to pass location for attacker/defenders
  - Defenders in passing lane
  - Defender closeness
- Pass geometry features: Distance, angular mismatch between body orientation and pass location, and previous pass
- Pass features: Duration of possession, footer/header
- Passer speed