Case Study 2: The Substitution That Changed Everything

Background

It is the 58th minute of a crucial late-season match. The home team, trailing 0-1 and struggling to break through a well-organized low block, makes a triple substitution. Within 15 minutes, the match is transformed: the home team scores twice and takes control of the game. Was this a tactical masterstroke, or was it luck? This case study uses the analytical tools from Chapter 22 to decompose the substitution's impact into measurable components.

Pre-Match Context

Team Profiles

Home Team (Team A): A top-four side playing a possession-based 4-3-3 system. Their tactical fingerprint shows high possession (z = +1.2), high pressing intensity (z = +1.4), and wide attacking patterns (z = +1.1). Their weakness is a relatively slow build-up (z = -0.3) and vulnerability to counter-attacks when both fullbacks push high.

Away Team (Team B): A mid-table side that excels at defensive organization. Their fingerprint shows low possession (z = -0.8), very compact defensive shape (z = +1.6), effective counter-attacking (z = +0.9), and high set-piece efficiency (z = +0.7).

Match State at 58 Minutes

The xG numbers reveal the story: Team A has dominated possession and created shot volume, but their shot quality has been poor -- they have been shooting from distance and into a packed defense. Team B's single goal came from a well-executed counter-attack with an xG of 0.42.

The Substitution Decision

Players Removed and Introduced

Formation Change

The substitution changed the formation from a 4-3-3 to a 3-4-2-1:

Before (4-3-3):                After (3-4-2-1):
         ST                           ST(R)
    LW       RW                  CAM(Q)    RW
      CM   CM                  LWB   CM   RWB
   CDM                           CB   CB   CB
  LB  CB  CB  RB                    GK
       GK

The key structural changes:

1. Left-back pushed to wing-back, providing width previously supplied by the winger.
2. Extra center-back dropped in, allowing the remaining center-backs to step higher.
3. Attacking midfielder occupied the half-spaces, creating numerical overloads between the lines.
4. New striker provided an aerial focal point, enabling direct supply to the box.

Quantitative Analysis

Formation Detection

Using the formation detection algorithm from Section 22.1, we track the detected formation at 5-minute intervals:

The transitional period (58-65 min) reflects the players adapting to new positions. By 65 minutes, the new formation is established.

Tactical Fingerprint Shift

The most striking change is in shot quality: the average xG per shot more than doubled. The team sacrificed some possession and pressing intensity in exchange for more direct, higher-quality attacking play.

xG Flow Analysis

The 58-75 minute window produced more xG for the home team than the entire first half. This represents a genuine tactical shift, not merely random variation.

Win Probability Evolution

Using the win probability model from Section 22.5:

The substitution's impact is visible not in the immediate win probability (which remained at 22%) but in the subsequent goal events that it facilitated. The 68th-minute equalizer increased win probability by 20 percentage points, and the 73rd-minute go-ahead goal by another 36 points.

Win Probability Added (WPA) by Substitution

To isolate the substitution's effect, we compare the observed outcome to a counterfactual simulation:

• Observed: Team A wins with probability 100% (actual result).
• Counterfactual (no substitution): Simulating 10,000 match completions from the 58th-minute state without formation change, Team A wins 28% of the time.
• WPA of substitution decision: The substitution is credited with approximately 72 percentage points of win probability, though much of this is mediated through the subsequent goals.

A more conservative attribution, crediting the substitution only for the tactical improvement (higher xG rate) but not for the stochastic goal outcomes:

$$\Delta \text{xG rate} = 0.82/17\text{ min} - 0.20/13\text{ min} = 0.048 - 0.015 = 0.033 \text{ xG/min}$$

This translates to approximately 1.06 additional xG over the remaining 32 minutes -- a substantial tactical improvement.

Counterfactual Analysis

What If the Substitution Had Not Been Made?

We simulate the remaining 32 minutes 10,000 times under two scenarios:

Scenario A (No change): Team A continues in 4-3-3, xG rate remains at pre-substitution levels.

Scenario B (Substitution made): Team A switches to 3-4-2-1, xG rate increases as observed.

The substitution shifted the expected outcome distribution by approximately 24 percentage points in favor of the home team.

Lessons Learned

1. Formation changes can create immediate tactical mismatches. Team B's low block was optimized against a 4-3-3; the switch to 3-4-2-1 created unfamiliar defensive problems, particularly in the half-spaces.

2. Shot quality matters more than shot volume. Team A's xG per shot more than doubled after the substitution, even though total shot volume did not dramatically increase.

3. Timing of substitutions is critical. The 58th-minute timing allowed enough match time for the tactical change to take effect while Team B's energy levels were declining.

4. Win probability models help quantify tactical decisions. Without WPA analysis, the substitution would simply be called "inspired" in post-match commentary. With it, we can quantify the improvement as approximately 24 percentage points of win probability.

5. Counterfactual reasoning separates skill from luck. The actual 2-1 scoreline could have occurred by chance even without the substitution. Counterfactual simulation allows us to estimate the substitution's causal effect, not just its association with the outcome.

Code Reference

The complete simulation and analysis code is available in code/case-study-code.py and code/example-03-game-state.py.