How far forward should we push? Attack, defence and the price of a rule

The football question

A new manager wants his team to be braver: more players in the box, full-backs pushing on. The board is nervous. "Fine," says the chairman, "but we're not conceding more than 42 goals this season."

Pushing forward brings goals at one end and costs them at the other. So how far forward should the team push? And what does the chairman's rule cost: nothing, a little, or a lot?

The concept

Part 1 had yes-or-no decisions, sign a player or don't, so we could check every combination. This decision is a dial. The team can push forward a little, a lot or anywhere in between, so there are endless settings to choose from. Calculus finds the best one without trying them all, the same way gradient descent finds the bottom of a valley.

Two new ideas come with it:

• A constraint is binding if it changes the answer, and not binding if the best answer obeys it anyway.
• A binding constraint has a shadow price: how much the objective would improve if the rule were loosened by one unit. Here, how many points a season one more goal conceded would buy.

A football example

Call the setting a: 0 means everyone behind the ball, 1 means all-out attack. Here are made-up curves for the goals scored and conceded in a season, scaled so that a middling setting (0.5) gives an average Scottish top-flight side: 51 goals at each end, about 1.34 a game, the average over 26 seasons of real data.

$$\begin{aligned} \text{scored} &= 30 + 54a - 24a^2 \\ \text{conceded} &= 36 + 12a + 36a^2 \end{aligned}$$

To turn goals into points we use the values from how many points is a goal worth?: each goal scored is worth about 0.65 points over a season and each goal conceded costs about 0.60. An average side takes about 52 points, so

$$\begin{aligned} \text{points} = 52 \;&+ 0.65 \times (\text{scored} - 51) \\ &- 0.60 \times (\text{conceded} - 51) \end{aligned}$$

The best setting

Points rise as the team pushes forward, reach a peak, then fall. At the peak, a little more pushing would add exactly as many points through goals scored as it loses through goals conceded. The slope of each curve, its derivative, says how many goals a small push brings, so the peak is where

$$\begin{aligned} 0.65 \times (54 - 48a) &= 0.60 \times (12 + 72a) \\ a &= 0.375 \end{aligned}$$

Add the chairman's rule

The best setting concedes 45.6 goals, more than the chairman's 42. So the rule is binding: it changes the answer.

When a rule binds, the best answer sits right on it. Pushing forward still adds points, so the team should push until it concedes exactly 42: a = 0.274, scoring 43.0 and taking 52.20 points. The rule costs 0.38 points a season.

A rule of 50 goals wouldn't bind at all: the best setting already concedes fewer, so the rule costs nothing. A rule only has a cost when it stops you doing what you'd otherwise do.

The price of a rule

Here's what different limits cost, on the same curves:

Two lessons are in that table.

The top of the hill is flat. Tightening the limit from 45.6 to 45 costs almost nothing, 0.01 points. Each further goal costs more than the one before, and a 38-goal limit costs 2.38 points, more than two draws. A gentle rule near the best setting is nearly free; a strict one gets expensive fast.

The last column is the shadow price. It says what loosening the rule by one goal would be worth:

$$\text{price} = \frac{\text{points a small push wins}}{\text{goals that push concedes}}$$

Part 1 priced the "must sign a centre-back" rule at 0.75 points by solving the problem with and without it. That works for a yes-or-no rule. A rule like the chairman's can be loosened one goal at a time, and the shadow price says what each goal of slack is worth before anyone has to solve the problem again.

Why it matters

Every club runs on rules: a wage ceiling, a limit on a player's minutes, a minimum number of home-grown players, "no more than 42 conceded". Each one that binds has a price, whether or not anybody works it out. Shadow prices say which rule is costing the most, and so which one is worth arguing about. A rule with a price of zero isn't hurting anyone, and a rule with a high price deserves a good reason. The next part in this series asks what to do when there's more than one objective, and no single "best".

Limitations

• The curves are made up. They're scaled to an average Scottish side, but nobody can measure how far forward a team "pushes" as one number, and our data has no tactical settings.
• One dial is a simplification. Real tactics have many settings at once. The method extends to them, with one slope per dial, as in machine learning.
• Goals aren't everything. The 0.65 and 0.60 points per goal are averages over many seasons, and points also depend on when the goals come.
• The price is for small changes. At the 42-goal limit, loosening it to 43 gains a little less than 0.24 points, because the price falls as the rule loosens. It's a guide, not an exact quote.

Try it yourself

Before reading the code, guess: if the chairman's limit were 40 goals, how far forward should the team push? Then change the curves in the code to suit a stronger team, one that scores more for each step forward, and see whether its best setting moves forward or back.

Reproduce the analysis

This finds the best setting two ways, by trying every setting and by calculus, checks they agree, and prints every number in the article. Nothing to download.

# Made-up curves for a season, scaled so a middling setting (0.5) gives an average Scottish top-flight side: 51 goals
# scored, 51 conceded and 52 points. a runs from 0 (everyone behind the ball) to 1 (all-out attack).
def scored(a):
    return 30 + 54 * a - 24 * a ** 2


def conceded(a):
    return 36 + 12 * a + 36 * a ** 2


def points(a):   # 0.65 points per goal scored, 0.60 per goal conceded (Research: how many points is a goal worth?)
    return 52 + 0.65 * (scored(a) - 51) - 0.60 * (conceded(a) - 51)


def slope(fn, a, h=1e-6):
    return (fn(a + h) - fn(a - h)) / (2 * h)


# Way 1: try every setting from 0 to 1 in steps of 0.0001 and keep the best one the rule allows.
grid = [i / 10000 for i in range(10001)]


def search(limit=None):
    return max((a for a in grid if limit is None or conceded(a) <= limit), key=points)


# Way 2: calculus. With no rule, the best setting is where points stop rising: 0.65 × (54 − 48a) = 0.60 × (12 + 72a).
# If that concedes too many, the best setting is the one that concedes exactly the limit: 36a² + 12a + 36 = limit.
free = (0.65 * 54 - 0.60 * 12) / (0.65 * 48 + 0.60 * 72)


def solve(limit=None):
    if limit is None or conceded(free) <= limit:
        return free
    return (-12 + (144 + 144 * (limit - 36)) ** 0.5) / 72


print(f"Parking the bus (0): scored {scored(0):.0f}, conceded {conceded(0):.0f}, points {points(0):.1f}")
print(f"Middling (0.5):      scored {scored(0.5):.0f}, conceded {conceded(0.5):.0f}, points {points(0.5):.1f}")
print(f"All-out (0.8):       scored {scored(0.8):.1f}, conceded {conceded(0.8):.1f}, points {points(0.8):.1f}")
print()
print("Rule             setting  scored  conceded  points  cost  price")
for limit in [None, 50, 45, 42, 40, 38]:
    a = solve(limit)
    assert abs(a - search(limit)) < 0.0001   # the two ways agree
    price = slope(points, a) / slope(conceded, a)   # points per extra goal the rule allows; 0 if it doesn't bind
    rule = "none" if limit is None else f"concede <= {limit}"
    print(f"{rule:<16} {a:7.3f}  {scored(a):6.1f}  {conceded(a):8.1f}  {points(a):6.2f}  {points(free) - points(a):4.2f}"
          f"  {price:5.2f}")

It prints:

Parking the bus (0): scored 30, conceded 36, points 47.4
Middling (0.5):      scored 51, conceded 51, points 52.0
All-out (0.8):       scored 57.8, conceded 68.6, points 45.9

Rule             setting  scored  conceded  points  cost  price
none               0.375    46.9      45.6   52.58  0.00   0.00
concede <= 50      0.375    46.9      45.6   52.58  0.00   0.00
concede <= 45      0.360    46.3      45.0   52.57  0.01   0.03
concede <= 42      0.274    43.0      42.0   52.20  0.38   0.24
concede <= 40      0.206    40.1      40.0   51.52  1.06   0.47
concede <= 38      0.122    36.2      38.0   50.20  2.38   0.91

The price column measures the slopes at the best setting rather than solving the problem again, which is how shadow prices are usually found in practice.

Further reading

• Lagrange multiplier, Wikipedia. The general method for optimising with a rule, and why the multiplier is a price.
• Shadow price, Wikipedia. The same number as economists use it.
• How many points is a goal worth? The Research piece behind the 0.65 and 0.60 points per goal.