4.8 How Many Runs for a Win?

“ten-runs-equal-one-win” rule of thumb

Earlier:

\[\widehat{W_{\text{pct}}} = 0.5 + 0.0006*RD\]

$RD = 0 \rightarrow W_{\text{pct}} = 0.5$
- Over 162 games: 81 wins
$RD = +10 \rightarrow W_{\text{pct}} = 0.506$
- Over 162 games: 82 wins

4.8.1 Calculus

Ralph Caola derived the number of extra runs needed to get an extra win in a more rigorous way using calculus

\[W = \frac{G \cdot R^{2}}{R^{2} + RA^{2}}\]

$$ \[\begin{array}{rcl} \frac{\partial W}{\partial R} & = & \frac{\partial}{\partial R}\frac{G \cdot R^{2}}{R^{2} + RA^{2}} \\ ~ & = & \frac{2 \cdot G \cdot R \cdot RA^{2}}{(R^{2} + RA^{2})^{2}} \\ \end{array}\]

calculus in R!

D(expression(G * R ^ 2 / (R ^ 2 + RA ^ 2)), "R")

## G * (2 * R)/(R^2 + RA^2) - G * R^2 * (2 * R)/(R^2 + RA^2)^2

4.8.2 Incremental Runs per Win

\[IR/W = \frac{(R^{2} + RA^{2})^{2}}{2 \cdot G \cdot R \cdot RA^{2}}\]

We can make a user-defined function (and assuming rate statistics “runs per game” and “runs allowed per game” to remove $G$):

IR <- function(RS = 5, RA = 5) {
  (RS ^ 2 + RA ^ 2)^2 / (2 * RS * RA ^ 2)
}

With two inputs, we will use a grid search to express different numbers of runs scored and runs allowed.

ir_table <- tidyr::expand_grid(RS = 1:7, RA = 1:7) |>
  mutate(IRW = IR(RS, RA)) |>
  pivot_wider(names_from = RA, values_from = "IRW",
              names_prefix = "RA=")

RS	RA=1	RA=2	RA=3	RA=4	RA=5	RA=6	RA=7
Incremental runs per win
as posed by Ralph Caola
1	2.0	3.1	5.6	9.0	13.5	19.0	25.5
2	6.2	4.0	4.7	6.2	8.4	11.1	14.3
3	16.7	7.0	6.0	6.5	7.7	9.4	11.4
4	36.1	12.5	8.7	8.0	8.4	9.4	10.8
5	67.6	21.0	12.8	10.5	10.0	10.3	11.2
6	114.1	33.3	18.8	14.1	12.4	12.0	12.3
7	178.6	50.2	26.7	18.9	15.6	14.3	14.0

table code

ir_table |>
  gt() |>
  cols_align(align = "center") |>
  data_color(columns = -RS,
             palette = "viridis") |>
  fmt_number(columns = -RS,
             decimals = 1) |>
  tab_header(title = "Incremental runs per win",
             subtitle = "as posed by Ralph Caola")