Planets example

Let's consider a same-costs market and plot the valuation curve, which gives $v(\mathcal{X}_h)$ as a function of $h$, where $\mathcal{X}_h$ is the optimal portfolio for each $h$ between $0$ and $m$.

The market data is as follows:

\[\begin{align} f &= (0.39, 0.33, 0.24, 0.24, 0.05, 0.03, 0.1, 0.12) \\ t &= (200, 250, 300, 350, 400, 450, 500, 550) \end{align}\]

Let's import the PrettyTables and UnicodePlots libraries for visualizations and read in the market data.

using PrettyTables
using UnicodePlots
using OptimalApplication
f = [0.39, 0.33, 0.24, 0.24, 0.05, 0.03, 0.1, 0.12]
t = [200, 250, 300, 350, 400, 450, 500, 550]
mkt = Market(f, t, 8)

SameCostsMarket{Int64}(8, [0.39, 0.33, 0.24, 0.24, 0.05, 0.03, 0.1, 0.12], [200, 250, 300, 350, 400, 450, 500, 550], 8, [1, 2, 3, 4, 5, 6, 7, 8])

Notice how we set $h = m = 8$ because we know from the nestedness property that by calling applicationorder_list(mkt), we can get a permutation of the schools that encodes all the optimal portfolios. Let's do that now:

X, V = applicationorder_list(mkt, verbose=true)

([4, 2, 8, 1, 7, 3, 5, 6], [84.0, 146.7, 195.096, 230.047488, 257.6427392, 281.513441792, 288.7777697024, 294.10643661132804])

$\mathcal{X}_h$ is given by the first $h$ entries of X, and $v(\mathcal{X}_h)$ is V[h]. Let's plot V.

lineplot(0:mkt.m, vcat(0, V), xlabel="h", ylabel="v")

         ┌────────────────────────────────────────┐ 
     300 │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠤⠔⠒⠒⠒│ 
         │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
   v     │⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         │⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
       0 │⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
         └────────────────────────────────────────┘ 
         ⠀0⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀8⠀ 
         ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀h⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ 

This graph is called a valuation curve. It gives the applicant's utility as a function of her budget $h$. The curve's concave shape is guaranteed by the nestedness property.

An alternative way of looking at the optimal portfolios is to take the inverse-permutation of X. The value of invperm(X)[j] gives the smallest value of $h$ for which school $j$ is in $\mathcal{X}_h$. This is called the schools priority number; if a school has priority number 1, it is the first school you should apply to, and so on.

This format lets us create the following table:

priority = invperm(X)
pretty_table(
    Any[f t priority V[priority]],
    header = ["f", "t", "priority", "valuation"],
    header_crayon = crayon"bold yellow",
)

┌──────┬─────┬──────────┬───────────┐
│    f │   t │ priority │ valuation │
├──────┼─────┼──────────┼───────────┤
│ 0.39 │ 200 │        4 │   230.047 │
│ 0.33 │ 250 │        2 │     146.7 │
│ 0.24 │ 300 │        6 │   281.513 │
│ 0.24 │ 350 │        1 │      84.0 │
│ 0.05 │ 400 │        7 │   288.778 │
│ 0.03 │ 450 │        8 │   294.106 │
│  0.1 │ 500 │        5 │   257.643 │
│ 0.12 │ 550 │        3 │   195.096 │
└──────┴─────┴──────────┴───────────┘