colors = plt.rcParams['axes.prop_cycle'].by_key()['color']

# ensure the notebook generates the same randomness

np.random.seed(1337)

rng = np.random.default_rng(1337)

```

We repeat an auction with 5 bidders for 100,000 times.

N = 5

R = 100_000

v = np.random.uniform(0, 1, (N, R))

v = rng.uniform(0, 1, (N, R))

# BNE in first-price sealed bid

bid_analytical = b_star(v_grid, N)

# Redraw valuations

v = np.random.uniform(0, 1, (N, R))

v = rng.uniform(0, 1, (N, R))

bid_simulated = [evaluate_largest(ii, v) for ii in v_grid]

fig, ax = plt.subplots(figsize=(6, 4))

We'll start by taking a look at a $\chi^2$ distribution with the help of the following Python code:

```{code-cell} ipython3

np.random.seed(1337)

v = np.random.chisquare(df=2, size=(N * R,))

rng = np.random.default_rng(1337)

v = rng.chisquare(df=2, size=(N * R,))

plt.hist(v, bins=50, edgecolor='w')

plt.xlabel('Values: $v$')

Now we'll get Python to construct a bid price function

```{code-cell} ipython3

np.random.seed(1337)

v = np.random.chisquare(df=2, size=(N, R))

rng = np.random.default_rng(1337)

v = rng.chisquare(df=2, size=(N, R))

# we compute the quantile of v as our grid

pct_quantile = np.linspace(0, 100, 101)[1:-1]

```

```{code-cell} ipython3

np.random.seed(1337)

v = np.random.chisquare(df=2, size=(N, R))

rng = np.random.default_rng(1337)

v = rng.chisquare(df=2, size=(N, R))

chi_squ_case = bid_price_solution(v)

```