We can check the accuracy of our `evaluate_largest` method by comparing it with an analytical solution.

We find that despite small discrepancy, the evaluate_largest method functions well.

Furthermore, if we take a very large number of auctions, say 1 million, the discrepancy disappears.

We find that the `evaluate_largest` method functions well

```{code-cell} ipython3

v_grid = np.linspace(0.3,1,8)

bid_analytical = b_star(v_grid,N)

v_grid = np.linspace(0.3, 1, 8)

bid_analytical = b_star(v_grid, N)

# Redraw valuations

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

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

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