write python code for this project you should answ...

python
import numpy as np
import inspect, sys, textwrap, os, math

# ---------------------------
# Part A – Finite‑horizon secretary via backward induction
# ---------------------------

def finite_secretary(N: int, *, verbose: bool = False):
    """
    Dynamic‑programming solution of the finite‑horizon secretary problem
    taken from Puterman Example 4.6.4.

    Parameters
    ----------
    N : int
        Total number of applicants (horizon length).
    verbose : bool, default False
        If True, prints the recursion verification for each t.

    Returns
    -------
    r          : int     – first interview index at which we are willing to hire
    p_success  : float   – optimal probability of choosing the best applicant
    u1, u0     : np.ndarray – optimal value functions u*_t(1) and u*_t(0)
    """
    # u1[t] = u*_t(1) – value if current candidate is record‑holder so far
    # u0[t] = u*_t(0) – value if current candidate is NOT record‑holder
    u1 = np.zeros(N + 2)
    u0 = np.zeros(N + 2)

    # Boundary conditions (Eq 4.6.4/4.6.5 in the book)
    u1[N] = 1.0   # last applicant is definitely the best among the first N
    u0[N] = 0.0

    # Backward induction (Algorithm 4.5)
    for t in range(N - 1, 0, -1):
        u0[t] = (1 / (t + 1)) * u1[t + 1] + (t / (t + 1)) * u0[t + 1]
        u1[t] = max(t / N, u0[t])

    # The decision threshold r is the first t for which accepting is optimal
    r = next(t for t in range(1, N + 1) if t / N >= u0[t])
    p_success = u0[1]          # start state: no record‑holder among first 0

    if verbose:
        ok = all(abs(u1[t] - max(t / N, u0[t])) < 1e-12 for t in range(1, N + 1))
        print(f"[verify] recursion holds for N={N}: {ok}")

    return r, p_success, u1, u0


def demo_part_a():
    print("=== Part A – finite‑horizon results ===")
    for N in [10, 50, 100, 1000]:
        r, p, *_ = finite_secretary(N)
        print(f"N={N:<4}  optimal r={r:<4}  r/N={r/N:6.3f}  P*= {p:7.4f}")
    print()


# ---------------------------
# Helper for the infinite‑horizon formulations
# ---------------------------

def _solve_threshold_equation(gamma: float, *, tol: float = 1e-12) -> float:
    """
    Solve θ = γ E[V] numerically, where for a threshold policy
        E[V] = (1 – θ²) / (2 · (1 – γ θ))

    The fixed‑point gives the unique optimal threshold.
    """
    theta = 0.5
    for _ in range(1000):
        a = (1.0 - theta ** 2) / (2.0 * (1.0 - gamma * theta))
        new_theta = gamma * a
        if abs(new_theta - theta) < tol:
            return new_theta
        theta = new_theta
    return theta


# ---------------------------
# Part B – Infinite‑horizon MDP with discount γ
# ---------------------------

def infinite_secretary_value_iteration(gamma: float = 0.99,
                                       grid: int = 1001,
                                       eps: float = 1e-10):
    """
    Value‑iteration implementation for the “continuous‑quality” infinite‑horizon
    secretary problem.  The applicant’s (observable) quality lies in [0,1].
    """
    xs = (np.arange(grid) + 0.5) / grid        # mid‑points
    V = np.zeros(grid)

    while True:
        mean_V = V.mean()
        accept = xs
        reject = gamma * mean_V
        V_new = np.maximum(accept, reject)
        if np.max(np.abs(V_new - V)) < eps:
            break
        V = V_new

    threshold_idx = np.argmax(accept >= reject)
    theta = xs[threshold_idx]
    p_success = (theta + 1) / 2.0               # E[x | x ≥ θ] = (θ+1)/2
    return theta, p_success


def infinite_secretary_policy_iteration(gamma: float = 0.99):
    """
    Exact policy‑iteration solution exploiting the analytical structure:
      – threshold policy ⇒ value function piece‑wise constant/linear
    """
    theta = 0.5
    for _ in range(1000):
        a = (1.0 - theta ** 2) / (2.0 * (1.0 - gamma * theta))   # E[V]
        new_theta = gamma * a                                    # improvement
        if abs(new_theta - theta) < 1e-12:
            break
        theta = new_theta
    p_success = (theta + 1.0) / 2.0
    return theta, p_success


def infinite_secretary_modified_policy_iteration(gamma: float = 0.99,
                                                 m_eval: int = 5,
                                                 grid: int = 1001):
    """
    Modified policy‑iteration: do `m_eval` partial policy‑evaluation
    sweeps between successive improvements.
    """
    xs = (np.arange(grid) + 0.5) / grid
    theta = 0.5
    V = xs.copy()

    for _ in range(100):                # outer loop
        # --- partial policy evaluation under π_θ ---------------------------
        for _ in range(m_eval):
            mean_V = V.mean()
            V = np.where(xs >= theta, xs, gamma * mean_V)

        # --- policy improvement -------------------------------------------
        mean_V = V.mean()
        new_theta = gamma * mean_V
        if abs(new_theta - theta) < 1e-12:
            break
        theta = new_theta

    p_success = (theta + 1.0) / 2.0
    return theta, p_success


def demo_part_b(gamma: float = 0.99):
    print("=== Part B – infinite‑horizon with γ = 0.99 ===")
    vi_theta, vi_p = infinite_secretary_value_iteration(gamma)
    pi_theta, pi_p = infinite_secretary_policy_iteration(gamma)
    mpi_theta, mpi_p = infinite_secretary_modified_policy_iteration(gamma)

    print(f"Value‑iteration      θ* = {vi_theta:7.4f}   P* = {vi_p:7.4f}")
    print(f"Policy‑iteration     θ* = {pi_theta:7.4f}   P* = {pi_p:7.4f}")
    print(f"Modified PI (m=5)    θ* = {mpi_theta:7.4f}   P* = {mpi_p:7.4f}")
    print()

    # Analytical fixed‑point for comparison
    fp_theta = _solve_threshold_equation(gamma)
    print(f"Analytical solution  θ* = {fp_theta:7.4f}")
    print()


# ---------------------------
# Part C – Randomly‑sized pool with continuation prob L
# ---------------------------

def random_length_secretary(L: float,
                            grid: int = 1001,
                            eps: float = 1e-10):
    """
    Optimal strategy when another applicant arrives with probability L
    (geometric distribution of pool size).
    """
    xs = (np.arange(grid) + 0.5) / grid
    V = np.zeros(grid)

    while True:
        mean_V = V.mean()
        accept = xs
        reject = L * mean_V          # continue with prob L, otherwise stop
        V_new = np.maximum(accept, reject)
        if np.max(np.abs(V_new - V)) < eps:
            break
        V = V_new

    theta = xs[np.argmax(accept >= reject)]
    p_success = (theta + 1.0) / 2.0
    return theta, p_success


def demo_part_c(L: float = 0.99):
    print(f"=== Part C – random pool, continuation L = {L} ===")
    theta, p = random_length_secretary(L)
    print(f"Optimal θ* = {theta:7.4f}   P* = {p:7.4f}")
    print("Note: identical to γ = L case because geometric tail acts as a "
          "discount factor.\n")


# --------------------------------------------------------------------------
# Run the demonstrations
# --------------------------------------------------------------------------
if __name__ == "__main__":
    demo_part_a()
    demo_part_b(0.99)
    demo_part_c(0.99)

    # ----------------------------------------------------------------------
    # Save a standalone copy of this module for download
    # ----------------------------------------------------------------------
    code_text = textwrap.dedent(inspect.getsource(sys.modules[__name__]))
    path = "/mnt/data/secretary_algorithms.py"
    with open(path, "w") as f:
        f.write(code_text)
    print(f"\nPython source saved to: {path}")

python
# Retry saving the code by reconstructing the source manually
module_source = """
# secretary_algorithms.py
# -------------------------------------------------------------
# Stand‑alone implementation of finite‑ and infinite‑horizon
# secretary (best‑choice) problems, featuring:
#   * Backward Induction (finite horizon)
#   * Value Iteration
#   * Policy Iteration
#   * Modified Policy Iteration
# -------------------------------------------------------------
import numpy as np

# ---------- Part A ----------
def finite_secretary(N: int):
    u1 = np.zeros(N + 2)
    u0 = np.zeros(N + 2)
    u1[N] = 1.0
    u0[N] = 0.0
    for t in range(N - 1, 0, -1):
        u0[t] = (1 / (t + 1)) * u1[t + 1] + (t / (t + 1)) * u0[t + 1]
        u1[t] = max(t / N, u0[t])
    r = next(t for t in range(1, N + 1) if t / N >= u0[t])
    p_success = u0[1]
    return r, p_success

# ---------- Helpers ----------
def _solve_threshold(gamma: float, tol: float = 1e-12):
    theta = 0.5
    for _ in range(1000):
        a = (1.0 - theta ** 2) / (2.0 * (1.0 - gamma * theta))
        new_theta = gamma * a
        if abs(new_theta - theta) < tol:
            break
        theta = new_theta
    return theta

# ---------- Part B ----------
def infinite_secretary(gamma: float = 0.99, algo: str = "value", m_eval: int = 5,
                       grid: int = 1001, eps: float = 1e-10):
    xs = (np.arange(grid) + 0.5) / grid
    if algo == "value":
        V = np.zeros(grid)
        while True:
            mean_V = V.mean()
            V_new = np.maximum(xs, gamma * mean_V)
            if np.max(np.abs(V_new - V)) < eps:
                break
            V = V_new
        theta = xs[np.argmax(xs >= gamma * V.mean())]
    elif algo == "policy":
        theta = _solve_threshold(gamma)
    elif algo == "mpi":
        theta = 0.5
        V = xs.copy()
        for _ in range(100):
            for _ in range(m_eval):
                mean_V = V.mean()
                V = np.where(xs >= theta, xs, gamma * mean_V)
            new_theta = gamma * V.mean()
            if abs(new_theta - theta) < 1e-12:
                break
            theta = new_theta
    else:
        raise ValueError("algo must be 'value', 'policy', or 'mpi'")
    p_success = (theta + 1.0) / 2.0
    return theta, p_success

# ---------- Part C ----------
def random_length_secretary(L: float, grid: int = 1001, eps: float = 1e-10):
    xs = (np.arange(grid) + 0.5) / grid
    V = np.zeros(grid)
    while True:
        mean_V = V.mean()
        V_new = np.maximum(xs, L * mean_V)
        if np.max(np.abs(V_new - V)) < eps:
            break
        V = V_new
    theta = xs[np.argmax(xs >= L * V.mean())]
    p_success = (theta + 1.0) / 2.0
    return theta, p_success
"""
with open("/mnt/data/secretary_algorithms.py", "w") as f:
    f.write(module_source)

print("Standalone code written to /mnt/data/secretary_algorithms.py")