Auctions

First price sealed bid auction as number of agents increases

def optimal_bid_first_price(v, n, valuation_distribution):

    """

    Calculates the optimal bid for a bidder with valuation v in a first-price sealed-bid auction

    with n bidders, based on the integral formula for the expected bid.

    Parameters:

        v (float): Bidder's valuation.

        n (int): Number of bidders.

        valuation_distribution: The valuation distribution (assumed to be a scipy.stats distribution).

    Returns:

        float: Optimal bid for the given valuation.

    """

    # Check for edge cases where cdf(v) might be zero to avoid division errors

    if valuation_distribution.cdf(v) <= 1e-8:

        return 0

    # Define the integrand function with error handling for potential issues at t=0

    def integrand(t):

        cdf_v = valuation_distribution.cdf(v)

        cdf_t = valuation_distribution.cdf(t)

        if cdf_v > 0:

            return (cdf_t ** (n - 1)) / (cdf_v ** (n - 1))

        else:

            return 0

    # Perform the integral with error handling for bad behavior

    try:

        integral_value, _ = quad(integrand, 0, v)

    except Exception as e:

        print(f"Integration error at valuation {v}: {e}")

        integral_value = 0

    # Calculate the optimal bid using the formula

    optimal_bid_value = v - integral_value

    return optimal_bid_value