Tipping Point

Author: Ryan Duecker

A lightweight, high-performance marketing intelligence module that uses machine learning and calculus to determine the exact inflection points of a media response curve.

Growth marketers and media buyers constantly ask two questions: "When are we out of the inefficient learning phase?" and "When should we stop scaling spend?" By fitting historical performance data to a continuous mathematical curve, this tool identifies the Minimal Marginal Cost Point (where efficiency peaks) and the Point of Diminishing Returns (where scaling is no longer profitable), defining your exact Optimal Scaling Zone.

🧠 Methodology

This project leverages the mathematical foundations of modern Marketing Mix Modeling (MMM)—specifically the techniques popularized by Google’s Meridian.

Instead of basic linear or logarithmic approximations, this module natively models media saturation using the Hill Function.

$$Return = \frac{\beta \cdot Spend^\alpha}{K^\alpha + Spend^\alpha}$$

• $\beta$ (Beta): The asymptote (maximum possible return/capacity).
• $\alpha$ (Alpha): The shape parameter. If $\alpha > 1$, the curve is S-shaped (featuring a warm-up phase). If $\alpha \le 1$, it strictly exhibits diminishing returns from the start (C-shaped).
• $K$ (Half-Saturation): The spend amount at which you achieve half of the maximum return.

The Calculus Engine

Once the Hill Curve parameters are found, the module uses exact calculus to provide strategic recommendations:

• Marginal ROAS ($f'(x)$): Represents the efficiency of the next dollar spent.
• Minimal Marginal Cost ($f''(x) = 0$): The inflection point of the S-Curve. This is the absolute peak of Marginal ROAS. You should spend at least this much to exit the inefficient warm-up phase.
• Point of Diminishing Returns ($f'(x) = Target_mROAS$): The exact spend level where the efficiency drops below your acceptable baseline unit economics (e.g., Target CPA or Target ROAS constraint).

🚀 Installation & Prerequisites

This module uses tinygrad for ultra-lightweight GPU-accelerated gradient descent, alongside standard scientific libraries.

pip install tippingpt

💻 Usage

1. Fitting from Historical Data

You can pass raw Spend and Return (Revenue, Conversions, etc.) arrays directly into the module. The PyTorch/Tinygrad backend will automatically find the optimal $\beta$, $\alpha$, and $K$ parameters.

import numpy as np
from tippingpoint import MarketingReturnCurve

# 1. Provide your historical marketing data (Spend vs. Revenue/Conversions)
spends = np.array([1200, 5000, 15000, 25000, 40000])
returns = np.array([200, 1500, 12000, 22000, 28000])

# 2. Fit the Curve (MLE)
model = MarketingReturnCurve.from_historical_data(
    spend_array=spends,
    return_array=returns,
    channel_name="YouTube",
    epochs=3000,
    lr=0.05
)

# 3. Fit the Curve (Bayesian MCMC)
# This provides posterior distributions and uncertainty intervals
model_bayesian = MarketingReturnCurve.fit_bayesian(
    spend_array=spends,
    return_array=returns,
    channel_name="YouTube (Bayesian)",
    n_samples=2000,
    chains=4
)

2. Extracting Intelligence & Inflection Points

Once the curve is fitted (or if you manually provide parameters from an existing Meridian model), you can extract actionable business intelligence.

target_mroas = 1.5      # We need at least $1.50 back on the marginal dollar
current_spend = 12000   # Our current daily/weekly budget

# Get precise inflection points
optimal_floor = model.get_minimal_marginal_cost_point()
spend_cap = model.get_diminishing_returns_point(target_mroas)

print(f"Start Scaling At: ${optimal_floor:,.2f}")
print(f"Stop Scaling At: ${spend_cap:,.2f}")

# Access Posterior Samples (if fitted via Bayesian method)
if hasattr(model_bayesian, 'posterior_samples') and model_bayesian.posterior_samples:
    print(f"Mean Alpha: {np.mean(model_bayesian.posterior_samples['alpha'])}")
    # Access full distribution
    beta_samples = model_bayesian.posterior_samples['beta']

# Get a text-based evaluation of your current strategy
model.evaluate_current_budget(current_spend, target_mroas)

Example Output:

--- Budget Evaluation: Paid Social ---
Current Spend: $12,000.00 | Current mROAS: 2.10
Status: OPTIMAL SCALING ZONE
Recommendation: You are operating within the highly efficient growth window.

4. Visualization

Generate an executive-ready, dual-axis chart mapping the Incremental Return curve against the Marginal ROAS curve. If fitted via Bayesian methods, it automatically includes 90% Credible Intervals.

model_bayesian.plot_response_curve(target_mroas=1.5, current_spend=12000)

🛠 Integrating with existing MMMs (Meridian)

If you already run Google Meridian, you do not need to use the from_historical_data method. You can simply extract the posterior mean parameters for a specific channel directly from your Meridian output and initialize the class:

# Assuming you extracted beta, alpha, and K from your Meridian posterior
model = MarketingReturnCurve(beta=100000, alpha=1.8, half_saturation_k=20000)