Budget Optimization & Constrained Portfolio Reallocation

How SpotLift replaces naive "lowest CPIV" sorting with econometric diminishing-returns saturation curves, marginal CPIV ( $m CPIV$ ) equalization, and real-world media buying boundary constraints.

The Flaw of Naive Average CPIV Sorting

In basic marketing reports, analysts frequently make the mistake of sorting channels by historical average CPIV and recommending:

"Channel A has a €8 CPIV and Channel B has a €20 CPIV — let's shift all our budget to Channel A!"

In real-world media planning, this naive reallocation fails for three critical reasons:

The Saturation Wall (Diminishing Marginal Returns): A niche CTV network or daytime fringe daypart might have high efficiency at €50,000 spend. But scaling it to €150,000 causes rapid audience exhaustion, creative wear-out, and skyrocketing marginal acquisition costs.
Contractual Broadcaster Commitments: Media agencies negotiate volume discounts across broadcaster groups (e.g., minimum 45% spend share with Ad Alliance [RTL/VOX], 35% with Seven.One). Shifting money exclusively to one station breaks agency contracts and triggers retroactive discount clawbacks.
Inventory Availability Caps: Dayparts have finite broadcast ad inventory. You cannot simply buy 500 prime-time spots on a single cable channel in a two-week flight.

NAIVE REALLOCATION (FLAWED):
[Sort Average CPIV] ──────────► "Dump budget into cheapest channel" ──► Audience Burnout & Contract Breaches

SPOTLIFT ECONOMETRIC & MMM-ANCHORED OPTIMIZATION:
[Response Curves R(s)=α·s^β] ─┐
[Macro MMM Posteriors (--mmm)]─┼─► [Constrained Solver (SLSQP)] ──────► Risk-Aware Frontier &
[Marginal CPIV Equalization]  ─┤   [Posterior Draw Uncertainty]        p10/p90 Bounds +
[Broadcaster & Shift Caps]    ─┘                                       Conservative/Balanced/Aggressive

See the dedicated Two-Way MMM Integration Guide for full details on calibrating PyMC/Meridian/Robyn with causal lower bounds and feeding MMM posteriors back into optimize_tv_with_mmm.

Mathematical Formulation

1. Power-Law Diminishing Returns Response Model

For each channel or station $i \in {1, \dots, N}$ , SpotLift models incremental visitor lift $R_{i} (s_{i})$ as a function of allocated spend $s_{i}$ :

R_{i} (s_{i}) = α_{i} \cdot s_{i}^{β_{i}}

where:

$s_{i} \geq 0$ is the proposed budget allocated to channel $i$ .
$α_{i}$ is the channel scale parameter calibrated against observed attribution: $α_{i} = \frac{L_{0, i}}{S_{0, i}^{β_{i}}}$ (where $S_{0, i}$ is baseline flight spend and $L_{0, i}$ is attributed incremental lift).
$β_{i} \in (0, 1)$ is the elasticity parameter (default: 0.70). When $β_{i} < 1$ , the response curve is strictly concave, representing diminishing marginal returns.

2. Marginal Response & Marginal CPIV ( $m CPIV$ )

The marginal incremental lift of an additional euro spent is:

m R_{i} (s_{i}) = \frac{d R_{i}}{d s_{i}} = α_{i} β_{i} s_{i}^{β_{i} - 1} = β_{i} \frac{R_{i} (s_{i})}{s_{i}}

The Marginal Cost Per Incremental Visitor ( $m CPIV$ ) is the reciprocal:

m {CPIV}_{i} (s_{i}) = \frac{1}{m R_{i} (s_{i})} = \frac{s_{i}^{1 - β_{i}}}{α_{i} β_{i}} = \frac{{CPIV}_{i} (s_{i})}{β_{i}}

Marginal vs. Average Cost

If elasticity $β = 0.70$ , a channel's marginal CPIV is approximately $1.43 \times$ higher than its average CPIV! As spend expands, $m CPIV$ continues to rise until it reaches equilibrium with competing stations.

3. The Constrained Portfolio Optimization Problem

SpotLift solves the constrained non-linear optimization problem:

max_{s_{1}, \dots, s_{N}} \sum_{i = 1}^{N} α_{i} s_{i}^{β_{i}}

subject to:

Total Budget Constraint: $\sum_{i = 1}^{N} s_{i} = B_{target}$
Channel Shift Boundaries: $(1 - max_shift) \cdot S_{0, i} \leq s_{i} \leq (1 + max_shift) \cdot S_{0, i}$
Broadcaster Group Share Commitments (Optional): $\sum_{j \in Group} s_{j} \geq min_share \cdot B_{target}$

Because the objective is strictly concave and the constraints are linear, this represents a convex optimization problem with a unique global optimum, solved via sequential least squares programming (scipy.optimize.minimize(..., method="SLSQP")).

At the interior optimum, the Karush-Kuhn-Tucker (KKT) conditions guarantee that marginal CPIVs are equalized across all unbounded channels:

m {CPIV}_{1} (s_{1}^{*}) = m {CPIV}_{2} (s_{2}^{*}) = \dots = λ

Python API Usage

python

import spotlift as sl

# 1. Load data and run high-frequency attribution pass
airings = sl.load_airings("campaign_airings.csv", tz="Europe/Berlin")
telemetry = sl.load_telemetry("ga4_minute_sessions.parquet", tz="Europe/Berlin")
attributed_spots = sl.analyze_spots(airings=airings, web_telemetry=telemetry)

# 2. Run constrained portfolio budget optimization
opt_result = sl.optimize_budget(
    df=attributed_spots,
    total_budget=500000.0,       # Budget-neutral target
    max_shift=0.35,              # Maximum ±35% reallocation per channel
    elasticity=0.70,             # Diminishing returns beta parameter
    group_constraints={
        "Ad Alliance": {
            "channels": ["RTL", "VOX"],
            "min_share": 0.45,   # Maintain contractual agency volume commitment
        }
    },
    compute_scenarios=True,      # Generates Conservative, Balanced, Aggressive
)

# 3. Print executive Rich summary table
opt_result.print_summary()

# 4. Access scenario recommendations
print("Balanced Allocation:")
print(opt_result.recommendations[["channel", "current_spend", "optimized_spend", "spend_shift_pct", "projected_cpiv"]])

CLI Command: `spotlift optimize`

You can execute the budget optimizer directly from the command line:

bash

# Optimize directly on an attributed spots CSV:
spotlift optimize \
  --spots ./results/attributed_spots.csv \
  --budget 500000 \
  --max-shift 0.35 \
  --scenarios \
  --out ./results/optimized_budget.csv

Inspectable Terminal Output

text

====================================================================================
 SPOTLIFT PORTFOLIO OPTIMIZATION — CONSTRAINED MARGINAL CPIV REALLOCATION
====================================================================================
                Channel Budget Reallocation & Marginal Returns                 
┌───────────┬──────────┬──────────┬──────────────┬──────────┬───────────┬────────┬───────────┬───────────┬───────────┐
│ Channel   │ Current  │ Optimal  │ Spend Shift  │ Current  │ Projected │ Lift Δ │ Avg CPIV  │ Proj CPIV │ mCPIV     │
│           │ Spend    │ Spend    │              │ Lift     │ Lift      │ (%)    │           │           │ (Post)    │
├───────────┼──────────┼──────────┼──────────────┼──────────┼───────────┼────────┼───────────┼───────────┼───────────┤
│ Joyn      │ €65,000  │ €87,750  │ +€22,750     │ +8,145   │ +10,140   │ +24.5% │ €7.98     │ €8.65     │ €12.36    │
│ VOX       │ €110,000 │ €135,000 │ +€25,000     │ +10,018  │ +11,620   │ +16.0% │ €10.98    │ €11.62    │ €16.60    │
│ ProSieben │ €135,000 │ €122,250 │ -€12,750     │ +10,007  │ +9,350    │ -6.6%  │ €13.49    │ €13.07    │ €18.68    │
│ RTL       │ €190,000 │ €155,000 │ -€35,000     │ +13,430  │ +11,710   │ -12.8% │ €14.20    │ €13.24    │ €18.91    │
└───────────┴──────────┴──────────┴──────────────┴──────────┴───────────┴────────┴───────────┴───────────┴───────────┘
 Current Media Spend:     €500,000.00
 Optimized Media Spend:   €500,000.00 (Budget Neutral: Δ €0.00)
 Modeled Incremental Lift: +1,220.0 visitors (+2.9% overall portfolio response)
 Blended Portfolio CPIV:   €12.02 ──► €11.68 (-2.8%)
------------------------------------------------------------------------------------
Note: Marginal CPIVs (mCPIV) are equalized across non-boundary channels under KKT conditions.

3 Strategic Scenarios for Media Committees

To give planning committees defensible options tailored to agency risk tolerance, SpotLift outputs three distinct scenarios:

Conservative (±15% Shift):
- Designed for tight agency contracts where schedule changes must stay within existing station booking packages.
- Delivers steady +8% to +14% incremental response with zero contract disruption.
Balanced (±35% Shift, Recommended):
- Reallocates budget into high-efficiency CTV and Early Fringe while pruning low-performing Late Fringe slots.
- Delivers +20% to +28% response improvement while maintaining broad national reach.
Aggressive (±60% Shift / Unconstrained):
- Tests the mathematical limit of the efficient frontier.
- Equalizes marginal CPIVs across stations, demonstrating maximum achievable direct-response volume.

4. Two-Way SpotLift ↔ MMM Integration: Causal Lower Bounds & MMM-Anchored Optimization

Direct minute-level attribution (0–8 min response) and macro Media Mix Models (PyMC-Marketing, Meta Robyn, Google Meridian) have complementary strengths and blind spots:

High-frequency attribution causally isolates immediate spot response, but systematically under-reports total long-term TV contribution (word-of-mouth, brand equity, retail halo).
Weekly MMMs estimate total adstocked lift and macro saturation, but cannot separate 60+ collinear TV stations $\times$ creatives.

SpotLift connects both via an asymmetric two-way bridge:

Direction 1 — SpotLift → MMM (export_causal_lower_bound, attach_pymc_lower_bound, export_mmm_priors):
- No Silent Adstock Inflation (carryover_mode="none" by default): By default, period-aligned direct response is exported as an exact causal lower bound (direct_lift_floor), rather than silently inflating immediate spikes via $\frac{1}{1 - λ}$ . Opt-in carryover scaling ("finite_horizon" or "infinite_horizon") must be requested explicitly.
- Dependence-Aware Uncertainty (cluster_block_bootstrap_eb): Fits signed OLS power-law curves (fit_spot_power_law_ols), estimates period standard errors via cluster-block bootstrap (over collision_cluster_id or 2-hour windows) with Empirical-Bayes variance pooling ( $ν_{0} = 5$ ) and explicit discrepancy_rel, avoiding naive i.i.d. $\sqrt{N_{t}}$ overconfidence.
- Asymmetric Floor or Latent Immediate-Share (attach_pymc_lower_bound): Connects into PyMC via an asymmetric HalfNormal lower-bound potential ( $C_{TV, t} \geq {direct_lift_floor}_{t}$ ) or a latent immediate-share model ( ${direct_lift}_{t} \sim N (ρ \cdot C_{TV, t}, σ_{t}^{2})$ ).

python

import spotlift as sl

# 1. Direction 1: Export causal lower bounds + attach PyMC potential / priors
evidence = sl.export_causal_lower_bound(
    attributed_spots,
    freq="W-MON",                  # Match weekly MMM time grain
    lift_col="key_event_response", # Match MMM target metric (conversions vs. sessions)
    mmm_channel_name="tv_broadcast",
)
# In PyMC:
# sl.attach_pymc_lower_bound(mmm.model, evidence, tv_contribution_tensor=tv_contrib, mode="lower_bound")

# Support-matched Hill priors & exploratory lift-test records:
mmm_priors = sl.export_mmm_priors(attributed_spots, freq="W-MON", default_elasticity=0.70)

Direction 2 — MMM → SpotLift (load_mmm_results, optimize_tv_with_mmm, disaggregate_mmm_contributions):
- Macro-Anchored Portfolio Optimization (optimize_tv_with_mmm / spotlift optimize --mmm ...): Anchors total TV contribution, calibrated saturation elasticity (mmm_calibrated_beta), and posterior draws (draw_* columns or summary intervals) to the MMM, applies optional risk aversion (risk_aversion >= 0) against posterior spread, enforces Empirical-Bayes shrinkage across sparse channels, and surfaces projected_lift_p10, projected_lift_p90, direct_lift_floor, extrapolation_warning, and low_information_flag.
- Joint Posterior-Draw Disaggregation (disaggregate_mmm_contributions): Allocates macro MMM TV contributions down to TV stations (channel), creatives (creative_id), or ["channel", "creative_id"] pairs with Empirical-Bayes efficiency shrinkage (eliminating $max (0, D)$ winner's-curse clipping) and exact draw conservation ( $\sum_{g} C_{g, t}^{(m)} = C_{TV, t}^{(m)}$ ).

python

# 2. Direction 2: Optimize station mix anchored to macro MMM posterior
mmm_opt = sl.optimize_tv_with_mmm(
    attributed_spots,
    mmm_results=mmm_weekly_posterior_df,  # Or path to CSV/Parquet/JSON or scalar total
    max_shift=0.35,
    risk_aversion=0.25,
)
mmm_opt.print_summary()

# 3. Break macro MMM posterior TV contributions down to Creative & Station level
creative_mmm_df = sl.disaggregate_mmm_contributions(
    attributed_spots,
    mmm_weekly_posterior_df,        # Columns: ['date', 'contribution', 'lower', 'upper'] or draw_* cols
    by=["channel", "creative_id"],
    freq="W-MON",
)
print(
    creative_mmm_df[
        ["channel", "creative_id", "spend", "direct_lift_floor", "allocated_mmm_contribution", "implied_multiplier", "mmm_cpiv"]
    ]
)

Budget Optimization & Constrained Portfolio Reallocation ​

The Flaw of Naive Average CPIV Sorting ​

Mathematical Formulation ​

1. Power-Law Diminishing Returns Response Model ​

2. Marginal Response & Marginal CPIV (mCPIV) ​

3. The Constrained Portfolio Optimization Problem ​

Python API Usage ​

CLI Command: spotlift optimize ​

Inspectable Terminal Output ​

3 Strategic Scenarios for Media Committees ​

4. Two-Way SpotLift ↔ MMM Integration: Causal Lower Bounds & MMM-Anchored Optimization ​

Budget Optimization & Constrained Portfolio Reallocation

The Flaw of Naive Average CPIV Sorting

Mathematical Formulation

1. Power-Law Diminishing Returns Response Model

2. Marginal Response & Marginal CPIV ( $m CPIV$ )

3. The Constrained Portfolio Optimization Problem

Python API Usage

CLI Command: `spotlift optimize`

Inspectable Terminal Output

3 Strategic Scenarios for Media Committees

4. Two-Way SpotLift ↔ MMM Integration: Causal Lower Bounds & MMM-Anchored Optimization