Ad Budget Allocation Optimization

Overview

Budget allocation distributes a total advertising budget across campaigns to maximize overall returns. Uses the equal marginal returns principle: allocate until the marginal CPA (or marginal ROAS) is equalized across all campaigns. Handles diminishing returns and budget constraints.

When to Use

Trigger conditions:

• Distributing a fixed budget across multiple campaigns or channels
• Identifying diminishing returns and optimal spend levels per campaign
• Rebalancing budget after performance changes

When NOT to use:

• When optimizing bids within a single campaign (use bidding strategy)
• When there's only one campaign (nothing to allocate across)

Algorithm

IRON LAW: Equal Marginal Returns Principle
Optimal allocation makes the MARGINAL return of the last dollar
equal across ALL campaigns. If Campaign A's marginal CPA is $5
and Campaign B's is $15, shift budget from B to A until they equalize.
Total budget constraint: Σ budget_i = total_budget.

Phase 1: Input Validation

Collect per-campaign: historical spend, conversions, revenue at multiple spend levels. Need at least 3 data points per campaign to fit response curve. Gate: Sufficient historical data to estimate response curves.

Phase 2: Core Algorithm

1. Fit response curve per campaign: conversions = f(spend). Common models: log curve, power curve, or S-curve
2. Compute marginal return curve: f'(spend) for each campaign
3. Allocate: use Lagrangian optimization or iterative greedy — assign next marginal dollar to campaign with highest marginal return
4. Apply constraints: minimum spend floors, maximum caps, channel-specific rules

Phase 3: Verification

Check: total allocation = total budget, no campaign below floor or above cap, marginal returns approximately equal at boundaries. Gate: Allocation sums to budget, constraints satisfied.

Phase 4: Output

Return allocation table with expected performance projections.

Output Format

{
  "allocation": [{"campaign": "Search-Brand", "budget": 50000, "expected_conversions": 200, "expected_cpa": 250}],
  "total": {"budget": 200000, "expected_conversions": 650, "blended_cpa": 308},
  "metadata": {"optimization_method": "lagrangian", "response_model": "log_curve"}
}

Examples

Sample I/O

Input: Budget: $100K, Campaigns: Search ($50K, 100 conv), Social ($30K, 60 conv), Display ($20K, 20 conv) Expected: Shift budget from Display (high marginal CPA) to Search (low marginal CPA). e.g., Search $60K, Social $30K, Display $10K.

Edge Cases

Input	Expected	Why
One campaign dominates	Most budget to winner	But maintain minimum floor for others
All campaigns saturated	Reduce total spend	Spending more won't help
New campaign, no data	Use minimum test budget	Need data before optimizing

Gotchas

• Response curve extrapolation: Don't optimize beyond observed spend ranges. The curve may change shape at higher spend levels.
• Attribution overlap: Users may see ads across campaigns. Last-click attribution double-counts, inflating high-funnel campaign CPA. Use multi-touch attribution.
• Diminishing returns assumption: Not all campaigns follow smooth diminishing returns. Some have step functions (e.g., reaching a new audience segment at a spend threshold).
• Time dynamics: Response curves shift seasonally and competitively. Refit curves monthly or use rolling windows.
• Minimum viable spend: Each campaign needs enough budget to exit the learning phase. Spreading too thin means no campaign gets sufficient data.

References

• For response curve fitting methods, see references/response-curves.md
• For multi-touch attribution integration, see references/attribution-integration.md