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

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