algo-price-bundle

# Bundle Pricing Strategy

## Overview

Bundle pricing sells multiple products together at a combined price, extracting consumer surplus by averaging valuations across products. Works when customers have heterogeneous, negatively correlated valuations. Three types: pure bundling (bundle only), mixed bundling (bundle + individual), unbundling.

## When to Use

**Trigger conditions:**
- Deciding whether to bundle products/services together
- Setting bundle price relative to individual prices
- Analyzing whether a current bundle should be unbundled

**When NOT to use:**
- When products have independent demand with no valuation correlation (bundling adds no value)
- When regulations prohibit tying arrangements

## Algorithm

```
IRON LAW: Bundling Increases Profit ONLY With NEGATIVELY CORRELATED Valuations
If ALL customers value the same items highly, bundling adds no surplus.
Bundling works when: Customer A values Product 1 high + Product 2 low,
while Customer B values Product 1 low + Product 2 high. The bundle
price captures both at a middle price neither would pay for their
low-value item alone.
```

### Phase 1: Input Validation
Collect: individual product valuations (or willingness to pay) per customer segment. Compute correlation of valuations across products.
**Gate:** Valuation data available, correlation is negative or mixed.

### Phase 2: Core Algorithm
1. Compute optimal individual prices: maximize Σ(revenue per product)
2. Compute optimal bundle price: find price that maximizes bundle revenue given joint valuation distribution
3. Compare: pure bundling revenue, mixed bundling revenue, individual pricing revenue
4. Mixed bundling: set bundle price < sum of individual prices; discount = bundle incentive

### Phase 3: Verification
Check: mixed bundling should weakly dominate both pure bundling and individual pricing (Adams & Yellen, 1976). If not, review valuation assumptions.
**Gate:** Mixed bundling profit ≥ max(pure bundling, individual pricing).

### Phase 4: Output
Return optimal pricing strategy with profit projections.

## Output Format

```json
{
  "recommendation": "mixed_bundling",
  "prices": {"product_a": 299, "product_b": 199, "bundle_ab": 399},
  "profit_comparison": {"individual": 45000, "pure_bundle": 48000, "mixed_bundle": 52000},
  "metadata": {"segments": 3, "valuation_correlation": -0.35}
}
```

## Examples

### Sample I/O
**Input:** Product A (WTP: Seg1=$80, Seg2=$30), Product B (WTP: Seg1=$30, Seg2=$70). Each segment has 100 customers.
**Expected:** Individual optimal: A=$80, B=$70, revenue=$15K. Bundle at $100: both segments buy, revenue=$20K. Bundling wins.

### Edge Cases
| Input | Expected | Why |
|-------|----------|-----|
| Perfectly positive correlation | Individual pricing wins | All customers value both high or both low |
| One product is free good | Bundle = premium + free | Common in software (free trial + paid add-on) |
| 10+ products in bundle | Mixed bundling complex | Too many combinations — use tiered bundles |

## Gotchas

- **Cannibalization**: The bundle may cannibalize high-WTP customers who would have bought individually at higher total. Mixed bundling mitigates this.
- **Perceived value**: Bundle discount must be salient. A $499 bundle of $299+$299 products (16% off) is better perceived than $499 for two $260 products.
- **Marginal cost matters**: Zero marginal cost products (software, digital) benefit most from bundling. Physical goods with high COGS have tighter margins.
- **Complexity cost**: Too many bundle options create choice paralysis. Limit to 2-3 bundle tiers.
- **Regulatory tying**: In some markets, forcing purchase of one product to get another is illegal (antitrust). Ensure bundle is a discount, not a requirement.

## References

- For Adams-Yellen bundling theory, see `references/bundling-theory.md`
- For multi-product pricing optimization, see `references/multi-product-pricing.md`