auctiora/docs/VALUATION.md

# Auction Valuation Mathematics - Technical Reference

## 1. Fair Market Value (FMV) - Core Valuation Formula

The baseline valuation is calculated using a **weighted comparable sales approach**:

$$
FMV = \frac{\sum_{i=1}^{n} \left( P_i \cdot \omega_c \cdot \omega_t \cdot \omega_p \cdot \omega_h \right)}{\sum_{i=1}^{n} \left( \omega_c \cdot \omega_t \cdot \omega_p \cdot \omega_h \right)}
$$

**Variables:**
- $P_i$ = Final hammer price of comparable lot *i* (€)
- $\omega_c$ = **Condition weight**: $\exp(-\lambda_c \cdot |C_{target} - C_i|)$
- $\omega_t$ = **Time weight**: $\exp(-\lambda_t \cdot |T_{target} - T_i|)$
- $\omega_p$ = **Provenance weight**: $1 + \delta_p \cdot (P_{target} - P_i)$
- $\omega_h$ = **Historical weight**: $\left( \frac{1}{1 + e^{-kh \cdot (D_i - D_{median})}} \right)$

**Parameter Definitions:**
- $C \in [0, 10]$ = Condition score (10 = perfect)
- $T$ = Manufacturing year
- $P \in \{0,1\}$ = Provenance flag (1 = documented history)
- $D_i$ = Days since comparable sale
- $\lambda_c = 0.693$ = Condition decay constant (50% weight at 1-point difference)
- $\lambda_t = 0.048$ = Time decay constant (50% weight at 15-year difference)
- $\delta_p = 0.15$ = Provenance premium coefficient
- $kh = 0.01$ = Historical relevance coefficient

---

## 2. Condition Adjustment Multiplier

Normalizes prices across condition states:

$$
M_{cond} = \exp\left( \alpha_c \cdot \sqrt{C_{target}} - \beta_c \right)
$$

**Variables:**
- $\alpha_c = 0.15$ = Condition sensitivity parameter
- $\beta_c = 0.40$ = Baseline condition offset
- $C_{target}$ = Target lot condition score

**Interpretation:**
- $C = 10$ (mint): $M_{cond} = 1.48$ (48% premium over poor condition)
- $C = 5$ (average): $M_{cond} = 0.91$

---

## 3. Time-Based Depreciation Model

For equipment/machinery with measurable lifespan:

$$
V_{age} = V_{new} \cdot \left( 1 - \gamma \cdot \ln\left( 1 + \frac{Y_{current} - Y_{manu}}{Y_{expected}} \right) \right)
$$

**Variables:**
- $V_{new}$ = Original market value (€)
- $\gamma = 0.25$ = Depreciation aggressivity factor
- $Y_{current}$ = Current year
- $Y_{manu}$ = Manufacturing year
- $Y_{expected}$ = Expected useful life span (years)

**Example:** 10-year-old machinery with 25-year expected life retains 85% of value.

---

## 4. Provenance Premium Calculation

$$
\Delta_{prov} = V_{base} \cdot \left( \eta_0 + \eta_1 \cdot \ln(1 + N_{docs}) \right)
$$

**Variables:**
- $V_{base}$ = Base valuation without provenance (€)
- $N_{docs}$ = Number of verifiable provenance documents
- $\eta_0 = 0.08$ = Base provenance premium (8%)
- $\eta_1 = 0.035$ = Marginal document premium coefficient

---

## 5. Undervaluation Detection Score

Critical for identifying mispriced opportunities:

$$
U_{score} = \frac{FMV - P_{current}}{FMV} \cdot \sigma_{market} \cdot \left( 1 + \frac{B_{velocity}}{B_{threshold}} \right) \cdot \ln\left( 1 + \frac{W_{watch}}{W_{bid}} \right)
$$

**Variables:**
- $P_{current}$ = Current bid price (€)
- $\sigma_{market} \in [0,1]$ = Market volatility factor (from indices)
- $B_{velocity}$ = Bids per hour (bph)
- $B_{threshold} = 10$ bph = High-velocity threshold
- $W_{watch}$ = Watch count
- $W_{bid}$ = Bid count

**Trigger condition:** $U_{score} &gt; 0.25$ (25% undervaluation) with confidence &gt; 0.70

---

## 6. Bid Velocity Indicator (Competition Heat)

Measures real-time competitive intensity:

$$
\Lambda_b(t) = \frac{dB}{dt} \cdot \exp\left( -\lambda_{cool} \cdot (t - t_{last}) \right)
$$

**Variables:**
- $\frac{dB}{dt}$ = Bid frequency derivative (bids/minute)
- $\lambda_{cool} = 0.1$ = Cool-down decay constant
- $t_{last}$ = Timestamp of last bid (minutes)

**Interpretation:**
- $\Lambda_b &gt; 5$ = **Hot lot** (bidding war likely)
- $\Lambda_b &lt; 0.5$ = **Cold lot** (potential sleeper)

---

## 7. Final Price Prediction Model

Composite machine learning-style formula:

$$
\hat{P}_{final} = FMV \cdot \left( 1 + \epsilon_{bid} + \epsilon_{time} + \epsilon_{comp} \right)
$$

**Error Components:**

- **Bid momentum error**:
  $$\epsilon_{bid} = \tanh\left( \phi_1 \cdot \Lambda_b - \phi_2 \cdot \frac{P_{current}}{FMV} \right)$$

- **Time-to-close error**:
  $$\epsilon_{time} = \psi \cdot \exp\left( -\frac{t_{close}}{30} \right)$$

- **Competition error**:
  $$\epsilon_{comp} = \rho \cdot \ln\left( 1 + \frac{W_{watch}}{50} \right)$$

**Parameters:**
- $\phi_1 = 0.15$, $\phi_2 = 0.10$ = Bid momentum coefficients
- $\psi = 0.20$ = Time pressure coefficient
- $\rho = 0.08$ = Competition coefficient
- $t_{close}$ = Minutes until close

**Confidence interval**:
$$
CI_{95\%} = \hat{P}_{final} \pm 1.96 \cdot \sigma_{residual}
$$

---

## 8. Bidding Strategy Recommendation Engine

Optimal max bid and timing:

$$
S_{max} =
\begin{cases}
FMV \cdot (1 - \theta_{agg}) & \text{if } U_{score} &gt; 0.20 \\
FMV \cdot (1 + \theta_{cons}) & \text{if } \Lambda_b &gt; 3 \\
\hat{P}_{final} - \delta_{margin} & \text{otherwise}
\end{cases}
$$

**Variables:**
- $\theta_{agg} = 0.10$ = Aggressive buyer discount target (10% below FMV)
- $\theta_{cons} = 0.05$ = Conservative buyer overbid tolerance
- $\delta_{margin} = €50$ = Minimum margin below predicted final

**Timing function**:
$$
t_{optimal} = t_{close} - \begin{cases}
5 \text{ min} & \text{if } \Lambda_b &lt; 1 \\
30 \text{ sec} & \text{if } \Lambda_b &gt; 5 \\
10 \text{ min} & \text{otherwise}
\end{cases}
$$

---

## Variable Reference Table

| Symbol | Variable | Unit | Data Source |
|--------|----------|------|-------------|
| $P_i$ | Comparable sale price | € | `bid_history.final` |
| $C$ | Condition score | [0,10] | Image analysis + text parsing |
| $T$ | Manufacturing year | Year | Lot description extraction |
| $W_{watch}$ | Number of watchers | Count | Page metadata |
| $\Lambda_b$ | Bid velocity | bids/min | `bid_history.timestamp` diff |
| $t_{close}$ | Time until close | Minutes | `lots.closing_time` - NOW() |
| $\sigma_{market}$ | Market volatility | [0,1] | `market_indices.price_change_30d` |
| $N_{docs}$ | Provenance documents | Count | PDF link analysis |
| $B_{velocity}$ | Bid acceleration | bph² | Second derivative of $\Lambda_b$ |

---

## Backend Implementation (Quarkus Pseudo-Code)

```java
@Inject
MLModelService mlModel;

public Valuation calculateFairMarketValue(Lot lot) {
    List&lt;Comparable&gt; comparables = db.findComparables(lot, minSimilarity=0.75, limit=20);

    double weightedSum = 0.0;
    double weightSum = 0.0;

    for (Comparable comp : comparables) {
        double wc = Math.exp(-0.693 * Math.abs(lot.getConditionScore() - comp.getConditionScore()));
        double wt = Math.exp(-0.048 * Math.abs(lot.getYear() - comp.getYear()));
        double wp = 1 + 0.15 * (lot.hasProvenance() ? 1 : 0 - comp.hasProvenance() ? 1 : 0);

        double weight = wc * wt * wp;
        weightedSum += comp.getFinalPrice() * weight;
        weightSum += weight;
    }

    double fm v = weightSum &gt; 0 ? weightedSum / weightSum : lot.getEstimatedMin();

    // Apply condition multiplier
    fm v *= Math.exp(0.15 * Math.sqrt(lot.getConditionScore()) - 0.40);

    return new Valuation(fm v, calculateConfidence(comparables.size()));
}

public BiddingStrategy getBiddingStrategy(String lotId) {
    var lot = db.getLot(lotId);
    var bidHistory = db.getBidHistory(lotId);
    var watchers = lot.getWatchCount();

    // Analyze patterns
    boolean isSnipeTarget = watchers &gt; 50 && bidHistory.size() &lt; 5;
    boolean hasReserve = lot.getReservePrice() &gt; 0;
    double bidVelocity = calculateBidVelocity(bidHistory);

    // Strategy recommendation
    String strategy = isSnipeTarget ? "SNIPING_DETECTED" :
                    (hasReserve && lot.getCurrentBid() &lt; lot.getReservePrice() * 0.9) ? "RESERVE_AVOID" :
                    bidVelocity &gt; 5.0 ? "AGGRESSIVE_COMPETITION" : "STANDARD";

    return new BiddingStrategy(
        strategy,
        calculateRecommendedMax(lot),
        isSnipeTarget ? "FINAL_30_SECONDS" : "FINAL_10_MINUTES",
        getCompetitionLevel(watchers, bidHistory.size())
    );
}
```
```sqlite
-- Core bidding intelligence
ALTER TABLE lots ADD COLUMN starting_bid DECIMAL(12,2);
ALTER TABLE lots ADD COLUMN estimated_min DECIMAL(12,2);
ALTER TABLE lots ADD COLUMN estimated_max DECIMAL(12,2);
ALTER TABLE lots ADD COLUMN reserve_price DECIMAL(12,2);
ALTER TABLE lots ADD COLUMN watch_count INTEGER DEFAULT 0;
ALTER TABLE lots ADD COLUMN first_bid_time TEXT;
ALTER TABLE lots ADD COLUMN last_bid_time TEXT;
ALTER TABLE lots ADD COLUMN bid_velocity DECIMAL(5,2);

-- Bid history (critical)
CREATE TABLE bid_history (
    id INTEGER PRIMARY KEY AUTOINCREMENT,
    lot_id TEXT REFERENCES lots(lot_id),
    bid_amount DECIMAL(12,2) NOT NULL,
    bid_time TEXT NOT NULL,
    is_winning BOOLEAN DEFAULT FALSE,
    is_autobid BOOLEAN DEFAULT FALSE,
    bidder_id TEXT,
    created_at TEXT DEFAULT CURRENT_TIMESTAMP
);

-- Valuation support
ALTER TABLE lots ADD COLUMN condition_score DECIMAL(3,2);
ALTER TABLE lots ADD COLUMN year_manufactured INTEGER;
ALTER TABLE lots ADD COLUMN provenance TEXT;

CREATE TABLE comparable_sales (
    id INTEGER PRIMARY KEY AUTOINCREMENT,
    lot_id TEXT REFERENCES lots(lot_id),
    comparable_lot_id TEXT,
    similarity_score DECIMAL(3,2),
    price_difference_percent DECIMAL(5,2)
);

CREATE TABLE market_indices (
    category TEXT NOT NULL,
    manufacturer TEXT,
    avg_price DECIMAL(12,2),
    price_change_30d DECIMAL(5,2),
    PRIMARY KEY (category, manufacturer)
);

-- Alert system
CREATE TABLE price_alerts (
    id INTEGER PRIMARY KEY AUTOINCREMENT,
    lot_id TEXT REFERENCES lots(lot_id),
    alert_type TEXT CHECK(alert_type IN ('UNDervalued', 'ACCELERATING', 'RESERVE_IN_SIGHT')),
    trigger_price DECIMAL(12,2),
    is_triggered BOOLEAN DEFAULT FALSE
);

```