Mathematical Models

Stochastic Warfare uses 11 mathematical and stochastic models to drive simulation behavior. Each model is grounded in established operations research, signal processing, or military science literature.

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1. Markov Chains

What It Models

State transitions where the next state depends only on the current state (memoryless property). Used for weather evolution and morale state transitions.

Key Formula

$$P(\text{next state}) = \text{transition_matrix}[\text{current state}]$$

The transition matrix T is a row-stochastic matrix where T[i][j] gives the probability of transitioning from state i to state j.

Weather Example

Weather has 4 states: CLEAR, OVERCAST, RAIN, FOG. Each tick, the weather engine samples from the current row of the transition matrix:

         CLEAR  OVERCAST  RAIN   FOG
CLEAR    [0.85   0.10    0.03   0.02]
OVERCAST [0.15   0.70    0.12   0.03]
RAIN     [0.05   0.20    0.70   0.05]
FOG      [0.10   0.15    0.05   0.70]

If current weather is CLEAR, there's an 85% chance it stays clear, 10% chance of overcast, etc.

Morale (Continuous-Time)

Morale uses a continuous-time Markov chain with 5 states: CONFIDENT, STEADY, SHAKEN, BROKEN, ROUTED. Transition rates (not probabilities) define how quickly units move between states based on combat stress, casualties, leadership, and cohesion.

Where Used

• environment/weather.py -- weather state evolution
• morale/state.py -- unit morale transitions

Key Parameters

• transition_matrix -- state transition probabilities (weather)
• rate_matrix -- continuous-time transition rates (morale)
• Morale modifiers: casualty rate, leadership quality, cohesion, suppression

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2. Monte Carlo Methods

What It Models

Statistical estimation of outcomes through repeated random sampling. Used for engagement validation and campaign outcome distributions.

Key Formula

Run N independent simulations, collect metric X from each:

$$\bar{X} = \frac{1}{N}\sum_{i=1}^{N} X_i, \quad \text{CI}_{95\%} = \bar{X} \pm 1.96 \frac{s}{\sqrt{N}}$$

Example

To validate the 73 Easting scenario, run 100 iterations with different seeds. Collect the exchange ratio (blue kills / red kills) from each. Compare the mean and 95% confidence interval against the historical outcome (~0.1 loss ratio for blue).

Where Used

• validation/monte_carlo.py -- MonteCarloHarness for batch runs
• validation/campaign_validation.py -- campaign-level MC validation

Key Parameters

• num_iterations -- number of independent runs (default: 100)
• base_seed -- starting seed (each iteration uses base_seed + i)
• max_ticks -- per-run tick limit

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3. Kalman Filter

What It Models

Optimal estimation of enemy position and velocity from noisy sensor measurements. Tracks a 4-state vector [x, y, vx, vy] (position + velocity in 2D).

Key Formulas

Predict step (between measurements):

$$\hat{x}{k|k-1} = F \hat{x} F^T + Q$$}, \quad P_{k|k-1} = F P_{k-1|k-1

Update step (when a measurement arrives):

$$K = P_{k|k-1} H^T (H P_{k|k-1} H^T + R)^{-1}$$ $$\hat{x}{k|k} = \hat{x}} + K(z_k - H\hat{x{k|k-1})$$ $$P$$} = (I - KH) P_{k|k-1

Where:

• F -- state transition matrix (constant velocity model)
• P -- state covariance (uncertainty)
• Q -- process noise covariance
• H -- measurement matrix (extracts position from state)
• R -- measurement noise covariance (sensor accuracy)
• K -- Kalman gain (how much to trust new measurement vs prediction)
• z -- measurement vector

Track Association

New measurements are associated with existing tracks using Mahalanobis distance gating:

$$d_M = \sqrt{(\mathbf{z} - H\hat{\mathbf{x}})^T S^{-1} (\mathbf{z} - H\hat{\mathbf{x}})}$$

where S = H P H^T + R is the innovation covariance. Measurements with d_M below the gating threshold are associated; others start new tracks.

Where Used

• detection/estimation.py -- KalmanTracker for all sensor types

Key Parameters

• process_noise_std -- expected target acceleration noise
• measurement_noise_std -- sensor measurement accuracy
• gate_threshold -- Mahalanobis distance for track association (default: 3.0)
• max_coasts -- ticks without measurement before track is dropped

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4. Poisson Process

What It Models

Random events occurring at a constant average rate. Used for equipment breakdown and maintenance scheduling.

Key Formula

Probability of at least one failure in time interval dt, given mean time between failures (MTBF):

$$P(\text{fail}) = 1 - e^{-dt / \text{MTBF}}$$

Example

A tank engine has MTBF = 500 hours. In a 1-hour tick:

$$P(\text{fail}) = 1 - e^{-1/500} = 1 - 0.998 = 0.002$$

So about 0.2% chance of breakdown per hour. Over a 72-hour campaign, the probability of at least one breakdown is:

$$P(\text{any fail}) = 1 - e^{-72/500} = 0.134$$

Where Used

• logistics/maintenance.py -- equipment failure scheduling

Key Parameters

• mtbf_hours -- mean time between failures (per equipment type)
• Modifiers: operating tempo, terrain difficulty, weather severity

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5. M/M/c Queue (Medical Evacuation)

What It Models

Priority-based service with multiple servers. Used for medical evacuation where casualties arrive stochastically and medical facilities have limited capacity.

Key Concept

• M/M/c: Markovian arrivals, Markovian service times, c servers
• Casualties are prioritized by severity (T1 urgent > T2 delayed > T3 minimal)
• When the queue exceeds capacity, treatment quality degrades
• Service time is exponentially distributed with mean depending on injury severity

Queue Dynamics

Arrival rate lambda depends on combat intensity. Service rate mu depends on facility type and staffing. With c treatment slots:

$$\rho = \frac{\lambda}{c \cdot \mu}$$

When utilization rho > 1, the queue grows and outcomes degrade.

Where Used

• logistics/medical.py -- medical evacuation and treatment

Key Parameters

• capacity -- number of treatment slots (c)
• service_rate -- patients per hour per slot
• Priority classes: T1 (immediate), T2 (delayed), T3 (minimal), T4 (expectant)
• Overwhelm degradation factor when utilization > threshold

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6. SNR-Based Detection (erfc)

What It Models

Unified probability of detection across all sensor types (visual, thermal, radar, acoustic, sonar). Uses signal-to-noise ratio (SNR) to compute detection probability via the complementary error function.

Key Formula

$$P_d = 0.5 \times \text{erfc}\left(-\frac{\text{SNR} - \text{threshold}}{\sqrt{2}}\right)$$

SNR is computed from:

$$\text{SNR}{\text{dB}} = S$$}} - L_{\text{path}} - N_{\text{noise}} + G_{\text{sensor}

Where:

• S_source -- target signature strength (visual, thermal, RCS, acoustic)
• L_path -- propagation loss (range, atmosphere, terrain)
• N_noise -- ambient noise floor (weather, clutter, electronic)
• G_sensor -- sensor gain (aperture, integration time, processing)

Example

A tank (thermal signature 25 dBW) at 3km, atmospheric loss 15 dB, noise floor 5 dB, sensor gain 10 dB:

$$\text{SNR} = 25 - 15 - 5 + 10 = 15 \text{ dB}$$

With threshold = 12 dB:

$$P_d = 0.5 \times \text{erfc}\left(-\frac{15 - 12}{\sqrt{2}}\right) = 0.5 \times \text{erfc}(-2.12) \approx 0.983$$

Where Used

• detection/detection.py -- unified DetectionEngine for all sensor types
• detection/sonar.py -- underwater acoustic detection

Key Parameters

• Signature profiles (per unit, per modality)
• Sensor specifications (sensitivity, range, FOV)
• Environmental conditions (weather, time of day, sea state)
• detection_threshold_db -- SNR threshold for detection

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7. Lanchester Attrition Models

What It Models

Analytical force-on-force attrition for COA (Course of Action) wargaming. Provides quick comparative estimates without running full tactical simulation.

Key Formulas

Square Law (aimed fire, modern combat):

$$\frac{dB}{dt} = -\beta \cdot R, \quad \frac{dR}{dt} = -\alpha \cdot B$$

Both sides attrit the opposing force proportional to their own strength.

Linear Law (area fire, indirect fire):

$$\frac{dB}{dt} = -\beta \cdot R \cdot B, \quad \frac{dR}{dt} = -\alpha \cdot B \cdot R$$

Attrition proportional to the product of both forces.

Lanchester Square Law Victory Condition

Blue wins if: $\alpha B_0^2 > \beta R_0^2$

The force ratio matters quadratically -- doubling your force is 4x as effective, not 2x.

Where Used

• ai/coa.py -- COA comparative wargaming
• Commander AI uses Lanchester outcomes to rank COA alternatives

Key Parameters

• alpha, beta -- per-unit attrition rates (derived from weapon effectiveness)
• B_0, R_0 -- initial force strengths
• Force type multipliers (armor vs infantry vs artillery)

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8. Wayne Hughes Salvo Model

What It Models

Naval missile exchange between surface combatants. Models the attacker's salvo against the defender's defensive systems to compute "leakers" (missiles that penetrate defenses).

Key Formula

$$\text{leakers} = \max(0, \alpha \cdot A - \beta \cdot D)$$

Where:

• alpha -- offensive missiles per attacking ship
• A -- number of attacking ships
• beta -- defensive intercepts per defending ship
• D -- number of defending ships

Each leaker that hits causes damage proportional to missile lethality. Ships absorb damage up to their staying power.

Example

4 attacking ships fire 8 missiles each (alpha=8). 3 defending ships intercept 6 missiles each (beta=6):

$$\text{leakers} = \max(0, 8 \times 4 - 6 \times 3) = \max(0, 32 - 18) = 14$$

14 missiles get through defenses.

Where Used

• combat/naval_surface.py -- surface naval engagement resolution

Key Parameters

• missiles_per_salvo -- offensive firepower per ship
• defensive_capacity -- intercepts per ship (SAMs + CIWS)
• missile_pk -- probability of kill per leaker
• ship_staying_power -- damage absorption before mission kill

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9. Boyd OODA Loop

What It Models

Commander decision-making as a finite state machine cycling through four phases: Observe, Orient, Decide, Act. The speed of this cycle relative to the opponent is a key combat multiplier.

Key Concept

Each phase has a base duration modified by:

$$t_{\text{phase}} = t_{\text{base}} \times f_{\text{echelon}} \times f_{\text{comms}} \times f_{\text{school}} \times X_{\text{lognormal}}$$

Where:

• t_base -- base phase duration (seconds)
• f_echelon -- echelon multiplier (higher echelons are slower)
• f_comms -- communications quality multiplier
• f_school -- doctrinal school modifier (e.g., Maneuver school is faster in DECIDE)
• X_lognormal -- stochastic friction (log-normal noise)

Decision Quality

The ORIENT phase produces a situation assessment that feeds into DECIDE. Assessment quality depends on:

• Intelligence completeness (fog of war)
• Commander personality traits (risk tolerance, aggression)
• Doctrinal school biases (what the commander considers important)

Where Used

• ai/ooda.py -- OODAEngine finite state machine
• ai/commander.py -- CommanderEngine personality effects
• ai/schools/ -- 9 doctrinal school implementations

Key Parameters

• Phase durations per echelon level
• Commander personality traits (risk_tolerance, aggression, adaptability)
• School-specific weight overrides for assessment factors
• Lognormal friction parameters (mu, sigma)

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10. Beer-Lambert Law (DEW)

What It Models

Atmospheric transmittance for directed energy weapons (lasers, high-power microwaves). Determines how much beam power reaches the target after atmospheric absorption and scattering.

Key Formula

$$\tau = e^{-\alpha \cdot R}$$

Where:

• tau -- transmittance (fraction of power reaching target, 0 to 1)
• alpha -- atmospheric extinction coefficient (per meter)
• R -- range to target (meters)

The extinction coefficient depends on:

• Wavelength
• Weather conditions (humidity, rain, fog, dust)
• Altitude (atmospheric density)

Laser Damage

Effective power on target:

$$P_{\text{target}} = P_{\text{source}} \times \tau \times \eta_{\text{optics}}$$

Damage occurs when power density exceeds the target's damage threshold for a sufficient dwell time.

Where Used

• combat/directed_energy.py -- DEWEngine for laser and HPM weapons

Key Parameters

• beam_power_kw -- source power in kilowatts
• extinction_coefficient -- atmospheric attenuation rate
• dwell_time_s -- time on target
• damage_threshold_kw_cm2 -- target material damage threshold
• Weather modifiers (rain increases extinction dramatically)

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11. Calibration Methodology (Dupuy CEV)

What It Models

Per-scenario calibration of combat effectiveness using Dupuy's Combat Effectiveness Value (CEV) concept from Numbers, Predictions, and War (1979). CEV captures the aggregate quality advantage of one force over another — encompassing training, morale, leadership, technology, tactical situation, and terrain advantage as a single scalar multiplier.

Key Concept

Each side in a scenario has a force_ratio_modifier in calibration_overrides. This is the CEV scalar:

calibration_overrides:
  french_force_ratio_modifier: 2.5   # French CEV
  coalition_force_ratio_modifier: 1.0  # Coalition CEV (baseline)

The ratio of CEVs (here 2.5:1) determines how effectively each side converts its numerical strength into combat power. A CEV of 2.5 means that side fights as if it had 2.5x its actual numbers.

Calibration Parameters

The full calibration toolkit includes:

Parameter	Purpose	Typical Range
force_ratio_modifier	Per-side CEV (Dupuy quality scalar)	0.4 – 3.0
target_size_modifier_{side}	Per-side vulnerability (formation density, exposure)	0.3 – 3.0
hit_probability_modifier	Global Pk scaling	0.5 – 1.8
destruction_threshold	Fraction of force lost to trigger force_destroyed	0.25 – 0.7
morale_degrade_rate_modifier	Speed of morale degradation	0.3 – 3.0
morale_casualty_weight	How much casualties affect morale	0.4 – 0.9
target_side	Which side's losses trigger force_destroyed	side name
count_disabled	Include DISABLED units in loss calculation	true/false

Per-Scenario CEV Table

Scenario	Winner CEV	Loser CEV	Ratio	Source
Austerlitz	2.5 (French)	1.0 (Coalition)	2.5:1	Chandler, Campaigns of Napoleon
Trafalgar	2.5 (British)	0.6 (French-Spanish)	4.2:1	Lambert, Nelson
Agincourt	3.0 (English)	0.4 (French)	7.5:1	Barker, Agincourt; Keegan, Face of Battle
Cannae	3.0 (Carthaginian)	0.5 (Roman)	6.0:1	Goldsworthy, Cannae
Salamis	3.0 (Greek)	0.4 (Persian)	7.5:1	Strauss, Battle of Salamis
Midway	3.0 (USN)	0.5 (IJN)	6.0:1	Parshall & Tully, Shattered Sword
Hastings	1.5 (Norman)	1.2 (Saxon)	1.25:1	Gravett, Hastings 1066

Note on Circular Calibration

CEV values are calibrated to produce correct historical outcomes — the same outcomes they are derived from. This circularity is acceptable for historical validation (confirming the engine can reproduce known results with plausible parameters) but does not constitute predictive validation. Predictive validation requires testing against engagements not used for calibration, which is done via the Monte Carlo regression suite with multiple seeds.

Where Used

• calibration_overrides in every scenario YAML
• simulation/calibration.py — CalibrationSchema pydantic model
• simulation/battle.py — applied during engagement resolution

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Summary Table

Model	Module	Purpose	Key Formula
Markov Chains	weather, morale	State transitions	P(next) = T[current]
Monte Carlo	validation	Statistical outcome estimation	N runs, aggregate stats
Kalman Filter	detection	Enemy state tracking	Predict-update with gating
Poisson Process	logistics	Equipment breakdown	P(fail) = 1 - exp(-dt/MTBF)
M/M/c Queue	logistics	Medical evacuation	Priority queue, overwhelm degradation
SNR Detection	detection	Unified sensor Pd	Pd = 0.5 * erfc(-(SNR-thresh)/sqrt(2))
Lanchester	ai/planning	COA wargaming	dF/dt = -b * E (square law)
Wayne Hughes	combat	Naval missile exchange	leakers = max(0, aA - bD)
Boyd OODA	ai	Commander decision cycle	phase_time * modifiers * lognormal
Beer-Lambert	combat	DEW transmittance	tau = exp(-alpha * R)
Dupuy CEV	calibration	Per-scenario force quality	force_ratio_modifier per side