Asymmetric Trust Dynamics
Why trust is hard to gain and easy to lose — the gain and loss formulas with real numerical examples.
Asymmetric Trust Dynamics
Trust in BASIS is deliberately asymmetric: it is hard to earn and easy to lose. This mirrors real-world trust — a colleague who is reliable for a year earns your trust gradually, but a single betrayal can destroy it overnight.
This article breaks down the gain and loss formulas with real numbers.
The Gain Formula
gain = gainRate × ln(1 + C - S) × ∛R
Parameters
| Symbol | Meaning | Value / Range |
|------------|--------------------------------|------------------------|
| gainRate | Base gain coefficient | 0.05 (canonical) |
| C | Observation tier ceiling | 600–1000 |
| S | Current trust score | 0–1000 |
| R | Risk multiplier of the action | 1–30 |
| ∛R | Cube root of risk multiplier | 1.0–3.11 |
How It Works
The gain formula has three deliberate properties:
1. Logarithmic decay: ln(1 + C - S) shrinks as S approaches C.
When the agent is far from its ceiling, gains are relatively large.
As it approaches the ceiling, gains diminish toward zero.
S = 200, C = 750: ln(1 + 750 - 200) = ln(551) = 6.31
S = 500, C = 750: ln(1 + 750 - 500) = ln(251) = 5.53
S = 700, C = 750: ln(1 + 750 - 700) = ln(51) = 3.93
S = 740, C = 750: ln(1 + 750 - 740) = ln(11) = 2.40
The closer you are to the ceiling, the harder every point becomes.
2. Sub-linear risk bonus: ∛R gives a bonus for riskier successful
actions, but sub-linearly. This prevents agents from seeking high-risk
actions to accumulate trust faster.
READ (R=1): ∛1 = 1.00
LOW (R=3): ∛3 = 1.44
MEDIUM (R=5): ∛5 = 1.71
HIGH (R=10): ∛10 = 2.15
CRITICAL (R=15): ∛15 = 2.47
LIFE_CRIT(R=30): ∛30 = 3.11
A CRITICAL success earns about 2.5× what a READ success earns — not 15×.
3. Ceiling cap: When S >= C, the ln(1 + C - S) term becomes ln(1)
or less, making gains effectively zero. The observation ceiling is a hard
stop.
Worked Example: Gain Calculation
Agent at score 400, GRAY_BOX observation (ceiling 750), completing a MEDIUM risk action (R=5):
gain = 0.05 × ln(1 + 750 - 400) × ∛5
= 0.05 × ln(351) × 1.71
= 0.05 × 5.86 × 1.71
= 0.50 points
Same agent at score 700:
gain = 0.05 × ln(1 + 750 - 700) × ∛5
= 0.05 × ln(51) × 1.71
= 0.05 × 3.93 × 1.71
= 0.34 points
At score 400, the agent gains 0.50 points. At score 700, the same action earns only 0.34 points. Trust builds slower as the score climbs.
The Loss Formula
loss = -P(T) × R × gainRate × ln(1 + C/2)
Parameters
| Symbol | Meaning | Value / Range |
|------------|----------------------------------|------------------|
| P(T) | Penalty ratio at tier T | 3× (T0) – 10× (T7) |
| R | Risk multiplier of failed action | 1–30 |
| gainRate | Same base coefficient | 0.05 |
| C | Observation tier ceiling | 600–1000 |
| ln(1+C/2)| Fixed midpoint reference | ~5.70–6.22 |
The Penalty Ratio: P(T)
The penalty ratio scales linearly from T0 to T7:
P(T) = penaltyRatioMin + (T/7) × (penaltyRatioMax - penaltyRatioMin)
= 3 + T // with defaults: min=3, max=10
| Tier | P(T) | Meaning | |------|-------|------------------------------------------| | T0 | 3× | Low stakes — limited damage possible | | T1 | 4× | Still learning — moderate penalty | | T2 | 5× | Operating — proportional penalty | | T3 | 6× | Standard — meaningful consequences | | T4 | 7× | Full operations — significant penalty | | T5 | 8× | Trusted — high responsibility | | T6 | 9× | Certified — severe consequences | | T7 | 10× | Autonomous — maximum accountability |
A T7 agent loses more than 3× what a T0 agent loses for the exact same failure. Higher trust means higher accountability.
Fixed Midpoint Reference
The loss formula uses ln(1 + C/2) — a fixed midpoint — instead of the
agent's current position. This is a critical design decision.
If the formula used the agent's current score (like the gain formula does), an agent could minimize loss exposure by hovering near its ceiling. The fixed midpoint eliminates this gaming strategy. A failure at score 900 costs the same as a failure at score 500 (same tier and risk level).
GRAY_BOX (C=750): ln(1 + 375) = ln(376) = 5.93
WHITE_BOX (C=900): ln(1 + 450) = ln(451) = 6.11
VERIFIED_BOX (C=1000): ln(1 + 500) = ln(501) = 6.22
Worked Example: Loss Calculation
T3 agent (P=6), GRAY_BOX observation (C=750), failing a MEDIUM action (R=5):
loss = -6 × 5 × 0.05 × ln(1 + 375)
= -6 × 5 × 0.05 × 5.93
= -8.90 points
Same failure for a T6 agent (P=9):
loss = -9 × 5 × 0.05 × ln(1 + 375)
= -9 × 5 × 0.05 × 5.93
= -13.35 points
The T6 agent loses 50% more than the T3 agent for the same failure.
Asymmetry in Practice
Let us compare gain and loss for identical risk levels:
MEDIUM risk (R=5), GRAY_BOX (C=750), agent at score 500 (T3, P=6)
Gain on success: 0.05 × ln(1 + 750 - 500) × ∛5
= 0.05 × 5.53 × 1.71
= 0.47 points
Loss on failure: -6 × 5 × 0.05 × ln(1 + 375)
= -6 × 5 × 0.05 × 5.93
= -8.90 points
Ratio: The agent loses 18.9× more than it gains for the same risk level. It takes roughly 19 successful MEDIUM actions to recover from a single MEDIUM failure.
HIGH risk (R=10), same agent
Gain on success: 0.05 × ln(251) × ∛10
= 0.05 × 5.53 × 2.15
= 0.59 points
Loss on failure: -6 × 10 × 0.05 × 5.93
= -17.79 points
Ratio: 30× asymmetry. One HIGH failure requires ~30 HIGH successes to recover.
CRITICAL risk (R=15), T5 agent at score 820 (P=8), WHITE_BOX (C=900)
Gain on success: 0.05 × ln(1 + 900 - 820) × ∛15
= 0.05 × ln(81) × 2.47
= 0.05 × 4.39 × 2.47
= 0.54 points
Loss on failure: -8 × 15 × 0.05 × ln(1 + 450)
= -8 × 15 × 0.05 × 6.11
= -36.66 points
Ratio: 67× asymmetry. A T5 agent needs approximately 68 CRITICAL successes to recover from a single CRITICAL failure. This is working as designed.
Why This Design?
The asymmetry serves three purposes:
1. Risk aversion is rational. An agent that understands the trust model will prefer lower-risk actions unless higher risk is necessary. Seeking risk to build trust faster is a losing strategy — the cube-root bonus is too small to offset the amplified losses.
2. Failures are expensive at the top. A T7 Autonomous agent has enormous capability. A failure at that level could cause real harm. The 10× penalty ratio ensures that T7 agents are held to the highest standard.
3. Position cannot be gamed. The fixed midpoint in the loss formula means that an agent cannot reduce its loss exposure by choosing where to sit on the scale. The only way to minimize losses is to not fail.
Try It: Build a Gain/Loss Table
import {
GAIN_RATE,
RISK_LEVELS,
OBSERVATION_TIERS,
PENALTY_RATIO_MIN,
PENALTY_RATIO_MAX,
} from '@vorionsys/basis';
function calculateGain(score: number, ceiling: number, riskMult: number): number {
const headroom = Math.max(0, ceiling - score);
return GAIN_RATE * Math.log(1 + headroom) * Math.cbrt(riskMult);
}
function calculateLoss(tier: number, riskMult: number, ceiling: number): number {
const penalty = PENALTY_RATIO_MIN + tier; // P(T) = 3 + T
return -(penalty * riskMult * GAIN_RATE * Math.log(1 + ceiling / 2));
}
// Compare gain vs loss for each risk level at T3 (score 500), GRAY_BOX
const ceiling = OBSERVATION_TIERS.GRAY_BOX.ceiling;
const score = 500;
const tier = 3;
for (const [name, risk] of Object.entries(RISK_LEVELS)) {
const gain = calculateGain(score, ceiling, risk.multiplier);
const loss = calculateLoss(tier, risk.multiplier, ceiling);
const ratio = Math.abs(loss / gain);
console.log(
`${name.padEnd(14)} | ` +
`gain: +${gain.toFixed(3)} | ` +
`loss: ${loss.toFixed(3)} | ` +
`ratio: ${ratio.toFixed(1)}×`
);
}
Key Takeaways
- Trust gains are logarithmic and capped by the observation ceiling.
- Trust losses use a fixed midpoint — position on the scale does not reduce exposure.
- The penalty ratio scales from 3× (T0) to 10× (T7).
- Risk-seeking is penalized: cube-root gain bonus is far outweighed by linear loss multiplier.
- Real-world asymmetry: 19× for MEDIUM, 30× for HIGH, 67× for CRITICAL.
Next Steps
- Risk Levels — the six risk tiers and their multipliers
- Cooldowns and Circuit Breakers — what happens after failures
- Observation Tiers — how the ceiling is determined