How It Works

🧠 Beacon Algorithm

Welcome to Kuqus, where decision-making is fast, intuitive, and scientifically precise! Our Beacon Algorithm revolutionizes preference ranking through quick pairwise comparisons—like choosing between "Rendang vs. Sushi." Instead of cumbersome list rankings, you make simple choices, and our algorithm constructs a robust preference graph. By leveraging transitive reduction, reaction time analysis, and smart scoring, Kuqus delivers accurate rankings with fewer comparisons while prioritizing your privacy. Let’s dive into how it works!

📈 The Science of Preferences

The Beacon Algorithm combines graph theory, psychophysical modeling, and statistical ranking to deliver accurate and efficient preference rankings. Here’s the breakdown:

Pairwise Comparisons & Transitive Reduction

When you choose one item (e.g., A) over another (e.g., B), the algorithm records this preference in a transitive graph. Using transitive reduction, we infer indirect preferences (e.g., A > B and B > C implies A > C), minimizing the number of comparisons needed.

Comparison Efficiency:

For $n$ items, a full pairwise comparison requires $\binom{n}{2} = \frac{n(n-1)}{2}$ comparisons.
The Beacon Algorithm achieves:
- Best case: $n-1$ comparisons (e.g., for a linear order like A > B > C).
- Average case: ~44% of $\binom{n}{2}$ for $n = 15$ (e.g., ~15-34 comparisons instead of 105).
- Smaller sets (e.g., $n = 3$ ): ~50% of $\binom{3}{2} = 2..3$ comparisons.
This efficiency is driven by a transitive reduction graph, which removes redundant edges while preserving the preference order:

\text{Edge}(i \to j) \text{ if } i \text{ is directly chosen over } j \\ \text{and no transitive path } i \to k \to j \text{ exists}

Aho, A. V., Garey, M. R., & Ullman, J. D. (1972). The Transitive Reduction of a Directed Graph. SIAM J. on Computing.

Scoring Preferences

After collecting comparisons, the algorithm assigns scores to items based on their rank in a topological sort of the transitive graph:

Initial Scoring: For $n$ items ranked A, B, C, … (e.g., A > B > C > D > E for $n = 5$ ), scores are assigned in fixed steps from 100 to 0:
$s_i = 100 \cdot \left(1 - \frac{i-1}{n-1}\right), \quad i = 1, 2, \dots, n$
Example: For A, B, C, D, E, scores are 100, 75, 50, 25, 0.
Last Item Adjustment: If the reaction time for the comparison between the second-to-last item (e.g., D) and the last item (e.g., E) exceeds the median reaction time of the session, the last item’s score is set to half the second-to-last item’s score to reflect implicit preference:
$s_n = \begin{cases} \frac{s_{n-1}}{2} & \text{if } t_{n-1,n} > t_{\text{median}} \\ 0 & \text{otherwise} \end{cases}$
Example: If D vs. E reaction time > median, E’s score is set to $\frac{25}{2} = 12.5$ .

Reaction Time & Uncertainty Adjustment

Inspired by Kiani et al. (2014), we use reaction time to adjust scores, reflecting decision certainty. Longer reaction times indicate higher uncertainty, reducing the score gap between items to capture “hard choices.”

Normalization with Weber-Fechner Law: Reaction times are normalized logarithmically to reflect the psychophysical scaling of perception (Weber-Fechner law), clamping times within a 10,000ms range: $t_{\text{norm}} = \frac{\log(t + c) - \log(t_{\text{min}} + c)}{\log(t_{\text{max}} + c) - \log(t_{\text{min}} + c)}$ Where:
$t$ : reaction time in milliseconds
$t_{\text{min}}$ : 20th percentile of reaction times
$t_{\text{max}}$ : 10,000ms
$c = 1.0$ : constant to avoid undefined logs

This logarithmic transformation (aligned with the Weber-Fechner law) provides a scaled representation of reaction time, where proportional differences in time (e.g., doubling the reaction time) are treated with more consistent impact. This scaled time is then used to derive a certainty score ( $w_t$ ) via an exponential function in the next step. This combined approach ensures that:

Faster reaction times (indicating high certainty) result in a very small penalty, thus minimally affecting the score gap between chosen items.
Slower reaction times (indicating lower certainty or harder choices) lead to a progressively larger penalty, which more significantly reduces the score gap between items to reflect the increased decision difficulty.

Certainty Score: The normalized reaction time is used to compute a certainty score using an exponential function, reflecting non-linear uncertainty growth:
$w_t = e^{-1.5 \cdot t_{\text{norm}}}$ $\text{penalty} = 1 - w_t$
Score Adjustment: For a pair where item A (score $s_A$ ) is chosen over item B (score $s_B$ ), A’s score is adjusted toward B’s score based on the penalty, with a buffer to preserve order:
$s_A' = s_A - \text{penalty} \cdot s_A$ $s_A' = \max(s_A', s_B + 1.0)$

This ensures $s_A' > s_B$ , maintaining the ranking while reflecting decision difficulty.

Kiani, R., et al. (2014). Choice Certainty Is Informed by Both Evidence and Decision Time. Journal of Neuroscience.

Theoretical Support: Reaction times are normalized logarithmically using the Weber-Fechner Law, reflecting the non-linear perception of decision certainty. An exponential certainty score ( $e^{-1.5 \cdot t_{\text{norm}}}$ ) adjusts scores minimally for fast decisions and more for slower, uncertain ones. While Stevens’ Power Law ( $t^k$ ) is an alternative, our logarithmic-exponential approach is robust and psychophysically grounded.

Final Scaling

To ensure scores are intuitive, all scores are scaled so the highest score is 1000:

s_i' = s_i \cdot \frac{1000}{\max(s_1, s_2, \dots, s_n)}

This stretches the score range while preserving relative differences and rankings.

Head-to-Head Probabilities

To display preference strength, we compute head-to-head probabilities using the Bradley-Terry model:

P(A > B) = \frac{s_A}{s_A + s_B}

Where $s_A$ and $s_B$ are the final adjusted scores. These probabilities are shown on the results page, highlighting how strongly one item is preferred over another.

Bradley, R. A., & Terry, M. E. (1952). Rank Analysis of Incomplete Block Designs: I. The Method of Paired Comparisons. Biometrika.

🔁 Smart Pair Selection

The algorithm selects pairs to minimize comparisons while maximizing information gain:

Priority: Pairs involving items with fewer comparisons or unclear transitive relationships.
Transitive Reduction: Skips pairs already resolved by transitive paths (e.g., if A > B and B > C, skip A vs. C).
Randomization: Ensures diversity by randomly selecting among viable pairs when needed.

This approach achieves logarithmic efficiency, requiring only ~44% of $\binom{n}{2}$ comparisons for $n = 15$ .

🛑 Early Stopping for Efficiency

The algorithm stops when rankings stabilize, typically after ~44% of possible comparisons for larger sets ( $n = 15$ ). Stability is assessed by checking if recent comparisons align with the current ranking, ensuring efficiency without sacrificing accuracy.

📊 Benchmarking Performance

We rigorously tested the Beacon Algorithm with thousands of simulations, assuming a ground truth where items are ranked alphabetically (A > B > C > …). Each comparison uses a fixed reaction time of 1000ms to simulate maximum certainty.

Evaluation Metric

We use a Misplacement Rate (MR) to measure ranking accuracy:

\text{MR} = \frac{\sum_{i=1}^n \mathbb{1}_{\sigma(i) \neq k_i}}{n}

Where:

$\sigma(i)$ : Key at position $i$ in the sorted order (by score, descending). (Note: Changed $i-1$ to $i$ for clarity, assuming 1-based indexing for positions, or keep $i-1$ if you stick to 0-based indexing for lists in explanation).
$k_i$ : Expected key at position $i$ (e.g., A at 1, B at 2).
$\mathbb{1}$ : Indicator function (1 if keys differ, 0 otherwise).

Key Performance Results

(Based on thousands of simulations for 3-15 items)

✅ Perfect Rank Accuracy: Our algorithm consistently delivers 100% correct ranking order (Misplacement Rate of 0%). This precision stems directly from user choices, ensuring no item is out of its true relative preference.
⚡ Unmatched Efficiency:
- Small Sets (e.g., 3 items): Achieve reliable rankings with just 1-2 comparisons, a remarkable 33-80% reduction from exhaustive methods.
- Larger Sets (e.g., 15 items): Get accurate results with only 14-51 comparisons, representing up to a 60% reduction in effort compared to traditional pairwise methods. This efficiency means you can gain valuable insights with significantly fewer comparisons per user, enabling faster feedback loops even with a limited number of survey participants (e.g., often just 1-30 individual responses are enough to uncover preferences within a given item set, depending on required confidence).
🧭 Deeper Insights: Beyond simple ranks, our system uses reaction time to gauge decision certainty, providing nuanced scores (e.g., 1000 vs. 740 for a close preference).
📊 Actionable Probabilities: Head-to-head probabilities (via Bradley-Terry) offer clear, market-ready insights into the strength of preferences, perfect for strategic decision-making.

👤 Privacy First: Pseudonymous Data

Your privacy is paramount. Kuqus ensures:

Anonymous session IDs (X-Anonymous-ID) using server-generated UUID + IP location estimation (no IP storage).
No personal profiling or hidden tracking.
Regional insights or private survey dashboards without linking to personal data.

🔒 Transparent Yet Powerful

We share the core logic of the Beacon Algorithm but protect key implementation details to prevent manipulation. Expect:

No dark patterns.
No tracking cookies unless explicitly accepted (see Cookie Policy).

Our mission: deliver fast, fair, and delightful decision tools.

🧪 Disclaimer These metrics are based on internal simulations and are designed for real-world reliability but are not peer-reviewed. For academic collaboration, contact us.

Kuqus blends cutting-edge science, user-friendly design, and privacy-first principles to deliver accurate, efficient, and engaging surveys. Try Kuqus today and experience the future of decision-making!