The shape of outcomes: bells, long tails, rare counts, and how much surprise to expect.
Bell-curve averaging: many small independent effects sum to a predictable center with thin tails.
Use → Use when errors aggregate. Don't use for rich-get-richer or contagious outcomes.
Starter • Reason · Predict · Communicate
Sample averages settle to normal; variation reduction pays until returns diminish.
Use → Quality programs, polling margins, A/B sizing.
Starter • Reason · Communicate
Rich-get-richer: a few huge winners, many tiny players. Straight line on log-log.
Use → City sizes, book sales, network degrees. Never average it like a normal.
Starter • Explain · Predict
Counts of rare independent events per window: calls, mutations, clicks.
Use → Staffing, queues, buffers. Don't use when events cluster or cascade.
Core • Predict · Design
Entropy scores surprise: equilibrium, cyclic, random and complex outcomes sit at different levels of uncertainty.
Use → Classify how predictable a system is, compare category mixes, measure diversity.
Core • Explain · Communicate · Predict
Lines, buckets, and bends — when effects add up, and when curvature flips the best move.
Best-fit line plus a causation audit. A big coefficient is not a big cause.
Use → Screen relationships, then hunt omitted variables and reverse causality.
Starter • Reason · Predict · Communicate
Buckets and cutoffs turn messy attributes into testable if-then rules.
Use → Segmentation, eligibility, triage. Watch boundary gaming.
Core • Reason · Predict · Communicate
Curved payoffs change optimal bets: diversify when concave, concentrate when convex.
Use → Portfolios, effort allocation, pricing tiers.
Core • Design · Act
How individual thresholds and mild preferences aggregate into stark group patterns.
Mild individual preference for similar neighbors aggregates to stark city-wide segregation.
Use → Housing, schools, platforms. Try the tolerance sandbox below.
Core • Reason · Explain · Predict
Each person has a join threshold; cascades depend on the distribution, not the average.
Use → Protests, adoption, bank runs. Map thresholds before messaging.
Core • Explain · Predict
Quality plus sightlines plus personal thresholds explain why crowds stand — or don't.
Use → Social proof, launches, applause dynamics.
Fun • Explain
Who connects to whom — hubs, bridges, and the tipping point where islands fuse.
Degree, clustering, bridges and formation logic predict what flows — and what breaks.
Use → Org design, virality, resilience. Prefer bridges over hubs for novelty.
Core • Reason · Design
New links favor already-linked nodes, manufacturing power laws from pure growth.
Use → Citations, followers, city pull. Seed early visibility deliberately.
Core • Explain · Predict
Sparse links suddenly gel into a giant component — bank networks and blackouts tip fast.
Use → Fragility audits, buffers, circuit breakers.
Advanced • Explain · Predict
How ideas, products, and germs spread: broadcast, word of mouth, and contagion.
One source reaches a fixed pool at a steady rate: fast early, then fading returns — the r-shaped curve.
Use → Ads, press releases, mandated rollouts. If growth is S-shaped instead, word of mouth is doing the work.
Starter • Predict · Communicate
Susceptible–Infected–Recovered with an R0 tipping point. Same math for flu and pop-star fever.
Use → Epidemics, memes, outages. Push R below 1 or segment the network.
Core • Explain · Predict · Act · Explore
Innovators plus imitators make the S-curve. Spend early on seeds, later on social proof.
Use → Launches, rollouts, feature adoption.
Core • Predict · Design
Systems that move: noise, memoryless chains, path dependence, feedback, and equilibrium.
Unpredictable steps accumulate; efficient markets leave mostly randomness behind.
Use → Separate skill from luck before forecasting.
Core • Reason · Predict
Next state depends only on now. Run it forward to get the long-run mix.
Use → Mobility, churn, brand switching. Check memorylessness first.
Core • Predict · Act · Explore
Early draws rewrite the odds — history locks in QWERTYs, VHSs and dominant cities.
Use → Audit increasing returns; intervene early or standardize late.
Advanced • Explain
Tiny local rules aggregate into gliders, blinkers and chaos. Emergence needs no boss.
Use → Prototype emergence; don't over-interpret any single run.
Core • Explore · Explain
Stocks, flows and reinforcing loops turn small pushes into addiction spirals or virtuous cycles.
Use → Map loops before adding pressure; find the balancing lever.
Core • Explain · Act · Explore
An energy-like function proves a system settles down — and bounds how fast.
Use → Stability checks for cities, markets, queues.
Advanced • Reason
Compounding, capital, and ideas — why some economies and fortunes pull away.
Compounding plus capital, labor and ideas explain why some economies stall.
Use → Growth diagnostics; ideas are the non-rival engine.
Core • Explain · Predict
When returns on capital outrun growth, wealth concentrates without any villain.
Use → Tax and redistribution design; pair with mobility data.
Advanced • Explain · Act
Strategic interaction: dilemmas, coordination, signals, fair splits, and rule design.
Solo incentives wreck the group; repetition, reputation, punishment and sorting rebuild it.
Use → Pricing, commons, teamwork. Change payoffs, not slogans.
Core • Reason · Design
Pure coordination games plus memory harden into culture as equilibrium.
Use → Norms, standards, onboarding rituals.
Core • Explain · Design
Split forces across fronts with no pure best move — win by being unpredictable.
Use → Campaigns, ad budgets, competitive bids.
Advanced • Act · Design
Costly actions reveal hidden quality when only the strong can afford them.
Use → Credentials, warranties, pricing, hiring. Ask what makes a signal too costly to fake.
Core • Explain · Design · Communicate
Credit each player by their average marginal contribution across every order of joining.
Use → Splitting revenue, attributing features (SHAP), measuring voting power.
Advanced • Reason · Design · Communicate
Shared pools survive with boundaries, monitors and graduated sanctions — no panacea.
Use → Fisheries, groundwater, shared infra.
Core • Design · Act
Write rules so truth-telling wins despite hidden actions and hidden information.
Use → Auctions, procurement, public projects.
Advanced • Design · Act
How populations and teams improve: replication, exploration, search, and diverse crowds.
Winning strategies reproduce faster; adaptation speed rises with variation.
Use → Learning systems, org routines. Protect variation to adapt faster.
Advanced • Explain · Explore
Balance pulling the best-known lever against testing new ones. Regret is the score.
Use → Trials, dosing, ad creative. Run small experiments, then exploit.
Core • Act · Explore
Interacting parts create many local peaks; hill-climbers get stuck, diverse searchers find higher ground.
Use → Product design, strategy, R&D portfolios. Add jumps and diverse heuristics when stuck.
Advanced • Explore · Design
Crowd error equals average error minus diversity. Diverse models beat the lone genius.
Use → Ensembles, forecasting teams, model committees.
Core • Predict · Design · Communicate
Choosing under uncertainty, and how individual choices aggregate into group outcomes.
Fold back probabilities; pay for information only if it changes the action.
Use → Product, medical and policy bets.
Core • Act · Design · Communicate
Rational individuals aggregate into irrational crowds — location and order matter.
Use → Voting, siting, assortment. Aggregate rules, not vibes.
Advanced • Reason · Predict