Shadow Futures

Interactive equation lab Live model

The Comparison Playground

Five companies—A through E—compete for 360 customers. Each customer is one round. Watch what changes when being genuinely better helps a company win, and when having already won starts helping it win again.

Back to story
The question behind the modelHow much did the leader win because it was better—and how much because it had already won?
See the two equations underneath
sit=eβxi(a+Ni(t))ρ  →  pit=sit∑jsjts_{it}=e^{\beta x_i}(a+N_i(t))^\rho \;\to\; p_{it}=\frac{s_{it}}{\sum_j s_{jt}}
εt=1−max⁡ipit  →  BT=∑t<Tεt\varepsilon_t=1-\max_i p_{it} \;\to\; B_T=\sum_{t<T}\varepsilon_t
Customer 1 of 360

The contest is open

Big Tech. Early customers create data, revenue, rankings, integrations, and attention. A small lead can grow into a steep ridge.

Leader nowCompany C1 customers · 37% latest chance
Cyan history that happenedPurple other possible historiesOrange contest still alive
Left to rightcompanies A–EFront to backcustomers 1–360Heightchance of winning next
How much of the contest is still alive
63%

Competition preserved. This is the share of an open benchmark’s comparison opportunities that remains.

How unevenly customers were divided
0.80Reward inequality · Gini coefficient

0 means an equal split; 0.80 means one of five firms won every customer.

Read it in 20 seconds

One market. Three layers of evidence.

  1. 01

    Five companies compete for 360 customers. A–E run left to right; history runs from the first customer at the front toward customer 360 at the back. Height is a company’s chance of winning the next customer.

  2. 02

    The cyan path is the history that actually happened. Each point records which company won that customer. In a reinforcing market, each win changes the next customer’s odds.

  3. 03

    The purple paths are shadow futures. They are other histories that could have happened with the same firms and qualities if a few early customers had chosen differently. A simulation can replay them; a real market cannot.

  4. 04

    The orange line asks how much contest remains. If the leader has a 25% chance, 75% is still open to rivals. At 98%, only 2% remains. Adding those open chances across customers gives the comparison budget.

Three ways the same market can evolve

Every customer is a new test—until earlier tests begin deciding the later ones.

Competitive Marketβ 1.00 · ρ 0.00 · η 0%

Each customer is a fresh contest.

Think USB cables or standardized parts: easy to compare and easy to switch.

Quality matters, but yesterday’s winner gets no extra boost. Companies stay visible and the surface remains relatively flat.

Ordinary competition; no special intervention is built into the model.
Big Techβ 1.00 · ρ 1.25 · η 0%

Winning helps you win again.

Think an AI assistant, search engine, app store, or social network.

Early customers create data, revenue, rankings, integrations, and attention. A small lead can grow into a steep ridge.

No protection for challengers; scale and lock-in reinforce the leader.
Big Tech with Regulationβ 1.00 · ρ 0.90 · η 8%

Challengers keep getting real chances.

The same tech market, with portability, open APIs, and neutral discovery slots.

Success can still compound, but the rules stop yesterday’s victory from completely deciding tomorrow’s contest.

Feedback is weaker and 8% of discovery is kept open for alternatives.