The AI capability trade-off model

Compute reallocation

Let C_R(t) be total frontier AI R&D compute and x the share of it reallocated to alignment, with 0 ≤ x ≤ 0.5 over the range the calculator exposes. Total R&D compute is exogenous and is held at its initial level across the modelled horizon.

Under the baseline nothing is reallocated, so the whole budget stays with capability work.

Under the policy the capability-directed budget falls by exactly the reallocated fraction.

The calculator implements all three. The policy enters every equation below through (1.3) alone: reallocation changes the compute available to capability research, and nothing else.²

§ 1 Compute reallocation 1 / 10

Software progress

Let S(t) denote the software — that is, algorithmic — progress index. Software progress follows an ideas-production function, run once for the baseline and once for the policy.

Both arms start from the same initial condition.

The policy enters through one factor. Comparing (2.2) with (2.1), reallocation multiplies the instantaneous rate by (1 − x)^β and changes nothing else.

§ 2 Software progress 2 / 10

Frontier AI capability

Frontier AI capability is modelled with a reduced-form Cobb–Douglas production function over training compute and software.

Since the policy reallocates only R&D compute, training compute is identical in both arms.

Capability cost is therefore the shortfall of the policy capability path against the baseline,

or equivalently, once the common training-compute term cancels,³

The cancellation in (3.4) is why α never needs a value. It is a real parameter of the model that the relative outputs happen to be insensitive to — not an omission.

§ 3 Frontier AI capability 3 / 10

AI risk

Rather than modelling alignment progress explicitly, AI risk is represented with a reduced-form function. Under the policy, risk decays away from its baseline path in proportion to the reallocated share and the time elapsed since the policy began.

The specification supplies no shape for R_B(t) and no empirical bounds for λ.⁴ This implementation therefore holds baseline risk at a normalized index of 100 and draws λ from an explicit working prior, λ ∼ Triangular(0.20, 0.40, 0.60). The AI-risk plate is a sensitivity surface over that assumption, not a risk forecast.

§ 4 AI risk 4 / 10

Safety benefit

Safety benefit is defined as the proportional reduction in AI risk relative to the baseline.

Substituting the risk function (4.1), the baseline path divides out,⁵

Increasing x therefore increases safety benefit, and increasing λ makes alignment compute more effective. Both follow from (5.2) alone. Note what (5.2) means for the calculator: the baseline path divides out entirely, so the safety-benefit plate is independent of the assumed shape of R_B(t) and depends only on λ, x and elapsed time. The normalization in § 4 constrains the risk plate; it does not constrain this one.

§ 5 Safety benefit 5 / 10

User inputs

The reader controls two inputs, and only two.

1. Compute reallocation share The share x of frontier AI R&D compute reallocated to alignment. The calculator exposes it as a slider over 0 %–50 %.
2. Policy impact level A categorical selection, z ∈ {Lower, Central, Higher}, which selects a parameter bundle.

The mapping in (6.1) carries four parameters, and the shipped bundles carry all four.⁶ β, γ and δ are drawn from the source’s own triangular ranges; λ is drawn the same way, but from a range this implementation chose, because the source supplies none.

§ 6 User inputs 6 / 10

Calibration of the bundles

The parameter bundles come from 10,000 × 4 seeded Monte Carlo draws on triangular distributions. Each parameter is drawn independently, sorted,⁷ and read at the marginal percentile its impact level names.

The bundles rank-align each parameter’s marginal percentile. Seeding the draw makes the same z selection reproduce the same bundle on every run, which is what allows a reader to check a figure against the source.

§ 7 Calibration of the bundles 7 / 10

Normalization and scope

Software begins at index 100. The baseline is normalized to index 200 in 2034, so the plates communicate policy divergence — not a dated forecast of absolute capability.⁹ The normalization absorbs A; unchanged training compute makes α cancel from relative outputs.

The horizon runs 2026–2034 at quarterly resolution. Because the normalization fixes the baseline endpoint, A and C_R(t) never appear as numbers: the calculator solves (2.1) for the growth constant that reaches the fixed endpoint, then applies the policy factor (1 − x)^β from (2.2) to obtain the policy arm.

§ 8 Normalization and scope 8 / 10

Parameter contract

Every symbol in the specification, its role, where its value comes from, and what the shipped model does with it. Every row is implemented; the one still marked is R_B(t), which the model supplies itself because the source does not.

Return to the calculator — the instrument this document specifies.

§ 10 Parameter contract 10 / 10