NOTATION.md

Notation conventions (Grad-IO)

Following Berry & Haile (2014). Adopted 2026-08 for the demand material; new decks should conform.

Choice probabilities and shares

Object	Symbol	Notes
Individual choice probability (model)	\sigma_{ijt} (or \sigma_{ij} in static settings)	Conditionals: \sigma_{ij|g}, nest-level \sigma_{ig}
Market-level choice probability / share function	\sigma_{jt}(\cdot), e.g. \sigma_j(\delta_t,\theta)	Inversion: \sigma_j^{-1}(\cdot)
Observed market share (data)	s_{jt} only	Vector: \symbf{s}_t. Empirical frequencies may wear hats: \hat{s}_{ij}
Estimation equation	\sigma_j(\delta_t,\theta) = s_{jt}	The model matches the data

Retired symbols (do not use): \mathfrak{s}_{jt}, \mathcal{S}_{jt}/\calS for shares, \tilde{s}_{jt}, P_{ij}, S_{ij}, and \pi as a choice probability.

Parameters and other objects

• \rho — nesting parameter (Berry 1994's $\sigma$; matches PyBLP). Where a deck deliberately contrasts parameterizations, keep McFadden's \lambda with $\rho = 1-\lambda$ and a remark that Berry (1994) writes $\sigma$.

• \pi_f — profits, and nothing else. Type/mixture weights are w_i / w_{it} (so s_{jt} = \sum_i w_{it}\,\sigma_{ijt}).

• \Pi, \Sigma — demographic-interaction and random-coefficient parameter matrices (PyBLP convention). Random coefficients use capital \Sigma whenever possible: write \beta_i = \beta + \Sigma \nu_i (with $\Sigma$ diagonal if need be) and refer to elements of $\Sigma$, rather than lowercase \sigma_x \nu_i / \sigma \nu_i. Never use bare \sigma_0 for an RC value at the truth (it now reads as the outside-good choice probability) — write \Sigma_0 etc.

• Prices are always lowercase p_{jt}; capital P is never a price.

• \Delta_{(j,k)}(\symbf{p}) = -\partial \sigma_j / \partial p_k — the demand-derivative matrix (not \Omega).

• Diversion ratios — two distinct aggregate objects, kept notationally separate:

  • D_{jk}(\symbf{p}) — small-price-change (derivative-based) diversion: $D_{jk} = \frac{\partial q_k/\partial p_j}{|\partial q_j/\partial p_j|}$.
  • D_{j \rightarrow k} — second-choice (choice-set-removal) diversion.

  Classify aggregate uses by the underlying experiment (marginal price change vs. removal of $j$). Individual-level diversion ratios are always plain D_{jk,i} (no arrow): $D_{jk,i} = \frac{\sigma_{ik}}{1-\sigma_{ij}}$ does not depend on the intervention — only the aggregation weights do.

• a — the T1EV/error scale parameter (formerly $\sigma$): $\sigma_{ij} = e^{V_{ij}/a}/\sum_k e^{V_{ik}/a}$, only $\beta/a$ identified.

• Probability statements: use \Pr(\cdot) or \prob{\cdot} (both render blackboard $\mathbb{P}$ via teaching_slides.sty), not literal \mathbb{P}.