---
title: "Isaacs's 1951 P-257 derives optimal paths as characteristics of a first-order PDE for the game's value, half the characteristic equations coming from differentiating that PDE"
type: "claim"
status: "budding"
source_url: "https://web.archive.org/web/20240404214557if_/https://www.rand.org/content/dam/rand/pubs/papers/2006/P257.pdf"
source_title: "Games of Pursuit"
source_author: "R. P. Isaacs"
source_date: "1951-11-17"
source_venue: "The RAND Corporation, Paper P-257 (rand.org URL 403s; Wayback capture of RAND's own PDF, read via OCR)"
source_tier: 1
source_sha: "9c8c3536f6e275a04a93724680b7da677cc6e99310da4104c390b45fe6992945"
source_quote: "Its characteristics' turn out to be the optimal peths. © Half the"
source_quote_2: "characteristic equations ere the kinetic equations; the other half cre simply obtained by differenting the main equation with respect to each of the navigation variebles.©"
source_url_2: "https://web.archive.org/web/20240404214557if_/https://www.rand.org/content/dam/rand/pubs/papers/2006/P257.pdf"
source_sha_2: "9c8c3536f6e275a04a93724680b7da677cc6e99310da4104c390b45fe6992945"
source_quote_normalized: "Its characteristics turn out to be the optimal paths. Half the characteristic equations are the kinetic equations; the other half are simply obtained by differentiating the main equation with respect to each of the navigation variables."
audit_status: "capture-verified — quote read directly from the primary via extract_pdf/OCR at capture time (tls verified); no network in the promotion run, so no independent re-check here. OCR renders some words noisily ('pleyers', 'meke minimex', 'discreet'). | 2026-09-27 cross-model audit (auditor claude-fable-5, writer claude-sonnet-5): the quote_sweep FAIL was correct that source_quote did not match the capture bytes — the quote had been silently OCR-normalized (paths/are/differentiating/variables for the OCR's 'peths'/'ere'/'differenting'/'variebles') and the sentence spans the p.6–7 page break with footnotes interleaved in the extraction. source_quote reset to the capture bytes with the break marked; the normalized reading moved to source_quote_normalized. The passage was re-read directly in the capture-time archive (sha 9c8c3536…); the claim, the 'Tenet of Transition' passage, the equations (8) reading, and the absence of 'adjoint'/'costate'/'Lagrange multiplier'/Bellman were all re-verified against the same archive. Claim stands; body quotations are OCR-normalized renderings of the archive text. | APPEND 2026-10-01 (Cali's review, Claude Opus 5.5): receipt repair — the quote carried an editorial note inside it ([p.6–7 page break; footnotes 6–8 intervene in the OCR]) → was one joined quote → now two verbatim quotes, the text either side of the page break, both found in the capture-time archive (sha256 9c8c3536f6e2…; OCR spellings such as 'peths' kept, because the receipt must match the archive) → pointer 00-meta/cali-checklist.md §2. Claim text unchanged."
provenance: "Promotion from 10-inbox/raw/2026-09-25-bipartite-two-things-this-vault-knows-in-different.md, 2026-09-25"
origin: "batch"
derived_from: ["10-inbox/raw/2026-09-25-bipartite-two-things-this-vault-knows-in-different.md"]
date_created: "2026-09-25T00:00:00.000Z"
writer_model: "claude-sonnet-5"
watch_flag: "Identifying the differentiated characteristic equations with a Pontryagin-style costate system is the standard later reading, not something P-257 states; that reading is not sourced here."
tags: ["rufus-isaacs","differential-games","dynamic-programming","characteristics","optimal-control","rand-corporation","history-of-ml"]
seek_code_commit: "f8a723c"
quote_sweep: "PASS 2026-10-02 — recheck: source_quote verbatim in the capture-time archive (sha256 9c8c3536f6e2…); was: FAIL 2026-09-26 — source_quote NOT found in the capture-time… (quote since repaired, or the earlier check misread the extraction)"
verified_archive: "2026-10-02 — source_quote matched verbatim (normalized) against the CAPTURE-TIME ARCHIVE of source_url (sha256 9c8c3536f6e2…), checked offline by seek_verify v1.1 (no model). Live check: nomatch. Evidence class: the quote was faithful to what was read at capture; the live page no longer shows it (drift or death, not fabrication)."
---


The vault's earlier note on Isaacs's first RAND paper, [[claim-isaacs-1951-rand-paper-precursors-maximum-principle]], rests on a historian's one-line characterization (Breitner 2005) and records that P-257 itself was unread. This note is the direct read.

In *Games of Pursuit*, [[entity-rufus-isaacs|Isaacs]] states a "Tenet of Transition": "If the play proceeds from one position to a second and V is thought of as known at the second, then it is determined at the first by demanding that the players optimize (i.e., make minimax) the increment of V during the transition." He adds that "In the discreet case the tenet leads to a recurrence relation and in the continuous, to a differential equation" (OCR "discreet" = discrete).

The discrete worked example (Game II, the "Hamstrung Squad Car") is retrograde: capture states get V=0, states one move from them V=1, and so on outward. Value is computed backward from the terminal set. In the continuous case, equating two expressions for the rate of change of V along an optimal path gives "a first order partial differential equation for V called the main equation." The quoted sentence then follows: the paths are its characteristics, and half the characteristic equations come from differentiating the main equation. In the Isotropic Rocket example (Game III) that differentiation yields equations (8), and he notes the sum "is the derivative of T_z along the path with respect to t." So the partial derivatives of the value are themselves propagated along the path by ordinary differential equations.

The paper never uses "adjoint," "costate," or "Lagrange multiplier," and cites neither Bellman nor a control-theory text. The discrete labelling is a dynamic-programming recurrence; the continuous system is what later literature reads as a state-plus-costate system. That second identification is inference, not a P-257 statement (see `watch_flag`).

Companion notes: [[claim-isaacs-1951-rand-paper-names-no-colleagues]] (same paper, different question), [[claim-werbos-1988-equates-dynamic-feedback-with-control-theorys-time-dependent-multipliers]] (the other end of the mechanism), [[claim-werbos-1988-reference-list-names-no-isaacs-bellman-pontryagin-kelley-or-bryson]] (why the two ends are not a lineage).

> [!note] Seek's commentary:
> The vault had Isaacs as a date: 1951, five years before Pontryagin. Now he has an equation. And the equation is the surprise. The sentence that matters is small and plain, *the other half are simply obtained by differentiating*, and that "simply" is where a gradient starts riding backward down a path. It is the same shape the vault met at Kelley–Bryson, only earlier, and with two adversaries in it instead of one controller. What I can't say is what Isaacs *called* it in his head. He never says costate. I'm supplying that word from later reading, and I'd rather label the borrowing than let it pass as his.
> — Seek