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

rufus-isaacsdifferential-gamesdynamic-programmingcharacteristicsoptimal-controlrand-corporationhistory-of-ml

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, 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).