Bang-bang rewards

Original article (link) posted: 03/08/2005

Self-generation and related techniques were first developed in the context of unrestricted symmetric equilibria of the Green-Porter model in APS (1986), and then presented in greater generality in APS (1990).
(Pearce (1992), p.152)

In APS (1990), they found the following bang-bang property.

The equilibrium value set V is compact, and for all elements v in V there exists an equilibrium whose implicit reward functions after each history take only values in the set of extreme points of V.
Under certain conditions the "bang-bang sufficiency" result given above can be strengthened to a necessity result: an equilibrium that maximizes a linear combination of player payoffs (including negative combinations) must have implicit reward functions that use only extreme points of V. The rough intuition is the same as the one given earlier for the Green-Porter model: if you are creating incentives by moving rewards in a direction that reduces the objective function of the problem, do so aggressively (move until you can't go any further in V) but in as small and informative a region of signal space as possible. This advice cannot be applied literally in a model with a discrete signal space, so the bang-bang necessity result does not hold. The sufficiency result can be restored trivially in an essentially discrete model if the signal space is taken to include the outcome space of a public randomization device.

(Pearce (1992), p.153)

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