### G-85-06

# Definition and Properties of Cooperative Equilibria in a Two-Player Game of Infinite Duration

## and BibTeX reference

A two-player multistage game, with an infinite number of stages is considered. The concepts of overtaking and weakly-overtaking pay-off sequences are introduced. The class of strategies considered consists of "memory-strategies" which are based on the past history of the control, and the initial state from where the game has been played. Weak equilibria are defined in this class of strategies. It is then shown how such equilibria can be constructed by composing into a trigger-strategy a nominal cooperative control sequence, and two threat strategies representing the announced retaliation by each player in the case where the other player does not play according to the nominal control.

When the threats consist in a feedback equilibrium pair the resulting cooperative equilibrium is "perfect". Another result shows that, if each player can use a "most effective threat" based on a saddle-point feedback strategy, then any weak equilibrium in the class of memory-strategies is in some sense related to this particular kind of equilibrium in the class of trigger-strategies.

Published **March 1985**
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11 pages