In game theory, the Helly metric is used to assess the distance between two strategies. It is named for Eduard Helly.
Contents
Consider a game
(in other words, if player I plays
The Helly metric
The metric so defined is symmetric, reflexive, and satisfies the triangle inequality.
The Helly metric measures distances between strategies, not in terms of the differences between the strategies themselves, but in terms of the consequences of the strategies. Two strategies are distant if their payoffs are different. Note that
If one stipulates that
The metric on the space of player II's strategies is analogous:
Note that
Conditional compactness
Notation (definition of an
A metric space
A game that is conditionally compact in the Helly metric has an
Other results
If the space of strategies for one player is conditionally compact, then the space of strategies for the other player is conditionally compact (in their Helly metric).