Consider a game , between player I and II. Here, and are the sets of pure strategies for players I and II respectively; and is the payoff function.
(in other words, if player I plays and player II plays , then player I pays to player II).
The Helly metric is defined as
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 does not imply but it does imply that the consequences of and are identical; and indeed this induces an equivalence relation.
If one stipulates that implies then the topology so induced is called the natural topology.
The metric on the space of player II's strategies is analogous:
Note that thus defines two Helly metrics: one for each player's strategy space.
Notation (definition of an -net). A set is an -net in the space with metric if for any there exists with .
A metric space is conditionally compact if for any there exists a finite -net in .
A game that is conditionally compact in the Helly metric has an -optimal strategy for any .
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).
N. N. Vorob'ev 1977. Game theory lectures for economists and systems scientists. Springer-Verlag (translated by S. Kotz).
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