Swap regret

Swap regret is a concept from game theory. It is a generalization of regret in a repeated, n-decision game.

Definition
A player's swap-regret is defined to be the following:


 * $$\mbox{swap-regret}=

\sum_{i=1}^n \max_{j \leq n}\frac{1}{T}\sum_{t=1}^T x^t_i \cdot (p^t_j-p^t_i).$$

Intuitively, it is how much a player could improve by switching each occurrence of decision i to the best decision j possible in hindsight. The swap regret is always nonnegative. Swap regret is useful for computing correlated equilibria.