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:

${\mbox{swap-regret}}=\sum _{i=1}^{n}\max _{j\leq n}{\frac {1}{T}}\sum _{t=1}^{T}x_{i}^{t}\cdot (p_{j}^{t}-p_{i}^{t}).$

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.

References

• Blum, Avrim; Mansour, Yishay (2007), "From external to internal regret", Journal of Machine Learning Research (JMLR), 8: 1307–1324, MR 2332433.