A regret-free mechanism incentivizes agents who want to avoid regret to report their preferences truthfully.
A mechanism or rule is a function f that gets as input the agents' preferences P1,...,Pn, and returns as output an outcome from X.
It is assumed that, once an agent observes the outcome of the mechanism, he feels regret if his report is a dominated strategy "in hindsight".
Each cutter knows all past choices of the chooser, and can potentially exploit this information in order to make a cut that will guarantee to him more than half of the cake.
There are protocols that use fewer cuts, depending on the number of dimensions of the cake: Cresto and Tajer[6] also study regret in fair cake-cutting among two agents, where the regret comes from a change in preferences: after one player sees the choice of the other player, his preferences may change.