Individual rationality ($u_i\ge 0$)

Individual rationality means participation yields at least the outside-option utility, normalized here to zero. A risk-neutral zero-value bidder with nonnegative payments and no positive-valued allocation must have zero expected payment and utility if participation is individually rational.

Ex post individual rationality requires nonnegative participation utility at every realized profile of types. A truthful threshold allocation with its critical-value payment satisfies this condition for nonnegative valuations.

Interim individual rationality requires a nonnegative expected participation utility for each private type, averaging over the other types with the player's conditional beliefs. It differs from requiring nonnegative utility at each realized profile, which is ex post individual rationality.