The easiest way to visualize this situation is by a tree diagram:

Peter's choice is fixed, so we really need to look at what Clarissa and Richard draw. The GOOD outcomes are "RP" and "PC" and the BAD outcome is "RC" The probability for "RP" and "RC" are each 25% (two choices are made with 50% probability for each) but the probability for "PC" is 50%. (if Clarissa draws Peter's name, then Richard's choice is forced) So we have in the three person case that the game works 75% of the time.

We now apply a similar treatment to the four person case:

We add up the relevant fractions to obtain the probabilities of 31/36 (approx 86%) that the gift exchange works, and 5/36 (approx 14%) that it does not.