For a sentence to be grammatically correct all the different elements on it need to agree genre and number, this includes words such as himself, themselves, myself that act as a reflexive pronoun and refer back to a noun or pronoun. Additionally, words such as either or each represent singular nouns despite including to nouns. This implies in the sentence presented the subject "Either Richard or Luke" is equivalent to the pronoun "he" rather than "they" as the expression "either...or" refers to only one of them at a time and according to this, the word that agrees in number and genre and therefore is the best option for completing the sentence is "himself".
