The more authority someone wields, the more fear of the Lord is required to rule responsibly.  ‘Fear of the Lord’ defined in context of chapter 16: ‘Taking God as seriously as His sovereignty demands’.  Each of the five references to the king in verses 10-15 imply the ideal, that is the perfect example of dominion.  You could place the word ‘perfect’ or ‘ideal’ before each mention of ‘king’ in this section to help clarify the sense of Solomon’s instruction. In addition, it is helpful to remember the wisdom literature device of ‘citing the maximal case’.  This way of speaking commends the application of these principles not exclusively to kings, but furthermore to all lesser dominion agents. Consider Prov 16 in light of the (Gen 1:28) original cultural mandate for all humanity. Given this dynamic, the reader may substitute ‘dominion agent’ for each reference to ‘king’ to clarify the scope of application.