Under a standard action resolution mechanic, a player rolls 3d6 and compares the total rolled to a Target Number (TN) assigned by the gamemaster, succeeding if the total meets or exceeds the TN. For example, if they rolled 1+3+2=6 or 1+3+1=5 against TN 5, they would succeed because their total was at least 5 in either case.
Under an alternative mechanic, a player rolls 3d6 and compares the single highest die rolled to a Die Rating (DR) assigned by the gamemaster, succeeding if the single highest die meets or exceeds the DR. For example, if they rolled max(1, 3, 4)=4 or max(1, 3, 3)=3 against DR 3, they would succeed because their highest die was at least 3 in either case.
Suppose the common TN's used for the standard mechanic are 5, 10, and 15. What DR value best approximates the probability of success for each common TN? In other words, if the player has a certain probability of success when rolling the total of 3d6 versus TN 5 (and so on) then they would have roughly the same probability of success when rolling the highest of 3d6 versus what DR (respectively)?
For reference, the standard mechanic described is from Risus for "professional"-ranked cliches using only the TN's that 3d6 can beat (the game actually allows rolling as many as 6d6 and the highest listed TN is 30), and the alternative mechanic described is an attempt to convert the "Best of Set" rule variant from the Risus Companion to work for unopposed rolls. I focused on only 3d6 to get the gist for the typical number of dice as a baseline.