Retrogression Rules

The USGA recommends settling ties with a playoff of at least three holes for stroke play. If the competition is still tied, a hole-by-hole (sudden death) playoff may be used.
…if the play-off is less than 18 holes, the percentage of 18 holes played should be applied to the players’ handicaps to determine their play-off handicaps. Handicap stroke fractions of one half stroke or more should count as a full stroke and any lesser fraction should be disregarded.

If a play-off of any type is not feasible, matching score cards is recommended. The method of matching cards should be announced in advance and should also provide what will happen if this procedure does not produce a winner. An acceptable method of matching cards is to determine the winner on the basis of the best score for the last nine holes. If the tying players have the same score for the last nine, determine the winner on the basis of the last six holes, last three holes and finally the 18th hole. If this method is used in a competition with a multiple tee start, it is recommended that the “last nine holes, last six holes, etc.” is considered to be holes 10-18, 13-18, etc.
From USGA Rules and Decisions, Appendix I – Local Rules; Conditions of the Competition, Part B, Section 10, Stroke Play

Matching scorecards tie breaker method:
Summarizing the non-playoff, “matching score cards” method…
Compare…
Gross score for last 9 holes minus one-half handicap; if still tied continue
Gross score for last 6 holes minus one-third handicap; if still tied continue
Gross score for last 3 holes minus one-sixth handicap; if still tied continue
Gross score for 18th hole minus one-eighteenth handicap

Example:
Assume Dan has a 10 handicap and Joe a 14 handicap.
Last 9 holes
Dan 39 – (10 / 2) = 39 – 5 = 34
Joe 41 – (14 / 2) = 41 – 7 = 34
Still tied…
Last 6 holes
Dan 27 – (10 / 3) = 27- 3.3 = 27 – 3 = 24
Joe 29 – (14/3) = 29 – 4.7 = 29 – 5 = 24
Still tied…
Last 3 holes
Dan 12 – (10/6) = 12 – 1.7 = 12 – 2 = 10
Joe 14- (14/6) = 14 – 2.3 = 14 – 2 = 12
Dan WINS!