Haterobics covered most of this already, but here is my spin:
Most discount strategies involve some degree of inconvenience and/or uncertainty. After all, if discounts were that easy to get, producers wouldn't be able to pay the bills!
For someone in your position, who comes into the city a couple of times a year, likes to pick his or her own seats, prefers close orchestra, and wants to book in advance, discount codes are probably the best way to go (as previous posters have suggested).
Another good idea for you might be to join Lincoln Center Theater for The King and I discounts, although I don't know how many good seats would be left at this point.
My usual bottom-feeding methods include:
1) TDF (you can't pick your seat location, plus you have to be a member)
2) Rushing (very uncertain, not able to book in advance)
3) Lotteries (ditto)
4) TKTS booth (uncertain, and not able to book in advance. Also, it's not all that cheap)
5) Special promotional deals (a case-by-case basis, usually involves waiting on line at an ungodly hour)
Enjoy your trip and have fun!
Updated On: 2/10/15 at 06:05 PM