I'm not sure of the exact size of the screw end of those posts, but what about something like this:

http://www.amazon.com/Hanson-53401-Spiral-Screw-Extractor/dp/B00004YOB2/ref=sr_1_1

They also have some drill bits which you run in reverse to try and work it out

http://www.amazon.com/MAXCRAFT-60406-Screw-Extractor-3-Piece/dp/B003BXOJSM/ref=sr_1_fkmr2_2

Or you could drill through it and switch to a threaded base 8^).

titanium drill bit..

drill through, and replace with machine post. nut on bottom...

-jon

Use a pin punch, and drive it through the other side of the playfield
Fill the hole with wood filler, and put a new post in

Blue tape off around the post, duct tape over that to protect the pf then slot it with a dremel and back it out with a flat blade screwdriver.
Remove tape, replace with new post.

Before you drill through the playfield, make sure nothing is on the back side of it. Also, Sears sells some screw extractors that I have had success with. They are basically a left hand drill bit on one side, then an extractor on the other. If you live in the Northern Virginia Area, I am happy to let you borrow my set.

Tom

i felt the same. just do it slow.

press straight down firmly so it doesn't slip, but no too much that it spins too quick, and know it's going to guide you if you focus it.

as shavings start to fill the hole, blow them out several times thru the process, and wipe the bit.

always reverse the drill to remove it from the wood, and let it carve slightly and cleanly on its way up. makes for a bigger hole to allow the machine screw to fit nicely in the newly formed hole.

trust me. it's the best solution.

-jon - reciprocal nyc

dont forget to tap or punch a starting point in the center of the broken screw. there are specific tools that do it.

i can't think of their names, just the sound they make when they do the job. but someone can prob chime in here.

don't rush it until you have this tool. else you risk slipping all over that thing.

-jon

Listen to John, don't waste your time trying anything else!

Center punch It will drive it through the playfield before it puts a dent in the screw.

I can't see from the pic if there is a bit of the post sticking up or not, if there is slotting it is the way to go, if not just punch it through, the bottom of the punch should look like this (<.
If there is room underneath use a machine thread post, nut and washer or a tnut if there isn't.
If it is broken off down in the PF chances are you won't get a drill to stay centered and you will mung the pf.
Listen to John on this one and punch it through, it may splinter the bottom a bit but an easy repair compared to off centering and munging the top of the PF.

First thing I thought of.
I'd stick with the screw extractor first, try drilling it out second, and the punching it out only as a last resort.
And if you do drill thru / punch it out, do like bigehrl suggested and replace it with a machine post and
T-nut.

when i used this fix on my getaway lane divider post...

the center punch basically created a small pin hole in the center of the broken-off screw, giving the drill bit a good guide.

i didn't experience it actually 'pushing' the metal down further into the pf so much.

-jon

Thanks for posting the resolution

Tom