My immediate thought is somewhat similar to Dave's. Also much less effort than aerating, reseeding, etc. etc., which IMO would not work.
Get a couple short sections of 2x4 lumber, longer than the distance between the ends of 2 adjacent chair legs (I would do 2 left legs, then the same with 2 right legs). Mark the position of those 2 leg ends on one piece of wood. Use a drill and a spade bit that is a little larger in diameter than the leg ends to drill holes in the wood where you marked it, going roughly half way through the wood.
Do the same with the opposing 2 chair leg ends and the second piece of wood. Put the chair leg ends into those holes. You will have the chair on stable footing, no sinking.