if his initial demand is too high, the absolute worst thing you can do is offer early. you can only undercut current demand by so much relative to personality and other relevant factors.
so, 'keeping him interested' in this way will only preclude you from a lower offer later. once you offer 'more' he won't be interested in less and you only upset them by withdrawing an offer. this falls under don't offer him early unless price is right, as above.
if you know your market, you should have a good idea of a reasonable price his demand will drop to. how risky you want to be about saving money is the only question and making sure you don't cross the line of how much you want to spend if a bidding war ensues, of course.
once in a while someone gets overpaid, and sometimes they get underpaid... i'm willing to forego the latter to prevent the previous. if they don't offer a good extension price, then i am looking for reasonable price in FA and won't push my luck. (extreme financial situation may force the opposite, but i don't let such things happen -- no forced decisions). the moment i think my bid will be accepted, i offer.
the AI offers are not bound by same rules as ours -- one example being that they can low-ball a bid on day one with no repercussions from that player being offended like us doing the same thing. so, if "3" other teams have offered and his demand keeps dropping tht tell you their bids are WAY below what he wants... your 'reasonable' bid is likely to win in that situation. (consider competition and their budget, be smart about it in every way the game gives us an opportunity relative to this context)
when someone outbids, don't just add $1 or the like. either beef it up or quit. adding some minimal amount is almost certainly going to fail. think about how IAFA work. you can up it 50-500k all day and all that will do is maximize what you spend when adding +1-1.5m to start may save you in long run by scaring bidders away. you really can increase the cost of a player by raising small..
i've raised knowing the ai would spend more and that i had already decided the price was too high, but that's a bit risky, lol. i'd do that to a human in an online league any opportunity too. push your competition's costs up