The comments clear it up a little - a coin flip breaks tied bids, and you're looking for an amount that would let you win even in the worst case. So you'd need

The comments clear it up a little - a coin flip breaks tied bids, and you're looking for an amount that would let you win even in the worst case. So you'd need to be able to pay 34/round to only allow your opponent two plays. It's worthwhile to note that if you win the first two turns, you can bid the house on the third turn and your opponent has to respond in kind - otherwise they lose. Then you can simply outbid them for two other spaces. Supposing you take the center and a corner, in the first two rounds, then you only need to have 51 to spend on the last space in order to win. Your opponent pays 51 and blocks, leaving them on 49. You can bid 25 to set up a fork, your opponent blocks (leaving them with 24), then you pay 25 each to play out a different win. Or if your opponent doesn't block, you have enough left to convert one of the two wins guaranteed. So, 119 points is sufficient to win if you suppose that your opponent lets you take those first two spaces for 34 points in each. You could probably get it even lower if you realize that you have two fork opportunities, so you could probably bid even lower than 51 and still force out a win. I'll leave determining how many points a win takes if your opponent does block one of your first two moves as an exercise for someone else.