

Test Your Intuition: What does it Take to Win Tic-Tac-Toe - ColinWright
http://gilkalai.wordpress.com/2013/03/15/test-your-intuition-17-what-does-it-take-to-win-tic-tac-toe/

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Someone
A first step that brings the limit down: with only 100 in hand, O cannot beat
you more than twice with a bet of 34 or more. So, bidding 33 on each turn is
sufficient to win. That definitely is doable with at most 5 turns that you win
for a total cost of 165.

However, the second square isn't worth as much to the one who already won the
first, as it leaks information about his strategy. If you won the first, it
may be better to let the other player win the second, draining his resources,
allowing you to get the other squares you need cheaper.

Because of that, it may be possible to get the maximum down from that 165 (my
intuition says you can, but I am too lazy to do the calculations now)

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jbri
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.

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gelisam
The way in which ties are resolved is very important. I had assumed that in
case of a tie, neither player won the bidding war, so they had to bet again.
Under that interpretation, the worst case is that O always bets the same as X,
trapping the game into a non-productive sequence of tied bids. The only way to
avoid this is to always bet more than O can possibly bet, which means we need
more than 300.

Fortunately, Gil Kalai has added in the comments that tied bids are resolved
with coin flips, so if you can find a winning strategy in both sub-trees, ties
are fine.

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dmxt
Apparently my ISP blocked that website for some reason. Corbina, RUS.

