Head Up Theory Question

[FishAndChips]

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If you anticipate having to fold at any point during the hand to one of your opponent's reraises, just don't raise yourself-- simply call. Then you at least get a showdown, and can not be bluffed out of a large pot.

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This argument isn't particularly persuasive because you could conceivably be costing yourself by not raising (even though you'd sometimes fold to a reraise).

Suppose that at a certain point in a certain game your opponent raises so that the pot is $95 and you could either fold, call $1, or raise $1 more. If you know that your opponent is playing an optimal strategy, you can put your opponent on a range of hands, and he should be bluffing with probability 1/96. It might be the case that you are holding the unique second-nuts, and if you raise, your opponent will call with the third-nuts, fourth-nuts, and fifth-nuts (each of which is an equally likely holding for him, at this point, as the nuts). But your opponent won't reraise for value again without the nuts (if you raise and he does reraise, there would then be a 1/100 probability he is bluffing).

If you just call, you'd be giving up EV. You need to raise for value because your opponent will call your raise with a worse hand three times as often as he will reraise with a better hand. So the question is, do you

(a) raise and always call a reraise;

(b) raise and always fold to a reraise; or

(c) raise and sometimes call, sometimes fold to a reraise.

The answer here is (c). Read the section "Using Game Theory to Call Possible Bluffs" in chapter 19 of TOP. In the scenario I've just described, your hand, the second-nuts, can only beat a bluff. If your strategy is to always call the reraise, your opponent could exploit it by never bluffing -- then you would be calling all the time with a certain loser. Of course since the pot is so big, you still have to call the vast majority of the time or else your opponent could exploit your strategy by bluffing more. When your opponent bluffs, he is risking $2 to win $97. Therefore your optimal mixed strategy at the end is to call 97/99 of the time and fold 2/99 of the time.

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Well I must return serve here, and say that your argument is not very persuasive either. There is no way you raise, and reraise, your opponent 94 times in $1 increments and then fold to another raise. You should stop raising BEFORE you get to the point where you raise and then fold. If you can't call a raise, why raise yourself. Yeah you may sacrifice slight EV on a value raise if he'd call three times out of four with a worse hand, but you lose a huge pot if you incorrectly fold the one time he raises.

Your proposed strategy could be greatly exploited. Look at it from your opponents perspective. If he knows your strategy, he is getting 94-1 on a bluff that will work 1 in 4 times!! I'd gladly sacrifice the EV of one additional raise 3 in 4 trys, if it meant never folding a winner in such a large pot.

As for your citing TOP ch. 19, it actually proves your strategy is flawed by showing that (as stated above) your opponent is getting 94-1 on a 3-1 bluff, and is therefore extremely -EV for you. You opponent shouldn't bluff 1 in 95 tries as you suggest in your reply, but should bluff 1 in 4. If you want your opponent to only bluff 1 in 95 times, you need to call 94 times out of 95! "If your opponent is getting 4-to-1 odds on a bluff, you must call randomly four out of five times to make bluffing unprofitable." (TOP pg.190)

Also, most examples about bluffing on the end deal with the "bluffer" knowing almost exactly what the other player's hand is, while the player deciding whether to call the possible bluff could not tell if the other player made his hand (and therefore was at a big informational disadvantage against the potential bluffer.) That is not the case here. Neither player has any exposed information, and no one's hand has changed value throughout the course of the betting. Without those things, bluffing here is completely different.