A Quick Briefing on the Poker Bluff

In poker, there are good hands, bad hands and hands in between. On the last betting round (1), obviously, you'd like to bet with your good hands, hoping for your opponent to call with a lower hand, thus making you money. However, you can't bet all of your hands, so you will check some of your hands. But if you bet only with your best hands, your opponent can adjust to this, and call only with hands slightly better than your worst betting hand. It wouldn't make sense for him to call with hands lower than your worst betting hand, since that would lose him money.

But if he wouldn't call with a hand lower than your worst betting hand, then why would you bet it in the first place? It doesn't win you any money when he has a lower hand, but it costs you money when he has a better hand. So, you wouldn't bet it. This narrowing of your betting range would lead your opponent to narrow his calling range as well, so that you'd have another worst betting hand that isn't worth betting, either. And so it goes on, until you don't bet any hand but the very best hand possible. To put it in game-theory terms, there is no equilibrium where you bet only a top range of your hands.

In order to find an equilibrium, you need to introduce the bluff to your strategy. By betting a top range of your hands, and a bottom range of hands, that is, some of your absolutely worst hands, you'll make it worthwhile for your opponent to call with a hand lower than your worst value-betting hand (value betting being betting with one of the top range hands, as opposed to bluff betting, which you do with the bottom range hands). It follows that betting with that very worst value-betting hand will be worthwhile to you, since now it makes you money when your opponent calls with a lower hand.

But do we really want to make it worthwhile for our opponent to call? That sounds beneficial to him, doesn't it? Well, if we bluff with the optimal frequency, that is, if our value-betting range stands in the right proportion to our bluffing range, he will break even from calling with those hands that can only beat a bluff. In game-theory terms, he's indifferent to calling. But even if he could choose to fold instead of calling, he must call with a proper fraction of those hands, in order not to open up for you to exploit his folding tendency by bluffing more.

There, that's a quick briefing on the poker bluff.
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Notes:
(1) On earlier betting rounds, when the lower hand sometimes has a chance of winning, due to drawing possibilities, things are a bit different. But this is outside of the scope of this entry.

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