Dara O’Kearney: Results-Based Thinking and the Fundamental Theorem of Poker

You’ll hear it all the time when you play live: unsophisticated players either castigating their play or that of others and offering “advice” on what they should have done differently. If their opponent chased hit their draw, they’ll say “you should have shoved the turn” (or the flop or preflop: whatever the street before the draw got there was).

If their opponent shows a bluff after you fold, you should have called. If their opponent shows the nuts, you should have folded. And so on: you get the idea.

Often when I’m coaching players nearer to the start than the end of their journey to poker mastery, I have to try to break down this kind of results based thinking before real progress is possible.

They tell me hands they lost expecting me to tell them what their “mistake” was, or they proudly tell me a hand they won thinking they played it perfectly (which they rarely did). It’s seldom enough to just point out it’s not a good way to think about poker, so I often end up asking a series of questions along the lines of:

Challenging Results-Based Thinking

If the player who had aces when you called their all in with kings hadn’t looked at their cards before they shoved, would you still think your call was a mistake? If not, why not, because they still had aces.

If you had hit your king, would your call still be a mistake given you won the hand?

If your opponent had missed his draw on the river, do you still think you should have shoved the turn?

The Fundamental Theorem of Poker

Yet while results based thinking is clearly sub optimal, it contains within it a useful kernel of truth, a kernel that the Fundamental Theorem of Poker, formulated by David Sklansky, focuses on. It states:

“Every time you play a hand differently from the way you would have played it if you could see all your opponents’ cards, they gain; and every time you play your hand the same way you would have played it if you could see all their cards, they lose.”

In essence, it emphasizes the importance of making decisions based on incomplete information (not knowing your opponents’ cards) and how deviations from optimal play (assuming perfect information) can impact your profitability in the long run. It underscores the strategic aspect of poker, where skillful players aim to make decisions that maximize their edge over opponents who are also making decisions based on incomplete information.

An example

Suppose we are playing Texas hold ’em and we are dealt 9♣ 9♠ under the gun 9 handed one hundred big blinds deep. Should you call, raise or fold?

The chances of being dealt any pair (including nines) are 220/1, or about 0.45%. There are five higher pairs than nines, so the chances that any individual player yet to act has a better hand than ours is about 2.25%. With eight players behind us, the chances that one of them has a higher pair is about 20%, so 80% of the time we have the best hand. Therefore we have a clear raise.

It folds to the big blind who calls.

The flop comes A♣ K♦ 10♦, and the big blind checks. Should we bet or check?

We now have a decision to make based upon incomplete information. We don’t know whether our hand is best or not, but it’s extremely unlikely we started with a worse hand than the big blind (they would usually reraise their bigger pairs). They will also hit the flop less than half the time if they started with two unpaired cards. So since we usually still have the best hand, we should bet.

We bet and now get raised. We again have a decision to make based upon incomplete information. In this particular circumstance, the correct decision is almost certainly to fold. Even if we have the best hand right now, there are too many turn and river cards that could kill our hand. Even if our opponent does not have an ace or a king, there are 3 cards to a straight and 2 cards to a flush on the board, and they could easily be on a straight or flush draw. When we are behind we are essentially drawing to 2 outs (another 9), and even if we catch one of these outs, our set may not end up being the best hand.

However, suppose we knew (with 100% certainty) that our opponent held 8♦ 7♦. In this case, it would be correct to raise an amount the denies our opponent the correct price to call. Therefore, by folding (or even calling), we have played our hand differently from the way we would have played it if we could see our opponent’s cards, and so by the fundamental theorem of poker, our opponent has gained.

We have made a “mistake”, in the sense that we have played differently from the way we would have played if we knew our opponent held 8♦ 7♦, even though this “mistake” is almost certainly the best decision given the incomplete information available to us.

Inducing Opponents to Make Mistakes

This example also illustrates that one of the most important goals in poker is to induce the opponents to make mistakes. In this particular hand, our opponent has executed a semi-bluff — they raised a hand knowing it’s the worst hand, hoping we will fold, but they still have outs even if he calls or raises. They have induced us to make a “mistake”.