Morton's theorem

An example

In a limit hold'em example credited to Morton, Arnold holds A♦K♣ on a flop of K♠9♥3♥, giving him top pair with best kicker. After flop betting, Arnold has two opponents: Brenda, who has the nut flush draw (e.g., A♥J♥, 9 outs), and Charles, who holds second pair with a random kicker (e.g., Q♣9♣, 4 outs). The turn is 6♦, and the pot size is P big bets. Arnold bets.

Brenda, getting correct pot odds, calls. Charles must decide whether to call or fold. With 42 remaining cards, Charles wins the pot 4/42 of the time when calling, losing one big bet otherwise. Setting his expectation for calling equal to zero (the value of folding) gives a pot size at which he is indifferent.

Arnold's expectation also depends on P. Comparing the two indifference points reveals a range of pot sizes where Charles should fold correctly, yet Arnold makes more money when Charles folds than when he chases. This paradoxical region is shown graphically, with pot sizes on the horizontal axis and thresholds indicating when Charles should fold or call and when Arnold wants Charles to call or fold.

Analysis

In the paradoxical region, Charles pays too high a price for his weak draw, but Arnold is not the sole benefactor of that price; Brenda takes Charles' money when she makes her flush. Compared to a heads-up situation, Arnold still risks losing the whole pot but no longer receives full compensation from Charles' loose calls. This middle region explains the standard strategy of thinning the field when holding the best hand, as even opponents with incorrect draws cost a player money because part of their calls goes to other drawing opponents.

The aggregate of Brenda and Charles gains from Charles' call, implying implicit collusion if they split profits. This differs from schooling, where many opponents correctly call against the best hand. One conclusion is that in a loose hold'em game, the value of suited hands increases because they benefit from implicit collusion.