What is Signified by High-Card Points?

by Andrzej Matuszewski

Over the last 50 years, no top expert has asked the title's question. Key words and phrases used in discussions of HCP include "simple," "natural," "very easy algorithm," and "useful and basic tool." From a scientific viewpoint, if one wants to treat a player's decisions as solutions of probabilistic problems, every part of the supporting reasoning must have a probabilistically-defined empirical meaning.

What do HCPs mean in the probabilistic language? In 1987, Richard Cowan published Hand Evaluation in the game of Contract Bridge (Applied Statistics: vol. 36, pp. 58-71). He used the only way to establish an empirical meaning of a given valuation system of high cards: through regression analysis. Two types of statistical variables enter the regression: the dependent variable is trick-taking potential (TTP), the independent variables are the numbers of aces (A), kings (K), queens (Q), and jacks (J). We want to find an environment that confirms the equation: TTP = 4*A + 3*K + 2*Q + 1*J. Cowan proved that there exists a "flat hands"-type of environment where TN = -1.54 + 0.282 * (5*A + 4*K + 3*Q + 2*J + 1*T), where TN is the number of tricks taken and T is the number of tens held.

For example, suppose North-South hold:

	NORTH ♠ A Q 5 3 ♥ J 7 6 2 ♦ Q J 10 ♣ 10 4
	SOUTH ♠ K 6 4 ♥ K 10 3 ♦ A 6 3 2 ♣ K Q 5

Substituting the values 2, 3, 3, 2, and 3 for A, K, Q, J and T yields TN = 9.18. This is an estimation (or average) of the number of tricks North-South will take at notrump. It suggests that a contract of three notrump is more than reasonable, as declarer will more often take nine or more tricks than eight or fewer.

I offer this explanation of the equation:

(1) Cowan considered only 4-3-3-3 and 4-4-3-2 distributions for the declaring side, North-South. All values in his equation refer to the combined North-South holdings. He assumed a notrump contract. TN is the expected number of tricks (an average) taken by North-South for a given pair of holdings; it usually has a non-integer value. The expected number of tricks can be calculated exactly. In any case, the algorithm for that calculation requires some normal assumptions and is nontrivial but also not especially sophisticated. Cowan's equation represents the best choices for the multipliers on the right-hand side. Another thing to keep in mind is that TN is not an exact average, because it incorporates "sampling error." This error occurs because regression simplifies reality in order to represent the situation with one linear equation.

(2) Regression (whether linear or not) is a statistical method for summarizes a set of observed data in one statement. Cowan generated a large set of combined holdings for North-South from which to calculate multipliers for A, K, Q, J and T, and a constant, that create the best approximation of TN using that form of equation. He then arranged the result to make the multipliers 5, 4, 3, 2, and 1 to enable such a point-count to be used in practice. This valuation technique yields 60 valuation points for the honors in a deal, compared to 40 HCPs from the 4-3-2-1-count.

(3) One can check that for very low values of A, K, Q, J and T, the equation gives TN a negative value. For very high values of A, K, Q, J and T, the equation gives TN a value higher than 13. Cowan demonstrated that those outcomes would arise very rarely, so such cases are statistically insignificant. When the outcome is negative or over 13, the strange result does not help estimate the number of tricks that can be taken.

(4) Because the calculation was restricted to flat hands and notrump contract, experienced players will feel that aces are under-valued and tens are over-valued, which gives some empirical support to HCPs and other methods of valuing honors. However, during the bidding the partners may discover that their holdings are flat and the contract will be in notrump, in which case 5-4-3-2-1-points may be very useful.

(5) In practice, few players will always calculate both 4-3-2-1 points and 5-4-3-2-1 points. However, Cowan's research suggests that with a balanced hand one should add half a point for each ten held (as the experiment shows that on average a ten has half the value of a jack).

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