On Queen-Jack-Low Opposite Ace-Fifth
by Mark Thompson
This note assumes that the reader has read the first installment of Mixing It Up by Jeff Rubens in the December 2005 issue of The Bridge World. It analyzes the play of card combinations similar to those described in that article; in particular, we assume that dummy (North), on lead, holds queen-jack-low in a suit where both sides know that declarer (South) needs to take five tricks with his five cards headed by the ace.
For the moment, assume that dummy's low card is the seven. From the sight of dummy alone, there are 185 (roughly equally likely) cases in which the issue is in doubt. Using x to represent any card from deuce through six, we have:
In a relatively complex game such as this, and in the absence of a non-computational argument that suggests a good strategy for either the maximizer (South) or the minimizer (East-West), two possible analytic approaches are: (1) brute force--calculating the entries in a (rather large) matrix that displays the number of times declarer will succeed (on average) for each possible straregy of South, when opposed by each possible strategy of East and West, then applying one of the usual game-theory techniques, such as linear programming, to solve the game it represents--unfortunately, the sneaky graphical trick Jeff used is trickier to apply here with more strategic options to be considered; (2) guess what seem to be intelligent (possibly mixed) strategies for each side, in the hope of being able to show that your guess for the maximizer guarantees at least a certain average success rate, let's call it v (for what we hope will be the value of the game), while your guess for the minimizer guarantees at most that same success rate, v. [Approach (2) is based on the fundamental Minimax Theorem of game theory, proved in 1928 by John von Neumann. Von Neumann showed that in games such as this — zero-sum (one side's loss is the other's gain) with two contending sides — there must be such a value v. If we can determine both a maximizer's strategy that will secure an average level of success v no matter what the minimizers do, and a minimizer's strategy such that, against it, the maximizer cannot achieve a higher average success rate than v, then the game is effectively solved (although we may not have discovered all strategies that achieve no worse than v for each side). If we are satisfied with the results of (2), we are spared the exhaustive analytic work that would be entailed by (1).]
In applying Approach (2), I was encouraged by a few clues that came to me with the problem: Dick Zeckhauser had conjectured that the situation was strategically similar to the one with queen-jack-eight in dummy. Jeff Rubens had conjectured that East should never play the eight (and also -- I guess this is a metaconjecture -- that no one he knew would ever try using brute force). Thus encouraged(?), I came up with what seemed a reasonable pair of strategies. Guided by the conjectures, I experimented until I came up with:
Strategy for East-West: East (as with queen-jack-eight in dummy) covers from king-low one-third of the time; East never plays the eight from king-eight or king-eight-low; after queen, low, low, ?, West plays the ten with ten-nine-low or ten-nine-doubleton, but plays low from ten-eight-low.
Against that East-West strategy, we can figure out South's strategy to maximize the number of declarer successes, which is:
(1) Holding Axxxx or A8xxx: after queen, low (or eight), low, ten (or nine or eight or low), to continue with dummy's seven; (2) Holding A9xxx: after queen, low (or eight), low, low (or eight), to continue with dummy's jack; after queen, king, ace, low, to play for East to have begun with either king-ten (dropping the ten next) or king-eight (pinning with a nine-lead); (3) Holding A98xx: after queen, low, low, low, to continue with dummy's jack; after queen, king, ace, low, to continue with either the finesse or the attempted drop.
This strategy achieved for South an expectation (that is, a long-run average) of 96 and two-thirds successes. To check if 96 and two-thirds is indeed the value of the game, South's strategy needs to be better specified (to include no "either . . . or . . ."'s) before seeing how well East-West can do against it. Further experimentation led to specifying that: (2') with A9xxx, after queen, king, ace, low, South next plays for the drop; and (3') with A98xx, after queen, king, ace, low, South should randomize and finesse five-sixths of the time, playing for the drop one-sixth of the time.
Against this strategy for South, even if East-West knew what it was, there is no way for the defense to prevent declarer from winning at least 96 and two-thirds (out of 185) of the time. These two results confirm that 96 and two-thirds is indeed the value of the game. The two specified strategies are optimal in the sense of achieving best possible results against best opposition. Against inferior opponents, one can deviate from these strategies in hope of doing better than the game value — but only at the risk of possibly doing worse.
A few other points of possible interest: (1) The same analysis works when dummy's low card is deuce through six. (2) As when dummy has queen-jack-eight, mixing is profitable. If East-West were restricted to pure strategies (i.e., without any randomization), the best East could do would be never to cover, after which South could succeed in an average of 105 cases. Alternatively, if South were restricted to a pure strategy, his best choice would be to finesse against the ten on round two with A98xx after the queen is covered, which would reduce the success rate to 95. (3) An unstated assumption is that the defenders will play spot cards known to be insignificant at random. This is meaningful, because if they instead always play lowest from all three-card holdings including two spot cards (while otherwise playing their optimal strategy), best declarer play boosts the expected success total to 113 and one-third.
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