The programmer of the game of hearts appears to have given perfect memory to opponents of the live player, but along with that he or she has given them imperfect strategy. This is to be expected if the programmer seeks to entertain, for who would want to play the game if opponents consistently won?

What follows is a diagnosis of the program for the game derived only from playing it, not from an analysis of the code, which is inaccessible. It is assumed that all dealing results from random shuffles in the Microsoft version of the game. (There are at least two exceptions to the assumption of random shuffles, as noted below.)

There are four players. The object of the game is to avoid taking points. Each heart taken counts one point against the recipient and the queen of spades counts thirteen. Game is concluded when any player reaches 100 points. If, however, a player takes all 26 points (all 13 hearts and the queen of spades), then each opponent is given 26 points. The player with the lowest score wins.

In the first deal, players are required, first, to pass any three cards to their left; in the second deal, they must pass three cards to their right; thirdly, three cards are passed across to the player opposite them; fourth, to pass no cards but keep what cards were dealt. In subsequent deals, these passing procedures are repeated. The opening lead is always the two of clubs.

It is a fact that a reasonably astute player, without perfect memory, can score an average of close to 2 (or even less), that is, coming in second in a field of 4, where 1 is first and 4 is last. Thus, after 149 games, the player here studied (here called Player with a capital "P") averaged a score of 2.15.

The mean, if all players are of equal skill, is of course, 2.5, the average of the sum of 1, 2, 3, and 4. (The standard deviation of a randomly distributed universe of scores is 1.118—the square root of the mean of squares of the deviations of each of four values from the mean of 2.5. The sampling error is the root of the quotient of the deviation, 1.118, divided by a sample size of 149 or .087. Hence, a mean score of 2.15 differs from the 2.5 mean by .35, more than four standard errors from the mean. The result is statistically significant well beyond the one-tenth percent level. That is, so high a score is unlikely to be due to chance and more likely to be due to relative skill.)

What are the defects in the strategy of opponents?

Opponents too frequently fail to sweep all 26 points, thereby failing to award 26 points to each other player. In 86 games in which there were successful sweeps, Player made 59 successful sweeps to only 27 sweeps made by all three opponents. (It would be expected, under an equal opportunity assumption, that each of the four players would successfully execute one-quarter of the successful sweeps. But Player recorded 68.6 percent of the successful sweeps. This is "off the chart" in terms of probability, too highly improbable to warrant an assumption of equal skills.)

Of course, failed attempts to sweep lower the average gain of anyone who tries too often. But one successful sweep that penalizes each of three opponents with 26 points is worth much more than a failed attempt in which somewhat fewer than 26 points are incurred by the one who fails. The remarkably high frequency with which Player can sweep suggests that perhaps favorable hands are more often dealt to him, thereby vitiating the assumption of random dealing.

Opponents usually fail to recognize when Player is sweeping, that is, trying to take all the hearts and the Queen (Q) of spades. They slough off, say, the King (K) and Q of hearts, thereby making good Player's Jack (J) and 10 (when, of course, he either holds the Ace or it has already been sloughed.) They make this same mistake when one of them attempts a sweep. They lead the Q of spades, which counts 13 points against them, when they have an alternative card to exit from their hands.

An opponent will make a sweep even when it insures that he or she will lose the game by driving one player to 100 points while leaving himself in second or third or fourth place. An analogy from football is the team that decides to kick the point after touchdown in the few seconds left to play rather than pass into the end zone even though the single point awarded for successfully kicking will lose the game while a successful pass will at least tie it.

Opponents see the advantage of passing honor cards in hearts but often fail to see the advantage of passing a low heart which may foil an attempt to sweep; holding a deuce of hearts will likely prevent a player from taking all the hearts except in the case where he holds an extremely long heart suit and so can draw all of opponents' hearts.

There are doubts that opponents will, when possible, hold off, say, sloughing the Q of spades (or a heart) until the most threatening opponent (the lowest scoring) is likely to take the trick. It is clearly advantageous to dump points on the leading opponent. Dumping on some opponent who is already near having 100 points while you are far behind, in third or fourth place, merely insures that you will not end up at or near the head of the pack.

It appears that only one time in five will the player be dealt the Q of spades when the hand is played as a "hold" rather than as a pass left, right, or across. This fact, of course, is an exception to the inference drawn, above, that each deal is random.

In passing three cards, opponents seem to fail to void themselves in clubs so that they may slough such dangerous cards as the Ace or King of spades on the opening lead. In four games, Player did this ten times and, as well as could be determined, opponents did it once. They also seem to slough low spades, a tactic that is almost never appropriate. They also pass some of the lowest hearts, which also seems inappropriate. Equally important, opponents will sometimes fail to pass the Q of spades when the opportunity presents itself.

The only real strength in the strategy of opponents seems to be a result of perfect memory of what cards have been played. Thus, for example, if a player is attempting a sweep and towards the end of play he holds, say, a losing club, an opponent will save a winning club and take the lead to capture a heart, by a slough from another player. Similarly, an opponent will, when holding a high heart, sometimes capture the second or third lead of a lower heart during an attempted sweep.

Conclusion—»