When to take a Safety Play?

Although I do have a preference for matchpoints and consider my overall style better suited for that type of game, safety plays are among my favorite kind of declarer plays. In today's hand, there are a couple of subtle options. South is in 4♥ after an uncontested auction and west leads the ♦Q.

North
♠	KJxx
♥	Jxx
♦	xxx
♣	Axx
South
♠	Axx
♥	AKQxx
♦	Ax
♣	xxx

You might be able to make it on some sort of strip and endplay but you don't know enough about the distribution and that's not really a possibility to consider. The only real way to make 5 is to draw trumps, play ace of spades and take a spade finesse. This line will make 5 when spades are 3-3 with the Q onside, an 18% chance. It will make only 3 when the suit is 4-2 and the Q is off (24%) or when RHO has 5 or 6 spades with the Q (7%). Combined, this is a 31% chance you'll go down with this line, and you'll make 4 51% of the time: 3-3 w/ Q off (18%), 4-2 w/ Q on (24%), 5 or 6 spades w LHO or stiff Q with RHO (9%). That means on average, you'll make 3.87.

Let us consider another line of play, a line that is only available on a non-club lead, for a club lead would cut out an entry to dummy. What are the percentages of playing the K and A of spades first and then leading low towards the jack? This will never get 4 spade tricks because if the Q falls, the suit cannot split 3-3, so it essentially gives up the chance of an overtrick, but does it fact make the contract more likely to succeed. This line will make 4 77% of the time: any 3-3 split (36%), any other time the Q is onside (32%), Q or QX offside (9%). You will go down 23% of the time: RHO with at least 4 spades including the Q. On average this makes 3.77 tricks, slightly worse then the previous line.

In other words, playing the AK first wins an extra trick (and makes the contract when the other way goes down) when east has Qx of spades (8%), and the finesse wins an overtrick when west has Qxx (16%).

So, how do you decide what line of play to take? In a matchpoint game? In imps? In matchpoints, you should consider how likely the field is to be in the same contract. In this case 4♥ seems to be an easy and normal contract to reach so you probably should try for the maximum number of tricks. In an average game, the field will take the straight up finesse and the matchpoints to be won by taking the safety play when it wins should be about equal to the matchpoints lost when it loses, relative to the straight finesse. If you decide that a significant portion of the field will be in a partscore or in 3NT making only 3, you would be even more concerned with making the contract for just making would already be a good score.

In imps, it's a little different. You don't care so much about the 1 imp you could win by making an overtrick but you care about not losing the 10 or 12 imps (Depending on vulnerability) when you go down and they make at the other table. If the probability of making an overtrick is 10 or 12 times as big as the potential gain from a safety play, then it's an even chance, but this time the ratio is 2:1. By taking the safety play as opposed to the normal finesse at the other table, you'll push the board 76% of the time, win 1 imp 16% of the time, and lose 10 or 12 imps 8% of the time, so in this form of scoring, it's clear to take the safety play.