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Sleeping on the Couch

The continuing misadventures of Dave playing bridge with his wife, Anne

By Dave Caprera
Denver, Colorado

 

Return on Equity

In poker, “equity” is a measure of your likelihood of success. Stated differently, it is the average amount of money that a particular hand would win if the specific situation were repeated a large number of times. Assume you are dealt two aces (the best possible starting hand) and your opponent is dealt 7-2 unsuited (the worst possible starting hand). You are at a great advantage. Your equity is large. Your opponent’s equity is small. What you actually win is the return on your equity. A poker player with a good hand will bet to protect his equity. A poker player with little equity will likely fold.
Bridge equity is no different. When you pick up a hand full of aces and kings, long suits, and lots of distribution, you should expect to obtain a large positive score. You won’t always do so, but if you get such hands a number of times you should end up with a significant positive balance. Bonuses for games and slams bid and penalties extracted from defeating doubled contracts represent ways in which a bridge player may realize a return on his equity.
The Milton Work point count (aces are four, kings are three, etc.) as popularized by Charles Goren is one, perhaps crude, means of estimating one’s equity. But bridge is a partnership game and it is not one’s individual equity that counts, rather it is the combined equity of you and your partner that is determinative. One purpose of bidding is to attempt to assess how much equity your side has in any particular deal. A second purpose of bidding is to attempt to thwart your opponents from determining how much equity their side has in any particular deal. Of course, this must all be conducted in the context of bidding for a contract, which represents the schedule of rewards for making or failing to make the winning bid.
One can normalize equity on a scale by treating a deal which is passed out as having a zero rate of return to both sides. While it may not be a strictly accurate use of the terms, “negative equity” can be used to refer to a hand where one’s expected return is below zero, and “positive equity” where one’s expected return is above zero.
I was recently presented with three hands where this concept of equity was relevant. First, you pick up ♠AJxxx ♥KQx ♦Qxx ♣Ax. You are vulnerable against not. What is your equity? We could run a computer simulation, deal one million hands, and evaluate the hands double dummy, but my guess is that on some small percentage you make slam, perhaps close to half the time you make game, often you will go plus, but occasionally your opponents will outbid you and make their contract or beat yours. Arbitrarily, let’s say that your initial equity is +400.
But this time your right-hand opponent was dealer and opened 4♣. You have three reasonable choices: Pass, 4♠, and Double. If you pass, even though the auction isn’t yet over, the chances are good that you will be defending 4♣ undoubled and collecting a small plus score; your pass is likely to be the final decision for your side. If you double, partner is now called upon to participate in the decision-making, which is generally a good idea; partner can pass for penalties, bid, or invite you to bid further by cuebidding or jumping. However, if your left-hand opponent has a strong hand, he is also invited to participate in the auction by redoubling or doubling the final contract. Your double has increased your possible reward substantially, but it has also added the potential for a penalty. While the variance has increased, your chance for a larger plus score has also increased. If you bid 4♠, you will have conveyed more information to partner, who may still participate in the decision, but you have also put most of your eggs into a single basket. It may be the best single basket, but it makes it unlikely your side will be able to play in an alternative, possibly superior, contract. You might guess that I favored double.
I viewed this as a tough decision. When this hand was presented to a number of good players, there was an almost even three way split. But what does this have to do with equity? I think passing is a long-run loser. Admittedly, your original equity estimate has been altered. But I can’t accept taking a meager +50 or +100 by defending 4♣ when I have the realistic possibility of much more, even though it assumes additional risk. For me this translates into simple two rules of thumb:

1. Try not to pass if you have what you believe to be the best hand at the table.
2. Make a bid that gives you a chance of receiving your expected equity.

I have left out an important wrinkle with respect to the given hand: We were playing matchpoints. At match points, “Plusses are good, minuses are bad.” There are a number of other factors that have to be taken into account, including the strength of the field, the strength of your particular opponent, and the likelihood that the rest of the field will face the same 4♣ bid. I would still double, but as Annie will confirm, I suck at matchpoints, so my advice may not be very reliable.
The second hand I was presented with occurred when I was coaching my squad of juniors vying to represent the USA in the world junior championships next summer. We were practicing on BBO when one of them picked up ♠J9xx ♥10x ♦Qxx ♣AJxx. At vulnerable versus not, his LHO was dealer and the auction proceeded:

LHO     Partner RHO        “Junior”
1♣        1♥         1NT        Pass
2♣        2♥         3♣           ?

“Junior” bid 3♥, the 1NT bidder doubled (a somewhat aggressive double, but junior bridge is not for the faint of heart), and the result with normal play was -500 into +50 at the other table for defeating 3♣ a trick. Here is the entire deal:

♠J9xx
♥10x
♦Qxx
♣AJxx

♠10xx                    ♠KQx
♥QJxx                    ♥--
♦Kx                       ♦AJ98xx
♣Q108x                 ♣K9xx

 ♠A87
♥AK98xxx
♦10x
♣x

There are a number of ways to think about the North hand. It is almost entirely a defensive hand and “Junior” should see that it is worth several more tricks on defense than it is worth on offense. South’s rebid showed six hearts and North had only two, so he was violating the law of total tricks by contracting for nine tricks with only eight known trump. West’s free bid of 1NT was a warning signal that the hearts may not be breaking normally. And South had gotten the opponents to the three level, which is what you want to do in most partscore battles.
But a different way to think about the North hand is in terms of your equity. On the auction, I think North should expect both sides to have approximately 20 high card points. Nor is North’s hand particularly distributional. North should expect that his side’s equity is zero, that is to say he should expect, on average, to have a small plus half the time and a small minus the other half. Bidding turned his plus into a minus, which meant he had a negative return on his equity. He should have been happy with his +50. (Yes, perhaps East should make 3♣ but at the other table he misguessed the club suit and got tapped out.)
My third hand looks at how equity can be used to evaluate preemption. Consider the following: You hold ♠x ♥xx ♦KQ10xxx ♣QJxx, not vulnerable versus vulnerable, IMP scoring, and you are dealer. Annie and I play our 2♦ opener as conventional, so it is not available to us as a Weak Two Bid in diamonds, but even if 2♦ were available I would chose to open 3♦.
What is my equity? Not much. I have below average high card points, I am not vulnerable (so the game and slam bonuses are less), and all my cards are in the minor suits where I am likely to be outbid by the opponents, who rate to hold the majors.
The auction proceeded 3♦ - (Dbl) - 5♦. How do I feel about that? I am loving it! Partner may be bidding 5♦ to make, but more likely she is putting maximum pressure on the opponents by taking up as much room as possible. There is a direct relationship between the amount of equity your side has and the value of bidding space. If the opponents have large equity, the bidding space is valuable to them. If we have little equity, the bidding space is not valuable to us. Bidding space allows a partnership to convey information in order to reach the best contract.
When we have little equity, we want to take their bidding space away. That is why we preempt. Conversely, when we have large equity, we want to conserve bidding space as much as we can until we have enough information to place the contract.
RHO doubled and it went all pass. The whole deal was:


♠xxxx
♥xx
♦AJxx
♣10xx

♠AKxx                   ♠QJxx
♥KJxx                    ♥AQxxx
♦x                          ♦xx
♣Axxx                   ♣Kx

♠ x
♥xx
♦KQ10xxx
♣QJxx

Partner was clearly trading on the vulnerability. On the opening lead of the ♠K, East played the queen (showing the jack). The defense tried to cash a second spade, lost their possible club ruff, and we escaped for -500 rather than a possible -1430. East-West might have bid differently, but neither did anything silly. They had been denied the bidding room to realize the full return on their equity.
This was obviously a story with a happy ending for us. It isn’t always that way, which is why I also believe in a different kind of equity. Make sure to invest liberally in a warm blanket and a fluffy pillow because sometimes you may end up sleeping on the couch.