Expected Value

Expected value is a term from statistics and probabilities. It means: What is the value (or gain) we expect to get if we do the same experiment over and over again?

How is that related to Bridge? I'll present a few examples below.

Let's say we play IMPs. Our partner makes a game invite and we are in doubt: bid 4♠ or Pass?



We have 16 points. Is that 16 points hand good enough for a raise?

Looking at NS cards we can count 1 loser in each suit, which means that making 10 tricks will be totally depending on a successful spade finesse. So we can estimate our chances to make the contract at 50%.

Assuming that indeed the contract has a 50% chance to succeed, what should we do? Should we pass or raise? Also, is vulnerability an issue in our decision here?

Let's say we're playing a speedball and we get the exact same bidding and problem on all 12 hands. We'll check what happens if we would pass on all 12 hands, and what happens if we would raise on all 12 hands.

Since we assumed 50% success, we'll say that on 6 hands we make 9 tricks and on other 6 we make 10.

Let's start with the Pass policy:

Contract: 3♠

Tricks	Score	Probability	Value
9	140	50%	70
10	170	50%	85

We can see that our total expected value here (70+85) is 155 (which is the average between 140 and 170) whether vulnerable or not.

Now let's see the raise to 4♠ policy, not vulnerable:

Contract: 4♠

Tricks	Score	Probability	Value
9	-50	50%	-25
10	420	50%	210

We can see that our expected value here is 185, which means that we will gain more if we bid game = We can gain more than we lose!

And what if we are vulnerable? Should we raise or be more careful now? Let's see:

Contract: 4♠

Tricks	Score	Probability	Value
9	-100	50%	-50
10	620	50%	310

We can see that our expected value here is 260, which means that we will gain MUCH more if we bid game = We gain MUCH more than we lose!

In other words, if we are vulnerable, it is much more profitable, on the long term, and if the chance of making is 50%, to bid a game as we have much more to gain than to lose.

To make it more clear: The score in IMPs is calculated in that way: Calculate the average of all scores and check how far you score from the average. This distance is translated to IMPs in a special IMPs table.

Let's check these distances:
When not vulnerable:

• 140 for making 3♠ and -50 for 4♠ -1 = 90. This means an average of 45 for the hand. That means that 3♠ made will gain 95 which is 3 IMPs, and 4♠ -1 will lose 95 which is a loss of 3 IMPs.
• 170 for 3♠ +1 and 420 for making 4♠ = 590. This means an average of 295 for the hand. That means that 3♠ + 1 will lose 125 which is 4 IMPs and 4 made will gain 4 IMPs.

When vulnerable:

• 140 for 3♠ and -100 for 4♠ -1 = 40. This is an average of 20 for the hand. That means that 3♠ made will gain 120 (140 -20) which is 3 IMPs and 4♠ -1 will lose 120 (-100 -20) which is a loss of 3 IMPs.
• 170 for 3♠ + 1 and 620 for making 4♠ = 790. This is an average of 395 for the hand. That means that 3♠ + 1 will lose 225 (395 - 170) which is 6 IMPs and 4♠ made will gain 6 IMPs!

To sum it up, lets translate the IMPs to dollars and translate the play above to "heads and tails" (flipping a coin).

We flip the coin 12 times and we always choose "heads", which has a 50% chance to come up. On the first 6 flips our award for winning is 4$ and our cost for losing is 3$. Let's assume that on these 6 flips we win 3 times and lose 3 times = we earned 3$.

On the last 6 flips we win 6$ for each "heads" and lose 3$ for each "tails". If it's "heads" 3 times and "tails" 3 times we will win 9$.

Back to our speedball: If in all 12 hands we will raise to 4♠, we are expected to win 12 IMPs in total for our strategy (9 for the 6 hands vul and 3 for the 6 hands not vulnerable). If we pass all 12 hands, we are expected to lose 12 IMPs for our strategy.

Did you hear the saying: "A slam on a finesse is a good slam"?

Let's check it out:



Partner's 4NT is a quantitative slam invitation. It is more than likely that if we bid a slam, it would depend on a successful finesse in one of the suits, like in the hand above: The slam is totally depending on the successful spade finesse. Assuming the success of bidding slam is 50%, should we bid the slam or pass 4NT? Do we gain more than we lose by raising anyway?

Let's check it out:

1. Not vulnerable:

a. Contract: 4NT

Tricks	Score	Probability	Value
11	460	50%	230
12	490	50%	245

b. Contract: 6NT

Tricks	Score	Probability	Value
11	-50	50%	-25
12	990	50%	495

The expected value of passing game is 475 and for bidding slam it's 470 = about the same.

2. Vulnerable:

a. Contract: 4NT

Tricks	Score	Probability	Value
11	660	50%	330
12	690	50%	345

b. Contract: 6NT

Tricks	Score	Probability	Value
11	-100	50%	-50
12	1440	50%	720

The expected value of passing is 675 and for bidding slam it's 670 = about the same.

Conclusion: There is no significant loss or gain if we always bid a slam here, whether vulnerable or not.

How about grand slam vs small slam? Is it worth risking a small slam that is sure to make, for a grand slam with a 50% chance to make?

Let's check:

1. Not vulnerable:

a. Contract: 6NT

Tricks	Score	Probability	Value
12	990	50%	495
13	1020	50%	510

b. Contract: 7NT

Tricks	Score	Probability	Value
12	-50	50%	-25
13	1520	50%	760

The expected value of staying in a small slam is 1005 and for bidding a grand slam is 735 = Not worth trying a grand (on a 50% chance)! Much more to lose than to gain.

To be exact: 735/(735+1005) = 0.42 and 1005/(735+1005) = 0.58.

Conclusion: The value would be equal if we had a 58% chance to make.

2. Vulnerable:

a. Contract: 6NT

Tricks	Score	Probability	Value
12	1440	50%	720
13	1470	50%	735

b. Contract: 7NT

Tricks	Score	Probability	Value
12	-100	50%	-55
13	2220	50%	1110

The expected value of staying in a small slam is 1455 and for bidding a grand slam is 1055 = Not worth trying a grand (on a 50% chance)! Much more to lose than to gain.

To be exact: 1055/(1055+1455) = 0.42 and 1455/(1055+1455) = 0.58. Here too, the value would be equal if we had a 58% chance to succeed.

There are many more aspects to discuss related to expected value, and this subject can be expanded to more decisions, also to those that are not 50%-50%. Some of our decisions will be made due to vulnerability, and there are also different decisions to be made if the score is IMPs or MPs.

For example:

Non vulnerable:


Partner's 2♣ overcall shows approx opening values (overcall at the 2nd level = 12-16 points).

At IMPs it's better to pass:

• Whether you make a part score or set opponents (+130, +110 , +100 , +50), OR
• Whether they make or you go down (-130, -110 , -100 , -50)

the result would be not too different (might be up to at most a 2 IMPs difference).

But at MPs there are more things to take into account:

• +50 (3♣-1) is likely to be a bad score if we can make +90 (or +110), also
• +100 (3♣-2) can be bad if we can make +110 (3♦) .

Expected value here = if we PASS, we lose. So, I recommend taking action. Dbl is best. Partner is likely to leave it (he didn't bid 3♦ and he didn't dbl, so he has no more to add). Say there is a 25% chance they make it = you will get 0 if they make it (instead of 20-25%), but high chance on other 75% times to score high. While if you PASS, you get a bad score in all the situations as you lose to ones staying on 2♦ or bid other part score.

The hand can be:


Notice that not many W would bid 3♣ with such a hand, but "our" West did. It is also likely that our partner has ♣xx as he didn't dbl nor did he bid 3♦. On the example above 3♣ goes 2 down (3 hearts 1 club and 2 diamonds for defense) while 3♦ makes (lose 3 spade and 1 clubs). If we let them play 3♣ we get +100 instead of +110 (many will be in 2♦+1 = 110. Some might even play part score in NT, with our cards, scoring 8 tricks = 120). +100 will get us a negative score at MPs, while +300 will surely be close to a top.

Let's change the hand a bit:


Now each side can make 8 tricks: 2♦ loses 3 spades, diamond and club. 3♣ loses 1 club 1 diamond and 3 hearts. If we pass 3♣, +50 will not give us a good score as many N-S would play 2♦, making +90. 3♣ dbled -1 will give us +100 which is a good score for sure as 1NT is made exactly and 2NT is going down which means that no part score will score higher than +100.

But what will happen if they make 3♣ doubled with, say
:


3♣ loses 3 hearts and 1 diamond. Say that we stick to our policy and double 3♣ and we get 0% (-470) for them making it. But remember that this is not us getting a bottom instead of a top score. If we pass 3♣, -110 will still be a bad score worth around 20-30% as most will likely be playing 2♦ on our side, going down 1 (-50). Notice! Here, at MPs, the gap between -110 (around 20-25% ) and -470, is not that big.

In other words if we double, estimating that opponents are most likely going down, we will upgrade our score from a bad score to a great score, while if opps make, our score will go from a very bad score to a bottom. But our expected value is to royaly win from our policy here.

What if this was IMPs scoring?

At IMPs, the calculations are completely different. Whether the score is 90, 100, 110, or 120, it makes no difference as it gives us the same IMPs. However, if they make 3♣ doubled, we can lose 9-10 IMPs, which will make us lose much more than we can gain. At IMPs - PASS!