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Playing in 10-table IMP team tournament in your local club, you and your partner employ some common-sense bidding to reach sound contract – small slam in spades. After heart lead, only real issue is how to play trump suit. As you can afford one trump looser, you naturally consider if there in some space for safety play. There indeed is, if RHO has all four trumps, only way to make this hand is first round finesse of ten of spades. Is there any risk for contract if finesse fails? Not really, if there is any risk that RHO might ruff something, opening lead would be different. So you duly go for your safety play, and first round finesse expectedly loses, but you still score your small slam in comfort. After end of the match, you learn that you lose an IMP on this board, as score in another room is 1010, six spades made with overtrick. Oh well, you are so used to it...

At the end you wonder if your safety play was really justified. If you were playing rubber bridge, safety play was clearly called for. But playing IMPs, it’s not so sure. So, I decide to do some math. As I have invested some effort in it, why not to share it with BridgeWinners. If you are not interested in calculation but just result, just skip boring parts…

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Let’s approach our calculation in small steps

Step 0; define what we are calculating.

We are interested in comparing two lines of play in the same contract, and we want to find out which one yields better results in IMPs in the long run. Let’s imagine that we are in position to play this same hand 1000 times (of course, just our and partner hand does not change) stubbornly playing safe, and some other guy in stubbornly playing percentage play for maximum number of tricks. We want to find out if our safe line is earning or losing us IMPs, and also how much.

Step 1: formulate two lines of play we are comparing:

- ‘safe line’: one which maximizes chance of making contract, i.e. small to the ten.
- ‘max line’: one which maximizes number of tricks we make in contract. In this case, ‘max line’ means playing high spade on first round, and also on second, unless honor falls from LHO, in which case we will make second round finesse, according to restricted choice principle.

Step 2: enumerate ways how four spades can be distributed between LHO and RHO.

There are 16 combinations, and we can assume that all of them are equally likely. That assumption is not 100% accurate, because of ‘vacant spaces’ effect, and some possibility that perhaps opponents would get active in bidding with bit more shape, like void in spades. But, for this particular hand, it seems reasonable assumption.

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Step 3: compare how each of two lines is performing in those 16 different spade splits:

- In 7 cases both lines are performing equally well, both producing same number of tricks, namely:

In one case, when If RHO is void , both lines are producing 11 tricks

If RHO has any singleton (4 cases) both lines are producing 12 tricks

In one case, when LHO has QJ, both lines are producing 12 tricks

In one case, when RHO has QJ, both lines are producing 13 tricks

- In 6 cases ‘max-line’ is producing 13, and ‘safe-line’ 12 tricks, namely if RHO has Q95, J95, Q9, Q5, J9 or J5
- In 2 cases, ‘max-line’ is producing 12, and ‘safe-line’ 13 tricks, when RHO holds QJ9 or QJ5
- In 1 case ‘max-line’ is producing 11, and ‘safe-line’ 12 tricks, when RHO hold all four trumps, i.e. QJ95

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Step 4: compare probabilistic expectancy of ‘earned IMPs’ between two lines.

Initial and simplest way is to just assume that you will play ‘safe-line’ and declarer on another table will play ‘max-line’ when playing in same contract (that’s what actually happened at the table this time).

- value of lost overtrick for ‘safe line’ results in swing of -30 points, -1 IMP
- value of earned overtrick is swing of +30 points, +1 IMP
- value of making slam instead of going down is 14 IMP’s (+1030 points)
- as we have assumed same contract on another table, there will be no swing irrespective if 11, 12 or 13 tricks are scored by both lines

Now we just have to calculate expectancy of earned IMPs for ‘safe line’ via summing up all cases with their frequency and earned value. Result is (7*0IMP + 6*(-1IMP) + 2* 1IMP + 1*14IMP)/16 = 10 IMP/16 = 0.625 IMP. Now, that’s not that small advantage you gain on long run for playing ‘safe line’ on that type of hand.

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Step 5: But, of course there is no certainty that same contract will be played at another table. What’s the effect of this fact to comparison of two lines? To objectively compare two lines, one would have to estimate likelihood of another table playing in game or in grand-slam level (obviously both is quite possible on this type of hand). Rather than guess percentages, I have simply consulted results from all other tables in tournament. It turned out that out of 20 tables, 10 has played in small slam, 7 tables didn’t reach slam, and 3 tables overbid to grand slam. So let’s just assume that these are real probabilities, i.e.:

- In 50% cases other table will play small slam
- In 35% cases they will stop in game
- In 15% they will attempt grand slam.

Honestly, in my experience with similar deals, where on surface it seems not that difficult to reach clearly correct small slam, still this kind of distribution of final contracts on large number of tables is to be expected.

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Step 6: now we can compare two lines in these cases where contract is played on different level in another table. We can safely assume that declarers playing either in game or in grand slam will correctly choose ‘max-line’, as ‘safe-line’ does not make much sense in such context.

If another table is playing game

- value of lost overtrick is still -1 IMP (we score +10 IMP for 470 points difference instead of 11 for +500)
- value of earned overtrick is 0 (same +11 IMPs for +530 and +500)
- but value of making slam instead of going down is huge 20 IMP’s (+530 instead of -500, i.e. +10 instead of -10 IMP

This means expected earned value of ‘safe-line’ vs. ‘max-line’ (if you would know that another table is resting in game) is (7 * 0 + 6*(-1) + 2*0 + 1*20)/16 = 0.875 IMP

If another table is playing grand slam

- value of lost overtrick is 0 IMP (we score same -11 IMPs for -530 and -500)
- value of earned overtrick is again 0 IMP (we score same +14 IMPs for +1030 or +1060, as grand slam is failing)
- but value of making slam instead of going down is 12 IMP’s (+1080 instead of +50, i.e +14 instead of +2 IMPo

This shows expected earned value of ‘safe-line’ vs. ‘max-line’ (if you would know that another table is pushing to grand slam) is (7*0 + 6*0 + 2*0 + 1*12)/16 = 0.75 IMP

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Step 7: Let’s now aggregate all this. Overall earned value of ‘safe-line’ against ‘max-line’ is 0.5 * 0.625 + 0.35*0.875 + 0.15*0.75 = 0,73 IMP. This looks like quite convincing result! When playing safe, we stand to gain 0.73 IMP on long run, nearly as much as we can maximally loose with using it (i.e. 1 IMP). That proves that if we choose to use this safety play, it is steadily earning IMPs for us. Of course, major earning of IMPs happened when we have bid to optimal contract, but this 0.73 is just adding (or insuring, if you like) to what we have already earned with good bidding.

Step 8: admit boundaries of calculation, Let’s admit that this calculation implies that frequency of chosen contract is independent from how spades are distributed between opponents. In real life this is not fully the case, as if one of opponents is very short in spades, he is bit likelier to bid something and thus give some indication that might also influence level or play in final contract. However, on this board its relatively OK assumption, as it seems quite possible that most of the tables tables will have unopposed auction which does not give any hints about spade distribution.

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Step 9: reuse calculation for MP tournamentHaving done this calculation, why not try to see what would be earned value of our ‘safe-line’ in large MP tournament. It’s fair to assume that frequency sheet at the end would not be too different from the one we had this time, i.e. 35% of field playing game, 50% playing small slam and 15% shooting into grand. Let’s also assume that everyone plays ‘max-line’ this time, except us. In that case:

- Value of lost overtrick is -25% of top ( we score 35% beating just pairs in game, instead of 60%, beating them and being in middle of 50% range for small slam bidders, yielding extra 25%)
- Value of earned overtrick is 25% of top ( we score 85% instead of 60%)
- Value of making slam instead of going down is 60% of top (we score 100% instead of 40%)

so earned value is 7 * 0% + 6*-25% + 2*25% + 1*60%) /16 = -2.5% of top

So, not surprisingly, ‘safe-line’ is not earning but spending MP’s on Mitchell tournament, with frequency sheet we used. But, if small slam would be that sound, but harder to reach, yielding, say, 60% of field playing in game and 40% in slam, then earned value would be 7 * 0% + 6*-20% + 2*20% + 1*100%) /16 = +1.25 % of top, so ‘safe-line’ would be called for even in MP tournament.

I hope I didn’t’ bore you too much with this calculation, and of course, viva math, viva safety

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