DEFENDING AGAINST DEALER'S PARTNER'S LONER

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DEFENDING AGAINST DEALER'S PARTNER'S LONER
You are sitting at 1st seat, QD is the upcard. You pass and S2 orders up and goes alone.
You have at S1 AC KC 9C TH 9D.
What is your best lead to stop a loner? (justify your answer)
You have at S1 AC KC 9C TH 9D.
What is your best lead to stop a loner? (justify your answer)

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 Location: Aurora, IL
I wouldn't normally lead my only ace against a loner, but because i'm so long in clubs here, i'm leading it with the hope that 2nd will either have to follow, or will trump too low, allowing my partner to overtrump. That's my gut feeling  i think it's worth the risk of my partner getting squeezed. I'd have to do the math though.

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I'd lead the 10H.

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 Joined: Thu Sep 16, 2021 6:56 pm
The convention says to lead the 10H, but when Irish posts a problem he usually has a clever analysis which shows that, in this particular instance, it's better to play a different card. If only people would be so skeptical and discerning in their life decisions, and not blindly follow rules / heuristics / popular notions which may be antiquated or not applicable!
Fortunately, this is an easy problem to analyze with my simulator. The only hitch is that I need to have S2 call alone with the proper hands. Beyond that, the further play of the hand is nearly formulaic. And with a random distribution of the other cards, I am in principal capturing the "conditional probability" [conditional on S2 calling alone] of the various scenarios where leading one card or the other may be best.
I have defined a dozen or so scenarios where S2 calls alone, ranging from always with 5 trump to sometimes with just 2 trump, but I'm not going to enumerate all those here. I've run lots of tests and am confident with these choices; any variance in opinion shouldn't change the overall outcome of this analysis. The one debatable point is this: if S2 is going alone and is void in the suit led by S1, 1st trick, they will play their 2nd highest trump.
I will let a few more people chime in with their thoughts before I reveal the results from my simulator.
Fortunately, this is an easy problem to analyze with my simulator. The only hitch is that I need to have S2 call alone with the proper hands. Beyond that, the further play of the hand is nearly formulaic. And with a random distribution of the other cards, I am in principal capturing the "conditional probability" [conditional on S2 calling alone] of the various scenarios where leading one card or the other may be best.
I have defined a dozen or so scenarios where S2 calls alone, ranging from always with 5 trump to sometimes with just 2 trump, but I'm not going to enumerate all those here. I've run lots of tests and am confident with these choices; any variance in opinion shouldn't change the overall outcome of this analysis. The one debatable point is this: if S2 is going alone and is void in the suit led by S1, 1st trick, they will play their 2nd highest trump.
I will let a few more people chime in with their thoughts before I reveal the results from my simulator.

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 Joined: Thu Sep 16, 2021 6:56 pm
Here are the results of my simulation.
First of all, I found that S2 called alone about 4.4% of the time, so not all that often. So it's important to know what sorts of hand my program assumes S2 has.
S2 will call alone with any 5 trump (about 0.8% of their lone calls);
S2 will call alone with 4 trump if: have R and the 5th card is a Q or higher OR have L and the 5th card is a K or higher OR have A of trump and the 5th card is an A OR have 9/10/Q/K of trump and the 5th card is an A and the A or R of trump is turned (about 8.7% of their lone calls);
S2 will call alone with 3 trump with a very good hand (too many cases to enumerate) (about 67% of their lone calls);
S2 will call alone with 2 trump with an extremely good hand (something like R + another good trump + all offsuits led by an ace) (about 23.5% of their lone calls).
I set the hand for S1 and the turn card, then ran 1,000,000 iterations, trying different leads for S1 (S2 called alone 44,285 times):\
[THESE RESULTS ARE FROM THE PERSPECTIVE OF S2!]
lead 10H: (17,197 / 25,513 / 1,575) [4 pts / 1 pt / euchred] EV = +2.06
lead AC: (19,232 / 22,613 / 2,440) EV = +2.14
lead 9D: (22,121 / 21,745 / 419) EV = +2.47
So it looks slightly better to lead the 10H (S2 scores less pts); and it's predictably worst to lead trump.
Note that my simulator has S2 play their 2nd highest trump if they are void in the led suit, and I wondered if the results would change if they instead played their lowest trump. So I ran that comparison and found that it changed the results (EV) by less than 0.02, so it didn't change the outcome.
Analysis:
The goal is clearly to get one trick and stop the loner. And there are three readily apparent scenarios:
1) S3 has a guarded L or A of trump [loner is stopped no matter which card is played, so not relevant here; could have a small effect on euchre rate, but that is a secondary consideration];
2) S3 is void in clubs and has a trump, and S2 either follows a club lead or trumps a club lead with something S3 can overtrump;
3) same story for hearts, OR S3 has AS + AH and saves the correct A to the end (because 10H is led), and wins with the AS in the end.
The problem is calculating the conditional probability of these two scenarios GIVEN the constraints  we are dealing with a small subset (4 1/2%) of all the possible random distributions of hands). I believe it is precisely in this sort of case that is useful to have a simulator which finds that subset of hands and does the analysis on only those hands. The caveat, as always, is that my simulator has to having S2 call lone calls with all the right hands.
So I would lead the 10H.
First of all, I found that S2 called alone about 4.4% of the time, so not all that often. So it's important to know what sorts of hand my program assumes S2 has.
S2 will call alone with any 5 trump (about 0.8% of their lone calls);
S2 will call alone with 4 trump if: have R and the 5th card is a Q or higher OR have L and the 5th card is a K or higher OR have A of trump and the 5th card is an A OR have 9/10/Q/K of trump and the 5th card is an A and the A or R of trump is turned (about 8.7% of their lone calls);
S2 will call alone with 3 trump with a very good hand (too many cases to enumerate) (about 67% of their lone calls);
S2 will call alone with 2 trump with an extremely good hand (something like R + another good trump + all offsuits led by an ace) (about 23.5% of their lone calls).
I set the hand for S1 and the turn card, then ran 1,000,000 iterations, trying different leads for S1 (S2 called alone 44,285 times):\
[THESE RESULTS ARE FROM THE PERSPECTIVE OF S2!]
lead 10H: (17,197 / 25,513 / 1,575) [4 pts / 1 pt / euchred] EV = +2.06
lead AC: (19,232 / 22,613 / 2,440) EV = +2.14
lead 9D: (22,121 / 21,745 / 419) EV = +2.47
So it looks slightly better to lead the 10H (S2 scores less pts); and it's predictably worst to lead trump.
Note that my simulator has S2 play their 2nd highest trump if they are void in the led suit, and I wondered if the results would change if they instead played their lowest trump. So I ran that comparison and found that it changed the results (EV) by less than 0.02, so it didn't change the outcome.
Analysis:
The goal is clearly to get one trick and stop the loner. And there are three readily apparent scenarios:
1) S3 has a guarded L or A of trump [loner is stopped no matter which card is played, so not relevant here; could have a small effect on euchre rate, but that is a secondary consideration];
2) S3 is void in clubs and has a trump, and S2 either follows a club lead or trumps a club lead with something S3 can overtrump;
3) same story for hearts, OR S3 has AS + AH and saves the correct A to the end (because 10H is led), and wins with the AS in the end.
The problem is calculating the conditional probability of these two scenarios GIVEN the constraints  we are dealing with a small subset (4 1/2%) of all the possible random distributions of hands). I believe it is precisely in this sort of case that is useful to have a simulator which finds that subset of hands and does the analysis on only those hands. The caveat, as always, is that my simulator has to having S2 call lone calls with all the right hands.
So I would lead the 10H.

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This is what occurred in a tournament with the Hand above, my partner at 1st seat. He led the AC, S2 trumped low and I had to follow suit. S1 tossed his trump then his other Clubs. Because he played the AC, I now assume he has another ACE as a potential stopper but got no Signal as to what he was NOT saving l NOTHING!
Sitting at 3rd Seat, Pone I have my KS guarded but also have the KH. I tossed my 9S on trick 3 saving both Kings and have to decide which King to save. I sloughed the KS on trick 4 saving the KH singleton. Trick 5, he had four trumps at S2 and plays the JS as his trailer for 4 points.
With the Hand that S1 had above l of course play the 10H to trick 1. Not because what happened in this situation but for any Universal Set of all possible hands. Although the Simulator concluded also the 10H was the better lead, I do no see it being as close as it indicated! For what reasons might that be?
Leading the AC, there are only three other clubs possible QC JC 10 and saving the KC and signaling ASAP tells S3 what what S1 is saving. Plus, it catches S2 trailer in clubs. The only reason to lead the AC is for S3 to Overtrump S2 on the AC lead. But to do that S3 must have a void in clubs  3 unknown  35% that S3 has a void. Not only that with the Upcard being the QD, most likely S3 has to have the KD,AD,JD (or JH) . . . point being a higher trump and that might be useless anyway. But what would S2 have  NO CLUE. S3 has to have KD or higher as QD was the upcard. There is only approximately 28% (for one higher trump card unknown  assuming) chance he has that higher trump card x the void in Clubs (28% x 35% = ~10%) chance this is successful. If S2 holds the AD/JH/JD  over trumping is not possible. Saving the AC will catch the other lower clubs so only this chance of Overtrump is feasible here. And that is where I that the Simulator does not take this into account of what is better to lead the AC vs 10H.
Now looking at the leading the 10H  what are those chances of Stopping the Loner?
1) Leading he 10H there are 4 other Hearts (JH is a trump card) so a void is 23.4% and if S2 has a heart any trump will do. If S3 is void in Diamonds with upcard QD, and S1 has 9D and S2 has 3 trumps there is only one trump card left so this is a low probability since there are 8 cards with S4/Stock. So leading the AC and S3 having that one trump is not a great choice. compared to leading 10H being about the same success rate  maybe 8 to 10% for both. If it is the same or no major advantage, why do it?
2) Leading the 10H also gives S3 a chance and the partnership an opportunity to eliminate one suit. S3 can sort his hand. S3 would have to follow suit and S2 trumps but S3 now saves his KS (assuming S1 signals with AC that he has KC and clubs covered on trick 3). Does the simulator do this? Anyway, S1/S3 can stop everything S2 has except 5 trumps or 4 trumps with AS at this point in the hand. What is critical and soundly makes leading the 10H a better decision in all the possible hands S2 might have is the chance to catch the trailer card. He might catch an Ace doubleton or a low singleton trailer.
So Squeezing your partner at S3 is the Issue here. Yes for sure the 10H lead is better and if with the Simulator one could sort out the trailer cards of QD/JC/10C one would be able to see that the difference is far greater than the posted results. Here I am guessing the difference is more than double for stopping the loner. S1 saving the 10H only stops the 9H but help your partner out is a bigger factor here.
Having the Left guarded or AD guarded (with the 10D) does not change for any lead and is a constant. For the posted results, leading the AC, S2 success rate was 43.4% (2.3 attempts to get 4 pts) vs 38.7% (2.58 attempts to get 4 pts) leading the 10H. Thus there must of been a lot of trailers with low Cubs to get this close (only a 4.6% better success rate leading 10H), IMO. I wonder if the Simulator is signaling he has Clubs covered by playing the AC and holding the KC for the 5th trick? The difference MUST be higher than 4.6% such that S3 knows to save Spades. I question this aspect of the simulator, as stopping Spades is 33% of the offsuit trailers possible. This is also critical. Only if S2 has exactly 5 trumps it won't matter and is such a small probability will not impact the results one way or another.
In addition no wonder the results is such as the Simulator plays its 2nd Highest trump card as noted? This to all depends on what S2 has as to what should be played. In general play the lowest. For example  if he has JH AH KH  play the KH; if he has JH JD 10H  play the 10H; if he has JH KH 10H  play the 10H. You play the lowest as this is not the same as S3 going alone against S2/S4. You play fort the best possible outcome of success. This aspect of the Simulator also would skew the results closer together and does not do justice for leading the 10H vs AC.
I conclude  no contest between the two leads. Lead the 10H!
That is my story and I am sticking to it!
IRISH
Sitting at 3rd Seat, Pone I have my KS guarded but also have the KH. I tossed my 9S on trick 3 saving both Kings and have to decide which King to save. I sloughed the KS on trick 4 saving the KH singleton. Trick 5, he had four trumps at S2 and plays the JS as his trailer for 4 points.
With the Hand that S1 had above l of course play the 10H to trick 1. Not because what happened in this situation but for any Universal Set of all possible hands. Although the Simulator concluded also the 10H was the better lead, I do no see it being as close as it indicated! For what reasons might that be?
Leading the AC, there are only three other clubs possible QC JC 10 and saving the KC and signaling ASAP tells S3 what what S1 is saving. Plus, it catches S2 trailer in clubs. The only reason to lead the AC is for S3 to Overtrump S2 on the AC lead. But to do that S3 must have a void in clubs  3 unknown  35% that S3 has a void. Not only that with the Upcard being the QD, most likely S3 has to have the KD,AD,JD (or JH) . . . point being a higher trump and that might be useless anyway. But what would S2 have  NO CLUE. S3 has to have KD or higher as QD was the upcard. There is only approximately 28% (for one higher trump card unknown  assuming) chance he has that higher trump card x the void in Clubs (28% x 35% = ~10%) chance this is successful. If S2 holds the AD/JH/JD  over trumping is not possible. Saving the AC will catch the other lower clubs so only this chance of Overtrump is feasible here. And that is where I that the Simulator does not take this into account of what is better to lead the AC vs 10H.
Now looking at the leading the 10H  what are those chances of Stopping the Loner?
1) Leading he 10H there are 4 other Hearts (JH is a trump card) so a void is 23.4% and if S2 has a heart any trump will do. If S3 is void in Diamonds with upcard QD, and S1 has 9D and S2 has 3 trumps there is only one trump card left so this is a low probability since there are 8 cards with S4/Stock. So leading the AC and S3 having that one trump is not a great choice. compared to leading 10H being about the same success rate  maybe 8 to 10% for both. If it is the same or no major advantage, why do it?
2) Leading the 10H also gives S3 a chance and the partnership an opportunity to eliminate one suit. S3 can sort his hand. S3 would have to follow suit and S2 trumps but S3 now saves his KS (assuming S1 signals with AC that he has KC and clubs covered on trick 3). Does the simulator do this? Anyway, S1/S3 can stop everything S2 has except 5 trumps or 4 trumps with AS at this point in the hand. What is critical and soundly makes leading the 10H a better decision in all the possible hands S2 might have is the chance to catch the trailer card. He might catch an Ace doubleton or a low singleton trailer.
So Squeezing your partner at S3 is the Issue here. Yes for sure the 10H lead is better and if with the Simulator one could sort out the trailer cards of QD/JC/10C one would be able to see that the difference is far greater than the posted results. Here I am guessing the difference is more than double for stopping the loner. S1 saving the 10H only stops the 9H but help your partner out is a bigger factor here.
Having the Left guarded or AD guarded (with the 10D) does not change for any lead and is a constant. For the posted results, leading the AC, S2 success rate was 43.4% (2.3 attempts to get 4 pts) vs 38.7% (2.58 attempts to get 4 pts) leading the 10H. Thus there must of been a lot of trailers with low Cubs to get this close (only a 4.6% better success rate leading 10H), IMO. I wonder if the Simulator is signaling he has Clubs covered by playing the AC and holding the KC for the 5th trick? The difference MUST be higher than 4.6% such that S3 knows to save Spades. I question this aspect of the simulator, as stopping Spades is 33% of the offsuit trailers possible. This is also critical. Only if S2 has exactly 5 trumps it won't matter and is such a small probability will not impact the results one way or another.
In addition no wonder the results is such as the Simulator plays its 2nd Highest trump card as noted? This to all depends on what S2 has as to what should be played. In general play the lowest. For example  if he has JH AH KH  play the KH; if he has JH JD 10H  play the 10H; if he has JH KH 10H  play the 10H. You play the lowest as this is not the same as S3 going alone against S2/S4. You play fort the best possible outcome of success. This aspect of the Simulator also would skew the results closer together and does not do justice for leading the 10H vs AC.
I conclude  no contest between the two leads. Lead the 10H!
That is my story and I am sticking to it!
IRISH
Last edited by irishwolf on Tue Nov 15, 2022 12:50 pm, edited 1 time in total.

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 Location: Aurora, IL
Oof. Ok, don't outthink yourself. Keep it simple.raydog wrote: ↑Tue Nov 15, 2022 9:05 amlead 10H: (17,197 / 25,513 / 1,575) [4 pts / 1 pt / euchred] EV = +2.06
lead AC: (19,232 / 22,613 / 2,440) EV = +2.14
lead 9D: (22,121 / 21,745 / 419) EV = +2.47
So it looks slightly better to lead the 10H (S2 scores less pts); and it's predictably worst to lead trump.

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 Joined: Thu Sep 16, 2021 6:56 pm
Irish, to address your questions about the simulator:
1) S2 plays 2nd highest trump (not highest) on the first trick. And I compared this with playing lowest trump instead, finding little difference. Playing their highest trump would be dumb.
2) If S3 has the hand you had in your tournament, S1 leads a H, and S2 trumps, the simulator will indeed have S3 save spades.
3) One of the reasons my simulator does not show more of a difference between the two leads (AC vs. 10H) is that the results are diluted by all the hands where the loner is stopped by S3 having a guarded L or A, or when S2 has 2 trump and S3 has 3. If S1/S3 win 1 or 2 tricks, it's the same  the loner is stopped. If this happens, say, 1/3 of the time that the loner is stopped, then the difference between the OTHER scenarios where the loner is stopped will grow by at least 50%. I think that is why I am not seeing more of a difference.
Ray
1) S2 plays 2nd highest trump (not highest) on the first trick. And I compared this with playing lowest trump instead, finding little difference. Playing their highest trump would be dumb.
2) If S3 has the hand you had in your tournament, S1 leads a H, and S2 trumps, the simulator will indeed have S3 save spades.
3) One of the reasons my simulator does not show more of a difference between the two leads (AC vs. 10H) is that the results are diluted by all the hands where the loner is stopped by S3 having a guarded L or A, or when S2 has 2 trump and S3 has 3. If S1/S3 win 1 or 2 tricks, it's the same  the loner is stopped. If this happens, say, 1/3 of the time that the loner is stopped, then the difference between the OTHER scenarios where the loner is stopped will grow by at least 50%. I think that is why I am not seeing more of a difference.
Ray

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 Joined: Sat Jun 02, 2018 12:50 pm
 Location: Aurora, IL
On a side note, I started collecting some data from play on Euchre 3D, and through a very small sample size (20 games), I have almost the same number for S2 loner attempt frequency.
Total Hands: 202
Seat 1  3.0% of hands (14.6% of all loner attempts)
Seat 2  4.0% of hands (19.5% of all loner attempts)
Seat 3  1.0% of hands (4.9% of all loner attempts)
Seat 4  12.4% of hands (61.0% of all loner attempts)

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RAY,
My point is this and has to do with the trailer card(s) of QC JC 10C. When S2 has one or more clubs S2 is stopped regardless of which is led. It masks the benefit of leading the 10H. We are trying to show the benefit of leading 10H vs AC.
Thus, what is that number (%) when the AC is led and just show when S3 overtrumps S2? And I am saying you will see the stops will be of benefit double of leading 10H vs AC.
All those other stop combinations are in both posted results masking the true benefit of leading the 10H, IMO. "Those" being when S2 has a club and/or S3 has guarded trump card. It's a Fact that leading the AC, S3 gets squeezed on Spade vs Heart  two possible stopper suits. It's no contest, i.e., not close as 0.08 (2.14 vs 2.06) which is not really Statistically Significant for 44,285 attempts.
IRISH
My point is this and has to do with the trailer card(s) of QC JC 10C. When S2 has one or more clubs S2 is stopped regardless of which is led. It masks the benefit of leading the 10H. We are trying to show the benefit of leading 10H vs AC.
Thus, what is that number (%) when the AC is led and just show when S3 overtrumps S2? And I am saying you will see the stops will be of benefit double of leading 10H vs AC.
All those other stop combinations are in both posted results masking the true benefit of leading the 10H, IMO. "Those" being when S2 has a club and/or S3 has guarded trump card. It's a Fact that leading the AC, S3 gets squeezed on Spade vs Heart  two possible stopper suits. It's no contest, i.e., not close as 0.08 (2.14 vs 2.06) which is not really Statistically Significant for 44,285 attempts.
IRISH

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I did some more analysis of this scenario, mostly to see how my simulator was playing. It turns out I found 3 different situations where it was making the wrong play, resulting in a missed opportunity to stop the loner or even euchre the opponent! I corrected these errors, but it turns out this didn't changed the overall results by much.
I also divided the initial hands into 3 different camps:
1) S2 has at least 1 club in their hand (about 15% of hands);
2) S3 can stop the loner by virtue of having trump (the R, or L guarded, or A guarded, or 3 trump when S2 only has 2 trump) (about 11% of hands);
3) all other hands (about 74% of hands).
1) when S2 has a C, the loner will be stopped when a C is eventually played. But leading the AC turned out to be slightly better than leading the 10H because of the extra euchres which were possible.
lead 10H: (0 / 6277 / 186) EV = +0.91 [from the perspective of S2!]
lead AC: (0 / 6182 / 281) EV = +0.87
here are a couple of hands where leading the AC leads to a euchre, but leading the 10H doesn't:
S2: AS + AJ10D + QC; S3: J9S + KJH + KD
S2: AKJD + QJC; S3: 9S + KQJ9H
2) when S3 has good trump (good enough to stop the loner), it also turns out to be slightly better to lead the AC, once again because of the better chance of euchring S2.
lead 10H: (0 / 3874 / 1042) EV = +0.36
lead AC: (0 / 3526 / 1380) EV = +0.16
here is a hand where playing the 10H is better:
S2: AKQS + J10D; S3: JS + JH + AKD + 10C
here is hand where leading the AC is better:
S2: A10S + AH + J10D; S3: KJH + AKD + JC
(so apparently hands like the 2nd one are more likely than hands like the first one)
3) when neither 1) nor 2) is true, it's a bit better to lead the 10H, and since this situation covers 74% of all cases, it makes leading the 10H better overall.
lead 10H: (16,811 / 14,781 / 502) EV = +2.52
lead AC: (18,850 / 12,422 / 822) EV = +2.69
here is a case hand where leading the 10H is better (stops the loner):
S2: AQ9H + KJD; S3: K109S + 10D + 10C
and here is case where leading the AC is better (stops the loner):
S2: A9S + AH + J10D; S3: J10S + KH + QJC
The overall results (found by running a different set of 1,000,000 random hands) are:
lead 10H: (17,030 / 24,919 / 1638) EV = +2.06
lead AC: (19,010 / 22,093 / 2484) EV = +2.14
(S2 calls alone 4.4% of the time)
These results are more significant than they may seem because they are the weighted average of 3 different scenarios which pull the results in opposite directions.
I don't dispute Irish's quick calculation of when a 10H lead is better and when a AC lead is better, I just believe there are many more situations which he DOESN'T look at (including important euchre scenarios), and it is hard to accurately calculate the true odds of all the possible hands where leading the AC is better and where leading the 10H is better. Fortunately, he has probably done a calculation on the most LIKELY situation, and therefore we arrive at the same conclusion: leading the 10H is best overall. But I don't share Irish's skepticism as to the magnitude of the difference I found (rather small difference).
Assuming my program is calling alone with the correct hands, and doesn't have some major errors in play I haven't uncovered...
I also divided the initial hands into 3 different camps:
1) S2 has at least 1 club in their hand (about 15% of hands);
2) S3 can stop the loner by virtue of having trump (the R, or L guarded, or A guarded, or 3 trump when S2 only has 2 trump) (about 11% of hands);
3) all other hands (about 74% of hands).
1) when S2 has a C, the loner will be stopped when a C is eventually played. But leading the AC turned out to be slightly better than leading the 10H because of the extra euchres which were possible.
lead 10H: (0 / 6277 / 186) EV = +0.91 [from the perspective of S2!]
lead AC: (0 / 6182 / 281) EV = +0.87
here are a couple of hands where leading the AC leads to a euchre, but leading the 10H doesn't:
S2: AS + AJ10D + QC; S3: J9S + KJH + KD
S2: AKJD + QJC; S3: 9S + KQJ9H
2) when S3 has good trump (good enough to stop the loner), it also turns out to be slightly better to lead the AC, once again because of the better chance of euchring S2.
lead 10H: (0 / 3874 / 1042) EV = +0.36
lead AC: (0 / 3526 / 1380) EV = +0.16
here is a hand where playing the 10H is better:
S2: AKQS + J10D; S3: JS + JH + AKD + 10C
here is hand where leading the AC is better:
S2: A10S + AH + J10D; S3: KJH + AKD + JC
(so apparently hands like the 2nd one are more likely than hands like the first one)
3) when neither 1) nor 2) is true, it's a bit better to lead the 10H, and since this situation covers 74% of all cases, it makes leading the 10H better overall.
lead 10H: (16,811 / 14,781 / 502) EV = +2.52
lead AC: (18,850 / 12,422 / 822) EV = +2.69
here is a case hand where leading the 10H is better (stops the loner):
S2: AQ9H + KJD; S3: K109S + 10D + 10C
and here is case where leading the AC is better (stops the loner):
S2: A9S + AH + J10D; S3: J10S + KH + QJC
The overall results (found by running a different set of 1,000,000 random hands) are:
lead 10H: (17,030 / 24,919 / 1638) EV = +2.06
lead AC: (19,010 / 22,093 / 2484) EV = +2.14
(S2 calls alone 4.4% of the time)
These results are more significant than they may seem because they are the weighted average of 3 different scenarios which pull the results in opposite directions.
I don't dispute Irish's quick calculation of when a 10H lead is better and when a AC lead is better, I just believe there are many more situations which he DOESN'T look at (including important euchre scenarios), and it is hard to accurately calculate the true odds of all the possible hands where leading the AC is better and where leading the 10H is better. Fortunately, he has probably done a calculation on the most LIKELY situation, and therefore we arrive at the same conclusion: leading the 10H is best overall. But I don't share Irish's skepticism as to the magnitude of the difference I found (rather small difference).
Assuming my program is calling alone with the correct hands, and doesn't have some major errors in play I haven't uncovered...

 Posts: 1321
 Joined: Tue Apr 24, 2018 9:33 pm
Lots of extra work but I do have some issues with your assumptions.
Starting here, you stated:
I also divided the initial hands into 3 different camps:
1) S2 has at least 1 club in their hand (about 15% of hands);
2) S3 can stop the loner by virtue of having trump (the R, or L guarded, or A guarded, or 3 trump when S2 only has 2 trump) (about 11% of hands);
3) all other hands (about 74% of hands).
I have an issue with the 15%  I will say not prudent to go alone with just the 10C (3 unknown clubs  QC JC 10C). Not at S2 even with 4 trumps, IMO. I say will score more points taking Dealer along unless way behind in score 3+. Just how I play. And if he had doubleton JC 10C or JC QC might be better to pass if one of those holdings is the AD. There are 13 off suit cards and your 15% is too high in my book.
Secondly #2 you have S3 with 3 trumps. There are Five unknown trumps. Terrible call going alone with any 2 trumps by S2 unless, again, way behind in score. So S3 could have just JD, or JH grd, or AD grd. so your program, no clue what it is doing. I am having difficulty with your statement when S3 has good hands? He can only I stated NEVER 3 trumps! So your 15% + 11% is too high  more like half. Program playing Optimally? If S2 did go alone with JD JH AX + AY dblt should not go and that could also be a pass hand. There is a time and place to stretch it and swing for the fence, and that all depends on score & situation. I could lay out the trump hands S2 should go alone with but not sure that adds anything to the discussion and NONE include two trumps except JD JH AX AY DBLT.
NOTE also S3 having Left guarded or the Right blocks a loner no matter what is led. Must be factored in #2 above and will not necessarily result in a euchre either.
You show S2 getting euchred with JD 10D  those are not lone hands, IMO.
So yes I disagree for the reasons already stated  lead the 10H. My concern is more likely to get squeezed here so stopping the loner is more of a benefit to me than euchring the opponent in general. I am okay with disagreeing here and appreciate your work. That issue probably boils down to S2 going alone with 2 trumps and getting Euchred a lot adding to to the EV you posted. Just maybe we are more in agreement than not!
Irish
Starting here, you stated:
I also divided the initial hands into 3 different camps:
1) S2 has at least 1 club in their hand (about 15% of hands);
2) S3 can stop the loner by virtue of having trump (the R, or L guarded, or A guarded, or 3 trump when S2 only has 2 trump) (about 11% of hands);
3) all other hands (about 74% of hands).
I have an issue with the 15%  I will say not prudent to go alone with just the 10C (3 unknown clubs  QC JC 10C). Not at S2 even with 4 trumps, IMO. I say will score more points taking Dealer along unless way behind in score 3+. Just how I play. And if he had doubleton JC 10C or JC QC might be better to pass if one of those holdings is the AD. There are 13 off suit cards and your 15% is too high in my book.
Secondly #2 you have S3 with 3 trumps. There are Five unknown trumps. Terrible call going alone with any 2 trumps by S2 unless, again, way behind in score. So S3 could have just JD, or JH grd, or AD grd. so your program, no clue what it is doing. I am having difficulty with your statement when S3 has good hands? He can only I stated NEVER 3 trumps! So your 15% + 11% is too high  more like half. Program playing Optimally? If S2 did go alone with JD JH AX + AY dblt should not go and that could also be a pass hand. There is a time and place to stretch it and swing for the fence, and that all depends on score & situation. I could lay out the trump hands S2 should go alone with but not sure that adds anything to the discussion and NONE include two trumps except JD JH AX AY DBLT.
NOTE also S3 having Left guarded or the Right blocks a loner no matter what is led. Must be factored in #2 above and will not necessarily result in a euchre either.
You show S2 getting euchred with JD 10D  those are not lone hands, IMO.
So yes I disagree for the reasons already stated  lead the 10H. My concern is more likely to get squeezed here so stopping the loner is more of a benefit to me than euchring the opponent in general. I am okay with disagreeing here and appreciate your work. That issue probably boils down to S2 going alone with 2 trumps and getting Euchred a lot adding to to the EV you posted. Just maybe we are more in agreement than not!
Irish

 Posts: 260
 Joined: Thu Sep 16, 2021 6:56 pm

 Posts: 1321
 Joined: Tue Apr 24, 2018 9:33 pm
Just a final comment on this:
One has to look at the 'probable' hands that S2 might have going alone:
Of those I see this:
1. 5 trumps;
2. 4 trumps + Ace
3. 4 trumps + trailer; (meaning less than an Ace off suit card)
4. 3 trumps + A/A;
5. 3 trumps + A dblt;
6. 3 trumps + A + trailer; (meaning another suit)
7. 3 trumps + K or Q or J dblt;
8. 2 trumps +_ A/A dblt.
I think that covers those opportunities for S2 going alone.
When you defend against a loner, unless S1 has a strong hand, you are not looking for Euchre. I contend those euchres and EV should be totally discounted as a factor, IMO. You are satisfied to just stop the loner. Thus, what is the best line of defense?
So looking at the list one can see leading the 10H is best for stopping the loner in numbers: 3. 5. 6. 7. & 8.
Leading the AC would be better in 1. 2.& 4., (actually 1., 2. & 4. only slightly favors AC lead  3C's & 4H's approximately equal with either lead at ~15% each) but the probability of having those is extremely low. Collectively, one can see leading the 10H is far better as it eliminates one suit, thus two bits of the apple to avoid Squeezing your partner. And if S3 has a void in Hearts he gets a chance to over trump.
And I only wished the simulator results could be separated for the results when leading the AC that S2 had a club vs S3 over trumping S2 as that would validate my analysis. Leading the AC you not really looking to the Catch S2 club trailer you are hoping S3 can win by having a void (35%) over trumping S2 (basically less than17.5% depending on what S2 does and S3 holds). A lot of wishing thinking  ifs and buts.
Yes, I am done and AMEN!
IRISH
One has to look at the 'probable' hands that S2 might have going alone:
Of those I see this:
1. 5 trumps;
2. 4 trumps + Ace
3. 4 trumps + trailer; (meaning less than an Ace off suit card)
4. 3 trumps + A/A;
5. 3 trumps + A dblt;
6. 3 trumps + A + trailer; (meaning another suit)
7. 3 trumps + K or Q or J dblt;
8. 2 trumps +_ A/A dblt.
I think that covers those opportunities for S2 going alone.
When you defend against a loner, unless S1 has a strong hand, you are not looking for Euchre. I contend those euchres and EV should be totally discounted as a factor, IMO. You are satisfied to just stop the loner. Thus, what is the best line of defense?
So looking at the list one can see leading the 10H is best for stopping the loner in numbers: 3. 5. 6. 7. & 8.
Leading the AC would be better in 1. 2.& 4., (actually 1., 2. & 4. only slightly favors AC lead  3C's & 4H's approximately equal with either lead at ~15% each) but the probability of having those is extremely low. Collectively, one can see leading the 10H is far better as it eliminates one suit, thus two bits of the apple to avoid Squeezing your partner. And if S3 has a void in Hearts he gets a chance to over trump.
And I only wished the simulator results could be separated for the results when leading the AC that S2 had a club vs S3 over trumping S2 as that would validate my analysis. Leading the AC you not really looking to the Catch S2 club trailer you are hoping S3 can win by having a void (35%) over trumping S2 (basically less than17.5% depending on what S2 does and S3 holds). A lot of wishing thinking  ifs and buts.
Yes, I am done and AMEN!
IRISH

 Posts: 260
 Joined: Thu Sep 16, 2021 6:56 pm
Irish, sorry, I didn't realize what you were asking before. I did now run the simulation I think you were looking for:
I set S1's hand + the turn card, then ran 1,000,000 random hands, of which S2 called alone, R1, on 43,512 occasions (4.35%).
I had S1 lead the AC, and kept track of how many times S3 won the first trick by overtrumping S2: 2,695 times.
I also counted how often S3 was "squeezed"  this was a bit more difficult. At a first approximation, S3 gets squeezed when they have both the AS and the AH, and at some point have to discard one of them, not knowing which one to keep to potentially stop the loner on the last trick. I see 3 ways this can occur:
1) S3 has 3 trump + both aces, and can't overtrump the first trick;
2) S3 has 2 trump + both aces, S1 wins the first trick with the AC then releads C, S2 can't overtrump S2 on the second trick;
3) S2 has 4 trump, and on the 4th trick S3 has to decide which A to discard. (I only counted this if it made a difference  if S2 still had a trump to lead on the last trick, the choice of discard was irrelevant and I didn't count it).
This is an approximation because S2 may also have a potential winner in a K (if S2 has Q offsuit), and may not know to save it.
Anyway, using my first approximation I counted S3 getting squeezed 541 times. So even if we triple this number, it is still far shy of the stops S1/S3 get by S3 overtrumping on the 1st trick. And also keep in mind that, even when S3 has to choose an A to discard at random, they will get it right 1/2 the time.
Finally, from my original data you can see that S1/S3 score about 850 more euchres when S1 leads the AC, and a euchre is worth +3 points  the same as stopping a loner.
I still find that it's a bit better to lead the 10H, but it doesn't seem like avoiding squeezing S3 is the reason why. And if all this data seems unreasonable to you, it's probably because we disagree on what hands S2 should call alone. A couple of examples:
1) with R+L+A+K trump + green J, I find it better for S2 to call with partner than alone (EV 1.57 vs 1.47). And I even find it better for S2 to discard their J on the first trick if S1 doesn't lead an A (EV 1.62).
2) with R+9 trump + AKQ green (9 turned), I find it better for S2 to call alone than with partner (EV 1.39 vs 1.23).
Perhaps since we are considering a different universe of hands that S2 holds, we are getting different results.
I set S1's hand + the turn card, then ran 1,000,000 random hands, of which S2 called alone, R1, on 43,512 occasions (4.35%).
I had S1 lead the AC, and kept track of how many times S3 won the first trick by overtrumping S2: 2,695 times.
I also counted how often S3 was "squeezed"  this was a bit more difficult. At a first approximation, S3 gets squeezed when they have both the AS and the AH, and at some point have to discard one of them, not knowing which one to keep to potentially stop the loner on the last trick. I see 3 ways this can occur:
1) S3 has 3 trump + both aces, and can't overtrump the first trick;
2) S3 has 2 trump + both aces, S1 wins the first trick with the AC then releads C, S2 can't overtrump S2 on the second trick;
3) S2 has 4 trump, and on the 4th trick S3 has to decide which A to discard. (I only counted this if it made a difference  if S2 still had a trump to lead on the last trick, the choice of discard was irrelevant and I didn't count it).
This is an approximation because S2 may also have a potential winner in a K (if S2 has Q offsuit), and may not know to save it.
Anyway, using my first approximation I counted S3 getting squeezed 541 times. So even if we triple this number, it is still far shy of the stops S1/S3 get by S3 overtrumping on the 1st trick. And also keep in mind that, even when S3 has to choose an A to discard at random, they will get it right 1/2 the time.
Finally, from my original data you can see that S1/S3 score about 850 more euchres when S1 leads the AC, and a euchre is worth +3 points  the same as stopping a loner.
I still find that it's a bit better to lead the 10H, but it doesn't seem like avoiding squeezing S3 is the reason why. And if all this data seems unreasonable to you, it's probably because we disagree on what hands S2 should call alone. A couple of examples:
1) with R+L+A+K trump + green J, I find it better for S2 to call with partner than alone (EV 1.57 vs 1.47). And I even find it better for S2 to discard their J on the first trick if S1 doesn't lead an A (EV 1.62).
2) with R+9 trump + AKQ green (9 turned), I find it better for S2 to call alone than with partner (EV 1.39 vs 1.23).
Perhaps since we are considering a different universe of hands that S2 holds, we are getting different results.

 Posts: 1321
 Joined: Tue Apr 24, 2018 9:33 pm
RAY,
Thanks again for all your effort on this. I agree with your concluding statement. On the R + 9 with A/K/Q. I went back and rereviewed this. I total agreement because when S1/S3 has trump of 0/0, 1/0, 0/1, or 1/1 trumps exceeds all other trump card configurations. Thus EV will be better. Also true for R + 1 (any size) and extending that to R + 1 with A/AK as well (probably with A/AQ). This expands my universe.
Of course R L A K (or R L A + ANY OTHER: Q/10/9) with 10 or 9 trailer. I have always looked at the Jack trailer as the breakeven point. But you simulation shows better to pass. ALL GOOD STUFF!
1) with R+L+A+K trump + green J, I find it better for S2 to call with partner than alone (EV 1.57 vs 1.47). And I even find it better for S2 to discard their J on the first trick if S1 doesn't lead an A (EV 1.62).
2) with R+9 trump + AKQ green (9 turned), I find it better for S2 to call alone than with partner (EV 1.39 vs 1.23).
Perhaps since we are considering a different universe of hands that S2 holds, we are getting different results.
IRISH
Thanks again for all your effort on this. I agree with your concluding statement. On the R + 9 with A/K/Q. I went back and rereviewed this. I total agreement because when S1/S3 has trump of 0/0, 1/0, 0/1, or 1/1 trumps exceeds all other trump card configurations. Thus EV will be better. Also true for R + 1 (any size) and extending that to R + 1 with A/AK as well (probably with A/AQ). This expands my universe.
Of course R L A K (or R L A + ANY OTHER: Q/10/9) with 10 or 9 trailer. I have always looked at the Jack trailer as the breakeven point. But you simulation shows better to pass. ALL GOOD STUFF!
1) with R+L+A+K trump + green J, I find it better for S2 to call with partner than alone (EV 1.57 vs 1.47). And I even find it better for S2 to discard their J on the first trick if S1 doesn't lead an A (EV 1.62).
2) with R+9 trump + AKQ green (9 turned), I find it better for S2 to call alone than with partner (EV 1.39 vs 1.23).
Perhaps since we are considering a different universe of hands that S2 holds, we are getting different results.
IRISH