https://worldofcardgames.com/#!replayer ... %3A1%7D%5D

All math arguments hinge on their assumptions so look closely at every assumption I make. If you disagree with any assumption let me know and give me another number to work with. Ideally all my assumptions will be conservative, I.E. they err on the side of the passing argument, but it's often hard to know if that's true.

First step of this argument is we have to figure out the calling frequency of S2, S3, and S4 after S1 passes. I don't want this post to get too bogged down so I will show my work on this front in another post. First I'm going to assume S2-S4 are aggressive competent players. This assumption is unrealistic as most euchre players are too passive and don't play the game that well but critically this assumption is conservative as it benefits the passing argument.

Assume S2 is calling with 3 trump, R+1, and L+1+an off ace when he doesn't block much in the 2nd round. S2 will be calling approx 13.84% of the time.

Assume S3 is calling with JJx and JJ+two off aces. That's 1.65% of the time.

Assume S4 is calling with 3 trump, 4 trump, R+1, and 2 trump + 2 off aces. That's 37.71% of the time.

So what's the calling frequency of S2/S4's team? Well you cannot simply add their calling percentages together becuz then you would be double counting hands. They can't have the same hands at the same time. A reasonable approximation would be S2's calling percentage + S4's calling percentage multiplied by 1 minus S2+S3's calling percentage:

**.1384 + .3771[1 - (.1384 + .0165)] = 45.71%**

Ok so when S1 passes his 3 trump, his opponents are calling approx

**45.71%**of the time.

Ok here's some more assumptions to look over:

**1)**If S1's opponents call S1's team will euchre them 30% of the time, so 70% of the time S1's opponents will get 1 pt. They can't get 2 pts given S1's hand.

**2)**If S1 passes, and both his opponents pass, whether S1 calls next or passes, he will lose approx 1 pt in the long run. This is a reasonable assumption IMO given how weak a Next call would be in this spot and given how dangerous it is to pass a hand that blocks nothing in this spot. It could easily be more than a 1 pt cost so I think this assumption is conservative.

**3)**If S1 calls in the first round, he will get euchred 40% of the time, make a point 50% of the time, and get 2 points 10% of the time.

Ok so here's what the EV equation looks like for passing:

.4571[(.30)(2) + (.70)(-1)] + (.5429)(-1) = -.5886

**So passing costs S1's team around .5886 points.**

Here's the EV equation for calling:

.4(-2) + .5(1) + .1(2) = -.1

**So calling will cost S1's team .1 pts.**

**-.1 is > .5886, therefore calling is best.**

Let's make assumption 3) more conservative and see what happens:

**3)**If S1 calls he will get euchred 50% of the time, make a point 45% of the time and get 2 points 5% of the time.

EV of passing is still -.5886

EV of calling: .5(-2) + .45(1) + .05(2) = -.45

**-.45 is still better than -.5886, therefore calling is still the better play.**

For the hell of it, let's make Assumption 1) more conservative and see what happens. Lets assume if S1's opponents call S1's team will euchre them 35% of the time, so 65% of the time S1's opponents will get 1 pt. They can't get 2 pts given S1's hand. The EV of passing would then be:

.4571[(.35)(2) + (.65)(-1)] + .5429(-1) = -.5200

**-.52 is worst than -.1 or -.45. So calling is still better.**

We can reach a point of indifference with these assumptions:

**1)**If S1's opponents call S1's team will euchre them 40% of the time, so 60% of the time S1's opponents will get 1 pt. They can't get 2 pts given S1's hand.

**2)**If S1 passes, and both his opponents pass, whether S1 calls next or passes, he will lose approx 1 pt in the long run. This is a reasonable assumption IMO given how weak a Next call would be in this spot and given how dangerous it is to pass a hand that blocks nothing in this spot. It could easily be more than a 1 pt cost so I think this assumption is conservative.

**3)**If S1 calls in the first round, he will get euchred 50% of the time, make a point 45% of the time, and get 2 points 5% of the time.

EV of passing: .4571[(.40)(2) + (.60)(-1)] + .5429(-1) =

**-.45.15**

EV of calling: .5(-2) + .45(1) + .05(2) =

**-.45**

I don't think those numbers are very realistic though. I don't think S1's marginal call is getting euchred 50% of the time. Yes this call is getting euchred more than we'd like but not that much imo. I also think S1 is not getting as many euchres on a S2-S4 call as he thinks. It's not like S2-S4 are randomly calling here. They are only calling when they have a hand, I.E. when they like hearts. Even with 3 trump, trying to euchre a range that likes hearts isn't that easy, especially with no off aces. Euchring them 40% of the time is way too optimistic imo.

Overall I think the first example captures reality best where the EV of passing was -.5886 and the EV of calling was -.1. IOW passing is around a half a point worse than calling. That's pretty significant. Also keep in mind this whole EV argument assumed S2-S4 we're competent aggressive players. The vast majority of the time this will not be the case, and the more S2-S4 passes the worse S1's passing strategy does.

For example, most S2 players are not calling with R+1+0 unless they have both bowers, and they're not calling with L+1+an off ace either. And most S4 players are not calling with R+1+0, and they're not calling with 2 trump 2 aces if the only trump they have is QhTh. Factoring that in, then S2-S4's team are only calling 35.96% of the time instead of 45.71%. Here's what the EV model looks like against these more passive and realistic opponents:

EV of passing:

.3596[(.30)(2) + (.70)(-1)] + (.6404)(-1) = -.6764

**So passing costs S1's team around .6764 points.**

EV of calling:

.4(-2) + .5(1) + .1(2) = -.1

**So calling will cost S1's team .1 pts.**

**-.1 is > -.6764, therefore calling is best.**