JJ's Blog

You can tell I'm missing poker.

So an interesting hand came up the other night playing euchre. I’m not a euchre player. I have absolutely zero skill at this game. But I thought we might be able to use some of the same concepts that we use in poker to analyze the correct play.

Quickie overview of Euchre
So we are playing 4 handed (which I guess is standard – even though there are 2 handed or 6 handed variations). Euchre is played with partners. The goal is to score more points than your opponent’s team (a race to 10 points is a game). You score points by taking “tricks” as in other suit game like hearts or spades. The game uses 24 cards (A through 9). The rank of the cards is the same as in poker except the Jack (aka right bower) is higher than an Ace for the trump suit. Another wrinkle is that the Jack that is the same color but different suit as the trump is the second highest card. This is the left bower. For example, when trump is hearts the ranks are Jh, Jd, Ah, Kh, Qh, 10h, 9h. The ranks for the other suits (non-trump) are standard poker ranks. For example when trump is hearts, the ranks for spades and clubs would be A,K,Q,J,10,9. The rank for diamonds would be A,K,Q,10,9 (notice that the J of diamonds is not used since it would be a left bower for trump).

You are dealt 5 cards each. The remaining 4 cards are put into a discard pile. Then, the top card from the discard pile is turned face up. Each player, in order, has the option of declaring that card’s suit as trump. If that card’s suit is declared as trump, the dealer puts that card into their hand and discards another card. If everyone passes (checks) on declaring trump, each player in turn has the option to declare any other suit as trump. If no one takes their “option” to declare trump, the dealer, who is last to act, must declare trump (this is the variation called “stick the dealer").

You can score points in the following ways:
* When you order trump and take 3 tricks or 4 tricks – 1 pt
* When you order trump and take 5 tricks – 2 pts
* When your opponents order trump and fail to take 3 tricks – 2 pts (called “getting euchred”)
* When you order trump and “go alone” (do not use a partner) and take 3 tricks or 4 tricks – 1 pt
* When you order trump and “go alone” and take 5 tricks – 4 pts
* When you order trump and “go alone” and fail to take 3 tricks – opponents score 2

The Setup
The opponent to my right (villain 1) is dealing and after it is passed around, she orders up hearts as trump when she picks the 10h off the pile. She then proceeds to take all 5 tricks when she plays 5 successive trump cards – no one else had trump. As it turns out, they whomped us 10 to 0 in that game.

Like poker, it is somewhat expected and traditional to berate a player for making a stupid play after they beat you. So me and my partner started give our opponent some shit for not going alone and trying to score 4 points. She defended by saying that since she had neither bower, the correct play is to use a partner.

Let’s take a look at the situation.

Since Villain 1 had 5 trump cards, there are only 2 trump remaining. These 2 remaining trump (the 2 bowers – highest ranking cards) are somewhere in my hand, my partners hand, my other opponent’s hand, or in the 3 cards left in the discard pile.

Like in poker, I think we can just construct a decision matrix with scenarios and probabilities. The way I see it, there are about 7 possible scenarios. I’ve summarized them below

Scenario 1: If both bowers (Jacks) were held by me or if both bowers were held by my partner, we would take 2 of the 5 tricks – since one person on my team would have the 2 highest ranked trump cards. In this situation, villain would score 1 point by ordering trump and going alone or by ordering trump and using a partner. There is no difference in the outcome. Going alone or taking a partner would net 1 point.

Scenario 2: If both I and my partner held a bower each, there could be 2 “2nd order” outcomes. One, my partner could lead with a non-trump card which would be trumped by my opponent. She would then lead with trump and would “draw” both bowers from me and my partner on the next card. That’s one trick for us. Then, our opponent would take the rest of the tricks (4 in total). Two, if my partner leads with his bower, it would draw my bower, and once again, we would take 1 of the 5 tricks. In either case, my opponents would take 4 tricks. So, going alone or using a partner in this case would result in our opponents scoring 1 point.

Scenario 3: If both of the bowers were held by Villain 2 (Villain1’s partner), they would take all 5 tricks. In this case, they would score 2 points for taking all 5 tricks and using a partner, but they would have scored 4 points by going alone. In this case the better play is to go alone.

Scenario 4: If both bowers were buried in the discard pile, villains would take all 5 tricks. In this case, they would have scored more points by going alone (4) vs. using a partner (2).

Scenario 5: Either me or my partner hold a bower and the other bower is buried. In this case, my team would take 1 trick. Our opponents would take 4 tricks. They would score 1 point either way (going alone or taking a partner)

Scenario 6: Either me or my partner hold the right bower and our opponent holds the left bower. In this case, my team would take 1 trick. Our opponents would take 4 tricks. They would score 1 point either way (going alone or taking a partner)

Scenario 7: Either me or my partner hold the left bower but our opponent holds the right bower. In this case my team would take 0 tricks. Our opponents would take 4 tricks by going alone and would score 1 point, but they would take all 5 tricks and score 2 points by using a partner.

Summary
Out of all 7 of the possible scenarios, there are 4 (#1, #2, #5, and #6) that there is no difference in the points that result from the choice to go alone or to take a partner. There are 2 scenarios (#3, #4) where going alone results in 2 more points than taking a partner. There is one scenario (#7) where taking a partner results in one more point than going alone. Let’s assume for now that each scenario has the same probability of occurrence (1 out of 7).
------------------------------------------------------------------------------------
Scenario, Probability, Alone Pts, w/Partner Pts, extra Pts
------------------------------------------------------------------------------------
.....1...............1/7...............1..............1................0
.....2...............1/7...............1..............1................0
.....3...............1/7...............4..............2................2
.....4...............1/7...............4..............2................2
.....5...............1/7...............1..............1................0
.....6...............1/7...............1..............1................0
.....7...............1/7...............1..............2...............-1
==========================================

Then, the equation for points (EV) for going alone would be.

4 out of 7 chances you score 0 extra points + 2 out of 7 chances you score 2 extra points + 1 out of 7 chances you score 1 less point
Or
(4/7 * 0) + (2/7 * 2) + (1/7 * -1) =

.43 pts – this would suggest the best play is to go alone when you hold 5 trump with no bowers.

In reality, the probabilities of each of these situations are not identical. I think we can simplify the problem by just determining the probability for scenario 3, 4 and 7.

For scenario 3, the probability that our villain (knowing the 5 cards that are in her hand) can assign to the likelihood that her partner holds both bowers is 5 out of 19 unseen cards multiplied by 4 out of 18 unseen cards. This is similar to the way we calculate the probability of catching runner-runner flush cards in holdem. In this case it is 6%.

For scenario 4, the probability that both bowers are in the discard pile would be 3/19 x 2/18 or 2%

For scenario 7, the probability that either me or my partner hold the left bower and our opponent holds the right bower is 10 chances out of 19 unseen cards x 5 chances out of 18 cards = 15%

So the revised probability matrix might look like this

------------------------------------------------------------------------------------
Scenario, Probability, Alone Pts, w/Partner Pts, extra Pts
------------------------------------------------------------------------------------
..1,2,5,6........77%.............1..............1................0
.....3...............6%...............4..............2................2
.....4...............2%...............4..............2................2
.....7...............15%.............1..............2................-1
==========================================

So the revised EV is
(77% * 0) + (6% * 2) + (2% * 2) + (15% * -1) = .01 extra pts - this would seem to suggest that there is no real difference in going alone or taking a partner when you hold 5 trump cards with no bowers in Euchre.

As always, let me know if I got this stuff right or wrong.

# posted by Jon J : 12:29 PM