Ordering up the upcard

As 1st Hand

1st hand is placed in the unique situation that he is disadvantaged by not yet having received any input from any other player having already passed during bidding. But he is advantaged by the fact that he gets first lead.

To order up as 1st hand, his cards must be sufficiently strong to defeat his opponents, who by the fact that they have been ordered up, will now have an extra trump. It is worthy to note that if you have only 2 trump, then there is a 47.7% chance that at least one of your opponents already has exactly 2 trumps -- excluding the upcard. There is a 3.25% chance that both your opponents already have 2 trumps, which by inferrence, means both that the dealer will end up with three, and your partner will have zero.

If you have 2 trumps, what can you expect for trump distribution? There are 3 trumps accounted for, the dealer will end up with the upcard trump. So there are 4 trumps left. There are also only 17 positions for the 4 remaining trump. 3 positions in the kitty, 5 in each of 2nd hand, and 3rd hand, 4 in the dealer's hand. The following probabilities apply accounting for the upcard being in the dealer's hand:

1st Hand Having 2 Trump

Location  # of trump  Probability
Kitty 042.06%
145.88%
211.47%
3 0.59%

2nd Hand 020.80%
146.22%
227.73%
3 5.04%
4 0.21%

Partner 020.80%
146.22%
227.73%
3 5.04%
4 0.21%

Dealer 130.04%
248.07%
319.66%
4 2.18%
5 0.04%

          1st Hand Having 3 Trump

Location  # of trump  Probability
Kitty 053.53%
140.15%
2 6.18%
3 0.15%

2nd Hand 032.35%
148.53%
217.65%
3 1.47%
 

Partner 032.35%
148.53%
217.65%
3 1.47%
 

Dealer 142.06%
245.88%
311.47%
4 0.59%
 

The probabilities are exclusive. That is, there is not a 75.80% (27.73% + 48.07%) chance that both 2nd hand and dealer will each have 2 trump. You can only ask questions about one player at a time. In any event, there is a 33% chance that the player to your left (who gets to play after you) has 2 or more trump. There is a 21.88% chance that the dealer will have more trump than you, and a 48% chance that he will have at least 2 -- which is a disadvantage if you want to lead trump. Because the dealer will play last, he will be in a good position to maximize a tenace holding in trump.

If you are certain of 2 trump tricks because you have both bowers, the risk is acceptable, since your partner has an equal chance of holding the top card in an offsuit. If, however you are highly expectant of both trump tricks because of a void, you still may not draw enough trump and risk losing one trump trick to each opponent by them trumping an offsuit.

It is reasonable to assume that if you have 3 trump, that the dealer has 2, and that both 2nd hand and partner each have no more than 1. So a reasonable course of play is to order up if your hand has other quality aspects and then immediately lead a high trump. The worst case scenario is that dealer wins and 2nd hand has an additional trump. In such a case, which is only only a 1 in 5 chance, you might be Euchred. My recommendations:
If you hold    Action

Both bowers with opposite colour doubleton Ace/xOrder up

Both bowers with opposite colour singleton KingOrder up

Both bowers with 2 AcesOrder up

Both bowers with any Ace and any 2 KingsOrder up

3 Trump, 2 of which are King or higherOrder up

3 Trump, with upcard showing 9 or 10Order up

3 Trump, with 2 voidsOrder up

3 Trump, with opposite colour singleton Ace and voidOrder up

3 Trump, with 2 offsuit (singleton) AcesOrder up

2 Trump, both Ace or higher, and an opposite colour singleton AceOrder up

2 Trump, both Ace or higher, and an Ace/King/x tripletonOrder up

2 Trump, at least one of which is greater than the upcard, and 3 singleton AcesOrder up

There are two caveats, of course, you need to know how to play each hand before making an aggressive bid. The second caveat is that you do not believe you have an acceptable suit bid if everyone passes. If you would be forced to pass on the second phase of bidding, 2nd hand gets an unfortunate opportunity. You might also wonder why you would ever order up if you have both bowers instead of waiting to bid the opposite coloured suit. If there is an advantage in your hand to bid the other upcard coloured suit, then by all means, you should not order up. If, however, there is no holding advantage to waiting, you can effectively communicate to your partner that your hand has other quality aspects. By trick 4, your partner can safely discard a non-Ace/King holding in an opposite colour suit which has already been played. This alone is worth the risk. For example, if clubs have already been played, on trick 4, partner can safely discard the Queen of clubs and retain a queen in another suit. This will win the necessary 3rd trick most of the time.