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Returning to the game after many years of not playing (home game) and ran into a renege last night, the solution to which I need help with.

On Wikipedia I found a rule regarding reneging which reads: "If the bidder reneges, they automatically take a double set and the amount of the bid is subtracted from their score."

When this happened during the game, the term "double set" caused confusion. Is it saying that double the bid amount is subtracted? The rule says only that the bid is subtracted, not double the bid.

Any help greatly appreciated.

Sorry, but what does it mean to "renege" in pinochle (or double deck pinochle)?
Renege means to play a card contrary to the rules.  When the rules say you must do A, but you do not-A.

Example, West leads  TS ;  North has a  AS but perhaps has mis-sorted it into his clubs.  So he plays a lower spade.  

Also applies if North ruffs, implying no spades (and spades are not trump).  

For reference, it looks like OP was looking at the rules on wikipedia.  I honestly don't know what they mean.  Nowhere is the notion of single or double set well-explained.  I also note, that section is in the Variants section, and right after talking about 3-handed pinochle.  The language seems to be addressing it in terms of 3-handed, too.

Better rules are over on pagat.com.  I'd encourage you to look there.

Renege rule...my suggestion is this, for a friendly game.

#1:  first try to agree that the renege did, in fact, occur.  When tricks get swept together, this is hard to do.
#2:  if the situation is simple...like...oh shoot, I screwed up 2 tricks ago...try to agree on a reasonable fix.  Perhaps the renege card becomes a dead card...it cannot win a trick, and it must be played at the next legal turn.  Something like this...

West leads  TS , North plays  JS which is a renege, East plays  AS and South plays a spade.  East plays, let's say,  AC  and all follow.  North goes ACK!!!  and realizes he reneged on the spade.  He shows it.  East now leads  TS  and South and West both follow with lower spades (legally).  North must play his  AS  BUT it does NOT win the trick because of the renege.  East scoops the trick.

If North gets the lead while still having an exposed card, that card must be led.  It is treated as a 9 of the suit led.  If that suit is trump and there are no other trumps in the other players' hands...well, life might get a bit messy in some cases.  If it's trick 20, and that card would win the trick...the opposing side can be awarded the trick, and the 2 points for last trick.

The notion here is on the assumption that the renege was an honest mistake, and it tries to make the penalty fit the crime.  It is, admittedly, situational.

The simpler, if harsher...

--if declarer or dummy reneges, his side is set as usual, and his score is decreased by the amount of the bid.  Defenders score their meld.
--if either defender reneges, defending side scores 0.  Declarer scores his bid.

Now, it's entirely plausible that this is not fair...it's not enough.  Say it went pass-pass-pass to dealer...who has double aces and an 8 card 19 point run.  The bid is 50 by the rules...but declarer's gonna score something like 150.  Similarly, say declarer bid total junk but with a bunch of meld.  Oh joy...each opponent melds aces, and each is licking their chops cuz...that's not all they have.  Defense lets the dummy in...oh gods, dummy has nothing either.  Declarer can see the defense is gonna score 35 in the play.  

SOoooo....another treatment would be this.  The offending player selects a card that could have been legally played (there can clearly be more than one);  that card becomes an exposed card, and must be played at the first legal opportunity to do so.  Play continues normally and points are counted, including last trick.  

--the offending side scores 0 (defenders) or goes set the bid (declaring side) automatically.
--the non-offending side scores its meld PLUS the trick points its side won.  NOTE:  if using the rule that you need 20 play points to save meld, it DOES NOT apply here.  And if the defense reneges...even if declarer would not normally make its bid because it didn't score enough...doesn't matter.  Declaring side does NOT go set.  They don't get the bid, they get the meld + play.

This is really not as complex as it sounds, but it's something you might need to work through.  So if what I'm suggesting isn't clear...and you *will not* find rules like this anywhere.  The basis for some of this is the rules of bridge.  It's still harsher than it needs to be.  Honest to gosh...I was playing in a friendly, lunch-time foursome, and the *only* renege I ever saw in live play happened when the guy (my opponent, and a very good, experienced player) did so *when he had a double run.*  I was all for restoring equity...cripes, he was gonna save.  He said no...the rule's the rule.  Adamantly.  I would have preferred a different result, but he clearly wasn't gonna listen.
After looking at Pagat and thinking about it some more, I'm thinking it might be explained like this:
1. Since the rule refers to a bidder renege, the term "double set" would mean 1) they didn't make their points, and 2) bidder reneged by misplaying.
2. If either situation had occurred alone it would be a set and would mean a subtraction of their bid from their score.
3. The rule itself says only that their bid is subtracted from their score, not double their bid.

So for now I'm going to explain it to my playing group that way, but continue researching it.