Why does a run of 4 in Cribbage score only 4?

Question

Cribbage scoring is fairly logical. There are card combinations worth a given number of points and you can score every unique subset of cards independently and total up the score. Except for runs of 4 or 5.

For example, a pair is worth 2. Three of a kind is three different pairs (3 choose 2 is 3) and scores 6 (called a "pair royal"). A run is 3 or more cards in a row (e.g. 4, 5, 6) and scores one point per card. It seems to me that in keeping with the other scoring rules, a 4-card run should score 6 (2 3-card runs each made by dropping the end cards) and a 5-card run should score 9. Instead, a run scores one point per card.

Is there a logical reason I'm missing or is this just an inconsistency in Cribbage?

(afterthought: a 5-card flush would score 20, which is a little too much)

Answer 1

First off, taken one way this might be an unanswerable question

I'm struggling with the question. It seems to be asking folks to guess on what they think the rationale might have been for scoring 4-card runs as 4-points instead of 6 (or some other number); however, since the game was invented in the 17th century its inventor is long-since dead, and absent some comprehensive history on the game and its inception that I'm not aware of the real answer to the question is not knowable. In other words, we can come up with some reasons that seem logical to us, but we will probably never understand the thought process of the game's inventor.

That said, here's why I like the rule as-is (and incidentally I like your question because I've had the same thought):

The best possible hand in cribbage is 29, which involves the Jack that counts for nobs and all of the 5s. Now although hands with lots of fives are always good, often retaining just one or two will only result in a mediocre score where you count a few 15s, a pair, and are done.

In contrast, a strategy that relies on runs, pairs, and 15s will have a lower maximum score, but will get more points if the up card is not exactly as desired.

For example if you're drawing for the best possible hand failing to get that last 5 as the up card will cost you 14 points, as you'll only score 15-8, a pair royal for 6 and 1 for his nobs for 15.

Consider now the runs, pairs, and 15s strategy, say for example, 6-7-7-8-9. Based on the rules this hand scores: 15-2 (7-8), 15-4 (7-8), 15-6 (6-9), run for 10 (6-7-8-9), run for 14 (6-7-8-9), and pair for 16 (7-7). A super solid hand. In the previous example there is only one card in the deck that gets you the extra 14 points. In this hand the best draw is one of the 4 nines, but the drop-off in points isn't that much if you draw another 8, or a 7, or a 6, or a 5. Less upside potential, but a much more robust hand statistically speaking.

Now here's why scoring runs differently breaks things.

If runs of 4 were scored as 6--or more precisely, if you got to count for 3 points every unique run of 3 that you could make, this hand scores much higher. The count to 15-6 is unchanged, but then: run for 9 (6-7-8), run for 12 (6-7-8), run for 15 (7-8-9), run for 18 (7-8-9), pair for 20 (7-7). Or perhaps a much simpler but more dramatic example is a hand of 5-6-7-8-9, where the score for just the runs would go from 5 points to 9.

If the value of runs is increased, this incentivises players to only go for the much more likely (and now also relatively more lucrative) run hands instead of the larger payoff but higher risk 15 hands. This would make the strategy of the game more myopic and because everyone would be playing the same way because the relative payoff of going for the knock-out blow would be much reduced.

I don't want to make too much of all this (too late probably) because ultimately a players freedom is highly restricted by their cards (you only get to discard 2 after all), but the incentives of changing the rule appear to be to be tilted in the wrong direction.

Answer 2

This is most likely, I think, a consistency thing between game and hand scoring of runs. During play you score for each card in a run, first player to get the run scores 3, then the second player for continuing it scores 4 - this rewards longer runs in game more without needing to ignore the last played card (to include the first sub-run of 3) which never happens in game. Now when we get to scoring in hand, we could allow counting sub-runs and not allow counting larger runs, but this then makes scoring harder because there are different rules for scoring the same thing between hand and game. While there are things not scored in one and scored in the other, like a flush, simpler scoring rules are better.

They could have scored longer runs differently, not per card, but this would make scoring harder to understand without thinking around it (why a run of 5 is worth 9) and would create the potential for some excessive single scores in long runs, Ace through 7 is the longest possible run, and in a 2 player game, scoring each card that finishes a longer run as 3 higher than the last would be 29 to one player (3+9+15 for runs and 2 for the 1,2,3,4,5 15) and 18 for the other player (6+12) for a total of 47 points between the players in a single go.

Answer 3

There isn't really a good reason why Cribbage does not score 2 3 4 5 as 234 345 (6 points) or flushes of five cards as 20. The objection above

"So yes, I agree with your afterthought, scoring a 5-card flush as 20 would be insane, given that it would be possible to have plenty of runs and 15s in there as well."

doesn't really hold water. A flush in poker is a very high hand - actually above a run. It should score a high proportion of points. A running flush is the highest hand and very very rare.

The only reason I can see for the rules as is is that it makes scoring runs in the playoff easier. But IMO it is a fault.

Answer 4

While writing a Cribbage app (BTO Cribbage), I had the same thought. When I wrote the scoring module, it came to me with the following rule:

For straights, you get a point per card considering every unique combination of straights of 3 cards or more.

Example:
2,3,4 -> only 1 combination of straight = 3 
2,3,4,4 -> 2 combinations of straights + 1 pair = 8 
2,3,4,5 -> only 1 combination straight = 4
2,3,4,5,5 -> 2 combinations of straights + 1 pair + fifteen = 10

Easy right? Now use the same logic for pairs / sets:

For sets (2,3,4) you get a point per card considering every unique combination of pairs.

Example (more detailed):
2C,2H -> 1 pair = 2 points
2C,2H,2D -> 3 pairs of 2C+2H, 2C+2D, 2H+2D = 6 points
2C,2H,2D,2S -> 6 pairs of 2C+2H, 2C+2D, 2H+2D, 2S+2C, 2S+2H, 2S+2D = 12 points

So in both cases, you score 1 point per card considering every combination. If you think about it this way, straights are scored just as logically as sets. Further, 15 scoring also works this way.