Do the vast majority of Rat-a-tat Cat rounds end very quickly?

Question

My son was gifted this game today. Given that we play a variety of complex games in our family (including Bridge), we're very surprised how much we like it and the various skills it brings into play (probabilistic assessment, poker face, memory, etc.). We're also surprised at how rounds usually end so quickly, as the rules provide for getting to the end of the deck. We have yet to go half way through a deck.

Is this typical, or are we missing something to the strategy?

Our probabilistic reasoning:

A deck consists of 4 each of 0 through 8, 9 9's, and 3 each of the power cards. So an average dealt hand has 20 points, with 10 points visible (I'm counting the power cards as 5's given that you'll have to draw randomly from the deck at the end for each of these).

A typical starting hand might have a low number and a high number known upon deal. If your first draw is a low number, you replace your high numbered card with that one and then you call for the end of a round. Why? Because you have, say, 5 points on your ends and an assumed 10 points (it's random) in the middle, for perhaps 15 total.

You have no knowledge of other hands but on average each will be 20, and with one more draw will be reduced by 4 points to 16 (on average). Not a huge edge, but definitely an edge. And this only widens as the sum of your two end cards drops further below 5.

Am I missing something? Or is it correct to call for the end of the round if the sum of your two end cards is 5 or fewer points after you draw the first card of the round?

The reasoning gets a little more complicated for subsequent rounds as other players' hands improve, but similarly if you know that 2 of your cards are very low, or better yet 3 within the first few rounds, then you call for an end, right?

Answer 1

Yes, you should call for and end or knock-out as it is commonly called.

Your reasoning is sound. With 54 cards in the deck, 9 of which are special non-point cards, and four cards each of 0-8 plus nine 9s, the average hand is 20 points. There in no way of knowing what your opponents hands are worth, except in the rare case that you draw 1 of 3 peek cards and then you draw 1 of 3 swap cards. (In the children's variant, all players two outermost cards cards are revealed).

Since you cannot know what your opponents have, it is best to call for the end of the round when you know that the sum of 2 of your cards is less than 7, or 3 of your cards are less than 12. (Edit: My initial calculations didn't take into account that your opponents get to look at one more card, and reduce there score as well.)

The only thing that might change this strategy slightly (and I am not certain if it does change the optimal strategy), is as the number of players increases the chances that one or more of your opponents draws a Swap card during the knockout round also increases. If your opponents know that you are using this strategy of only knocking out when you have < 7, or < 12 when you replace a hidden card with a drawn card, they will be sure to Swap cards with you (most likely they will swap with the card that you drew to replace your other card). They will know that they have a better chance of getting a '4' or less from you, then from drawing from the deck.

As for conclusion that each player would take less than six turns on average, I am not certain. I haven't worked out the math for it, but the number of turns on average would differ depending upon the number of players. If there are more players playing with the < 10 points sum total of known cards, then the chances that at least one of those players has a low hand increases.

This analysis was only performed for the first turn. I suppose you could calculate the AVG number of points that you would reduce your score per turn, and use that to figure out when to go out on later rounds. For the first round, with only looking at 3 cards, you will reduce your points on average by 2.63 points. This result was arrived at by looking at a million random 3 cards from the deck (with wilds removed, so the odds might be slightly different), and choosing the lowest 2 cards to keep for a single player. I did not examine what happens as the number of players increase, but my guess would be that as the number of players increase, the chances that at least one other player is able to reduce their score by more than the average increases (i.e. with two players, it is probable that one player reduces their score by more than 2.63 and the other by less than 2.63. For three players or more, the odds that someone is able to drop a '9' or other high number for a '3' or other low number increases, and that lucky person will have on average less than your total.