As already discussed in Do the vast majority of Rat-a-tat Cat rounds end very quickly?, optimal play leads to rounds ending quickly with a small number of players. Is this true when playing with 5-6 players? I just played my first 5 player game yesterday and did not enjoy a similar level of success when employing that strategy. Consider:

For one thing, the chance that an opponent will draw a power card is no longer very small, which matters a little for a draw 2 and matters a lot for a swap. But more importantly, if someone else is likely to call for the round to end, why not just wait and get your extra turn? Of course, if you were thinking to call for the round to end, you may not have visible cards that can be improved all that much anyway . . .

  • Why not wait for someone else to call for the end of the round? - every turn you don't call for the end of the round, you give your opponents another turn to improve their score (reduce). You should call for the end of the round when you believe that your opponents are reducing their score on average more than you are reducing your score on average. The problem with determining when this is, is quite difficult with so much hidden information. Even after going through the entire deck once, with 5 players you will still have between 18-15 cards that you have no one has any idea who has them – user1873 Mar 18 '12 at 19:36
  • @user1873 Agree that the large number of hidden cards (combined with 4 other people taking turns after you call for an end) makes the determination difficult. Which is why I asked the question. It's surprising how much this is to this seemingly simple game aimed at kids less than 10 years old. Significantly more complex than most other little kid card games I've seen. – Joe Golton Mar 18 '12 at 20:35

This isn't my answer yet, just some observations. I might take a stab at creating a computer simulation that can be easily customized with different numbers of players, different player strategies, and possibly different rule sets (children's variant where the outermost cards are public information, and perhaps a reverse of the normal rules where the outermost cards are only known to player whose hand they are in, but the innermost cards are known to every other player).

When the entire draw deck has been gone through once:

  • In a 2-player game, the highest known card for each player will be a '1'. This will occur if both players have '0's in their hidden cards. All 54 cards (minus 8) will be drawn when the draw pile is exhausted, and players will always discard either a known higher card for a lower card, or a random unknown card for a (0-4). This isn't counting the 3 peek cards being drawn either, but it is likely that players will discard unknown random cards at least when a drawn card is higher than their known cards but 4 or less, since an unknown card is worth '5' on average. This unknown random card is discarded face up for their opponent to take, so eventually all low cards will be claimed by the end of the draw deck.

  • In a 5-player game, the highest know card for each player will be a '4'. With 5 players, you can use the same reasoning as in the 2-player game. By the end of the draw deck, all the low cards will have been taken into players hands (4 zeros, 4 ones, 4 twos, 4 threes, and 4 fours).

  • From the reasoning above, we can see that Power Cards in hand are actually worth more than '5' on average as then number of cards are drawn. In a 5 player game, if the discard pile gets reshuffled into a new draw pile a hidden Power Card is worth 7.4 on average.

  • From the reasoning above, I believe it is also true that at least one player will knockout after the draw pile has been exhausted (or slightly before). At that point, a player should know exactly which cards are in the draw pile, so they will know if they can reduce their score further (they will also know how many swap cards are in the draw deck).

When a player discards a known card, we know that the card that the face up card they discarded was greater in value than the card they kept. This might be helpful in determining how many points we think an opponent has. (i.e. If an opponent discards a '9' card, we only know that the card that they kept is less than or equal to 9 and their total known score could be as high as 9+9=18. If they discard a '5' though we know that the card that they kept was less than or equal to 5 and their 5 was most likely their highest value card (discarding the highest known card, or an unknown card if the difference between the known card and kept is less than the AVG difference between an unknown card and kept card is optimal play. We cannot rely on this information since a player may not be playing optimally, but it should be the best move in most games.). Their total known score is no higher than 5+5=10). If we track exactly which cards are being discarded, and each card that is kept we can gain extra information about our opponents.

The game can end in several different ways, depending upon the End Game condition.

  • Play for a certain number of rounds. - Hopefully, the number of rounds is divisible by the number of players. The first player has the greatest incentive of knocking out, since if they knock out each other player has only taken as many turns as they have. The last player has the least incentive of knocking out, since if they do, each other player will have taken one more turn than they have. Strategy for this end condition will vary slightly as the number of rounds until the last round gets nearer.

  • Play for a specific length of time. - This is a poor choice of ending condition. Players in the lead can effectively stall the game to keep their lead.

  • Play to stay in the game and not reach 100 points. When a player reaches 100 points, he is out of the game. The last player in the game is the winner. - The Strategy for this end condition will vary slightly as a players point total nears 100 points. A player will probably not knockout when they are more likely than not to be out of the game. It would also depend upon what happens when all remaining players lose at the same time (all players have 100+ points). Do all players draw? Does the player with the fewest points win? Do the players play a final hand? The strategy will differ according to the rules.

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  • I appreciate the start and in a spirit of cooperation I offer some refinement to your observations. First: your assumption on discarding a known card. This is a poker-like game of facial expressions. There is a swap card. So there can be good reason to replace a 9 with a 9. I pulled it once against my son, with an expression of delight. As luck would have it, he did draw a swap card, and because he had two good outer cards, he swapped an unknown inner card to me which turned out to be a 1, and he got my 9 in return. Then he called for an end. – Joe Golton Mar 19 '12 at 14:52
  • My son called it "cheating" but the rules do not prohibit this sort of false-carding combined with faked facial expressions. In fact, over the course of large numbers of games, it can be good strategy to sometimes even worsen the card a little so that it makes people think twice about swapping. However, if you decide to set up a simulation, you'll need to make assumptions so perhaps your assumption is worth making just to keep the simulation simple enough that it won't take hundreds of hours of your time. – Joe Golton Mar 19 '12 at 14:55
  • Swap cards also mess up your conclusions for what happens if 2 players go through an entire deck, rationally. Because swaps will insure that some higher cards get slotted into peoples' hands. – Joe Golton Mar 19 '12 at 14:59
  • We never come close to exhausting the draw pile. But you reasoning made me realize that hidden information begins to be revealed as the draw pile gets close to exhausted, for those who precisely track all cards played. Not very many people can do this, and kids are even less likely to. But theoretically if there are only 3 cards left in the draw pile in a 2 player game, it means that you have seen nearly all the cards already, and can make some good guesses about the opponents hidden cards whose values you don't already know (especially if the remaining 3 undrawn cards are all high cards). – Joe Golton Mar 19 '12 at 15:04

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