How does Rat-a-tat-cat strategy change as the number of players increases?
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.