This weekend I played Innovation a few times at the games day I arranged. It is a card game with surprisingly good replay factor considering how small a package it comes in; it reminds me of a regular pack of cards in that there is tremendous room for novelty and ... innovation? ... in a very tight design. It illustrates very well the difficulty in scaling up a game from a two player game to three players or more.
The fundamental mechanic of the game is advancing in science to get to the later and more powerful technologies. You do this by drawing cards off of a stack; each stack has a number from 1 to 10 and if a given stack is gone you can draw from the next higher stack. You start out only having access to stack 1 and work your way up from things like Clothing and Writing to things like The Internet and Robotics. In the two player game it is possible to run a stack out and have to draw from a higher stack but it rarely goes very far - generally you have to work at it to be able to draw from higher numbered stacks. In the three player game the dynamic totally changes because it is easy for three players to draw all the cards in the stacks and players can draw from very high numbered stacks without even working at it. What often seems to happen is that every card from every stack numbered 5 or less is gone so I can draw from stack 6 regardless of whether or not I have been pushing to tech up or score so a lot of the strategy goes out the window.
One of the key mechanics that keeps the game interesting in two player is the choice between teching up to get power later and scoring points to right away. You can't do everything so you have to make tradeoffs and interesting strategies emerge but those strategies often vanish in three player mode because everybody techs at the same rate. This is the sort of problem I found when I tried to make FMB into a game that could accommodate two, three, or four players. I could make the rules work for all the games but I had to trim away some ideas that were problematic and accept that balance was out of whack somewhere. I often had to make the two player game worse in order to make the three player game better and that always felt wrong. What I really wanted to do was make one perfect game, rather than a more flexible game, so I ended up trashing the three and four player rules.
Puerto Rico is another good example of a game that works very well for some numbers of players and very poorly for others. I can't decide whether I like three or four player PR better but both have their good points; three is more personal and you have more control but four has better builder / shipper balance because you can have two on each stream. The one thing I know for sure and which everyone seems to agree on is that five players is poor. All kinds of basic things break down and the game ends up feeling very random with the player not having much ability to make impactful decisions.
Personally I own lots of games. I don't need games to have bad settings like three player Innovation or five player Puerto Rico if those settings compromise the regular game in any way. I am quite willing to buy more games for different numbers of people if that makes the games better. I don't know that either Innovation or PR really sacrificed much to add on their additional player options but I know for sure that when I make a game I will make it the best it can be at one thing. If other options are possible, so be it, but the game should not sacrifice quality to increase player options. When publishing a game it probably helps to have 2-4 players on the box instead of 2 players but that is a bridge I am not likely to cross for a long time, if ever.