In many games, such as Liar's Dice or On The Underground, there is an advantage to be had by sitting immediately after (in the turn order) the player who is least experienced or just plain bad at the game. Are there any widely applicable ideas for resolving this problem? Or, is there some house rule or variant that improves this aspect of a particular game?
Assign seats randomly. Card draw, dice roll, etc.
A part of strategy in games is knowing the situation you are in and adjusting your game play. If you get the unlucky draw of being to the right of this person, you'll need to adjust your strategy knowing the person to your left may not make the play an experienced player may make. Many times I make a play I normally wouldn't make when I sit the right of an inexperienced player if I cannot afford for the person to the left of the player to make specific play.
So in my opinion situational awareness should dictate your strategy.
After every game we play, we re-assign seats randomly.
I think you can't have a serious, cutthroat game with a wide range of skills, so relax. Gently coach the new players, but no more than they're willing to receive. Play a couple of dummy rounds for the new players and then restart if needed.
If you're really serious, you can investigate ways to handicap the game, but that's really hard to do without ruining the balance of most games.
Board game skill definitely transfers from game to game. If you really don't want to play Puerto Rico with an unskilled player because the person to their left will have an advantage, then play different games with them that do not have this problem.
For example. Power Grid has turn order that changes based on how many cities a player has. Of course good players will purposefully hold back on purchasing houses to get better position in the next round, but it is still a great mechanism to make sure that inferior players have limited ability to affect the game in a negative way.
Other games that solve this problem are Caylus, Agricola, Citadels, and pretty much any other game where turn order can be bid for, and is not set in stone. Also, many game sthat have limited player interactivity, such as pure racing games, do not suffer because turns are effectively, if not actually, simultaneous. Also, this problem obviously does not occur in two-player games or co-op games.
I generally highly discourage any modification to games. Players are not game designers, and without incredibly thorough play-testing by professionals, any so-called house rule you come up with will probably just ruin the game, even if you don't realize it.
If you absolutely feel compelled to modify a game, here is a general method to follow. First, determine if it is better to go first or last in any given round. For the sake of example, let us say it is advantageous to go first. In the first round of the game proceed as normal. Then using some quantifiable measure, rank the players as to who is in the lead. Since going first is advantageous, have the person in last place go first, second to last place go second, and so on. Redo the turn order at the beginning of every round by this same measure.
This modification will in all likelihood ruin your game, as skilled players will take advantage of the system to go first/last when they need to. It also allows for increased chance of error, when someone no doubt accidentally takes their turn out of order. Yet, it will fix the problem of sitting to the left of the weakest player under most circumstances.
We long-ago made a set of 'random position' counters for games where player order matters; one with the name of each player in our group. Then we deal out the seating for the game at random. This doesn't solve the problem in any individual game, but on average everyone comes out even. Getting the worst player as in a good place for your or the best as your most-likely attacker becomes just another random factor of the game - in the long run, luck evens out.
I don't know any games offhand that are absolutely ruined by this kind of position difference, unless the gap in skills is so great as to be a problem no matter what - in which case you might as well admit that and make it a relaxed teaching game.
When it's clear that one player has an advantage over all the others, my group often has fun by conspiring against that player, and if people are playing the game for enjoyment it can end up being pretty fun. Once my family noticed that I tended to win more than any of them at Settlers of Catan, they started to be very careful about what trades the offered me and, if I was winning, they'd all combine forces to make sure that robber was on me at all times. The same thing happens in Puerto Rico.
Experienced player to new player before me:
"No don't pick craftsman! That means he'll get to ship his goods first and get more victory points than us! Pick Builder so he doesn't get that stack of $$$!"
New player thinks about this fact and reconsiders. Picks the Builder, thwarting my plan. The other players had fun conspiring against me, I had fun being conspired against, and the unfair advantage I would've had was gone. What's not to like?
Now I don't win as much any more, and when I do it's far more of a treat. My opponents have fun trying to thwart my plans and I have fun scheming to win and laughing when I don't. If you have a new player at the table you probably won't have the super-cutthroat experience you might have if all the players were of equal skill level, but that doesn't mean you can't have fun. :D
I think it really depends on the game and the situation.
If I am new to a game, it doesn't really matter where I sit, until I become experienced enough to understand the positives and negatives of a seat position.
If I'm just a bad player, well, then I am not in any better position no matter where I am.
This being said, we try to shuffle seating regularly when playing several rounds of a single game. Depending on the game, we will also sit according to final score/placement, winner getting the first seat, then on down to the left. This helps with games like Puerto Rico and some other games that tend be a bit harder for those going first.
If there is too much of a difference between players such that winning is not an option for the "lesser" players, try to focus on having fun. The players won't return if they always lose.
Another option is to team up. Make a team with a couple of an experienced players and a new player, so the new player can learn from the pro without the need to be beaten.
This question makes me wonder:
How many games where turn order is the same from round to round would be "ruined" if you selected turn order randomly every round, with the caveat that no player could go twice in a row?
In other words, what if you re-organized turn order at the end of every round?
Note that I don't think this is a very practical solution, since the act of re-ordering turns at the end of each round would add at least another two minutes per round, but it's conceptually interesting.