Several of my gaming friends are victim to more or less chronic analysis paralysis, which can drag down even the simplest of games.
We've occasionally used an hourglass to combat this, but it tends to aggravate the slow-playing contestants and doesn't work well in interactive games (such as when trading in Settlers of Catan).
What kinds of countermeasures to this common issue you have found to be effective?
Gentle prodding is probably the best suggestion, but if that doesn't do the trick sit them down after the game and let them know what they're doing. A player that over analyzes every move generally isn't much fun to play with, and telling them that (nicely) can help.
If that doesn't work, try practicing with some simple games (for example poker). Once they get used to making faster decisions, it often translates into other games, especially when combined with gentle prodding.
As a final resort, try playing timed games, where the impulse to think long and hard about a decision is countered by a running clock. Good examples are chess/go with a timer, outburst, trivial pursuit, and cranium.
Hum the Jeopardy tune once you feel someone has gone too far!
Seriously, it works :)
We've tended to favor games that have either cooperative decision making or simultaneous decision making, instead of sequential decision making where your turn will rely heavily on what the previous players did.
In our group we have the same problem. Usually we urge everybody to think ahead. So when it's your turn you don't have to think of everything at that time. That usually works pretty well. Of course things can change during the other person's turn, but in most games, thinking ahead and keeping everyone's attention on the game definitely will speed things up.
Another problem we sometimes have is that someone constantly has to look stuff up because he is not 100% sure of the rules. Playing a certain game on a regular basis will solve that problem though.
A simple solution not so far mentioned: Allow takebacks.
It doesn't always help completely, but it often helps a lot, and is the approach written into the official rules of Mage Knight. Simply allow players to retract their moves so long as no information point has passed and encourage them to play out their action rather than engage in an extended internal debate, and things will often go better.
Not a solution for everybody, but Andy Looney (of Looney Labs fame, inventor of Fluxx and Icehouse pyramids) solves his frustration with waiting for other people's turn by playing Andy Vs. Everybody. He plays multiple games against dozens of opponents at once. Everyone else sits in a ring of tables around him, and raises a flag when it's his turn. He in turn runs madly around to raised flags, takes his turn, puts the flag down, and then moves on to another table. The rest of the players can have all of the analysis paralysis they want; he's got plenty of other games to play!
I think once people get experienced with a game, they get better at the game and take less time. If people understand the rules, they're going to be much better at decided what to do. Be polite about nagging them to move. If you badger someone to much, they're not going to want to play again. Some games, like chess are naturally going to be slow. If the other players in the group are fast, hopefully they'll learn to step up pace after a few playthroughs.
If you're teaching someone a new game, let them ask you questions and don't make the environment so competitive where they have to make the best possible decision each time.
Although this doesn't offer a magic solution to your problem, I'd like to share a coping strategy:
I got "bushwhacked" by a few AP players at games I played at BGG Con in November. I was too tired to politely handle the situation by attempting to change the AP players' behavior, so I did what I could to make it easier for myself: I carefully analyzed the board, got out my notepad, wrote down my next move(s), then spent extra time having friendly conversation with unoccupied players, figuring out additional advance strategies, and jotting down assorted ideas in my notebook.
I suppose my answer to your problem is (if you can't get the AP players to speed up, and you don't have enough game options that don't cause AP) to use the time to plot your next move, and spend the time chatting with the other players. After all, you're (ostensibly) friends, so you should have plenty of stuff to talk about.
One other option: Set up a simple side-game, like Zombie Dice or M:tG, and play it while you're waiting. If you do this, I strongly recommend jotting down ideas for your next move so you don't end up slowing the main game down.
I'm a slow player myself, and I often used to play games with a person who is a very quick decision maker. It would driver her completely bonkers, which wound me up, etc. Eventually we settled on a solution where she would read a book while I was thinking. Of course it's not ideal; I wouldn't recommend it to everyone and it won't work for groups of more than two people. But for us it rescued gaming.
I think part of the answer is that you have to play the same games with different styles of game play (see below, speed games). We all succumb, at one time or another, to the min-max problem. Said another way, in many games, there actually is an optimal solution that can be calculated decisively at certain points (especially near the end). But calculating it in ones head can be a slow process for some.
The best solution (in my opinion) in those situations is to just work with the player. You'll all learn more about the game and be able to make the decision faster next time. They'll have the satisfaction of knowing they scored the best they could (or at least others agreed with them). In fact, if done right, everyone can have a maximum score. (Doing this gets much harder as the number of players increases -- consider it a group challenge. And, recall, it's just a game and it'd be silly for someone to win or lose because of a mere accident or true mistake.) (Some games are well suited towards this style of group play. Highly recommended with the significant other, too. ;) )
As for the type of game, mix in play of "speed games" where everyone has to take fast moves. Compare to a 2 minute speed chess game -- it's a completely different game when played that way.
This can balance full, thoughtful, strategy play with intense, and sometimes silly, action play.
If someone is the only one affected by the "rules of speed" they'll feel picked on or just won't have as much fun. Everyone, at some time or another, probably takes a long turn. Another solution is to just have a limited number of "long turns" -- this way, the slow thinker or optimizing player can plan for one turn (or so) a game and the rest of the group can go refill their drinks and snacks.
Finally, as others have said, unless it's chess, game familiarity will almost always speed everyone along. Give new players a chance. Play "open discussion" games. Open it up so the "slow" player can ask for advice and receive fair, honest answers. If it's a constant problem, the person is probably just unsure of their moves and boosting their confidence will go a long way.
Most people say that it is rude to ask slow players to play more quickly. The opposite is actually true - slow players are the rude ones. They selfishly take extra time so that they can gain an advantage. AP players ruin the fun of the game, and it should be recognized as such. Your desire to win or play well is less important than the right of everyone else to experience a game that flows well and that finishes in a reasonable time.
In my main group, one of the players has DGT Cube, a device that records how much time each person takes. (Each side has a color that corresponds to common player colors. You flip it to your color when it's your turn and The Cube counts your time.) At the end of the game, you can definitively show a slow player how much more time he has taken than everyone else. As this guy says, if one person takes as much time as the rest of the players combined, that person is too slow and has to change or be uninvited to gaming. The Cube inspires everyone to play faster, which means we can play more games.
Studies have shown that people make better decisions when 1. they have fewer options to choose form and 2. they allow their emotions to play a part in the process. Therefore, AP players should be counseled to narrow down their choices to two or so options, even if it's done arbitrarily. If they still can't decide, they should be counseled to go with their gut. Tell them that if the choice turns out to be suboptimal, they will learn for next time.
In my circle of friends, our Settlers of Catan games started sprawling uncontrollably, so we started playing with a clock that gave, I believe, one minute per turn and saved our spare time to use in future turns. (It also handled the switchback start for initial placement. We're all software engineers so writing our own wasn't especially difficult.) After playing with the clock for a few games we got the hang of how to speed the game along, especially thinking and planning ones' next move between turns.
(It also helped that we all had roughly the same level of experience with Catan, so no one felt especially overwhelmed by the idea of jumping unprepared into the game.)
Anyways, I will add my bit to the list of answers already. After reading this question I realized that I suffer from this as well specifically when there is a speculation determining the cost of opportunity of doing (or not doing something).
Just being aware of it already had a positive impact in a couple of games. In recent games of Mage Knight - a rule intensive, unpractical game - it helped a lot because making a decision considering the rules/possibilities I knew didn't compare to the rules/possibilities I did not know.
I guess It's because I added my freinds' enjoyment to the decision making formula. So tellng this to players who are not aware, helps.
So a good solution for this has been known for a while. The "Chess clock". For any old board game do this:
Determine the length of the game you'd like to play. Divide by the # of players. Double the time. Each player gets that much time for the game.
If the time is exceeded, that player loses the game. If the game would be 'ruined' by him leaving in the middle, he can either play out his turns (without a chance of winning) or other players can take turns finishing his turns (round robin or random player takes his remaining turns)
This solution is superior to a turn timer because it allows you to take more time when necessary, but really results in faster turns since people want to 'save up' more than use the whole turn time.
There are several smart-phone apps that can be used for this purpose, even the default timers will do.
Fingertip drumming, tapping your feet and frequently asking "Is it my turn yet?" will help resolve this issue in future game sessions.
