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Well, thanks to all your help, I havn't lost, or at least not because of my overexpansion, in a while. This may have something to do with the fact that we declare a truce after a game has gone on a certain amount of time to prevent the game from taking all evening (our time that we can play is limited). However, a new and infuriating aspect has appeared that causes our games to end in draws: the WWI-Style Trench War/Stalemate.

It starts like this, 

Europe: Heavily Contested.

North America: Dominated by my uncle. 100%

South America: Contested, usually in my favor.

Africa: Heavily Contested.

Asia: Neutral/Contested. (We use a third neutral color as a buffer. It can't attack. For it's army placement, we alternate.)

Australia: Contested, Usually in my favor.

After maybe 20 or fewer turns, this is the layout:

Europe: Contested, backwater, usually with my uncle holding the advantage.

North America: Dominated by my uncle, 100%.

South America: Dominated by me. 100%

Africa: Contested, usually in my uncle's favor. Africa is really one of the few areas that's anyone's game.

Asia: Heavily contested, usually slightly in my favor. South held by me, north by my uncle.

Australia: Dominated by me. 100%.

At this point, it turns into something resembling the Western Front of World War I: Largely unmoving borders, occasional major shifts, but those are usually negated by counter-offensives. Massive troop buildup on both sides. What will also happen is that a large amount of fighting will occur in one area, then after we wear out our forces there, we will shift the brunt of the fighting to another area, and repeat. My uncle's slight turnly troop increase advantage is usually negated by my better use of cards, so amount of forces available is about the same either side. The only progress that ever gets made is when the center of the war's gravity is changed quickly enough and subtly enough that one person will not notice until it is too late.

How do I reduce the number of times where this happens? It's frustrating and eats up valuable time.

I have sympathy with you as I know how hard it could be when a game doesn't move forward. A number of solutions I can think of are:

• Change your configuration. Start from North America and Europe and let your uncle choose somewhere else. This may disrupt your natural expansion's pace and lead to a quicker game.
• Increase the rate of card sequence. It is well known that the flatter the incease is, the longer it takes to finish the game. So you can use seqeunces such as 2,4,6,8,..,20,25,30,35,.... or even 3,6,9,... This reduces the importance of continents quickly and forces you to use your cashes cards quickly before it is too late.
• If you want to keep your current configuration, then change the value of continents that war usually takes place in. For example if you fight each other in Africa most of the time, increase the continents's bonus to 4 or 5. That gives you more incentive to hold it and fight somewhere else, again disrupting the natural flow.
• You can also increase continents' bonuses based on a trigger later on. For example, when cards reach 30, Africa and Asia become twice as valuable.

Risk is a game of balance and a lot depends on the rules and the numbers. Try some of these variations or their combinations and expect the pace of the game to change. You could also use these variations to reduce the length of the game if you are short of time.

Let us know what works in the end

I have two sets of comments.  First for the scenario set above and about dealing with expansion problems.

How to handle a stalemate?  I would agree with Mr. Honary, you need to shake things up.  I personally love to play each Risk differently than I played the last to keep my opponents on edge and to try out different strategies.  After you play the same scenario over and over again, there is only so much you learn.  By trying different strategies and tactics, you expand your horizons and become a better player as a result. 

Try starting your conquest in Europe or Africa, places that sound like nobody goes to at first.  And then be sure to adjust your strategy to take into account the strengths and weakness of your new starting position.  For example, in South America, your first priority should be your borders and keeping your continent.  But when you are in Europe, you can't worry about that in the early going.  You need to attack a lot and devlop a strong base of territories to use for additional armies that way.  Be sure your opponent never gets a continent himself and always keep him guessing.  Mass your troops and be on the lookout for opportunities. 

An important point to consider: you and your uncle have achieved a type of Nash equilibrium in your games, a sort of evolutionary stable strategy.  What this usually means is that when one person decides to shift strategy, it usually leads to disaster for the one changing strategy which is why the game is at a standoff.  So when you adopt your new strategy, it should also include tactics that will change what your opponent does and the choices are available to him.  Failure to do this will just bring you back to where you started. 

When dealing with expansion problems, I would just like to refer you back to one of Dr. Honary's articles about creating a plan, and devising a strategy to accomplish that goal.   Plan, act, evaluate.  Once you are done, evaluate where you are and how you got there and then decide what you'll need to fix and what you'll keep.  If you set clear, definable goals that don't overstretch your resources, then you shouldn't have to worry about over expansion. 

After my last game with just my uncle (two days ago) I got completely beaten. This time, it ended up with a different configuration mid-game, but we were still fighting over Europe (we seem to have an affinity for the place. Every single game, without fail, we will fight over it) and so we diverted more armies than we probably should have, and I allowed him to get all of Asia relatively unopposed, so that doomed me, especially as I lacked Australia, and North America was divided (Africa and South America were both mine.) No cards got used, so his turnly advantage eventually doomed me, despite most of my best efforts. We talked about it after the game, and he said one of the big things was that I allowed him to get Asia without much of a fight, while we both pinned down lots of armies in Europe, he had more armies to go around.

But I never over-expanded myself! :)

On the otherhand, last night, in a spirit of family togetherness, we persuaded my sister to join us, and in the threeway game we ended up trucing (it was 1:30 in the morning by that point) and by territory and army count, she ended up winning. I got dead last, because I got ganged up on, and at one point i was reduced to holding only Egypt and the Middle East. Eventually, I marched across poorly defended Asia when my uncle diverted troops to defend against my sister's attack, and I took Australia and saved myself. However, at the height of my power, my sister held North America, I held Europe, Africa, and South America, and my uncle held Asia and Australia. However, the two of them ganged up on me (as my sister and I had done to my uncle and as my uncle and I would later do to her) and I first lost South America, then Europe, and I barely held on to Africa. Then my sister attacked Africa with a huge mass of armies and reduced me to Egypt and the Middle East, and then what was outlined above happened, then the game ended.