Guard everything, and advance your king, expect endgame before move twenty, and atleast have a one or two tempi lead
Dealing with exchange nuts
Learn how to play endgames, study players like Capablanca and Rubenstein, and out play them. Strong players are not opposed to exchanges if they gain something from them.
Here, I played one myself : ) I was White.
I called for a draw because I've been drinking coffee all day and was too scared to calculate out where this was going... but you see the idea. Starting from move 13, any time a 1:1 exchange was available, I went for it. End result: the two nearly symmetrical lines of pawns. (I wasn't expecting him to trade Rs at the end)
One of my computer games trades every change it gets. I agree with WestofHollywood, in that, learn to play the endgame. Playing chess is dealing with whatever your opponent throws at you.
I don't like playing with people who simply trade or try to force trades for no reason other than to rid the board of pieces. This leads to boring and somewhat purposeless chess. But, if I do run into to such an opponent, I try to engineer trades to my own advantage such as retaining Bishops over Knights, or trading a Rook for two minor pieces, keeping strong pawn islands and trading in ways that brings my King more towards the center. But what I never do is to shy away from the possibility of an exchange - that is, I would never retreat to an inferior position just to avoid an exchange. Try to ensure that when the endgame begins it's you who has an advantage, however slight, and not your opponent. I really hate validating such non-play by losing (though that does happen sometimes).
I agree with batgirl, in that, try to engineer trades to your own advantage. People that like to exchange for exchange sake usually do not think about how their opponents can use that desire to work against them.
No one who has ever watched a granmaster like Rubinstein or Capablanca extract a win from a simplified position would ever call it "boring and somewhat purposeless."
It's doubtful that either Capablanca or Rubinstein would equate non-strategic exchanging with simplification or clarification. Almost every exchange made by a master has either an immediate or a long-range purpose. People that exchange for the sake of exchanging probably, and from my experience usually, have no valid strategic plan (purposeless) and the games quickly become uninteresting (boring).
Well yeah, at my rank I am quite certain the players have no idea other than clearing the board. Which makes it very frustrating to lose to this: they drag you down to their level, then 50/50 beat you in the roulette of a 12-pawns endgame.
So far learning some good pawn endgames and making sure to get the king in play before the opponent seem like the best ideas.
I think I see it... things seem pretty equal until 26... Na4 where B can get a material advantage (and make a hole in the pawn line) in a bit, then keeping the K closer to the a pawn than W can get wins the day.
EDIT: still looks a bit of a race to me at the end, but I guess if you know you can win it... : )
Capablanca's opponent in post #14 was no mug.
Vera Menchik was to be Hastings and St.Leonards Chess Club champion (1930) and Women's World champion (1934). Vera remains one of our most famous former members and we still have a large framed photo of her in the playing room.
http://www.hastingschessclub.co.uk/veramenchik.html
Same tournament:

can be a little frustrating so mabe try keep a power piece for the end and swipe pawns as u go. But i wouldve probably offered a draw r resigned
Trading pieces is part of the game... you got to live with it.
Just improve your endgame technique. Capa's games are a great start, then Shereshevsky's "Endgame Strategy" monumental work, and maybe some recent commented games by Aronian, Kramnik and Mamedyarov (probably the strongest endgame magicians currently).
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I've had this happen in a couple of games: a match against an opponent who trades away pieces right from the start. They simply throw away all their pieces in even exchange, very quickly, until we are both depleted to pawns and it turns into a crapshoot for the win.
Any general strategy on how to deal with this?