Ok, here is an interesting idea: Retreats. How do they work?
Let's say you and your opponent start a game. You have an extremely bad hand, so you tell your opponent you would like to retreat. What this does is allows your opponent to salvage two cards from your deck regardless of the outcome of the match. If they win, they get to salvage those cards on top of what they would normally salvage. If they lose, they just get those cards.
The details could be hashed out, such as how many retreats allowed per player per game/match/team, how many cards salvaged for each retreat, and how many turns after the beginning of a game can pass before a player can't retreat from that battle. For instance, if you salvage 6 cards from your opponent via retreating, you win the match.
It fits perfectly into the nature of the event. Retreating is a common strategy in war.
Edit: You could also go a step further with this. During deck building, each deck could be assigned as offensive or defensive. If you choose offensive, you may not retreat, but you salvage an extra card per retreat. If defensive, you may retreat, but only salvage the normal amount per retreat.
Edit2: Just in case it's not clear, retreating causes that game to restart. If you retreat in the second game of your match, you restart that second game.