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Game theoretic approaches, the Nash strategies in particular, have often been criticized as being ineffective in competitive multi-team target assignment problems when compared to random or greedy targeting strategies. In this paper, we attempt to show that this is not the case. Using an attrition model composed of two teams of non-homogeneous fighting units simultaneously targeting each other, we compare the outcomes of various combinations of four targeting strategies used on each side: (1) A random strategy where each unit selects targets randomly, (2) A unit greedy strategy where each unit chooses the target that it is best suited to attack (3) A team optimal strategy where the units coordinate their choice of targets so as to optimize the overall team performance without considering possible adversarial strategies, and (4) A Nash strategy which guarantees that the other team's performance will deteriorate if it does not also use a Nash strategy. We compare the results for all possible combinations of these targeting strategies and show that for each team the Nash strategy outperforms all other strategies no matter what is the strategy employed by the other side.