 We present bounds on the number of buyers that, when added, cause BTR in the augmented market to achieve welfare at least as high as the optimal in the original market.
<clarity>  We present bounds on the number of buyers that, when added, cause BTR in the augmented market to achieve welfare at least as high as the optimal in the original market.
 We present bounds on the number of buyers that, when added, cause BTR in the augmented market have welfare at least as high as the optimal in the original market.