In Bury My Heart at Wounded Knee, why did the Kiowas lose the Buffalo war?
The answer to this question can be found in Chapter 11, which narrates how the Buffalo War started as a result of the way in which the American settlers were destroying the buffalo hordes on which the indigenous Americans depended so greatly for their livelihoods. The first attack that that Kiowas and Comanches make in the Buffalo War is not successful, and the Indians as a result retreat to Palo Duro, therefore breaking the Indian Bureau's command that they remain on their demarcated reservations. The Army sent out troops in response to this, and, because of their superior numbers and military technology, definitively stamp out any furhter resistance. In Palo Duro, for example, they kill 1000 horses and also many Indians throughout the rest of 1874. The conflict is only ended when Lone Wolf and his remaining men surrender at Fort Sill in February of 1875.
The Buffalo War then was lost because of the lack of technology and numbers of the Indians. As with so much of the history of Native Americans, supremacy is shown to be gained by the white settlers, who have more numbers and greater weapons and technology to draw upon.