Why does the affinity of haemoglobin for oxygen decrease at high altitudes?
The answer to this question is yes, a decrease in oxygen affinity will decrease the oxygen taken up by the haemoglobin (Hb), but it is an appropriate response because it will have a greater effect in increasing the release of oxygen to the tissues. 

This is not intrinsically obvious. It happens because of the (sigmoid) shape of the oxygen binding curve, and can only really be appreciated if you examine the curves for the normal situation and that in which there is increased 2,3 Bisphosphoglycerate (2,3-BPG) producing the change in oxygen affinity. I have devised a figure to demonstrate this, although it is not based on actual data, so should only be regarded as illustratative.




At sea level the lungs pick up oxygen at 1 (say 100% saturation of
Hb) and when the oxygen pressure drops to 40 mm Hg in the tissues (2)
the Hb will be 55% saturated. Hence in this example they have
released 45% of a full complement of oxygen.
If we look at the same curve for high altitude, in this example the oxygen pressure at the lungs (1') will be such that the Hb is only 80% saturated. Thus at 40 mm Hg in the tissues (2) when Hb is only 55% saturated it will only have released and made available 25% of a full complement of oxygen.
Now consider the effect of high altitude acclimatization with increased 2,3-BPG production changing the equilibrium and the oxygen binding curve (red). At the lungs (3) the Hb will be less charged with oxygen — only 70% saturation — but at 40mm Hg in the tissues (4) it will be much less saturated than on the black curve — 30%. Thus it will have made available 40% of a full complement of oxygen.


This is not a perfect solution, which is why over a longer term there is increased production of red blood cells to provide more haemoglobin to compensate for the smaller amount of oxygen it can bind.