My class 12 NCERT book says, Pg 226
The body compensates low oxygen availability by increasing red blood cell production, decreasing the binding affinity of haemoglobin and by increasing breathing rate.
Why should the haemoglobin binding capacity decrease at high altitude?
I think it should increase for better oxygen transfer and uptake from air. The concentrtion of oxygen in the atmosphere decreases with height. Hence, if the haemoglobin binding increases, we will be able to draw more oxygen from the air and transport it to the cells.
Answer
Haemoglobin's job is to transport oxygen and not store it. Therefore it should also be able to release oxygen effectively. When the differences in partial pressure of oxygen between the tissues and blood are low then oxygen will not be transported to the tissues from blood, leading to hypoxia.
2,3 Bisphosphoglycerate (2,3-BPG) stabilizes the T- (taut; oxygen unbound) form of haemoglobin thereby reducing its affinity to bind to oxygen. 2,3-BPG is found to be elevated in people living at high altitudes. The production of 2,3-BPG is controlled by a negative feedback (Mulquiney et al., 1999; also see wikipedia) so that it does not overdo its job.
David's answer explains the dynamics in details and addresses the non-linear (sigmoidal) nature of Hb-Oxygen binding which is a critical point for understanding how the effect of 2,3-BPG is actually favourable.
However, haemoglobin content also increases so that higher amounts of oxygen can be captured. This is a fine balance and as you may notice (even if you do not read it to great depth) in the linked paper, there are many mechanisms that are working towards this.
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