Typically, we consider three perspectives:
1. how quick you can make a good decision
2. how good that you can collaborate with other teams and leverage each individual.
3. Do this person take the leadership
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Sometimes counter offer is just a procedure established by the employer, it doesn't have to mean that they want to keep you, they just follow the procedures.
Our classes and the discussion inputs from our classmates really demystify a lot of topics which we (currently or will) encounter in our careers.
If you can, try to make your HR compensation analyst your best friend. 
make friends with your FP&A, your finance colleagues. They know everything.
Academia is quite different from industry. I am working in a research institute. Postdoc salary is much lower compared with similar positions in the industry. That being said, I know quite little for the industry side.
I agree with no ask no raise. I had a previous coworker who pushed her manager really hard for promotion and she did perform well. Then she get promoted faster than other people who are at same level of performance.
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