Nash equilibrium - mistake in proof of paper?
The objective is to show that, as long as $f'(n\hat\theta)\ne \alpha$, a firm can always engineer a package $(p',\hat\theta')=(p+\Delta p,\hat\theta+\Delta \hat{\theta})$ such that (i) a caring consumer strictly prefers this package, and (ii) the firm makes positive profits. 

1) Your maths are correct: the inequalities are always false when f is increasing and strictly concave. In that case, the proof in the paper is unclear.
The package $(p',\hat\theta')=(p+f'(n\hat\theta)\Delta \hat{\theta},\hat\theta+\Delta \hat{\theta})$ does not satisfy (i).

2) You can show the proposition with the following reasoning.
Assume $f'(n\hat\theta)&gt;\alpha$. Take any $h$ within the interval $(\alpha,f'(n\hat\theta))$.
For $\Delta \hat{\theta}$ small enough and positive, a firm that offers the package $(p',\hat\theta')=(p+h\Delta \hat{\theta},\hat\theta+\Delta \hat{\theta})$ satisfies (i) and (ii). With a first-order appoximation, we show (i): $$b-p'+f(\hat\theta'+(n-1)\hat\theta)&gt;b-p+f(n \hat\theta),$$
and (ii):
$$\Delta p - \alpha \Delta \theta&gt;0.$$

Assume $f'(n\hat\theta)&lt;\alpha$, and take $h$ within the interval $(f'(n\hat\theta),\alpha)$.
For $\Delta \hat{\theta}$ small enough and negative, a firm that offers the package $(p',\hat\theta')=(p+h\Delta \hat{\theta},\hat\theta+\Delta \hat{\theta})$ satisfies (i) and (ii).