I am reading a paper, Kessier & Lulfesmann 2006, and find that its main result totally depends on that the model setting is based on a Rubinstein game with outside option rather than a Nash bargain.
The main difference (as far as I learn from the paper) is that the outside options act as a lower bound to a party’s payoffs in negotiations but does not act as threat points. For example, in Nash Bargain the wage under negotiation is $w_{2}^{*}=\alpha(v_{2}(\cdot)-w^{E}-\pi^{E})$ where $\alpha$ measures the relative bargaining power of the worker and the superscript $E$ means outside option, while in Rubinstein game it is $$w_{2}^{*}=\left\{\begin{array}{ll} \alpha v_{2}(\cdot) & \text { for } \alpha v_{2}(\cdot) \geq w^{E} \\ w^{E} & \text { otherwise. } \end{array}\right.$$
Questions 1. I haven't seen the case of Rubinstein game with outside option before and have no idea why in this case outside options does not form threat options. The paper says "As the literature on non-cooperative bargaining has shown, this property will prevail if a quit (or layoff) effectively terminates the negotiations and forgoes all future gains from cooperation so that the worker cannot credibly threaten to quit in such a situation." I don't understand the statement here.
Questions 2. More importantly, given the distinguished model implication due to the assumption on wage bargaining, I wonder whether Nash bargain or Rubinstein game with outside option is more often used in the literature, and in which empirical case one would be more relevant than the other? (My prior is to use Nash bargain whenever there is a wage bargain as I haven't seen other assumptions used (at least for benchmark model). But I am afraid my reading sample is very limited and thus biased.)
