Is this a Subgame-perfect Nash Equilibrium of the Two-round Rubinstein Ultimatum game

Question

Suppose a firm and a employee bargain over the distribution of $\pi$ units of money into $\omega$(the employee's wage) and $\pi - \omega$ (the firm's profit) in the following 2-round game:

1. The Firm proposes a distribution $(\omega, \pi - \omega)$. If the employee accepts the game ends and $\pi$ is distributed accordingly. If the employee rejects the offer

2. the firm proposes a distribution again. If the employee accepts the game ends and $\pi$ is distributed accordingly. If the employee rejects the second offer both sides get zero.

Both the firm and the employee have the same discounting factor $\delta$.

At the lecture yesterday my professor insisted that in all subgame-prefect Nash equilibria of this game the employee gets a wage $\omega = 0$. I attempted to object but he cut me off.

Let's represent by $s^F = (s_1^F, s_2^F)$ the firm's wage offers in periods 1 and 2 respectively and by $s^E = (s_1^E(s_1^F), s_2^E(s_2^F))$ the employee's response functions in both periods.

I think that the following strategy profile $(s^F,s^E)$ is a SPNE: $s^F = ((1-\delta)\pi, 0)$, $s^E = (s_1^E(s_1^F), s_2^E(s_2^F))$ with

$s_1^E(s_1^F) = \begin{cases} \mathit{accept} & \text{if $s_1^F \geq (1-\delta)\pi$}\\ \mathit{reject} & \text{if $s_1^F < (1-\delta)\pi$}. \end{cases}$

$s_2^E(s_2^F) = \mathit{accept} \hspace{.2cm} \forall s_2^F \in [0, \pi].$

I found this using backward induction and the observation that a pay-off of $\pi$ in period two is worth only $\delta \times \pi$ to the firm in period 1. Therefore $(\omega, \pi - \omega) = (1-\delta)\pi, \delta \pi)$ after period one and $(\omega, \pi - \omega) = (0, \pi)$ after period two give the firm the same utility. It certainly is a Nash equilibrium - there are no profitable unilateral deviations. But I don't see why it is not subgame-perfect.

Answer 1

This strategy profile is not a subgame-perfect Nash equilibrium because the employee’s threat to reject an offer less than $(1-\delta)\pi$ in the first period is not credible. The employee knows that she will get $0$ if she follows through on this threat, and thus profits by deviating unilaterally and accepting a lower offer. A subgame-perfect Nash equilibrium must be a Nash equilibrium for all subgames, including the branches that would not actually be taken in applying the strategy profile. Otherwise the concept wouldn’t serve the purpose of eliminating non-credible threats.