# Alternating offer bargaining game

In an infinite-period alternating offer bargaining game if we suppose the discount factor is $\delta=1/2$, and the increment is $0.01$, what will be the subgame perfect Nash Equilibria?

It seems that the "proposer proposes $0.66$ and responder accepts" will be a subgame perfect Nash Equilibrium, but are there any other equilibria? How about the case when the proposer offers $0.67$?

Consider the following strategies. In any period a player makes an offer, they offer a share of $0.33$ to the other player, keeping $0.67$ for themselves. In any period a player accepts or rejects an offer, they accept any offer that gives them at least a share of $0.33$ and rejects anything worse.

Clearly, the outcome of these strategies will lead to the immediate acceptance of the offer. All that remains is to check that these strategies form an equilibrium. We do so by appealing to the one-shot deviation principle.

Consider the perspective of a player deciding to accept or reject. They will not want to deviate from the prescribed strategies only if

$$0.33 \ge 0.67 \delta \iff \delta \le \frac{33}{67}$$

However, this violates the assumption that $\delta = \frac{1}{2}$. Thus, the given strategies cannot form an equilibrium.

For characterising other equilibria, see van Damme, Selten and Winter (1990).

• This is a nice answer. I do not upvote it because the question is off-topic, as it is a homework/self-study question with 0 effort shown. (Also see OP's network profile.) Current consensus recommends downvoting the answer. – Giskard Feb 18 '17 at 8:45
• @denesp Fair enough -- I'm not fussed about deleting the answer if need be. However, I'm not sure it's clear this is the type of 'elementary homework' question we're trying to discourage (unlike, say, this one). The alternating-offer bargaining model is far from elementary. I also made sure to leave out enough detail in my answer for the OP to work for it. :) – Theoretical Economist Feb 18 '17 at 20:01
• @TheoreticalEconomist bear in mind that you can go on to meta if you think we need another discussion about where the "homework" line gets drawn. These policies are made by community consensus. – Ubiquitous Feb 21 '17 at 16:52
• @Ubiquitous Thanks for the suggestion -- it may be worth discussing. However, I'm not sure if it actually is (worth a discussion, that is). Let me think about it. In the meantime, I'm sure anyone else who is more certain of it than I am can start their own post on the meta. – Theoretical Economist Feb 21 '17 at 17:07