Dynamic Bertrand competition when players take turns

Consider the following game:

• There are two players, $i\in\{1,2\}$
• Time is discrete and runs to infinity during periods $t=\{1,2,\ldots\}$
• At eat point in time, players have a price $p_i(t)\in\mathbb{R}_+$
• Initialise the game with $p_1=p_2=p(0)$.
• In odd-numbered periods, player 1 can change his price to any $p_1\in\mathbb{R}_+$. Player 2 cannot change his price.
• In even-numbered periods, player 2 can change his price to any $p_2\in\mathbb{R}_+$. Player 1 cannot change his price.
• For each price, there is a demand $D(p)$, which is a decreasing function. The firm with the lowest price at the end of each period captures the whole demand, receiving payoff $D(p_i)p_i$ for that period. The firm with the highest price receives a payoff of zero for the period. If prices are equal then each firm gets a payoff of $pD(p)/2$.
• Players discount the future at common rate $0\leq\delta\leq1$ so a payoff of $\pi$ that occurs $t$ periods in the future has present value $\pi\delta^t$.
• Write $p^*$ for the monopoly price that maximises $D(p)p$.

The question is: can we identify a complete characterization of the set of Nash equilibria of this game?

Note that a valid strategy is a complete contingent plan, which specifies the choice of action for any history of the game.

This question follows the discussion here: What determines the outcome of a price war, and why isn't that outcome reached instantaneously?

• The word characterization is problematic. Saying that the strategies you are looking for are the strategies that constitute a Nash-equilibrium would be a tautology, but it is a characterization nonetheless. Such a tautology can be made less trivial by giving an alternate definition of Nash-equilibrium. I am not sure this is what you are looking for but you could say something about the range of payoffs: It seems to me that literally any payoff vector between the competitive solution and the cooperative solution is attainable in equilibrium using grim strategies. – Giskard Sep 30 '15 at 19:32
• @densep By characterization I mean a description of every Nash equilibrium. As you note, there are infinitely many equilibria just considering grim trigger strategies, so it is impossible to individually enumerate them all. But those equilibria can be concisely 'characterised' as "play $p$ unless someone ever played a $p'\neq p$ in which case play 0", which should work for $\delta$ large enough. – Ubiquitous Sep 30 '15 at 20:27
• Yes but the grim strategies are just a subset of all possible strategies. And what you describe is actually just a subset of all grim strategies, the condition to go grim can be quite ridiculous: "Play $p$ unless the other players last six prices were not $\pi, \pi, 2, 2, \sqrt{2}, \sqrt{2}$ in which case play 0." This strategy can be part of an equilibrium (given some restrictions on $\delta$). So a characterization of the kind you describe would not be a characterization of all equilibria. – Giskard Oct 1 '15 at 6:44
• @denesp You're right. The more I think about it, the more the premise of the question doesn't make much sense. – Ubiquitous Oct 1 '15 at 16:57
• Is the Maskin-Tirole ECTA 1988 (the second paper, oligopoly) what you are looking for? – Bayesian May 23 '19 at 20:29