<p>Consider the following game:</p>

<ul>
<li>There are two players, $i\in\{1,2\}$</li>
<li>Time is discrete and runs to infinity during periods $t=\{1,2,\ldots\}$</li>
<li>At eat point in time, players have a price $p_i(t)\in\mathbb{R}_+$</li>
<li>Initialise the game with $p_1=p_2=p(0)$.</li>
<li>In odd-numbered periods, player 1 can change his price to any $p_1\in\mathbb{R}_+$. Player 2 cannot change his price.</li>
<li>In even-numbered periods, player 2 can change his price to any $p_2\in\mathbb{R}_+$. Player 1 cannot change his price.</li>
<li>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 $pD(p)/2$.</li>
<li>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 $\pi\delta^t$.</li>
<li>Write $p^*$ for the monopoly price that maximises $D(p)p$.</li>
</ul>

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

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

<p>This question follows the discussion here: <a href="https://economics.stackexchange.com/questions/8398/what-determines-the-outcome-of-a-price-war-and-why-isnt-that-outcome-reached-i?noredirect=1#comment10737_8398">What determines the outcome of a price war, and why isn&#39;t that outcome reached instantaneously?</a></p>

CC BY-SA 3.0
2015-09-30T19:32:49.717Z
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
CC BY-SA 3.0
2015-09-30T20:27:22.947Z
@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
CC BY-SA 3.0
2015-10-01T06:44:21.097Z
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
CC BY-SA 3.0
2015-10-01T16:57:36.710Z
@denesp You're right. The more I think about it, the more the premise of the question doesn't make much sense.
Ubiquitous
CC BY-SA 4.0
2019-05-23T20:29:37.303Z
Is the Maskin-Tirole ECTA 1988 (the second paper, oligopoly) what you are looking for?
Bayesian
<game-theory><oligopoly><repeated-games>
Dynamic Bertrand competition when players take turns
Ubiquitous
# Dynamic Bertrand competition when players take turns

(Asked by: username_0 on 2015-09-30)

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?](https://economics.stackexchange.com/questions/8398/what-determines-the-outcome-of-a-price-war-and-why-isnt-that-outcome-reached-i?noredirect=1#comment10737_8398)

**Tags:** game-theory, oligopoly, repeated-games

> **Comments:**
> **username_1**: 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.  
> **username_0**: @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.  
> **username_1**: 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.  
> **username_0**: @denesp You're right. The more I think about it, the more the premise of the question doesn't make much sense.  
> **username_2**: Is the Maskin-Tirole ECTA 1988 (the second paper, oligopoly) what you are looking for?  

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## 0 answers

