<p>Hint on the non-existence of mixed strategy NE: </p>

<p>Suppose the row player uses a mixed strategy that plays C with probability <span class="math-container">$p$</span> and DC with probability <span class="math-container">$1-p$</span>. What's the column player's best response to such a strategy? Does the response depend on the value of <span class="math-container">$p$</span>? Based on this, what can you say about the optimality of <em>any</em> mixed strategy that assigns positive probability to C? </p>

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2019-11-12T21:56:20.787Z
2019-11-12T21:56:20.787Z
Herr K.
<p>1 - The normal form does not match the word problem. The profit of the DC firm when the other firm plays C is 1, not 2.</p>

<p>2 - Say <span class="math-container">$p$</span> is the probability given to C by player 1, then in equilibrium, player 2 is indifferent between his two strategies. Hence,</p>

<p><span class="math-container">$p - (1-p) = p$</span> </p>

<p>which only has one solution.</p>

<p>3 - You need to specify what you mean by cooperation. Do you mean cooperate as long as the other cooperates or cooperate no matter what the other one has done in the past? You can look up the grim trigger strategy as a possibility.</p>

<p>4 - Write the extensive form and do backwards induction. You'll find indifference in both sub games. </p>

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2019-11-12T22:52:24.803Z
Sorry, I fixed the wording of the problem. Does the normal form make sense now.
Web Typer
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2019-11-12T22:55:11.623Z
Regarding question 2, do you mean to tell me that there is also a mixed strategy NE in addition to the pure strategy NE of (DC, DC)??
Web Typer
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2019-11-12T22:56:30.927Z
For problem 3, we are cooperating no matter what the other one has done in the past.
Web Typer
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2019-11-12T23:00:22.020Z
2) p = 1 is the only solution (no mixing)
thematblais
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2019-11-12T23:06:00.010Z
3) delta does not matter then and each player is indifferent between the equilibrium outcome and the deviation payoff. This is a odd problem. Usually, these are structured in way to convey the idea of a trade off between an immediate deviation and an infinite stream of equilibrium payoff. In this case, they are the same.
thematblais
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2019-11-12T23:07:04.647Z
So p=1, therefore we have only pure strategies. So is my answer correct about the unique NE of (DC,DC) here?
Web Typer
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2019-11-12T23:09:30.070Z
3) Then maybe I understood wrongly. It is supposed to be greater than 1/2. i.e. $\delta > 1/2$. So basically, both need to be cooperating.
Web Typer
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2019-11-12T23:11:36.853Z
1) is the normal form correct now after fixing the wording of the problem?
Web Typer
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2019-11-12T23:39:37.957Z
2) (DC,DC) is the only Nash since mixing would mean p - (1-p) = 2p, which leads to 0 = -1 (contradiction). 3) $\delta$ does not matter if the players have no "punishment" built into their strategies. If player 2 commits to C for infinity, player 1's best response is to play DC forever regardless of $\delta$.
thematblais
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2019-11-12T22:49:51.977Z
2019-11-12T22:49:51.977Z
thematblais
<p>Firms Alpha and Beta serve the same market. They have constant average costs of <span class="math-container">$2
per unit. The firms can choose either a high price ($</span>10) or a low price (5) for their output.
When both firms set a high price, total demand = 10,000 units which is split evenly between
the two firms. When both set a low price, total demand is 18,000, which is again split evenly.
If one firm sets a low price and the second a high price, the low priced firm sells 15,000 units,
the high priced firm only 2,000 units.
Analyse the pricing decisions of the two firms as a non-co-operative game.
1. In the normal from representation, construct the pay-off matrix, where the elements of
each cell of the matrix are the two firms’ profits.
2. Derive the equilibrium set of strategies.
3. Explain why this is an example of the prisoners’ dilemma game.</p>

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2019-11-12T19:10:13.480Z
If you only want to verify your answers to 2 (the only boldfaced part of your post), please don't include questions 3 and 4 as they are irrelevant. If you want to get help on 3 and 4, please include your own attempt and point out where exactly your difficulty is.
Herr K.
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2019-11-12T20:28:33.933Z
Could you check on 1 and 2? I am currently working on 3 and 4. So any help will be helpful.
Web Typer
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2019-11-12T21:44:26.827Z
"If one cooperates and the other doesn't, then the firm that does not cooperate will earn a profit of **[two?]** million dollars"
Herr K.
CC BY-SA 4.0
2019-11-12T22:51:20.537Z
@HerrK. yes, he firm that does not cooperate will earn a profit of $2 mil
Web Typer
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2019-11-13T05:28:21.440Z
I'm voting to close this question as off-topic because we don't really know what OP intends to ask; this question has been [(re-)edited by OP](https://economics.stackexchange.com/posts/32724/revisions) to reflect at least three distinct game theory problems.
Herr K.
<econometrics>
Nash Equilibrium and SPNE
Web Typer
# Nash Equilibrium and SPNE

(Asked by: username_0 on 2019-11-12)

Firms Alpha and Beta serve the same market. They have constant average costs of $2
per unit. The firms can choose either a high price ($10) or a low price (5) for their output.
When both firms set a high price, total demand = 10,000 units which is split evenly between
the two firms. When both set a low price, total demand is 18,000, which is again split evenly.
If one firm sets a low price and the second a high price, the low priced firm sells 15,000 units,
the high priced firm only 2,000 units.
Analyse the pricing decisions of the two firms as a non-co-operative game.
1. In the normal from representation, construct the pay-off matrix, where the elements of
each cell of the matrix are the two firms’ profits.
2. Derive the equilibrium set of strategies.
3. Explain why this is an example of the prisoners’ dilemma game.

**Tags:** econometrics
> **Comments:**
> **username_1**: If you only want to verify your answers to 2 (the only boldfaced part of your post), please don't include questions 3 and 4 as they are irrelevant. If you want to get help on 3 and 4, please include your own attempt and point out where exactly your difficulty is.  
> **username_0**: Could you check on 1 and 2? I am currently working on 3 and 4. So any help will be helpful.  
> **username_1**: "If one cooperates and the other doesn't, then the firm that does not cooperate will earn a profit of **[two?]** million dollars"  
> **username_0**: @HerrK. yes, he firm that does not cooperate will earn a profit of $2 mil  
> **username_1**: I'm voting to close this question as off-topic because we don't really know what OP intends to ask; this question has been [(re-)edited by OP](https://economics.stackexchange.com/posts/32724/revisions) to reflect at least three distinct game theory problems.  

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## 2 answers

### Answer by username_1 on 2019-11-12. Score: 1

Hint on the non-existence of mixed strategy NE:

Suppose the row player uses a mixed strategy that plays C with probability $p$ and DC with probability $1-p$. What's the column player's best response to such a strategy? Does the response depend on the value of $p$? Based on this, what can you say about the optimality of *any* mixed strategy that assigns positive probability to C?


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### Answer by username_2 on 2019-11-12. Score: 0

1 - The normal form does not match the word problem. The profit of the DC firm when the other firm plays C is 1, not 2.

2 - Say $p$ is the probability given to C by player 1, then in equilibrium, player 2 is indifferent between his two strategies. Hence,

$p - (1-p) = p$

which only has one solution.

3 - You need to specify what you mean by cooperation. Do you mean cooperate as long as the other cooperates or cooperate no matter what the other one has done in the past? You can look up the grim trigger strategy as a possibility.

4 - Write the extensive form and do backwards induction. You'll find indifference in both sub games.

> **Comments:**
> **username_0**: Sorry, I fixed the wording of the problem. Does the normal form make sense now.  
> **username_0**: Regarding question 2, do you mean to tell me that there is also a mixed strategy NE in addition to the pure strategy NE of (DC, DC)??  
> **username_0**: For problem 3, we are cooperating no matter what the other one has done in the past.  
> **username_2**: 2) p = 1 is the only solution (no mixing)  
> **username_2**: 3) delta does not matter then and each player is indifferent between the equilibrium outcome and the deviation payoff. This is a odd problem. Usually, these are structured in way to convey the idea of a trade off between an immediate deviation and an infinite stream of equilibrium payoff. In this case, they are the same.  
> **username_0**: So p=1, therefore we have only pure strategies. So is my answer correct about the unique NE of (DC,DC) here?  
> **username_0**: 3) Then maybe I understood wrongly. It is supposed to be greater than 1/2. i.e. $\delta > 1/2$. So basically, both need to be cooperating.  
> **username_0**: 1) is the normal form correct now after fixing the wording of the problem?  
> **username_2**: 2) (DC,DC) is the only Nash since mixing would mean p - (1-p) = 2p, which leads to 0 = -1 (contradiction). 3) $\delta$ does not matter if the players have no "punishment" built into their strategies. If player 2 commits to C for infinity, player 1's best response is to play DC forever regardless of $\delta$.  

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