Collusion and Nash equilibrium question. Please help

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RetaliationDonut
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#1
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In tacit collusion, when a firm deviates, this mean it cheats on the other firm by producing more than the agreed quantity. This is always profitable in the short run as they get more profit than the colluded profit. However in the long run, the other firm will notice this deviation and retaliate to punish them. After that, both firms will produce a quantity with a smaller profit than the colluded profit, which they could've receive if one didn't retaliate in the first place. As a result, the possibility of retaliation will prevent the firm from deviating (cheating) from the first place.

My question next is

1) If no firm wishes to deviate, collusion will result a Nash Equilibrium of the game.

2) If ALL firm wishes to deviate, collusion will result a Nash Equilibrium of the game.

Which one is correct and can you explain why?


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TeeEff
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(Original post by RetaliationDonut)
In tacit collusion, when a firm deviates, this mean it cheats on the other firm by producing more than the agreed quantity. This is always profitable in the short run as they get more profit than the colluded profit. However in the long run, the other firm will notice this deviation and retaliate to punish them. After that, both firms will produce a quantity with a smaller profit than the colluded profit, which they could've receive if one didn't retaliate in the first place. As a result, the possibility of retaliation will prevent the firm from deviating (cheating) from the first place.

My question next is

1) If no firm wishes to deviate, collusion will result a Nash Equilibrium of the game.

2) If ALL firm wishes to deviate, collusion will result a Nash Equilibrium of the game.

Which one is correct and can you explain why?


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The first one, as a Nash equilibrium by definition exists when neither side can benefit from changing strategies in a game. If there is no benefit in doing so, neither side will wish to deviate.
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GTHargs
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(Original post by The Financier)
The first one, as a Nash equilibrium by definition exists when neither side can benefit from changing strategies in a game. If there is no benefit in doing so, neither side will wish to deviate.
Either, because the reason that cartels are inherently unstable is because it is NOT a Nash equilibrium. It can be though if there is a credible threat to punish defectors, however this makes it more complex and its hard to show in a single payoff matrix- its if the game has multiple rounds and involves lots of maths. For A level you assume that a collusive arrangement where there is short term gain from cheating is not a nash equilibrium as there is always an incentive to defect. However, you can evaluate taking about multi-round games and credible threats (e.g. OPEC - when smaller countries were cheating the big boys decided to punish them by flooding the market to push oil prices down to below their breakeven points).
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TeeEff
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(Original post by GTHargs)
Either, because the reason that cartels are inherently unstable is because it is NOT a Nash equilibrium. It can be though if there is a credible threat to punish defectors, however this makes it more complex and its hard to show in a single payoff matrix- its if the game has multiple rounds and involves lots of maths. For A level you assume that a collusive arrangement where there is short term gain from cheating is not a nash equilibrium as there is always an incentive to defect. However, you can evaluate taking about multi-round games and credible threats (e.g. OPEC - when smaller countries were cheating the big boys decided to punish them by flooding the market to push oil prices down to below their breakeven points).
I do not see how this answers the question (which is a little poorly worded). A key part of the Nash Equilibrium definition is that the opponent's strategy is constant. If they are able to change their strategy (e.g. by reorganising collusion) the condition no longer applies as it becomes a cooperative game, which the Nash Equilibrium is not a relevant concept in. If all firms wish to deviate, then there is clearly a benefit to unilateral deviation for each player hence it isn't a Nash Equilibrium condition.
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GTHargs
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(Original post by The Financier)
I do not see how this answers the question (which is a little poorly worded). A key part of the Nash Equilibrium definition is that the opponent's strategy is constant. If they are able to change their strategy (e.g. by reorganising collusion) the condition no longer applies as it becomes a cooperative game, which the Nash Equilibrium is not a relevant concept in. If all firms wish to deviate, then there is clearly a benefit to unilateral deviation for each player hence it isn't a Nash Equilibrium condition.
If no firms wish to deviate then collusion is not an equilibrium as there is an incentive for firms to expand their output in order to gain additional supernormal profit.

If all firms wish to deviate then it isn't collusion.
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TeeEff
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(Original post by GTHargs)
If no firms wish to deviate then collusion is not an equilibrium as there is an incentive for firms to expand their output in order to gain additional supernormal profit.

If all firms wish to deviate then it isn't collusion.
I think it's important to state here that I'm interpreting the question as being grammatically incorrect, in that I believe the actual options are:

1) If no firm wishes to deviate, the state represents a nash equilibrium in the game
2) If all firms wish to deviate, the state represents a nash equilibrium in the game

If interpreting the question strictly, collusion (that is, players can coordinate their actions) in either case is not a valid Nash Equilibrium. Again, it's a model for a non-cooperative game so would invalidate either answer. The whole point of NE is that you are not reliant on the benevolent goodwill of a player. It's the best strategy you have given how others will act.
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GTHargs
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(Original post by The Financier)
I think it's important to state here that I'm interpreting the question as being grammatically incorrect, in that I believe the actual options are:

1) If no firm wishes to deviate, the state represents a nash equilibrium in the game
2) If all firms wish to deviate, the state represents a nash equilibrium in the game

If interpreting the question strictly, collusion (that is, players can coordinate their actions) in either case is not a valid Nash Equilibrium. Again, it's a model for a non-cooperative game so would invalidate either answer.
Precisely, in the short-term collusion is never a Nash equilibrium. However, what the question is trying to ask is, if firms think that cheating will cause them to lose out in long run as they are punished by rivals does this create a short term equilibrium? And in response to that question, I said that it is beyond the scope of the course and is very complex.
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