Nash equilibrium of Cournot’s game with small firms - Mathematics Stack Exchange
Topic 4: Duopoly: Cournot-Nash Equilibrium. We now turn to the situation when there are a small number of firms in the industry and these firms have the option. In game theory, the Nash equilibrium, named after the late mathematician John Forbes Nash Jr. However, Nash's definition of equilibrium is broader than Cournot's. .. Although it would not fit the definition of a competition game, if the game is modified so that the two players win the named amount if they both choose the. Nash equilibrium of Cournot's game with small firms . _) > p _ then Q ∗ is not the total output of the firms in a Nash equilibrium, because in each case at least one firm can deviate and increase its profit. Bertrand competition - pure strategy Nash equilibrium Validation rule - created date DATE.
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In a game theory context stable equilibria now usually refer to Mertens stable equilibria. Occurrence[ edit ] If a game has a unique Nash equilibrium and is played among players under certain conditions, then the NE strategy set will be adopted. Sufficient conditions to guarantee that the Nash equilibrium is played are: The players all will do their utmost to maximize their expected payoff as described by the game.
The players are flawless in execution. The players have sufficient intelligence to deduce the solution. The players know the planned equilibrium strategy of all of the other players. The players believe that a deviation in their own strategy will not cause deviations by any other players. There is common knowledge that all players meet these conditions, including this one. So, not only must each player know the other players meet the conditions, but also they must know that they all know that they meet them, and know that they know that they know that they meet them, and so on.
Where the conditions are not met[ edit ] Examples of game theory problems in which these conditions are not met: The first condition is not met if the game does not correctly describe the quantities a player wishes to maximize. In this case there is no particular reason for that player to adopt an equilibrium strategy.
Intentional or accidental imperfection in execution. For example, a computer capable of flawless logical play facing a second flawless computer will result in equilibrium. Introduction of imperfection will lead to its disruption either through loss to the player who makes the mistake, or through negation of the common knowledge criterion leading to possible victory for the player.
An example would be a player suddenly putting the car into reverse in the game of chickenensuring a no-loss no-win scenario. In many cases, the third condition is not met because, even though the equilibrium must exist, it is unknown due to the complexity of the game, for instance in Chinese chess.
The criterion of common knowledge may not be met even if all players do, in fact, meet all the other criteria.
This is a major consideration in " chicken " or an arms racefor example. Where the conditions are met[ edit ] In his Ph. One interpretation is rationalistic: This idea was formalized by Aumann, R. Brandenburger,Epistemic Conditions for Nash Equilibrium, Econometrica, 63, who interpreted each player's mixed strategy as a conjecture about the behaviour of other players and have shown that if the game and the rationality of players is mutually known and these conjectures are commonly know, then the conjectures must be a Nash equilibrium a common prior assumption is needed for this result in general, but not in the case of two players.
In this case, the conjectures need only be mutually known.
A second interpretation, that Nash referred to by the mass action interpretation, is less demanding on players: What is assumed is that there is a population of participants for each position in the game, which will be played throughout time by participants drawn at random from the different populations.
If there is a stable average frequency with which each pure strategy is employed by the average member of the appropriate population, then this stable average frequency constitutes a mixed strategy Nash equilibrium. For a formal result along these lines, see Kuhn, H. Due to the limited conditions in which NE can actually be observed, they are rarely treated as a guide to day-to-day behaviour, or observed in practice in human negotiations.
However, as a theoretical concept in economics and evolutionary biologythe NE has explanatory power. The payoff in economics is utility or sometimes moneyand in evolutionary biology is gene transmission; both are the fundamental bottom line of survival.How to Solve a Cournot Oligopoly Problem
Researchers who apply games theory in these fields claim that strategies failing to maximize these for whatever reason will be competed out of the market or environment, which are ascribed the ability to test all strategies.
This conclusion is drawn from the " stability " theory above.
In these situations the assumption that the strategy observed is actually a NE has often been borne out by research. The blue equilibrium is not subgame perfect because player two makes a non-credible threat at 2 2 to be unkind U. The Nash equilibrium is a superset of the subgame perfect Nash equilibrium. Suppose that one of our two firms decides to break their collusive arrangement and to act independently, while the other firm chooses to follow that arrangement.
The situation with respect to the deviant firm is presented in Figure 2 below.
The deviant firm's demand curve must pass through the collusive demand curve at the collusive price and quantity equilibrium. As shown in cournot.
Its collusive partner, whose output is assumed to remain unchanged, suffers a reduction in its profits from the previous monopoly level of The deviant firm increases its profits by At this point, it becomes reasonable for the colluding partner firm to also break with the collusive arrangement and produce its most profitable output, given the output of its break-away partner.
At that point, the break-away partner can increase its profits by adjusting its output to the most profitable level given the new level of output of the other firm. The two firms will continue to adjust their outputs in this fashion until neither firm can gain by further adjusting its output. The resulting equilibrium is called the Cournot equilibrium, after Antoine Augustin Cournotand is presented in Figure 3 below which, given our assumption that the two firms are identical, represents the equilibrium of each of them.
To obtain this equilibrium we assume that each firm adjusts its output to maximize its profits, which are equal to 5. R and denoted as MC in the three Figures above. The difference between this Cournot equilibrium and the collusive one is that each firm adjusts its output independently of the other firm's output to maximize its profit, whereas under collusion it adjusts its output in conjunction with an agreed-upon equivalent adjustment of the other firm's output.
Of course, we can not take very seriously the magnitudes of the numbers in the example above.
That example is based solely on the assumptions that the average total cost curve is U-shaped, that the industry demand curve is negatively sloped, and that the two firms are identical. These assumptions are consistent with a wide variety of possible numerical resultsall that is important is the direction of the effects that arise from one or both firms breaking the collusive arrangement. An important tool used to analyse the interaction of firms under conditions where collusion is possible but such arrangements can be easily broken is game theory analysis.
The classic example for the duopoly analysis here is the prisoner's-dilemma game which can be described as follows. Suppose that there are two criminals jointly guilty of a serious crime who have been arrested by the police and are being interviewed separately and simultaneously.