EQUILIBRE COURNOT PDFJune 24, 2020
Then we show, in the same example, that the Cournot-Walras equilibrium converges by replication to the Walras equilibrium. [fre] Equilibres de Cournot- Wakas. non coopdratif resultant de l’echange est appele un equilibre de Cournot. Il introduire le concept d’equilibre de Cournot-Walras dans le cadre d’un modele. f ‘Sur l’equilibre et le mouvement d’une lame solide’ and Addition’, Em, 3, = W, (2)8, [C: Cournot c.] g ‘ ‘Cauchy, pere’, in.
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In terms of game theory, if each player has chosen a strategy, and no player can benefit by changing strategies while the other players keep theirs unchanged, then the current set of strategy choices and their corresponding payoffs constitutes a Nash equilibrium. Stated simply, Alice and Bob are in Nash equilibrium if Alice is making the best decision she can, taking into account Bob’s decision while Bob’s decision remains unchanged, and Bob is making the best decision he can, taking into account Alice’s decision while Alice’s decision remains unchanged.
Likewise, a group of players are in Nash equilibrium if each one is making the best decision possible, taking into account the decisions of the others in the game as long as the other parties’ decisions remain unchanged.
Nash showed that there is a Nash equilibrium for every finite game: Game theorists use the Nash equilibrium concept to analyze the outcome of the strategic interaction of several decision makers.
In other words, it provides a way of predicting what will happen if several people or several institutions are making decisions at the same time, and if the outcome for each of them depends on the decisions of the others.
The simple insight underlying John Nash’s idea is that one cannot predict the result of the choices of multiple decision makers if one analyzes those decisions in isolation. Instead, one must ask what each player would do, taking into account the decision-making of the others. Nash equilibrium has been used to analyze hostile situations like war and arms races  see prisoner’s dilemmaand also how conflict may be mitigated by repeated interaction see tit-for-tat.
It has also been used to study to what extent people with different preferences can cooperate see battle of the sexesand whether they will take risks to achieve a cooperative outcome see stag hunt.
It has been used to study the adoption of technical standards[ citation needed ] and also the occurrence of bank runs and currency crises see coordination game. Other applications include traffic flow see Wardrop’s principlehow to organize auctions see auction theorythe outcome of efforts exerted by multiple parties in the education process,  regulatory legislation such as environmental regulations see tragedy of the Commons natural resource management,  analysing strategies in marketing,  and even penalty kicks in football see matching pennies.
A version of the Nash equilibrium concept was first known to be used in by Antoine Augustin Cournot in his theory of oligopoly. However, the best output for one firm depends on the outputs of others.
A Cournot equilibrium occurs when each firm’s output maximizes its profits given the output of the other firms, which is a pure-strategy Nash equilibrium. Cournot also introduced the concept of best response dynamics in his analysis of the stability of equilibrium.
However, Nash’s definition of equilibrium is broader than Cournot’s. It is also broader than the definition of a Pareto-efficient equilibrium, since the Nash definition makes no judgements about the optimality of the equilibrium being generated.
The modern game-theoretic concept of Nash equilibrium is instead defined in terms of mixed strategieswhere players choose a probability distribution over possible actions. However, their analysis was restricted to the special case of zero-sum games.
They showed that a mixed-strategy Nash equilibrium will exist for any zero-sum game with a finite set of actions.
The key to Nash’s ability to prove existence far more generally than von Neumann lay in his definition of equilibrium. According to Nash, “an equilibrium point is an n-tuple such that each player’s mixed strategy maximizes his payoff if the equiljbre of the others are held fixed.
Thus each player’s strategy is optimal against those of the others. Since the development of the Nash equilibrium concept, game theorists have discovered that it makes misleading predictions or fails to make a unique prediction in certain circumstances.
They have proposed many related solution concepts also called ‘refinements’ of Nash equilibria designed to overcome perceived flaws in the Nash concept. One particularly important issue is that some Nash equilibria may be based on threats that are not ‘ credible ‘.
In Reinhard Selten proposed subgame perfect equilibrium as a refinement that eliminates equilibria which depend on non-credible threats. Other extensions of the Nash equilibrium concept have addressed what happens if a game is repeatedocurnot what happens if a game is played in the absence of complete information. However, subsequent refinements and extensions of the Nash equilibrium concept share the main insight on which Nash’s concept rests: Informally, a strategy profile equilibree a Nash equilibrium if no player can do better by unilaterally changing his or equuilibre strategy.
To see what this means, imagine that each player is told the strategies of the others. Suppose then that ciurnot player asks themselves: If any player could answer “Yes”, then that set of strategies is not a Nash equilibrf.
But if every player prefers not to switch cohrnot is indifferent between switching and not then the strategy profile is a Nash equilibrium. Thus, each strategy in a Nash equilibrium is a best response to all other strategies in that equilibrium. The Nash equilibrium may sometimes appear non-rational in a third-person perspective. This is because a Nash equilibrium is not necessarily Pareto optimal.
The Nash equilibrium may also have non-rational consequences in sequential games because players may “threaten” each other with non-rational moves. For such games the subgame perfect Nash equilibrium may be more meaningful as a tool of analysis. Note that the payoff depends on the strategy profile chosen, i.
A game can have a pure-strategy or a mixed-strategy Nash equilibrium.
Nash equilibrium – Wikipedia
In the latter a pure strategy is chosen stochastically with a fixed probability. Nash proves that if we allow mixed strategiesthen every game with a finite number of players in which each player can choose from finitely many pure strategies has at least one Nash equilibrium. Nash equilibrium need not exist if the set of choices is infinite and noncompact.
An example is when two players simultaneously name a natural number with the player naming the larger number wins. However, Nash equilibrium exists if the set of choices is compact with continuous payoff. The coordination game is a classic symmetric two player, two strategy game, with an example payoff matrix shown to the right. The players should thus coordinate, both adopting strategy A, to receive the highest payoff; i.
If both players chose strategy B though, there is still a Nash equilibrium. Although each player is awarded less than optimal payoff, neither player has incentive to change strategy due to a reduction in the immediate payoff from 2 to 1. A famous example of this type of game was called the stag hunt ; in the game two players may choose to hunt a stag or a rabbit, the former providing more meat 4 utility units than the latter 1 utility unit.
The caveat is that the stag must be cooperatively hunted, so if one player attempts to hunt the stag, while the other hunts the rabbit, he will fail in hunting 0 utility unitswhereas if they both hunt it they will split the payload 2, 2. The game hence exhibits two equilibria at stag, stag and rabbit, rabbit and hence the players’ optimal strategy depend on their expectation on what the other player may do. If one hunter trusts that the other will hunt the stag, they should hunt the stag; however if they suspect that the other will hunt the rabbit, they should hunt the rabbit.
This game was used as an analogy for social cooperation, since much of the benefit that people gain in society depends upon people cooperating and implicitly trusting one another to act in a manner corresponding with cooperation. Another example of a coordination game is the setting where two technologies are available to two firms with comparable products, and they have to elect a strategy to become the market standard. If both firms agree on the chosen technology, high sales are expected for both firms.
If the firms do not agree on the standard technology, few sales result. Both strategies are Nash equilibria of the game. Driving on a road against an oncoming car, and having to choose either to swerve on the left or to swerve on the right of the road, is also a coordination game. For example, with payoffs 10 meaning no crash and 0 meaning a crash, the coordination game can be defined with the following payoff matrix:.
In this case there are two pure-strategy Nash equilibria, when both choose to either drive on the left or on the right. If we admit mixed strategies where a pure strategy is chosen at random, subject to some fixed probabilitythen there are three Nash equilibria for the same case: Imagine two prisoners held in separate cells, interrogated simultaneously, and offered deals lighter jail sentences for betraying their fellow criminal.
They can “cooperate” with the other prisoner by not snitching, or “defect” by betraying the other. However, there is a catch; if both players defect, then they both serve a longer sentence than if neither said anything. Lower jail sentences are interpreted as higher payoffs shown in the table.
The prisoner’s dilemma has a similar matrix as depicted for the coordination game, but the maximum reward for each player in this case, a minimum loss of 0 is obtained only when the players’ decisions are different. Each player improves their own situation by switching from “cooperating” to “defecting”, given knowledge that the other player’s best decision is to “defect”.
The prisoner’s dilemma thus has a single Nash equilibrium: What has long made this an interesting case to study is the fact that this scenario is globally inferior to “both cooperating”.
That is, both players would be better off if they both chose to “cooperate” instead of both choosing to defect. However, each player could improve their own situation by breaking the mutual cooperation, no matter how the other player possibly or certainly changes their decision. An application of Nash equilibria is in determining the expected flow of traffic in a network.
Consider the graph on the right. If we assume that there are x “cars” traveling from A to D, what is the expected distribution of traffic in the network?
The “payoff” of each strategy is the travel time of each route. Thus, payoffs for any given strategy depend on the choices of the other players, as is usual. However, the goal, in this case, is to minimize travel time, not maximize it. Equilibrium will occur when the time on all paths is exactly the same. When that happens, no single driver has any incentive to switch routes, since it can only add to esuilibre travel time.
Every driver now has a total travel time of 3. Notice that this distribution is not, actually, socially optimal. If the cars agreed that coufnot travel via ABD and the other 50 equuilibre ACDthen travel time for any single car would actually be 3. This is also the Nash equilibrium if the path between B and C is removed, which means that adding another possible route can decrease the efficiency of the system, a phenomenon known as Braess’s paradox.
This can be illustrated by a two-player game in which both players simultaneously choose an integer from 0 to 3 and they both win the smaller of the two numbers in points. In addition, if one player chooses a larger number than the other, then they have to give equilbire two points to the other.
This game has a unique pure-strategy Nash equilibrium: Any other strategy can be improved by a player switching their number to one less than that of the other player. In the adjacent table, if the game begins at the green square, it is in player 1’s interest to move to the purple square and it is in player 2’s interest to move to the blue square.
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 same number, and otherwise win nothing, then there are 4 Nash equilibria: There is an easy numerical way to identify Nash equilibria on a payoff matrix.
It is especially helpful in two-person games where players have more than two strategies. In this case formal analysis may become too long. This vournot does not apply to the case where mixed stochastic strategies are of interest. The rule goes as follows: Indeed, for cell B,A 40 is the maximum of the first column and 25 is the maximum of the second row. For A,B 25 is the equilibge of the second column and 40 is the maximum of the first row.