Nash Vs Walras Equilibrium (Term Paper Sample)

Nash versus Walras Equilibrium
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Nash versus Walras Equilibrium
Introduction
This research paper sought to carry out a comparative analysis of the relationship, in terms of the similarities and differences, which exists between the Nash equilibrium and the Walras equilibrium. A detailed literature review has been done on these two forms of equilibrium, followed by a detailed independent examination of the nature of each of them, which have helped in conducting a comparative analysis between these two forms of equilibrium. Nash equilibrium is a concept in the game theory in which optimal outcome is associated with the condition that no any given player has an incentive of deviating from his or her strategy after he or she has considered the opponents move or choice. Walras equilibrium, which is also known as Walras law of general equilibrium, on the other hand, is the law of general equilibrium that asserts that when there is an excess demand in a given market, it will be matched with the excess supply in another market. On the contrary, if there is equilibrium in all the other markets, it is often taken by default that there is equilibrium in the market that is being examined.
This paper has been organized into five sections whereby section 1 provides a brief introduction of the study and the definitions of the concepts of the Nash equilibrium and the Walras equilibrium. Section 2 gives literature review of the works that had been carried out by other scholars on the concepts of the Nash equilibrium and the Walras equilibrium. Section 3 gives the analysis of the concept of the Nash and Walras equilibrium with illustrations in each case. Section 4 gives details of the analysis of the comparison in terms of similarities and differences between the Nash equilibrium and the Walras equilibrium, and lastly, section 5 gives the conclusion and the summary of the entire paper.
Literature Review
Nash Equilibrium:
As pointed out in the introduction section, Nash equilibrium can be defined as the concept in the game theory in which optimal outcome is associated with the condition that not any given player has an incentive of deviating from his or her strategy after he or she has considered the opponents move or choice (Hurwicz, 1979). Hurwicz (1979) asserts that the inventor known as John Nash founded the Nash equilibrium principle. This concept has been incorporated in a number of disciplines including ecology and economics. He further argues that testing of the Nash equilibrium involves revealing of the strategies of each of the players to all the players in which the Nash equilibrium will exist if none of the players can deviate from their strategies having learnt of the strategies adopted by their opponents.
According to Dubey and Geanakoplos (2003), Nash equilibrium is one of the fundamental concepts of that game theory that has been most widely applied as a method for the prediction of the strategic interaction’s outcome in the social sciences. They argue that a game is made up of players, a number of pure strategies (actions) that are available to all the players and a payoff or reward function for each of the players (utility). Dubey & Geanakoplos (2003) defined a payoff functions as functions that represent the preferences of the players over a given list of actions (strategies), and this makes a pure strategy Nash equilibrium to be the optimum point or a strategy at which none of the players in the game can deviate unilaterally and receive a higher payoff than the others receive.
In explaining in details the concept of pure strategy Nash equilibrium, Dubey and Geanakoplos (2003) used an example of a first game that involved two players with each of the players having an option of choosing two sets of actions (action A and action B). In this game, some rules of the game were adopted and they include; if the players coordinate well such that both of them choose action A, they both get a payoff of two units each; and if they coordinate such that both of them chose action B, they both earn a payoff of one unit each. However, if they fail to coordinate such that they choose different actions, they both get a payoff of zero units each. In this case, the Nash equilibrium is given by the action profile (B, B) and action profile (A, A) in which any unilateral deviation by any player from the equilibrium action does not give the player that has deviated a comparative advantage over the other player, but rather makes both of them to lose. The players will restrict their actions to the equilibrium point and this scenario represents a pure strategy Nash equilibrium.
Other than pure strategy Nash equilibrium, there exists a mixed strategy Nash equilibrium (Maheswaran & BaÅŸar, 2003). This is a type of Nash equilibrium strategy, in which the players consider the probability distributions of a given set of actions, rather than simply choosing an action in a game, that is, there is randomization of actions. The concept of the mixed strategy Nash equilibrium can be explained by considering an example of a game that consists of two players that have two sets of actions to choose from. Assuming that the set of actions that the two players in this game have is either choosing a tail (T) or a head (H), in which if the first player chooses an action that is different from the action that the second player has chosen, then he earns 2 points from the second player. If he chooses an action that is similar to the action that has been chosen by the second player, then he loses 2 points to the second player. In this case, both the players will not simply choose an action, but will also consider the probability distribution of the sets of action that the opponent is likely to take.
According to Dubey and Geanakoplos (2003), Nash equilibrium can in some cases present outcomes that may be preferred and feasible, but are not efficient to the players in the game. This is because there are some better outcomes that would have otherwise been preferred by both the players. In explaining this concept further, Dubey and Geanakoplos (2003) used the prisoners ‘dilemma in which the players either are to corporate or defect. In prisoners’ dilemma, a defection is unique mutual Nash equilibrium that is worse than cooperation. For instance, consider a prisoners’ dilemma where if both the prisoners agree to have committed an offence, they both serve a jail term of 2 years each, and if they both deny of having committed an offence, they both serve a jail term of one year each. However, if one of the players agree that they committed the offence that they are being prosecuted on while the other player denies, the prisoner who agreed is released while the one who denied the offence serves a jail term of 3 years. In this case, mutual cooperation is a better deal for both the prisoners since the number of years that one will serve as a jail term if he agrees to have committed the offence is less as compared to denial of the offence. This is because when a prisoner agrees that they committed the offence, he can serve a maximum of 2 years or being released, while if he denied to have committed the offence he risks serving a jail term of 3 years if the other prisoner agrees. This implies that in whatever circumstance, cooperation will be the best option for both the prisoners and it represents the pure strategy Nash equilibrium. Cooperation is a pure strategy Nash equilibrium
There has been generalization of the Nash equilibrium to suit situations where there are an incomplete information concerning a game (Maheswaran & BaÅŸar, 2003). This is because there are some cases where mutual rejection is a better strategy for the players involved in a game especially when the game is infinitely repetitive. In a situation where the players are drawn from the same environment such that all the players have, knowledge on the kind of probability distribution that is attached to various actions then there is a Bayesian form of Nash equilibrium. In this case, pure strategy Nash equilibrium will be represented by the deviation of a player to a set of action that he or she feels maximizes his or her utility or payoff.
Walras Equilibrium:
Many economists have described Walrasian general equilibrium as a complex dynamics whose explanation can be perfectly done by the evolution of the game theory (Hurwicz, 1979). It is the law of general equilibrium, which asserts that when there is an excess demand in one given market, it will be matched with the excess supply in another market, and if there is equilibrium in all the other markets, it is taken by default that there is equilibrium in the market that is being examined.
According to Hurwicz (1979), Walras equilibrium came into existence because of the development of a model in a competitive market exchange. This model suggests that Walrasian equilibrium is determined by the set of prices where the firms that exist in an economy maximize their profit and in which the households maximize their utility subject to budget constraint. The budget constraint of the consumers or households is determined by their resource endowments and Walras equilibrium occurs at a point where there is no excess demand or supply of goods and services in the economy.
Walras (2013) observes that Wald (1951) provided a simplified version or model of Walras equilibrium. He also pointed out that Debreu and Arrow (1954), Debreu (1952) and Feiwel (1987) were inspired by the existence of finite games in the Nash equilibrium that was adopted by the John Nash in 1950 to come up with a more general proof for the existence of Walras general equilibrium. He also argues that, central to the studies that were carried out by Hurwicz and Arrow (1960), Hurwirz, Block and Arrow (1959),Uzawal and Nkaido (1960), ...