# Nash Equilibrium

Note: Please I need a thorough explanation.

1. Explain the meaning of a Nash Equilibrium when firms are competing with respect to price. Why is the equilibrium stable? Why don't the firms raise prices to the level that maximizes joint profits? Also discuss and critique what strategies firms could use to attempt to maximize profits and discuss problems associated with such strategies given the economic and legal environment.

2. Using well explained graphical and verbal analysis, describe the market structure situation confronting professional sports (baseball, football, etc.) labor negotiations and whether you could predict the outcome of those negotiations.

3. Using well developed and thoroughly explained graphical and mathematical relationships, and verbal analysis, demonstrate that a general equilibrium can be achieved under conditions of perfect competition and that under perfect competition economic welfare is maximized. Conclude your analysis by explaining how monopoly would violate the maximization of economic welfare conditions and discuss how far you believe policy should go to try to avoid monopoly and aim for perfect competition. In the answer, discuss whether or not perfect competition is desirable, even though it has the potential for attaining welfare maximization.

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#### Solution Preview

If there is a set of strategies with the property that no player can benefit by changing her strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium.

Following a long tradition in economics, we will think of two companies selling "widgets" at a price of one, two, or three dollars per widget. the payoffs are profits -- after allowing for costs of all kinds -- The general idea behind the example is that the company that charges a lower price will get more customers and thus, within limits, more profits than the high-price competitor. (This example follows one by Warren Nutter).

Table 3

Acme Widgets

price = $1 price = $2 price = $3

Wiley Widgets price = $1 0,0 50,-10 40,-20

price = $2 -10,50 20,20 90,10

price = $3 -20,40 10,90 50,50

Industry profits in this example depend on the price and thus on the strategies chosen by the rivals. Profits may add up to 100, 20, 40, or zero, depending on the strategies that the two competitors choose. We can also see fairly easily that there is no dominant strategy equilibrium. Widgeon company can reason as follows: if Acme were to choose a price of 3, then Widgeon's best price is 2, but otherwise Widgeon's best price is 1 -- neither is dominant.

Let's apply that definition to the widget-selling game. First, for example, we can see that the strategy pair p=3 for each player (bottom right) is not a Nash-equilibrium. From that pair, each competitor can benefit by cutting price, if the other player keeps her strategy unchanged. Or consider the bottom middle -- Widgeon charges $3 but Acme charges $2. From that pair, Widgeon benefits by cutting to $1. In this way, we can eliminate any strategy pair except the upper left, at which both competitors charge $1.

We see that the Nash equilibrium in the widget-selling game is a low-price, zero-profit equilibrium. Many economists believe that result is descriptive of real, highly competitive markets -- although there is, of course, a great deal about this example that is still "unrealistic."

In fact, any dominant strategy equilibrium is also a Nash Equilibrium. The Nash equilibrium is an extension of the dominant strategy equilibrium.

Consider a duopoly, with each of two firms choosing a strategy. The strategy pair chosen is a Nash equilibrium if firm A's choice maximizes its profits, given firm B's choice and firm 2 maximizes its profits given firm 1's choice.

In non-cooperative oligopoly theory it is necessary to model the manner in which firms choose strategies, given the fact that their decisions will affect their rivals.

The most common assumption is that each firm chooses its strategy so as to maximize profits, given the profit-maximizing decisions of other firms.

In most instances the definition is derived from statistical standards developed by international organisations such as the IMF, OECD, Eurostat, ILO.

Strategies refer to the decisions firms make. Strategies may involve quantities, prices, or any other relevant decisions (such as R&D, investment, or location). The choice will depend on the nature of the problem.

When the strategy analyzed involves quantities, the resulting equilibrium is termed a Cournot (Nash) equilibrium. When the strategy involves prices, it is called a Bertrand (Nash) equilibrium.

2. Using well explained graphical and verbal analysis, describe the market structure situation confronting professional sports (baseball, football, etc.) labor negotiations and whether you could predict the outcome of those negotiations.

Labor negotiations:

The Nash bargaining game

A two-player game where two players attempt to divide a good

Each player requests an amount of the cake.

If their requests are compatible, each player receives the amount requested

If not, each player receives nothing.

Assume the utility function for each player to be a linear function in what they get.(utility function: A mathematical representation of a player's preferences for the outcomes of the game. John von Neumann and Oskar Morgenstern proved that if a player's preferences over a set of outcomes satisfy a relatively small set of axioms, it is possible to define a function f from the set of outcomes into the real numbers such that the player prefers A to B if and only if f(A) > f(B). One should not speak of "the" utility function of a player since utility functions are only determined up to multiplication by a nonnegative constant and a constant term. I.e., if f is a utility function for a player, then so is g = af + b, where a and b are real numbers and a > 0.)

An infinite number of Nash equilibria exist for this game. ...

#### Solution Summary

Define the meaning of a Nash Equilibrium when firms are competing with respect to price.

Dominant Strategy Equilibrium and Nash Equilibrium

Suppose two firms, Firm Y, Inc., and Firm X, Inc., are locked in a bitter pricing struggle in the bottled water industry. In strategy A, pricing payoff matrix, Firm Y can choose a given row of outcomes by offering A price ("up") or B price ("down"). Firm X can choose a given column of outcomes by choosing to offer A price ("left") or B price ("right"). Neither firm can choose which cell of the payoff matrix to obtain; the payoff for each firm depends upon the pricing strategies of both firms.

Please see the attached document for the table with data and then consider:

a. Is there a dominant strategy equilibrium in this problem? If so, what is it? Please provide a detail explanation.

b. Is there a Nash equilibrium in this problem? If so, what is it? Please provide a detail explanation.