Question

Suppose voters based their decisions on the ratio of utilities received from two candidates that is, Equation 25.6 would be Show that the results from a game involving net value platforms would in this case maximize the Nash Social Welfare function \\[ s w=Y l u_{l} \\]

Short answer

## Short Answer The game involving net value platforms can maximize the Nash Social Welfare function under specific assumptions. By introducing the concept of net value platforms, which represents the ratio of utilities received from two candidates, we can express the Nash Social Welfare function and take its second derivative with respect to voter preferences. If the second derivative is non-positive, then the function is concave and can be maximized. Under the given scenario, we assume that the utility function satisfies these requirements, and thus the game maximizes the Nash Social Welfare function.

Step-by-step solution

Understanding the Nash Social Welfare Function

The Nash Social Welfare function is defined as the product of individuals' utilities, as given by:\\[ SW = \prod_{i=1}^n u_i \\]In our particular case, we have two candidates: `Y` and `l`. The utility of candidate `Y` is denoted as `u_Y`, and the utility of candidate `l` is denoted as `u_l`.

Introducing Net Value Platforms

The net value platform refers to the difference in utility gained by voters when voting for one candidate over another. This difference allows voters to form a preference for one candidate over another. The net value platform, denoted as `v`, is given by the ratio of the utilities provided by the two candidates: \\[ v = \frac{u_Y}{u_l} \\]

Expressing the Nash Social Welfare Function in terms of Net Value Platforms

Now let's rewrite the Nash Social Welfare function using the net value platform concept. In this case, we have: \\[ SW = Ylu_l \\]According to the exercise, we've been given the equation as: \\[ s w=Y l u_{l} \\] This is the same equation as our derived Nash Social Welfare function.

Proving the game maximizes the Nash Social Welfare Function

To prove that the game involving net value platforms maximizes the Nash Social Welfare function, we need to demonstrate that the function is concave and the maximum point can be achieved. Taking the natural logarithm of the Nash Social Welfare function, we get: \\[ \ln SW = \ln (Ylu_l) = \ln(Y) + \ln(l) + \ln(u_l) \\] Now, we can take the second derivative with respect to `l`: \\[ \frac{\partial^2 \ln SW}{\partial l^2} = -\frac{1}{l^2} - \frac{\partial^2}{\partial l^2} \ln(u_l) \\] If the second derivative is negative, then the function is concave, and we can find the maximum point. Since `l` is positive, the first term \\(-\frac{1}{l^2}\\) is negative. For the solution to hold, we require that the second term, \\(- \frac{\partial^2}{\partial l^2} \ln(u_l)\\), is non-positive. This would depend on the specific utility function. Given the constraints of the exercise, we may assume that the utility function is such that the second derivative with respect to `l` is non-positive. Thus, the game involving net value platforms can maximize the Nash Social Welfare function under the given assumptions.

Key concepts

Understanding Microeconomic Theory

Microeconomic theory provides an intricate framework to examine how individuals and firms make decisions to allocate scarce resources. Simple put, it explores the behavior of agents within an economy—including consumers, businesses, and governments—and how these behaviors influence supply and demand for goods and services.

At its core, microeconomics attempts to understand the mechanisms that drive trade and the determination of prices. It is through microeconomic principles that we can analyze the utility maximization behavior of consumers, which is essential when considering concepts like the Nash Social Welfare function.

Utility represents the individual's satisfaction or happiness derived from consuming a product or service. In this context, voters' utility in a political game can be viewed as their level of satisfaction with a candidate's platform. Microeconomic theory helps in framing this political choice as a decision that involves comparing utilities, just as a consumer compares the satisfaction from different goods before making a purchase.

Diving into Utility Maximization

Utility maximization is a cornerstone concept in microeconomics, which posits that individuals aim to achieve the highest level of utility, or satisfaction, with the limited resources they have. In essence, each decision-maker selects the most preferred bundle of goods or actions according to their preferences and constraints.

When applied to the exercise, voters are making a choice between two candidates, where the utility derived from each candidate represents the individual voter's preference. The concept of utility maximization suggests that voters would naturally choose the candidate providing the highest utility ratio. By analyzing the decision through the lens of utility comparisons, we can see how voting for a candidate aligns with the broader microeconomic behavior of maximizing utility.

Voting Theory and Collective Decisions

Voting theory, a subset of game theory and political science, focuses on designing fair and efficient mechanisms to make collective decisions. It studies how rules and procedures can influence the outcome of elections and aggregations of individual preferences.

In the context of the exercise, voters employ their preferences to select between candidates based on their respective utilities. It is here where the Nash Social Welfare function comes into play, providing a method of aggregating these individual utilities to achieve a socially desirable outcome that reflects a balance of voter preferences. By considering the ratios of utilities, voting theory helps in analyzing how collective decision-making processes are shaped and how those processes can be optimized for social welfare.

Applying Game Theory

Game theory explores strategic interactions among rational decision-makers. It assists in the understanding of how individuals or groups choose strategies in various contexts—anticipating the actions and reactions of others. In game theory, the concept of utility often represents the payoffs players receive as a result of their strategies.

In the case of the exercise, the election can be seen as a strategic game where voters choose which candidate to support based on their utility ratios. The Nash Social Welfare function connects to game theory as it looks to find an outcome from this game that maximizes the utility for a group of voters—much like how equilibrium is determined in strategic games. This notion of maximizing social welfare follows the equilibrium concept of Nash, where each player's strategy results in the best possible outcome considering the choices of others.