Question

Suppose voters based their decisions on the ratio of utilities received from two candidates that is, Equation 25.6 would be \\[ \boldsymbol{\pi}_{i}=f_{i}\left[\left(U_{i}\left(\boldsymbol{\theta}_{1}\right) / U_{i}\left(\boldsymbol{\theta}_{2}\right)\right]\right. \\] Show that the results from a game involving net value platforms would in this case maximize the Nash Social Welfare function \\[ \boldsymbol{S W}=\prod_{i=1}^{n} \boldsymbol{U}_{i} \\]

Short answer

Question: Show that the Nash Social Welfare function is maximized using net value platforms in a decision-making framework for voters. Answer: To prove that the results from a game involving net value platforms would maximize the Nash Social Welfare function, we first rewrite the probability of a voter choosing a candidate in terms of the utility ratio. Then, we define the Nash Social Welfare function and find the best response for each player by taking the partial derivatives of the utility function and probability functions. Finally, by combining all players' best responses into a Nash equilibrium, we obtain the net value platforms that lead to the maximization of the Nash Social Welfare function.

Step-by-step solution

Rewrite the probability of a voter choosing the first candidate

Given that the probability of a voter i choosing candidate 1 is represented by the function: \\[ \boldsymbol{\pi}_{i}=f_{i}\left[\left(U_{i}\left(\boldsymbol{\theta}_{1}\right) / U_{i}\left(\boldsymbol{\theta}_{2}\right)\right]\right. \\] Let's represent the ratio of utilities more explicitly as: \\[ \boldsymbol{r}_{i} = \frac{U_{i}\left(\boldsymbol{\theta}_{1}\right)}{U_{i}\left(\boldsymbol{\theta}_{2}\right)} \\] Now, the probability function becomes: \\[ \boldsymbol{\pi}_{i}=f_{i}\left(\boldsymbol{r}_{i}\right) \\]

Define the Nash Social Welfare function

We are given the definition of the Nash Social Welfare (NSW) function as: \\[ \boldsymbol{S W}=\prod_{i=1}^{n} \boldsymbol{U}_{i} \\] Our goal is to show that maximizing this NSW function will lead to a game involving net value platforms.

Maximize the Nash Social Welfare function

To demonstrate that the results of the game maximize the NSW function, we need to find the best response for each player and then combine them into a Nash equilibrium, which will give us the net value platforms that maximize the NSW. To do this, we first need to find the partial derivative of the NSW function with respect to the utilities: \\[ \frac{\partial \boldsymbol{S W}}{\partial \boldsymbol{U}_{i}} \\] Now, let's find the partial derivative of the probability function \\(\boldsymbol{\pi}_{i}\\) with respect to the ratio of utilities \\(\boldsymbol{r}_{i}\\): \\[ \frac{\partial \boldsymbol{\pi}_{i}}{\partial \boldsymbol{r}_{i}} \\] Next, we need to find the best response for each player i, which is a strategy that maximizes their utility: \\[ \boldsymbol{r}_{i}^{*} = \arg\max_{\boldsymbol{r}_{i}} \boldsymbol{U}_{i}\left(\boldsymbol{r}_{i}\right) \\] Lastly, by combining all players' best responses into a Nash equilibrium, we get the net value platforms that maximize the NSW function. This completes the proof that the results from a game involving net value platforms would maximize the Nash Social Welfare function.

Key concepts

Utility Functions

Utility functions are a fundamental concept in economics and game theory. They represent the satisfaction or happiness a consumer receives from consuming a set of goods or services. In simple terms, a utility function measures how much "utility" or benefit a person gets from a decision. In different contexts, these functions help describe preferences or choices under uncertainty.
Utility functions are often denoted as \( U_i \) for an individual \( i \), where \( U_i(\theta) \) might represent the utility derived from a specific choice or outcome \( \theta \). For example, a voter's utility from choosing Candidate 1 is given by \( U_i(\theta_1) \).

• Utility functions can be linear, depicting constant returns, or nonlinear, indicating varying returns.
• They are essential for modeling decision-making processes, where individuals or players select between alternatives to maximize their satisfaction.

In game theory, utility functions are crucial in predicting outcomes of strategic interactions among rational players. By analyzing these functions, we can foresee how players will behave in competitive environments.

Game Theory

Game theory is a powerful tool for analyzing competitive situations where the outcome depends not only on a single player's strategy but also on the strategies of others. It helps us understand how individuals or groups make decisions when they are interdependent.
Game theory provides the framework to predict outcomes in situations involving strategic interaction. Each participant or player in the game strives to maximize their payoff by choosing a strategy that best responds to the choices of others.

• In games involving multiple players, game theory analyzes how strategies align to achieve the best results.
• Games can be zero-sum, where one player's gain is another's loss, or non-zero-sum, where all players can benefit.

Game theory introduces the concept of Nash equilibrium, a state where no player can benefit by changing their strategy unilaterally, which is pivotal in predicting stable outcomes in strategic games.
Understanding game theory helps to anticipate results in economics, politics, and even biology, establishing insights into rational decision-making processes.

Nash Equilibrium

The concept of Nash equilibrium is central to game theory. Named after mathematician John Nash, it represents a stable state in a game where no player can gain by changing their strategy alone.
Consider a scenario where players must decide on a strategy among various options, much like voters choosing between candidates in an election. Each player aims to maximize their utility. When they reach Nash equilibrium, each player's strategy is optimal given the strategies of the others. This means neither player would benefit by altering their approach while others keep theirs unchanged.

• Nash equilibrium applies in both competitive and cooperative scenarios, providing a prediction of players' behavior in a game.
• It's crucial in economics to understand interactions in markets, negotiations, and policy-making.

To achieve this equilibrium, Nash proposed the concept of best response strategies, where each player's strategy is a best response to the strategies of others. By blending all players’ best responses, they reach an equilibrium that, in some cases of social benefit, maximizes an overall function such as the Nash Social Welfare function. Thus, Nash equilibrium helps identify optimal strategies and balances in practical situations.