Question

Prove that, for any structure function \(\phi\), $$ \phi(\mathbf{x})=x_{i} \phi\left(1_{i}, \mathbf{x}\right)+\left(1-x_{i}\right) \phi\left(0_{i}, \mathbf{x}\right) $$ where $$ \begin{aligned} &\left(1_{i}, \mathbf{x}\right)=\left(x_{1}, \ldots, x_{i-1}, 1, x_{i+1}, \ldots, x_{n}\right) \\ &\left(0_{i}, \mathbf{x}\right)=\left(x_{1}, \ldots, x_{i-1}, 0, x_{i+1}, \ldots, x_{n}\right) \end{aligned} $$

Short answer

To prove that the given equation holds true for any structure function \( \phi \), we consider the two cases for \( x_i \), which can take values in {0, 1}. For the case \( x_i = 1 \), the equation simplifies to \( \phi(\mathbf{x}) = \phi\left(1_{i}, \mathbf{x}\right) \), and since the \( i \)-th element of \( \mathbf{x} \) is already 1, the equation holds true. For the case \( x_i = 0 \), the equation simplifies to \( \phi(\mathbf{x}) = \phi\left(0_{i}, \mathbf{x}\right) \), and since the \( i \)-th element of \( \mathbf{x} \) is already 0, the equation holds true. Hence, we can conclude that the given equation is valid for any structure function \( \phi \).

Step-by-step solution

A structure function, denoted by \( \phi \), is a function that maps the elements from a cooperative game (players and coalitions) to their corresponding values. It is defined such that \( \phi : \{0,1\}^{n} \to \mathbb{R} \), where \( \phi(\mathbf{x}) \) represents the value assigned to a coalition with the binary vector \( \mathbf{x} = (x_{1}, \ldots, x_{n}) \), and \( x_{i} \in \{0,1\} \) indicates whether player \( i \) belongs to the coalition or not. #Step 2: Analyze the given equation#

In order to prove the equation, we will analyze the terms involved. The given equation is $$ \phi(\mathbf{x}) = x_{i} \phi\left(1_{i}, \mathbf{x}\right) + \left(1 - x_{i}\right) \phi\left(0_{i}, \mathbf{x}\right) $$ Here, \( (1_{i}, \mathbf{x}) \) denotes the vector \( \mathbf{x} \) with the \( i \)-th element equal to 1, and \( (0_{i}, \mathbf{x}) \) denotes the vector \( \mathbf{x} \) with the \( i \)-th element equal to 0. #Step 3: Consider the two cases for \( x_i \)#

Since \( x_i \) can take values in {0, 1}, we can consider the two cases separately and verify that the given equation holds true for both cases. Case 1: \( x_i = 1 \)

In this case, the given equation becomes: $$ \phi(\mathbf{x}) = 1 \cdot \phi\left(1_{i}, \mathbf{x}\right) + \left(1 - 1\right) \phi\left(0_{i}, \mathbf{x}\right) $$ which simplifies to: $$ \phi(\mathbf{x}) = \phi\left(1_{i}, \mathbf{x}\right) $$ Here, since the \( i \)-th element of \( \mathbf{x} \) is already 1, both the vectors \( \mathbf{x} \) and \( (1_{i}, \mathbf{x}) \) are the same. Therefore, the equation holds true for the case \( x_i = 1 \). Case 2: \( x_i = 0 \)

In this case, the given equation becomes: $$ \phi(\mathbf{x}) = 0 \cdot \phi\left(1_{i}, \mathbf{x}\right) + \left(1 - 0\right) \phi\left(0_{i}, \mathbf{x}\right) $$ which simplifies to: $$ \phi(\mathbf{x}) = \phi\left(0_{i}, \mathbf{x}\right) $$ Here, since the \( i \)-th element of \( \mathbf{x} \) is already 0, both the vectors \( \mathbf{x} \) and \( (0_{i}, \mathbf{x}) \) are the same. Therefore, the equation holds true for the case \( x_i = 0 \). #Step 4: Conclusion#

Since we have shown that the equation holds true for both cases of \( x_i \) (0 and 1), we can conclude that the given equation holds true for any structure function \( \phi \), as required: $$ \phi(\mathbf{x}) = x_{i} \phi\left(1_{i}, \mathbf{x}\right) + \left(1 - x_{i}\right) \phi\left(0_{i}, \mathbf{x}\right) $$

Key concepts

Understanding Probability Models

Probability models are fundamental to comprehending various phenomena in fields like statistics, economics, and engineering. Essentially, a probability model is a mathematical representation that describes the likelihood of different outcomes in an experiment. It is composed of a sample space, which encompasses all possible outcomes, events that are subsets of the sample space, and a probability function that assigns a numerical value to each event.

In real-world scenarios, these models can predict trends, assess risks, and assist in decision-making processes. One classic example is the toss of a fair coin, which has a simple probability model with a sample space of {heads, tails} and an equal likelihood for each outcome. More complex models, like those used in cooperative game theory, not only consider the probability of individual outcomes but also the interactions between different entities, such as the strategies of players within a game.

Cooperative Game Theory Basics

Cooperative game theory is a branch of mathematics that deals with the study of cooperative behavior among a group of players. The main goal is to understand how groups, or coalitions, can form and how the collective gains can be fairly divided among the players. This theory differs from non-cooperative game theory in that it focuses on the collaboration between players rather than competition.

One crucial aspect of cooperative game theory is the use of structure functions, like the one described in the exercise with the mathematical representation \( \phi(\mathbf{x}) = x_{i} \phi(1_{i}, \mathbf{x}) + (1 - x_{i}) \phi(0_{i}, \mathbf{x}) \). This structure function interprets the value of a coalition when a player is either a part of it \( x_i = 1 \) or not \( x_i = 0 \). This mathematical approach helps in analyzing different coalition formations and determining fair payouts, which can be highly beneficial in various strategic and economic applications.

Binary Vector Representation Explained

Binary vector representation is a powerful mathematical tool used across different disciplines, including computer science and game theory. In simple terms, it uses vectors whose elements are binary digits, 0 and 1, to represent states or configurations.

Within the context of cooperative game theory, as the exercise solution outlines, binary vectors represent the presence or absence of players in a coalition. Each element in the vector corresponds to a player: a '1' indicates the player’s participation, while a '0' signifies non-participation. The structure function \(\phi(\mathbf{x})\) assigns a value to every possible coalition, and the binary vector \(\mathbf{x} = (x_{1}, \ldots, x_{n})\) precisely describes the setup of that coalition. This concise and clear method of representation is vital for computational efficiency, especially in cases involving many players, because it allows for a streamlined analysis of coalition values.