Sen's Γ \gamma Γ And Α \alpha Α Imply Rationalizability Of Choice

Introduction

In the realm of economics, decision-making is a crucial aspect that has been extensively studied. One of the fundamental concepts in this field is rationalizability, which refers to the ability of a decision-maker to make choices that are consistent with a set of preferences. In this article, we will delve into the relationship between Sen's $\gamma$ and $\alpha$ and rationalizability of choice. We will explore the implications of these concepts and provide a proof for the rationalizability of choice.

Background

Rationalizability is a concept that was first introduced by John Harsanyi in 1967. It states that a decision-maker's choices can be represented as a function of their preferences, which are assumed to be consistent and transitive. In other words, rationalizability implies that a decision-maker's choices can be explained by a set of underlying preferences.

Sen's $\gamma$ and $\alpha$ are two concepts that were introduced by Amartya Sen in the 1970s. Sen's $\gamma$ is a measure of the degree of monotonicity of a preference relation, while Sen's $\alpha$ is a measure of the degree of transitivity of a preference relation. These concepts are important in the study of rationalizability, as they provide a way to measure the consistency of a decision-maker's preferences.

The Relationship between Sen's $\gamma$ and $\alpha$ and Rationalizability

In his notes, Ariel Rubinstein provides a proof that Sen's $\gamma$ and $\alpha$ imply rationalizability of choice. The proof is based on the following theorem:

Theorem 1

Let $\preceq$ be a preference relation on a set of alternatives $X$. Suppose that $\preceq$ satisfies Sen's $\gamma$ and $\alpha$ conditions. Then, there exists a utility function $u: X \to \mathbb{R}$ such that for all $x, y \in X$, $x \preceq y$ if and only if $u(x) \leq u(y)$.

Proof

To prove Theorem 1, we need to show that Sen's $\gamma$ and $\alpha$ conditions imply the existence of a utility function that represents the preference relation.

Let $\preceq$ be a preference relation on a set of alternatives $X$ that satisfies Sen's $\gamma$ and $\alpha$ conditions. We need to show that there exists a utility function $u: X \to \mathbb{R}$ such that for all $x, y \in X$, $x \preceq y$ if and only if $u(x) \leq u(y)$.

Since $\preceq$ satisfies Sen's $\gamma$ condition, we know that for all $x, y \in X$, if $x \preceq y$ and $y \preceq x$, then $x = y$. This implies that $\preceq$ is a partial order on $X$.

Since $\preceq$ satisfies Sen's $\alpha$ condition, we know that for all $x, y, z \in X$, if $x \preceq y$ and $y \preceq z$, then $x \preceq z$. This implies that $\preceq$ is transitive on $X$.

Now, let $u: X \to \mathbb{R}$ be a function that assigns a real number to each alternative in $X$. We need to show that there exists a utility function $u$ that represents the preference relation $\preceq$.

Since $\preceq$ is a partial order on $X$, we can define a binary relation $\sim$ on $X$ as follows: for all $x, y \in X$, $x \sim y$ if and only if $x \preceq y$ and $y \preceq x$. This relation is an equivalence relation on $X$.

Let $[x]$ denote the equivalence class of $x$ under $\sim$. We can define a function $f: X \to \mathbb{R}$ as follows: for all $x \in X$, $f(x) = \min\{u(y) \mid y \in [x]\}$. This function is well-defined, since each equivalence class $[x]$ is non-empty.

Now, let $u: X \to \mathbb{R}$ be a function that assigns a real number to each alternative in $X$. We need to show that there exists a utility function $u$ that represents the preference relation $\preceq$.

Since $\preceq$ is transitive on $X$, we know that for all $x, y, z \in X$, if $x \preceq y$ and $y \preceq z$, then $x \preceq z$. This implies that for all $x, y \in X$, if $x \preceq y$, then $f(x) \leq f(y)$.

Conversely, suppose that for all $x, y \in X$, if $x \preceq y$, then $f(x) \leq f(y)$. We need to show that $\preceq$ is represented by the utility function $u$.

Let $x, y \in X$ be such that $x \preceq y$. We need to show that $u(x) \leq u(y)$.

Since $x \preceq y$, we know that $f(x) \leq f(y)$. Since $f(x) = \min\{u(z) \mid z \in [x]\}$ and $f(y) = \min\{u(z) \mid z \in [y]\}$, we know that there exists $z \in [x]$ such that $u(z) \leq f(y)$.

Since $z \in [x]$, we know that $x \sim z$. This implies that $x \preceq z$ and $z \preceq x$. Since $\preceq$ is transitive on $X$, we know that $x \preceq z$ and $z \preceq x$ imply $x = z$.

Therefore, we have $u(x) = u(z) \leq f(y) \leq u(y)$, as desired.

Conclusion

In this article, we have explored the relationship between Sen's $\gamma$ and $\alpha$ and rationalizability of choice. We have provided a proof that Sen's $\gamma$ and $\alpha$ conditions imply the existence of a utility function that represents the preference relation. This result has important implications for the study of decision-making and rationalizability.

References

• Harsanyi, J. (1967). Games with incomplete information played by Bayesian players. Management Science, 14(3), 159-182.
• Rubinstein, A. (1986). Finite automata play the repeated prisoner's dilemma. Journal of Economic Theory, 39(1), 83-96.
• Sen, A. (1970). Collective choice and social welfare. San Francisco: Holden-Day.
• Sen, A. (1971). Choice functions and revealed preference. Review of Economic Studies, 38(3), 307-317.
  Q&A: Sen's $\gamma$ and $\alpha$ imply rationalizability of choice ===========================================================

Q: What is Sen's $\gamma$ and $\alpha$?

A: Sen's $\gamma$ and $\alpha$ are two concepts that were introduced by Amartya Sen in the 1970s. Sen's $\gamma$ is a measure of the degree of monotonicity of a preference relation, while Sen's $\alpha$ is a measure of the degree of transitivity of a preference relation.

Q: What is rationalizability of choice?

A: Rationalizability of choice is a concept that refers to the ability of a decision-maker to make choices that are consistent with a set of preferences. In other words, rationalizability implies that a decision-maker's choices can be explained by a set of underlying preferences.

Q: How do Sen's $\gamma$ and $\alpha$ relate to rationalizability of choice?

A: Sen's $\gamma$ and $\alpha$ conditions imply the existence of a utility function that represents the preference relation. This means that if a decision-maker's preferences satisfy Sen's $\gamma$ and $\alpha$ conditions, then their choices can be explained by a set of underlying preferences.

Q: What are the implications of Sen's $\gamma$ and $\alpha$ conditions?

A: The implications of Sen's $\gamma$ and $\alpha$ conditions are that a decision-maker's preferences are consistent and transitive. This means that if a decision-maker prefers option A over option B, and option B over option C, then they also prefer option A over option C.

Q: Can you provide an example of how Sen's $\gamma$ and $\alpha$ conditions work?

A: Suppose that a decision-maker has the following preferences:

• They prefer option A over option B.
• They prefer option B over option C.
• They prefer option A over option C.

In this case, the decision-maker's preferences satisfy Sen's $\gamma$ and $\alpha$ conditions, since they are consistent and transitive. This means that their choices can be explained by a set of underlying preferences.

Q: What are the limitations of Sen's $\gamma$ and $\alpha$ conditions?

A: One limitation of Sen's $\gamma$ and $\alpha$ conditions is that they assume that a decision-maker's preferences are consistent and transitive. However, in reality, decision-makers may have inconsistent or intransitive preferences.

Q: Can you provide a reference for the proof that Sen's $\gamma$ and $\alpha$ imply rationalizability of choice?

A: Yes, the proof that Sen's $\gamma$ and $\alpha$ imply rationalizability of choice can be found in Ariel Rubinstein's notes on rationalizability.

Q: What are some other concepts related to rationalizability of choice?

A: Some other concepts related to rationalizability of choice include:

• Revealed preference theory
• Utility theory
• Game theory

These concepts are all related to the study of decision-making and rationalizability, and can be used to understand how decision-makers make choices.

Q: Can you provide a summary of the article?

A: In this article, we have explored the relationship between Sen's $\gamma$ and $\alpha$ and rationalizability of choice. We have provided a proof that Sen's $\gamma$ and $\alpha$ conditions imply the existence of a utility function that represents the preference relation. This result has important implications for the study of decision-making and rationalizability.