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tags: micro, lecture_note, book
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# State

The following is based on Rubinstein(2006).

## State-dependent Utility

### State-depedent bet
**Definition** Let $S=\{s\}$ be a finite set with $k$ elements, in the end, only one outcome in $S$ will realize. A bet $b$ is a vector $(x_1, ..., x_k) \in \mathbb{R}^k$, which means the agent can get $x_i$ dollar if $s_i$ outcome realize. The set of all possilble bets is $\mathbb{R}^k.$ We use $(\pi_1, ..., \pi_k)$ to denot the probability of each state, so $\sum_{i=1}^k \pi_i=1$.

### Extended Expected Utility Representation
**Definition** Given a probabilty $(\pi_1, ..., \pi_k)$, the preference relation $\succeq$ has an extended expected utility representation if for every $s \in S$, there is a function $u_s: \mathbb{R}^k \to \mathbb{R}$ such that for any $(x_1, ..., x_k) \in \mathbb{R}^k$ and $(x_1', ..., x_k') \in \mathbb{R}^k$:

$$(x_1, ..., x_k) \succeq (x_1', ..., x_k') \text{ iff } \sum_s \pi_s u_s(x_s) \ge \sum_s \pi_s u_s(x_s') .$$

### Extended Independence Axiom

**Definition** The preference relation $\succeq$ on $\mathbb{R}^k$ is independent if for any $b, b',b'' \in \mathbb{R}^k$ and $a \in (0,1)$, we have

$$b \succeq b' \text{ iff } ab +(1-a) b'' \succeq ab' +(1-a)b''.$$

### Existence of Expected Utility Function
To apply the vNM theorem, we need to restrict the domain of the bets. That is, $m \le x_i \le M$. Additionally, we assume that $(M, ...,M)$ is the best bet and $(m,...,m)$ is the worst bet. Based on these two assumptions, if $\succeq$ is continuous and independent, the preference can be represented by an extended expected utility representation.

### Subjective Probability Theory

Let $S=\{s\}$ be a finite set with $k$ elements, in the end, only one outcome in $S$ will realize. A bet $b$ is a vector $(x_1, ..., x_k) \in \mathbb{R}^k$, which me the agent can get $x_i$ dollar if $s_i$ outcome realize. The set of all possilble bets is $\mathbb{R}^k.$

**Definition**
Given two bets $(x_1, ...,x_k)$ and $(y_1, ...,y_k)$, a preference is **weak monotonicity** if $x_k>y_k$ for all $k$ implies $x \succ y$.

Given two bets $x$ and $y$, a preference is **additive** if $x \succeq y$ implies $x + z \succeq y + z$ for all $z\in \mathbb{R}^k.$

**Claim**
If an agent's preference $\succeq$ over $\mathbb{R}^k$ satisfies continuity, weak monotonicity, and additive, there exists a probability $(p_1,...,p_k)$ such that the

$$(x_1, ..., x_k) \succeq (x_1', ..., x_k') \text{ iff } \sum_{i=1}^k p_i x_i \ge \sum_{i=1}^k p_i x_i' .$$

**Proof**

Suppose $\succeq$ satisfies all three properties. Conside two sets $U = \{x | x\succeq 0\}$ and $D = \{x | 0 \succ x\}$. By weak monotonicity, both sets are not empty. By the continuity, $U$ is closed and $D$ is open. Since $U \cap D = \emptyset$, we can apply seperating hyperplane theorem. That is, there is a vector non-zero vector $q \in \mathbb{R}^k$ and a real number $r$ such that $q \cdot x \ge r$ for any $x \in U$ and $q \cdot x < r$ for any $x \in D$. By the weak monotonicty, $r=0$ and $q_i \ge 0$ for $1 \le i \le k$.

Based on $q$, we can construct a vector $p$ by $p_i = \frac{q_i}{\sum q}$, which is the subjective probability. By the addivity, $x \succeq y$ iff $x -y \succeq 0$ and iff $px \ge py$.