---
# System prepended metadata

title: Chapter 22. Markets and Information

---

# Chapter 22. Markets and Information
[toc]

- How markets aggregate exception or information?
- Exogenous or Endogenous?
## 22.1 Markets with Exogenous Events
- *Exogenous events*: the probabilities of the events are not affected by the outcomes in the market
- How markets aggregate opinions about exogenous events?
- *Prediction market*: individuals bet on the outcome of some event
- Forecasting of election results
    - Participants could buy or sell a contract, and if the Democrat won, they can get $1; otherwise, they got nothing.
    - The corresponding contract: paid $1 if a Republican won
    - ![](https://i.imgur.com/5ZU6Wx4.png)
    - we can view the price as an average of their beliefs
    - usual interpretation of price: an average prediction about the probability of the event occurring.

## 22.2 Horse Races, Betting, and Beliefs
- Betting on a two-horse race
    - Two horses: $A$ and $B$
    - Bettor has $w$ dollars
    - $r$: the fraction of his wealth will be bet on $A$
    - a bettor’s choice of bet will depend on
        - belief of bettor
            - bettor believes that horse $A$ will win with probability $a$,<br> and that horse $B$ will win with probability $b = 1 − a$.
        - odds
            - $A$: $o_A$
            - $B$: $o_B$
        - A bettor’s reaction to risk

### Modeling Risk and Evaluating the Utility of Wealth
- player evaluates each strategy according to the expected value of its payoff
- utility function $U(·)$:<br> when a bettor has a wealth $w$, his payoff $= U(w)$.
- simplest utility function:$U(w) = w$
    - Suppose that a bettor’s wealth is $w$, and he is offered a gamble in which he gains $w$ dollars with probability $\frac12$ , and loses $w$ dollars with probability $\frac12$.
    - expected utility $= \frac12 U(2w) +\frac12 U(0) = \frac12 × (2w) + \frac12 × 0 = w$
- Assume that the bettor’s utility grows at a decreasing rate
    - e.g. $U(w) = w^{1/2}$ or $U(w) = ln(w)$
    - if $U(w) = w^{1/2}$,<br> expected utility of previous example $= \frac12 U(2w) +\frac12 U(0) = \frac12 × (2w)^{1/2} + \frac12 × 0$ $=  2^{−1/2} × w^{1/2}<U(w)$

### Logarithmic Utility
![](https://i.imgur.com/2PkLttM.png)
- the increase in utility from doubling your money<br>$ln(2w) − ln(w) = ln(2w/w) = ln(2)$

### The Optimal Strategy: Betting Your Beliefs
- expected utility$=a\space ln(rwo_A) + (1 − a)\space ln((1 − r)wo_B )$
- The bettor wants to choose $r$ to maximize above expression.
- unpack:<br>$a\space ln(r) + (1 − a)\space ln(1 − r) + a\space ln(wo_A) + (1 − a)\space ln(wo_B )$
- To choose $r$ to maximize
$a\space ln(r) + (1 − a)\space ln(1 − r)$
![](https://i.imgur.com/OpLIWuT.png)
- The derivative of the expression<br>$\frac{a}r-\frac{1-a}{1-r}$
- $r = a$ is the maximum point.
    -  interpretation: the bettor bets his beliefs
## 22.3 Aggregate Beliefs and the “Wisdom of Crowds”
- Consider systems with multiple bettors.
    - $N$ bettors: $1, 2, 3, . . . , N$
    - Each bettor $n$ believes 
        - a probability of $a_n$ that horse $A$ will win.
        - a probability of $b_n = 1 − a_n$ that $B$ will win
    - Bettor $n$ has wealth $w_n$
        - total wealth $w = w_1 + w_2 + · · · + w_N$
    - utility function: $ln(w)$
    - optimal betting strategy for bettor $n$ with belief $a_n$ is $r_n = a_n$<br>$\Rightarrow$ Bettor $n$ will bet $a_nw_n$ on horse $A$ and $(1 − a_n)w_n$ on horse $B$<br>$\Rightarrow$ total amount bet on $A$ is $a_1w_1 + a_2w_2 + · · · + a_Nw_N$ <br> and  the total amount bet on $B$ is $b_1w_1 + b_2w_2 + · · · + b_Nw_N$.

### The Odds Determined by the Racetrack.
- Assume racetrack has no cost and makes no proﬁt
- If $A$ wins, the total amount owed to the bettors is <br> $a_1w_1o_A + · · · + a_N w_N o_A=w$. <br> $\Rightarrow \frac{{a}_{1}w_1}{w} + · · · + \frac{a_N w_N}{w} = o_A^{−1}$
- Let $f_n = w_n/w$,<br>$a_1f_1 + · · · + a_N f_N = o_A^{−1}$
- Similarly, $b_1f_1 + · · · + b_N f_N = o_B^{−1}$
- If $A$ wins, a bet of $o_A^{−1}$ dollars on horse $A$ will result in a payment of $$1$.

### State Prices
- Denote $ρ_A = {o_A}^{−1}$ for the event that $A$ wins and $ρ_B = o_B^{−1}$ for the event that $B$ wins. <br>They are usually called *state prices*.
- state prices are weighted averages of the bettors’ beliefs
- state prices can be interpreted as the *market’s beliefs*

## 22.4 Prediction Markets and Stock Markets
### Prediction Markets
- Individuals trade claims to a one-dollar return conditional on the occurrence of some event.
- In prediction markets and horse races, the prices reﬂect an averaging of the beliefs of the participants in the market.
- Example: 2008 U.S. Presidential election
    - Let $f_n$ be the share of the total wealth bet that is bet by trader $n$.
    - Let $a_n$ and $b_n$ be trader $n$’s probabilities of a Democrat and a Republican winning, respectively.
    - The market price for the contract on a Democratic winner is $ρ^D = a_1f_1 + · · · + a_N f_N$.
    - For the period 1988–2004, the Iowa Electronic Markets did a signiﬁcantly better job of predicting the outcome of U.S. Presidential elections than was done by an average of the major national polls.
### Stock Markets
- More complex
- The money that owner can get is determined by the value of the stock in future states.
- Investors should be willing to pay for the stock conditional on each state<br> = the value of the stock in that state $\times$ the price of a dollar in that state.
### State Prices in the Stock Market
- Determining state prices from stock prices and vice versa
- Two companies: $1$ and $2$
- Two possible states
    - $s_1$: company 1 does well
        - stock in company $1$ is worth two dollars
        - stock in company $2$ is worth one dollar
    - $s_2$: company 2 does well
        - stock in company $1$ is worth one dollar
        - stock in company $2$ is worth two dollars      
- price of stock
    - $v_1$ for company $1$
    - $v_2$ for company $2$ 
- Write $ρ_1$ and $ρ_2$ to denote the state prices for states $s_1$ and $s_2$.
- The price of a share of stock in company $1$ is the value now of the future worth of the company
    - Thus, <br>$v_1 = 2ρ_1 + ρ_2$<br>$v_2 = ρ_1 + 2ρ_2$
    - for the state prices $ρ_1$ and $ρ_2$.<br>$ρ_1 = \frac{2v_1 − v_2}3$<br>$ρ_2 = \frac{2v_2 − v_1}3$ 

## 22.5 Markets with Endogenous Events
- Some events are *endogenous*: <br>whether they come true depends on the aggregate behavior of the individuals themselves. 
- Example
    - Join a particular social networking site
    - Used Car
    - common: notion of *self-fulﬁlling expectations*
- Asymmetric information
    - Example of used car market
        - Seller knows something about  car.
        - But potential buyers do not know. 
        
[//]: <> (In all of these cases, uninformed traders need to form expectations about the value of the good being traded, and these
expectations should take into account the behavior of better-informed traders.)

## 22.6 The Market for Lemons
- Used cars
    - good cars and bad cars
    - Each seller knows the quality of his or her own car.<br> Buyers do not know the quality of any car.
    - values for used cars
        - Sellers value good cars at $10$ and bad cars at $4$.
        - Buyers value good cars at $12$ and bad cars at $6$.
    - Suppose that a fraction $g$ of used cars are good cars, and hence a fraction $1 − g$ are bad cars (Everyone knows $g$)
    - Suppose that there are more buyers than used cars(sellers).

<!-- ### The Market with Symmetric Information 
- If the type of each car is known to everyone, every car could be sold to some buyer. -->

- Since cars are indistinguishable to buyers, there can only be one price for a used car.
- The fraction of good cars in the population **for sale** is $h$ (might not be $g$).
- The value that any buyer places on a used car is<br>$12h + 6(1 − h) = 6 + 6h$.
- Thus, to know how much they should  pay, buyers need to predict $h$.
- We need to find a $h$ is self-fulﬁlling<br> i.e., if each buyer expects a fraction $h$ of the cars to be good, then indeed an $h$ fraction of the cars on the market will be good.

### Characterizing the Self-Fulﬁlling Expectations Equilibria
- If $h$ = $g$
    - buyers would be willing to pay $6 + 6g$ (call it $p^∗$) for a car.
    - A seller with a good(bad) car would offer it for sale at $p^∗$ provided that <br>$p^∗ = 6 + 6g ≥ 10$ ($g ≥ \frac23$)
    - So, if $g ≥ \frac23$ , there is a self-fulﬁlling expectations equilibrium in which all cars are offered for sale.
    - if $g < \frac23$, $p∗ = 6 + 6g < 10$. So, owners of good cars would not be willing to sell
    - Thus, there cannot be a self-fulﬁlling expectations equilibrium in which $h = g$.
- For any $g$, there is always a self-fulﬁlling expectations equilibrium in which $h = 0$ (only bad cars are sold)
    - Buyers expect that only bad cars are sold<br>$\Rightarrow$ Buyers are willing to pay $6$ for a car.<br>$\Rightarrow$ Only the sellers of bad cars would be willing to sell<br>$\Rightarrow$ The market would consist only of bad cars<br>$\Rightarrow$ This is a self-fulﬁlling expectations equilibrium with $h = 0$.

### Complete Market Failure
- Three types of used cars: good cars, bad cars, and lemons
- Suppose
    - The quantity of each is the same ($\frac13$)
    - Sellers value good cars at $10$, bad cars at $4$, and lemons at $0$.
    - Buyers value good cars at $12$, bad cars at $6$, and lemons at $0$.
    - There are more buyers than there are used cars
- With asymmetric information, we need to consider the possible self-fulﬁlling expectations equilibria.
- There are three candidates for an equilibrium:<br>(a) all cars are offered for sale <br> (b) only bad cars and lemons are offered for sale, <br> \(c\) only lemons are offered for sale
    - Note that option \(c\) represents the complete failure of the market, since all items on the market would have value $0$.
    - (a) Suppose buyers expect all cars to be on the market.<br>$\Rightarrow$ the expected value of a car to a buyer would be $\frac{12 + 6 + 0}3 = 6$.<br>This is less than the value that sellers of good cars places on their cars<br>$\Rightarrow$ they would not put them on the market<br>Hence, this is not an equilibrium.
    - (b) Suppose buyers expect bad cars and lemons to be on the market.<br>$\Rightarrow$ the expected value of a car to a buyer would be $\frac{6 + 0}2 = 3$.<br>This is less than the value that sellers of bad cars place on bad cars<br>$\Rightarrow$ They would not put them on the market<br>Hence, this is not an equilibrium.
    - \(c\) It is clearly an equilibrium if buyers expect only lemons to be sold. <br>In this case, their expected value for a car is $0$, and this is what they are willing to pay, then the market will consist completely of lemons.
- The market has been subverted by a kind of chain reaction
### Summary: Ingredients of the Market for Lemons
- Review the key features that led to market failure
    - (i) The items have varying qualities.
    - (ii) For any given type, the buyers value the items at least as much as the sellers do
    - (iii) Asymmetric information
    - (iv) Because of (iii), the items all must be sold for the same uniform price
- The market does not necessarily fail when these ingredients are present.<br> It depends on whether there is an equilibrium

## 22.7 Asymmetric Information in Other Markets
### The Labor Market
- people seeking jobs play the role of the sellers, <br>and companies seeking employees play the role of the buyers.
-  Two types of workers: productive and unproductive
    -  Half and half
    -  Each productive worker will generate $$80,000$ of revenue per year for the ﬁrm
    - Each unproductive worker will generate $$40,000$ of revenue per year for the ﬁrm.
    - self-employed
        - productive worker: $$55,000$ per year
        - unproductive worker: $$25,000$ per year
- Firm cannot determine the type of each worker.<br>$\Rightarrow$The ﬁrm has to offer a uniform wage of $w$
### Equilibria in the Labor Market
- If the ﬁrm expects all workers to be on the job market
    - expected revenue per employee will be $\frac{80, 000 + 40, 000}2 = 60, 000$
    - At wage $60,000$, both types of workers will be willing to accept, so we have an equilibrium.
- If the ﬁrm expects only unproductive workers to be on the job market
    - It expects to make only $$40,000$ per year per employee
    - At this wage, only unproductive workers will be willing to accept jobs, so we have an equilibrium.
- Suppose that only $\frac14$ of the workers are productive and $\frac34$ are unproductive.
    - If the ﬁrm were to expect all workers to be on the market
        - expected revenue per  employee would be $\frac14 \times 80000 + \frac34\times 40000=50000$ 

### The Market for Insurance
- Insurance companies usually do not know the health of the insured person better than the insured person
- Some feature
    - individuals in a given risk category can be more or less costly to insure
    - The buyers of health insurance have more information.
- For any risk category, the insurance company has to charge a uniform price for the insurance that is sufﬁcient to cover the average cost.
- The healthiest individuals are being charged a price that is greater than the expected cost for them, and so they may be unwilling to buy insurance. <br>$\Rightarrow$ The average healthy of the remainder goes down<br>$\Rightarrow$  The insurance company would need to set a higher price for remainder.<br>$\Rightarrow$ ...

## 22.8 Signaling Quality
:::info
[**Signaling**](https://en.wikipedia.org/wiki/Signalling_(economics)):  one party credibly conveys some information about itself to another party
:::

How to alleviate the information asymmetry?

* Signaling mechanism
    * Used-cars
        * Warranty
        * quality assurance
    * Labor Market
        * Performance in school

## 22.9 Quality Uncertainty On-Line: Reputation Systems and Other Mechanisms

### Reputation Systems

e.g., eBay, Amazon ...

1. After each purchase, buyer provide an evaluation of the seller
2. The evaluations turn into *reputation score* by some algorithm
3. A seller's **reputation score** evolves over time
    * Favorable evaluations cause the score to go up
    * Unfavorable evaluations cause the score to go down
4. Reputation score serves as a signal of seller quality

### Ad Quality in Keyword-Based Advertising

Ranking of an ad should not be based purely on the bid offered by the advertiser
but on the **ad quality**

**ad quality**

* click rate
* Relevance of the landing page to the ad
    * Users don't like to be fooled (redirect to a page that has nothing to do with the ad)
    * If users get tricked by the low-quality pages often, they won't click on ads anymore
    * Revenue will be lowered even advertiser provide a good price

Since only advertiser knows the quality of his ad (information asymmetry)
**ad quality** is a good way to rank the ads

## 22.10 Advanced Matrerial: Wealth Dynamics in Markets

As the time goes on,
market selects people whose decisions are closest to optimal,
their wealth shares would increase, </br> and as a result their overall effect on the aggregate marketprice would increase.

We provide a mathemactical analysis that a participant apply **Bayes' Rule** to make decision by information he has at the time.

:::info
Recall. ([Chapter 16. Information Cascades](/3vf6_kdOSHe7YD75_FqN_g))
**Bayes' Rule**:
$$P[A|B] = {P[A]P[B|A] \over P[B]}$$
(can be used to predict the decision of a participant by information he has.)
:::

### A. Bayesian Learning in a Market

We now analysis how a person would update his decision over time in a market </br> and we'll see how the decisions make by participants changes over time

A simple horse race example.

* Suppose 
    * two horses $A$ and $B$ will run a race every week
    * the outcomes of these races are independent
    * $A$ wins each one with probability $a$, $B$ with probability $b = 1-a$
* A person does not know the values of $a$ and $b$
    * He begins with a set of $N$ possible hypotheses for the probabilities, </br> say $(a_1, b_1), (a_2,b_2), ...,(a_N,b_N)$
    * Each hypothesis is associated with a [*prior probability*](https://en.wikipedia.org/wiki/Prior_probability), </br> say $f_1, f_2, ... , f_N$
    * A person's initial predicted probability of $A$ winning is </br> $a_1f_1+a_2f_2+...+a_Nf_N$
* After $T$ weeks, we observe a sequence $S$ of outcomes of these race
  horse $A$ wins $k$ times in total, horse $B$ wins $\ell$ times in total
  * Using **Bayes' Rule**, the $n$-th [*posterior probability*](https://en.wikipedia.org/wiki/Posterior_probability) is
  
  \begin{aligned}
  P[(a_n,b_n)|S] &= {{f_n \cdot P[S|(a_n, b_n)]}\over{P[S]}} \\
  &= {{f_n \cdot P[S|(a_n, b_n)]}\over{f_1 \cdot P[S|(a_1, b_1)] + f_2 \cdot P[S|(a_2, b_2)] + ... + f_N \cdot P[S|(a_N, b_N)]}} \\
  &= {{f_na_n^kb_n^\ell}\over{f_1a_1^kb_1^\ell+f_2a_2^kb_2^\ell+...+f_Na_N^kb_N^\ell}}
  \end{aligned}
  
  * Now, the person's predicted probability on horse $A$ is $a_1P[(a_1,b_1)|S] + a_2P[(a_2,b_2)|S] + ... + a_NP[(a_N,b_N)|S]$
  
### Convergence to the Correct Hypothesis.

Suppose that one of the hypotheses is correct 
and thus $\exists (a_c, b_c) \in \{(a_1,b_1), (a_2,b_2), ..., (a_N, b_N)\}$ s.t. $(a_c, b_c) = (a, b)$

:::info
Define *ratio* $$R_n[S] = {{P[(a_c,b_c)|S]}\over{P[(a_n,b_n)|S]}} = {{f_ca_c^kb_c^\ell}\over{f_na_n^kb_n^\ell}}$$
:::

1. For $n \neq c$
$$R_n[S] = {{f_ca_c^kb_c^\ell}\over{f_na_n^kb_n^\ell}}$$

2. take log both sides
$$\ln(R_n[S]) = \ln({f_c \over f_n}) + k\ln({a_c \over a_n}) + \ell\ln({b_c \over b_n})$$

3. divide both sides by the number of weeks $T$
$${1 \over T}\ln(R_n[S]) = {1 \over T}\ln({f_c \over f_n}) + {k \over T}\ln({a_c \over a_n}) + {\ell \over T}\ln({b_c \over b_n})$$

4. when $T \rightarrow \infty$

\begin{aligned}
{1 \over T}\ln(R_n[S]) &=  {1 \over T}\ln({f_c \over f_n}) + {k \over T}\ln({a_c \over a_n}) + {\ell \over T}\ln({b_c \over b_n}) \\
                       &=                                    a\ln({a_c \over a_n})           + b\ln({b_c \over b_n}) \\
                       &=                                    a\ln(a_c) + b\ln(b_c) - [a\ln(a_n) + b\ln(b_n)] \\
                       &=                                    a\ln(a_c) + (1-a)\ln(1-a_c) - [a\ln(a_n) + (1-a)\ln(1-a_n)] \\
                       &> 0 && \text{since} (a_c, b_c) = (a, b) \\
                       & && \text{as T}\rightarrow \infty
\end{aligned}

5. So $R_n[S] \rightarrow \infty$, it implies that 
    * $P[(a_c,b_c)|S] \rightarrow 1$
    * $P[(a_n,b_n)|S] \rightarrow 0$
    
Therefore, we can conclude that the correct hypothesis will survive. 
Moreover, the predicted probability on horse $A$ is converging to $a$

### Convergence without a Correct Hypothesis

Suppose that no hypothesis is correct
and choose a $(a_c, b_c)$, where $(a_c, b_c) \in  \{(a_1,b_1), (a_2,b_2), ..., (a_N, b_N)\}$

:::info
Define *relative entropy* between $(a_n, b_n)$ and $(a,b)$ to be

$$D_{(a,b)}(a_n,b_n) = a\ln(a) + b\ln(b) - [a\ln(a_n) + b\ln(b_n)]$$

If the value is smaller it means that it is closer to the truth
:::

1. Go back to ${1 \over T}\ln(R_n[S])$
\begin{aligned}
{1 \over T}\ln(R_n[S]) &=  {1 \over T}\ln({f_c \over f_n}) + {k \over T}\ln({a_c \over a_n}) + {\ell \over T}\ln({b_c \over b_n}) \\
                       &=                                    a\ln(a_c) + b\ln(b_c) - [a\ln(a_n) + b\ln(b_n)] \\
                       &= D_{(a,b)}(a_n,b_n) - D_{(a,b)}(a_c,b_c) && \text{as T} \rightarrow \infty
\end{aligned}

2. If we have that $(a_c, b_c)$ is closer than any other hypothesis in relative entropy, then

$$D_{(a,b)}(a_n,b_n) > D_{(a,b)}(a_c,b_c)$$

and hence

\begin{aligned}
{1 \over T}\ln(R_n[S]) &= D_{(a,b)}(a_n,b_n) - D_{(a,b)}(a_c,b_c) \\
                       &> 0 && \text{as T} \rightarrow \infty
\end{aligned}

3. So $R_n[S] \rightarrow \infty$, it implies that 
    * $P[(a_c,b_c)|S] \rightarrow 1$
    * $P[(a_n,b_n)|S] \rightarrow 0$
    
Therefore, the predicted probability on $A$ will converge to the best hypothesis $a_c$

### B.Wealth Dynamics

An example.

* Suppose
    * two horses $A$ and $B$ run a race every week
    * there are $N$ bettors
    * bettor $n$ has fixed belief that horse $A$ will win with probability $a_n$, $B$ with $b_n$
    * total wealth of all bettors is $w$
    * bettor $n$ has initial wealth of $w_n$, wealth share $f_n = w_n / w$
    * before the $t^{\text{th}}$ race, 
        * market determines odds $o_A^{\langle t \rangle}$ and $o_B^{\langle t \rangle}$ on horses $A$ and $B$
        * each bettor has a current wealth $w_n^{\langle t \rangle}$, wealth share $f_n^{\langle t \rangle}$

1. As we saw in Section 22.2, 
it is optimal for bettor $n$ put a bet $a_nw_n^{\langle t \rangle}$ on $A$ and $b_nw_n^{\langle t \rangle}$ on $B$
    * If $A$ wins, $w_n^{\langle t+1 \rangle} = a_nw_n^{\langle t \rangle}o_A^{\langle t \rangle}$, $f_n^{\langle t+1 \rangle} = a_no_A^{\langle t \rangle}f_n^{\langle t \rangle}$
    * If $B$ wins, $w_n^{\langle t+1 \rangle} = b_nw_n^{\langle t \rangle}o_B^{\langle t \rangle}$, $f_n^{\langle t+1 \rangle} = b_no_B^{\langle t \rangle}f_n^{\langle t \rangle}$
    
2. Consider the $t^{\text{th}}$ race and two bettors $m, n$
    * If $A$ wins, the ratio of wealth share of $m$ and $n$ changes from $f_m^{\langle t \rangle} / f_n^{\langle t \rangle}$ to $a_mf_m^{\langle t \rangle} / a_nf_n^{\langle t \rangle}$
    i.e., the ratio is multiplied by $a_m/a_n$
    * If $B$ wins, the ratio of wealth share of $m$ and $n$ changes from $f_m^{\langle t \rangle} / f_n^{\langle t \rangle}$ to $b_mf_m^{\langle t \rangle} / b_nf_n^{\langle t \rangle}$
    i.e., the ratio is multiplied by $b_m/b_n$
    
3. After $T$ weeks, over a sequence of races $S$ in which $A$ wins $k$ times and $B$ wins $\ell$ times
the ratio of wealth shares of $m$ and $n$ end up with ${f_ma_m^kb_m^\ell} \over {f_na_n^kb_n^\ell}$

4. It's similar to the ratio we defined above, so we have similar result
the bettor with the best belief will acquire a wealth share of $1$ in the limit

Overall conclusion, 
the market selects for the people with the most accurate beliefs, 
and asymptotically prices assets according to these beliefs.