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 $a$, 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 $U(·)$: 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 $U(w) = w^{1/2}$, 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 $ln(2w) − ln(w) = ln(2w/w) = ln(2)$ ### 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: $a\space ln(r) + (1 − a)\space ln(1 − r) + a\space ln(wo_A) + (1 − a)\space ln(wo_B )$ - 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 $\frac{a}r-\frac{1-a}{1-r}$ - $r = a$ 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 $r_n = a_n$ $\Rightarrow$ Bettor $n$ will bet $a_nw_n$ on horse $A$ and $(1 − a_n)w_n$ on horse $B$ $\Rightarrow$ total amount bet on $A$ is $a_1w_1 + a_2w_2 + · · · + a_Nw_N$ 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 $a_1w_1o_A + · · · + a_N w_N o_A=w$. $\Rightarrow \frac{{a}_{1}w_1}{w} + · · · + \frac{a_N w_N}{w} = o_A^{−1}$ - Let $f_n = w_n/w$, $a_1f_1 + · · · + a_N f_N = o_A^{−1}$ - 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. 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 = 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, $v_1 = 2ρ_1 + ρ_2$ $v_2 = ρ_1 + 2ρ_2$ - for the state prices $ρ_1$ and $ρ_2$. $ρ_1 = \frac{2v_1 − v_2}3$ $ρ_2 = \frac{2v_2 − v_1}3$ ## 22.5 Markets with Endogenous Events - Some events are *endogenous*: 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. 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).  - 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 $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 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 $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 $\Rightarrow$ Buyers are willing to pay $6$ for a car. $\Rightarrow$ Only the sellers of bad cars would be willing to sell $\Rightarrow$ The market would consist only of bad cars $\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: (a) all cars are offered for sale (b) only bad cars and lemons are offered for sale, $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. $\Rightarrow$ the expected value of a car to a buyer would be $\frac{12 + 6 + 0}3 = 6$. This is less than the value that sellers of good cars places on their cars $\Rightarrow$ they would not put them on the market Hence, this is not an equilibrium. - (b) Suppose buyers expect bad cars and lemons to be on the market. $\Rightarrow$ the expected value of a car to a buyer would be $\frac{6 + 0}2 = 3$. This is less than the value that sellers of bad cars place on bad cars $\Rightarrow$ They would not put them on the market Hence, this is not an equilibrium. - $c$ It is clearly an equilibrium if buyers expect only lemons to be sold. 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. 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, 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: $$25,000$ per year - Firm cannot determine the type of each worker. $\Rightarrow$The ﬁrm has to offer a uniform wage of $w$ ### Equilibria in the Labor Market - If the ﬁrm expects all workers to be on the job market - expected revenue per employee will be $\frac{80, 000 + 40, 000}2 = 60, 000$ - At wage $60,000$, both types of workers will be willing to accept, so we have an equilibrium. - If the ﬁrm expects only unproductive workers to be on the job market - It expects to make only $$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. $\Rightarrow$ The average healthy of the remainder goes down $\Rightarrow$ The insurance company would need to set a higher price for remainder. $\Rightarrow$ ... ## 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, 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 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, 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), say $f_1, f_2, ... , f_N$ * A person's initial predicted probability of $A$ winning is $a_1f_1+a_2f_2+...+a_Nf_N$ * 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.