Price Distortions and Public Information: Theory, Experiments, and Simulations

Abstract

This paper studies the effects on the asset price of the introduction of a public signal in the presence of asymmetric private information in a decentralized market. We introduce an artificial market model populated by bounded rational agents with heterogeneous levels of reasoning: sophisticated and naive traders. The model captures the main impacts of public information analyzed in the laboratory experiments reported by Ruiz-Buforn et al. (Higher-order beliefs and overweighting of public information in a laboratory financial market, 2019). Public information, when correct, coordinates market activity, improving price convergence to the fundamentals. By contrast, unwarranted public information pushes prices away from fundamentals. This strong influence of public information on prices is primarily driven by its common knowledge property.

Appendices

7 Public Signal Scenario

1.1 7.1 Sophisticated Proposers

The expected payoff function of a sophisticated proposer has two components: (i) the expected payoff if the offer is accepted multiplied by the probability of acceptance and (ii) the expected payoff if the offer is rejected, i.e., $D_i$. Submitting sell offers:

$$\begin{aligned} \pi ^S(a|D_i)=\sum _j \left( a \; Pr[a<D_j|D_i] + D_i \; Pr[a\ge D_j|D_i] \right) \;. \end{aligned}$$

Submitting buy offers:

$$\begin{aligned} \pi ^S(b|D_i)= \sum _j \Big ( (2 D_i-b) \; Pr[b>D_j|D_i] + D_i \; Pr[b\le D_j|D_i] \Big ) \;. \end{aligned}$$

1.2 7.2 Sophisticated Receivers

This section provides some illustrative examples to clarify the computation of expected payoffs when a sophisticated trader receives an offer. Table 3 lists all inferences that a sophisticated trader can make observing a particular offer, assuming all offers are submitted by naive traders.

Table 3 Sophisticated receivers’ inference about the expected dividend of the proposer

Receiving Buy Offers: An Example ²¹

Let us suppose that a sophisticated trader $S_L$, whose expected dividend is $D_L$ observes a bid. She updates her beliefs and decides whether accepting or rejecting the offer. For instance, in case she observes a bid $b_{H^-}=D_H-\varepsilon$, she infers the type of proposer is a naive whose expected dividend is $D_H$.²²

$$\begin{aligned} \pi ^S( b_{H^-},D_L) = {\left\{ \begin{array}{ll} b_{H^-} &{} \text { accepting the bid} \\ \sum _j D_{Lj} \; Pr[D_j|D_j>b_{H^-}] &{} \text { rejecting the bid} \end{array}\right. } \end{aligned}$$

where $D_j=D_H$ since a naive trader with a high expected dividend is the only trader submitting this offer without incurring in losses. Thus $D_{Lj}=D_{LH}$ refers to the updated expected dividend, where subindex L means her prior expected dividend and H is the guessed proposer’s expected dividend. Her updated expected dividend is

$$\begin{aligned} D_{LH} \equiv E[D=1|x_L,x_H,y]=\frac{1}{1+\left( \frac{1-p}{p}\right) ^{-2+2}\left( \frac{1-q}{q}\right) ^{y}} \;. \end{aligned}$$

In case she observes a bid $b_{M^-}=D_M-\varepsilon$, the type of proposer might be M or H.

$$\begin{aligned} \pi ^S( b_{M^-},D_L) = {\left\{ \begin{array}{ll} \begin{aligned} b_{M^-} \end{aligned} &{} \text { accepting the bid} \\ \begin{aligned} D_{LM} \; Pr[D_M|D_M>b_{M^-}]\\ {} +D_{LH} \; Pr[D_H|D_H>b_{M^-}] \end{aligned} &{} \text { rejecting the bid} \end{array}\right. } \end{aligned}$$

where the updated expected dividend is given by

$$\begin{aligned} D_{LM}\equiv E[D=1|x_L,x_M,y]=\frac{1}{1+\left( \frac{1-p}{p}\right) ^{-2+0}\left( \frac{1-q}{q}\right) ^{y}} \end{aligned}$$

and

$$\begin{aligned} D_{LH}\equiv E[D=1|x_L,x_H,y]=\frac{1}{1+\left( \frac{1-p}{p}\right) ^{-2+2}\left( \frac{1-q}{q}\right) ^{y}} \; . \end{aligned}$$

The probability assigned to a proposer of type M given that the receiver has an expected dividend $D_L$ is computed by

$$\begin{aligned} \begin{aligned} Pr[D_M|D_M>b_{M^-}]=\frac{Pr[b_{M^-}|D_M] \;\; Pr[D_M|D_L]}{Pr(b_{M^-}|D_L)}\\ =\frac{\frac{1}{4}2pq[(1-D_L)+D_L]}{\frac{1}{4}[D_L(p^2+2pq)+(1-D_L)(q^2+2pq)]} \end{aligned} \end{aligned}$$

Conversely, she cannot update her beliefs when she observes a bid $b_{L^-}=D_L-\varepsilon$ because any type of trader could submit that offer.

$$\begin{aligned} \pi ^S( b_{L^-},D_L) = {\left\{ \begin{array}{ll} \begin{aligned} b_{L^-} \end{aligned} &{} \text { accepting the bid} \\ \begin{aligned} D_{L} \end{aligned} &{} \text { rejecting the bid} \end{array}\right. } \end{aligned}$$

8 Common Signal Scenario

The analysis of the common signal scenario follows the same structure as the case of PS scenario. The main difference with the PS scenario lies in the sophisticated traders’ strategies. Nonetheless, the lack of common knowledge does not change naive traders’ behavior since they evaluate signals according to their precision about fundamentals. This Appendix explains the main differences in the CS scenario and the results of the model.

1.1 8.1 Sophisticated Traders

Sophisticated traders consider the distribution of information in order to assess market demand. However, contrary to public signal, the common signal does not allow them to better characterized the potential market demand. They estimate the potential demand assuming each trader possesses three independent private signals $\{x_i,y_i\}$ because they are not aware that $y_i$ is identical to all traders. We must redefine, therefore, the expected dividend for a trader of type i as:

$$\begin{aligned} E[D=1|x_i,y_i]=\frac{1}{1+\left( \frac{1-p}{p}\right) ^{x_i}\left( \frac{1-q}{q}\right) ^{y_i}} \end{aligned}$$

(10)

where $x_i=\{-2,0,2\}$ refers to private signals and $y_i=\{-1,1\}$ refers to the common signal. Notwithstanding the common signal is unique for all traders in the market, the sophisticated traders classify traders in four groups according to the four possible expected dividends $\{D_H,D_{\overline{M}},D_{\underline{M}},D_L\}$, corresponding to all the possible combinations of $\chi _i=(x_i+y_i)$.²³ We introduce the notation $\overline{M}$ and $\underline{M}$ to denote the low and high intermediate levels. The variable i takes the values $\{H,\overline{M},\underline{M},L\}$. It is important to stress, however that only three are the levels effectively present in the market. For instance, if the common signal is $y_i=1$, existing types of traders are $\{H,\overline{M},\underline{M}\}$ and the types of traders are $\{\overline{M},\underline{M},L\}$ when common signal is $y_i=-1$. The optimal offer is computed by following the process explained in Sect. 2.2.

In case a sophisticated trader receives an offer, it provides her with new information to be updated. Unlike markets in the PS scenario, she identifies four possible type of proposers $j\in \{H,\overline{M},\underline{M},L\}$, although one of them does not actually exist.

1.2 8.2 Transactions

Tables 4 and 5 list the market transactions when the dividend is $D=1$ and the common signal is correct or incorrect, respectively. The first column denotes the proposer’s type according to his level of reasoning and expected dividend. The second and the third columns show the optimal offer of each trader while the last column shows the counterpart of every transaction.

In order to compare the results between common and public signal, one should consider that when the common signal indicates dividend 1, $j=\overline{M}$ corresponds to the M and $j=\underline{M}$ corresponds to L. If the common signal indicates dividend 0, $j=\overline{M}$ corresponds to the H and $j=\underline{M}$ corresponds to M. We rename the type of traders and offers for each prediction of the common signal $y_i =\{-1,1\}$ for an easier comparison between markets with common signal and markets where the released signal is public. Considering only private signals, the possible types of traders are $\{H,{\overline{M}}, {\underline{M}}\}$ if the common signal predicts dividend 1 (Table 4); otherwise $j\in \{{\overline{M}},{\underline{M}}, L\}$ (Table 5). Considering the previous changes, we define a vector of market prices following the proposer’s type offer in Table 4, $\mathbf {P}=(D_M,D_L,D_L,D_M,D_H,D_M,D_L)$. The vector of transaction prices when the common signal predicts dividend 0 is $\mathbf {P}=(D_M,D_H,D_H,D_M,D_H,D_M,D_L)$, which is listed in Table 5.

Finally, the expected number of transactions per unit of time is listed in Table 6. The mean price is computed by Eq. (6).

Table 4 Transactions when the common signal is 1

Table 5 Transactions when the common signal is $-1$

Table 6 Expected number of transactions per unit of time for every type of trader, given $D=1$ in CS scenario

9 Robustness: Does Market Configuration Matter?

This subsection aims at testing the relevance of the distribution of signals in markets with public information. Intuitively, the proportion of informed traders in the aggregation and dissemination of information matters. For example, an incorrect public signal might largely distort prices when the proportion of informed traders is small. However, an incorrect public signal should be harmless when most of the traders are informed. Since the most concerning case is the impact of an incorrect public signal, we restrict our attention to the PS scenario to assess the importance of market configuration. We define three market configurations based on observed distributions of information across traders in the laboratory experiment. (i) Config. 1, markets are populated by 5 uninformed and 10 informed traders. (ii) Config. 2, markets are populated by 1 misinformed trader, 7 uninformed traders and 7 informed traders. (iii) Config. 3 where markets are populated by 2 misinformed, 5 uninformed and 8 informed traders.

Figure 6 shows that mean prices change depending on the distribution of private information. When the public signal is correct, one can see that the computational mean takes similar values to the theoretical prediction in markets where uninformed and misinformed traders have a large presence (Config. 2 and Config. 3). For the markets with an incorrect public signal, the public signal always dominates the mean price. The impact is larger when the proportion of informed traders is small (Config. 2 and Config. 3).

Fig. 6

Mean price of the market configurations assuming dividend $D=1$. Shaded area shows 1 standard deviation of the Monte Carlo simulations

Altogether, we can claim that the market configuration can generate systematic deviations from the theoretical prediction, however “not too large”, i.e., the general conclusions still hold. A special case seems to be the configuration where there is absence of misinformed traders. The mean price is noticeably higher than the other market configurations, independently of the prediction of the released signal $y=\{1,-1\}$. Besides, it is interesting to note that there are no transactions when $\theta =1$. Therefore, if a market where all traders are sophisticated and none is misinformed, we have no transactions.