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A theory of information inefficiency of markets

Conventional wisdom is that markets are information efficient. Alas, a simple game-theoretical model illustrates that value traders only have an incentive to invest in research and information if (i) information cost is low enough, (ii) the overall market is sufficiently clueless, and (iii) market makers do not suspect value traders of being well informed. This leaves ample scope for the overall market to remain inefficient, even in the long run, with undesirable consequences for society as a whole.

“Informational and Allocative Efficiency in Financial Markets with Costly Information”
Arina Nikandrova
Birkbeck Working Papers in Economics & Finance 1403, March 2014

The below are excerpts from the paper. Cursive lines and emphasis have been added.

Understanding the basics of the model

“This paper studies costly information acquisition in a version of sequential trading model first introduced by Glosten and Milgrom (1985)… The model constitutes a sequential game of incomplete information.””The model considers a market for a single risky asset with a binary liquidation value [meaning that physical net assets may be worth either USD1 or nothing, with market participants not knowing for sure]. It is assumed that all trades are mediated by a competitive market maker who sets the prices on the basis of the order flow.””There is a fixed proportion of value traders who, prior to trading, have a choice of whether or not to obtain a costly signal [which has an accuracy somewhere between 50% and 100% in indicating the correct liquidation value]. The remaining traders are liquidity traders who are price sensitive and do not care about the fundamental. [Each liquidity trader values the asset somewhere between USD0 and 1]””Once a signal is acquired, its cost is sunk and does not affect the trading decision of a value trader. Hence, the optimal trading decision for a value trader who observes signal…is to place a buy order if the expected value of the asset conditional on [the signal] weakly exceeds the current ask price…Upon acquiring a signal, value traders are always active in the market, i.e., depending on the value of the observed signal, a value trader entering the market in period t places either a buy or a sell order.”

“Any trader who has observed a signal is better informed than the market maker, and the market maker makes an expected loss on a trade with this trader. Therefore, in equilibrium, the market maker sets bid and ask prices so that the expected profits on trades with liquidity traders are just sufficient to compensate for the expected losses on trades with informed payoff maximisers.”

Understanding information inefficiency

“In the presence of fixed costs to information acquisition, some valuable information never reaches the market and hence prices cannot be fully informationally efficient, i.e. prices cannot converge to the liquidation value of the underlying asset. This outcome echoes the idea of Grossman and Stiglitz (1980) that there must be some equilibrium level of inefficiency in the market or otherwise equilibrium may fail to exist. In particular, in equilibrium, information must generate returns just sufficient to compensate for the costs associated with the gathering of this information.”

“Each value trader bases his decision of whether to acquire additional information on the expected payoff from signal acquisition, which is equal to the expected benefit of a signal minus its cost. While the cost of a signal is fixed…the expected benefit depends on the current public belief and the current bid and ask prices. These prices, in turn, depend of the market maker’s belief regarding the information acquisition decisions of value traders…

  • For every belief of the market maker [of the probability that value traders have acquired a signal] signal acquisition [by the value traders] can be profitable if and only if the public belief [regarding the expected liquidation value of the asset] is sufficiently close to USD0.5 [implying that the public does not have a strong view]…[Signal acquisition also] requires that the cost of signal acquisition be not too large…
  • From the perspective of the market maker, when value traders acquire signals with high probability, any buy or sell order has high informational content. Consequently, when [market makers expectation of value traders being informed] is high, the bid and ask spread is high as well.”

“With fixed costs to information acquisition, value traders find it optimal to stop obtaining signals when the asset price is sufficiently close to one of the extremes [of the liquidation value being USD1 or zero] and the benefit of a signal is outweighed by its costs. Thus there are two threshold values of the public belief about the likely asset value at which signal acquisition stops. As soon as the public belief reaches one of these thresholds, trades become uninformative and the market maker stops updating the price. As a result, the price gets trapped at a given level that may or may not be close to the liquidation value of the asset.”

“Conditional on the true asset value, the probability of price getting stuck far away from the liquidation value of the asset is strictly positive, indicating that costs to information acquisition may bring about informational inefficiency thereby prices do not reveal the value of the fundamental, even in the long-run.”

“Information acquisition does not stop abruptly. Instead, as the public belief regarding the asset value moves closer to the extreme values of 0 or 1, the probability with which value traders acquire signals decreases gradually and eventually falls to zero… as the public belief moves closer to the extremes, the bid and ask spread also becomes narrower.”

The consequences of information inefficiency

“Information about the fundamental is socially valuable. However, value traders, when they decide whether to acquire informative signals, do not take into account the positive externality their decisions exert on the payoff of liquidity traders. This suggests that in the long-run, market solution may underprovide information relative to a social optimum.”


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