Endogenous information acquisition and robustness of selection in the Condie–Yoo market-selection economy with noise traders

Condie and Yoo (2011) extended the Mailath–Sandroni (2003) market-selection economy with asymmetric information by making the event that equilibrium prices reveal a privately observed signal endogenous to the wealth distribution: with noise traders holding a sufficient wealth share, prices can be confounded and fail to reveal. They proved a uniform positive lower bound on the conditional probability of non-revelation and showed that an investor who receives unique private signals infinitely often drives out an investor who receives them only finitely often. However, the information environment itself is exogenous: which agents receive signals in which periods, the specific noise-trader primitives (independent uniform beliefs; wealth redrawn from a continuous full-support CDF), and the binary symmetric-precision signal structure are all imposed. The open problem addressed here is to extend this framework so that information acquisition is itself an equilibrium object — agents choosing whether and when to acquire signals at a cost, with acquisition responding to wealth and prices — and to determine whether the uniform non-revelation bound and the signal-dominance selection results survive endogenous acquisition, alternative noise-trader belief and wealth processes, and richer signal/state structures.

Background

Whether financial markets shift wealth toward traders with more accurate beliefs — the market selection hypothesis — is a classical question of dynamic asset-pricing theory. In the complete-markets economies of Blume and Easley (1992) and Sandroni (2000), investors trade each period and allocate consumption across future states via Arrow securities — assets that pay one unit of consumption in a single future state — and under logarithmic utility each trader bets in the proportions of their own subjective beliefs. A trader's wealth share then grows at a rate governed by how close their beliefs are to the truth, so in the long run the better predictor accumulates all wealth.

Mailath and Sandroni (2003) added asymmetric information to this selection question. One investor ("informed") privately observes a signal about next period's state, an otherwise identical "uninformed" investor does not, and a third group of noise traders — agents whose demands are driven by random beliefs (independent uniform draws on $(0,1)$) and random wealth endowments rather than by maximizing behavior, in the spirit of De Long et al. (1990) — trades alongside them. Equilibrium prices of the Arrow securities are then only partially revealing: depending on the noise traders' realized beliefs and wealth, the same price vector can be consistent with several signal realizations, so the uninformed cannot fully infer the private signal from prices. Mailath and Sandroni showed that whether the informed investor is selected for or against depends on how often, over the infinite horizon, prices fail to reveal the signal — but they imposed this revelation frequency as an exogenous condition rather than deriving it from the economy.

Condie and Yoo (2011) closed part of that gap. In a two-state, two-signal version of the economy (states $s\in\{1,2\}$ and signals $z\in\{1,2\}$ with symmetric precision $x\in(0,1)$, i.e. $\Pr(s=z)=\frac{1}{2}(1+x)$, all independent over time), they derived the revelation event endogenously from the wealth distribution. Their findings include: (a) a partially revealing equilibrium at date $t$ is possible only if the noise traders' wealth share exceeds a threshold,

$r_t^N > \frac{x}{1+x},$

(b) the ex-ante probability (before the noise-trader belief draw) that prices fail to reveal the signal is $1-\frac{r_t^I}{r_t^N}x$, where $r_t^I$ is the informed traders' wealth share; (c) because noise-trader wealth is redrawn each period from a continuous CDF with support $(0,\infty)$, the conditional probability that a received signal is obscured is uniformly bounded below by some $\epsilon>0$ along every history; and (d) consequently an investor who receives unique (exclusive) signals infinitely often drives out an investor who receives them only finitely often, with a multi-agent extension built on the transitive signal-dominance relation $i>_S j$, defined by $S_i\cap S_j^c$ infinite and $S_j\cap S_i^c$ finite, where $S_i$ is the set of periods in which agent $i$ receives a private signal.

What remains exogenous in Condie and Yoo is the information environment itself: the sets $S_i$ (who is informed when), the specific noise-trader primitives (independent uniform beliefs; independent wealth draws from a continuous full-support CDF), and the binary symmetric signal structure. This matters because of the Grossman and Stiglitz (1980) impossibility argument: since information acquisition is costly, fully revealing prices cannot be an equilibrium, so acquisition should itself respond to prices and wealth — the very input of the selection process (who is informed how often) should be an equilibrium object rather than data.

Problem Statement

Extend the Condie–Yoo (2011) economy — the Mailath–Sandroni noise-trader selection framework with endogenous information revelation — so that the information environment is itself an equilibrium object, and determine which of its selection conclusions survive. Concretely, formulate and analyze a general class of economies in this framework that jointly satisfies all three requirements below (one model class general along all three dimensions, not alternative special cases):

1. Endogenous acquisition. The signal-arrival sets $S_i$ — the periods in which agent $i$ privately observes the signal — are choice variables: in each period, agents decide whether (and possibly at what precision) to acquire the private signal, subject to a specified acquisition cost or technology, with acquisition decisions responding to wealth shares and equilibrium prices.
2. General noise-trader primitives. The noise traders' beliefs and wealth shares are arbitrary adapted stochastic processes (possibly history-dependent and non-stationary, defined on a filtered probability space), not necessarily iid uniform beliefs on $(0,1)$ or iid wealth draws from a continuous CDF with support $(0,\infty)$.
3. General signal structure. The state space and signal space are arbitrary finite or measurable sets with arbitrary signal precisions (possibly asymmetric and time-varying), not necessarily binary ($s,z\in\{1,2\}$) with symmetric precision $x\in(0,1)$.

Within this class the acquisition technology and costs, the noise-trader belief and wealth processes, and the signal and state spaces are free parameters: the task is to determine how each conclusion below depends on them, not to verify a single fixed parameterization. Attention is restricted to nondegenerate economies, in which each agent's signal-acquisition set $S_i$ is infinite with positive probability and the noise traders' wealth share stays bounded away from zero along every history; individual periods in which an agent's per-period acquisition probability is zero carry no conditioning event and are excluded from the bound question, and degenerate economies in which prices reveal fully because the noise traders are asymptotically inactive, or in which no agent ever acquires, are outside the intended scope.

For such an economy, the question is whether analogues of the paper's central results hold:

• Uniform non-revelation bound: does there exist $\epsilon>0$ such that, for every period $t$ and along every history, the conditional probability $\Pr(A_t\mid\mathcal{F}_{t-1})$ that a received private signal is not revealed in equilibrium prices exceeds $\epsilon$? (Here $A_t$ is the event that the signal acquired by a given agent at date $t$ is not revealed in that date's equilibrium prices, conditional on that agent acquiring a signal at $t$ — the natural analogue of the paper's per-signal obscuring probability — and $\mathcal{F}_{t-1}$ is the public information — prices and realized states observed so far — available before date $t$.)
• Selection results: does an agent who obtains unique (exclusive) information infinitely often drive out an agent who obtains it only finitely often — i.e., does the driven-out agent's equilibrium wealth and consumption share vanish asymptotically — and does the multi-agent signal-dominance conclusion extend when the dominance relation itself is generated by endogenous acquisition choices?

The two questions comprise three sub-questions: the uniform non-revelation bound; the two-agent selection result, that an agent who obtains unique information infinitely often drives out an agent who obtains it only finitely often (the driven-out agent's equilibrium wealth and consumption share vanishes asymptotically); and the extension of the multi-agent signal-dominance relation to dominance generated by endogenous acquisition. A proof or counterexample that settles only a special parameterization resolves that sub-question and leaves the general characterization open; the general characterization over the class above is the target.

The scientific objective is to characterize long-run market selection when the information environment is endogenous and the primitives general, rather than exogenous and specific, and to establish which of the original assumptions are essential — that is, to identify the exact primitive conditions (acquisition technology and costs, noise-trader belief and wealth processes, signal and state spaces) under which each conclusion holds or fails.

References

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