Polluted Information Equilibrium

• Polluted Information Equilibrium is an equilibrium concept where decisions are based on signals that are both informative about fundamentals and contaminated by noise or strategic interventions.
• It spans diverse models—from Bayesian coordination games to noisy information-sharing and AI markets—demonstrating how endogenous information pollution alters agent beliefs and coordination.
• The concept reveals that polluted signals can reverse standard selection logic, leading to multiple equilibria and influencing welfare, policy design, and market efficiency.

Polluted Information Equilibrium denotes equilibrium objects in which decisions are conditioned on information that is simultaneously informative about an underlying state and contaminated by noise, indirect observation, strategic mediation, or endogenous amplification. In recent arXiv work, the term is used in Bayesian coordination games with noisy observation of others’ actions, in global games with noisy information sharing, and in an AI-transformed information economy; a closely related information-fusion game studies the same structural issue through competing noisy senders and a Bayesian receiver (Grafenhofer et al., 2021, Mahdavifar et al., 2015, Zhang et al., 17 Sep 2025, Brody, 14 Jun 2026). Taken together, these formulations suggest a family of equilibrium concepts rather than a single canonical definition.

1. Conceptual scope and formal variants

The common feature across these models is that equilibrium cannot be characterized solely from a clean signal of fundamentals. The informational environment itself is endogenous or contaminated: observing aggregate action reveals something about fundamentals while also affecting coordination; receiving others’ signals through a noisy network distorts higher-order beliefs; competing senders alter the effective signal-to-noise ratio faced by a receiver; and AI changes the relative cost of producing low- and high-quality information, thereby altering the equilibrium composition of the information environment.

Setting	Polluted object	Equilibrium object
Bayesian coordination game	Observation of aggregate action that also reveals fundamentals	Steady-state aggregate attack function $A(\theta)$
Noisy information-sharing global game	Shared private observations contaminated by relay noise	Threshold function $h(y_i)$ solving a fixed point
Two-sender information game	Effective fused signal from correlated noisy channels	Nash equilibrium on $[0,1]^2$
AI information market	Content ecosystem with excessive low-quality supply	SPNE of platform, producers, consumers

This breadth matters because “pollution” does not have a single technical meaning. In some papers it refers to noisy or contaminated signals at the agent level; in others it refers to a market-level composition effect in which the information environment is diluted by low-quality content. A plausible implication is that the term marks a structural departure from benchmark models in which information enters only as exogenous private noise.

2. Polluted action observation in Bayesian coordination games

In "Observing Actions in Global Games" (Grafenhofer et al., 2021), a unit mass of agents $i\in[0,1]$ choose $a_i=1$ or $a_i=0$, with aggregate action

$A=\int_0^1 a_i\,di.$

The status quo fails if $A>\theta$, where $\theta$ is the strength of the status quo. Attackers obtain $1-c$ if the attack succeeds and $h(y_i)$0 if it fails, with $h(y_i)$1, so expected utility is

$h(y_i)$2

The prior over $h(y_i)$3 is uniform uninformative.

The benchmark Morris–Shin signal is

$h(y_i)$4

with precision $h(y_i)$5. Under a cutoff strategy $h(y_i)$6 iff $h(y_i)$7, the induced attack mass is

$h(y_i)$8

and the paper states Proposition 0: there exists a unique equilibrium $h(y_i)$9. This reproduces the standard global-games uniqueness result.

The paper’s extension adds action observation. In the simplest one-signal formulation,

$[0,1]^2$0

with $[0,1]^2$1. In the more general two-signal formulation, each player observes

$[0,1]^2$2

with joint density $[0,1]^2$3. The best response is to attack iff

$[0,1]^2$4

Equilibrium is a consistency condition for the aggregate attack schedule $[0,1]^2$5: given conjectured $[0,1]^2$6, the fraction of agents whose posterior success probability exceeds $[0,1]^2$7 must reproduce that same aggregate attack.

The substantive result is that polluted action information can reverse the global-games selection logic. In the one-signal normal case, aggregate attack satisfies

$[0,1]^2$8

and the paper proves that if private information is precise enough, $[0,1]^2$9, agents can coordinate on multiple equilibria. More generally, if $i\in[0,1]$0 and $i\in[0,1]$1, there exists a continuum of equilibria; with independent symmetric densities $i\in[0,1]$2, the paper also provides sufficient inequalities under which a continuum of equilibria exists. The candidate family is indexed by thresholds $i\in[0,1]$3, with

$i\in[0,1]$4

The mechanism is explicit. Information about fundamentals tends to support uniqueness, whereas information about others’ actions supports coordination and can restore multiplicity. The paper’s applications—bank runs, currency crises, recessions, riots and revolutions, and coordinated projects—are all environments in which agents monitor one another closely. The central point is that observing the crowd is not merely another signal of $i\in[0,1]$5; it is also direct coordination-relevant information.

3. Noisy information sharing and generalized threshold equilibria

"Global Games with Noisy Information Sharing" (Mahdavifar et al., 2015) studies a static $i\in[0,1]$6-agent coordination game in which agent $i\in[0,1]$7 chooses $i\in[0,1]$8 and receives payoff

$i\in[0,1]$9

In the standard version, each agent observes $a_i=1$0 with i.i.d. Gaussian noise, and the conventional result is a symmetric threshold equilibrium based on the private estimate of $a_i=1$1.

The paper alters the information structure by allowing agents to share signals in a noisy environment. Agent $a_i=1$2’s information is

$a_i=1$3

where

$a_i=1$4

and $a_i=1$5 are i.i.d. and independent of the $a_i=1$6’s. The aggregate information received from others is

$a_i=1$7

A key lemma gives the posterior representation

$a_i=1$8

where $a_i=1$9 is independent of $a_i=0$0, and hence

$a_i=0$1

The first major result is negative. Theorem 1 states that there do not exist threshold values $a_i=0$2 such that the intuitive policy

$a_i=0$3

is a Bayesian Nash equilibrium. The reason given is that best responses depend not only on the agent’s posterior over $a_i=0$4, but also on beliefs about others’ beliefs and actions under polluted signal exchange.

The equilibrium concept is therefore broadened to threshold functions of the full information vector: $a_i=0$5 where $a_i=0$6 is continuous, strictly decreasing in each argument, and symmetric in the specified sense. The equilibrium boundary condition is

$a_i=0$7

and the fixed-point condition is

$a_i=0$8

Theorem 2 states that such an $a_i=0$9 leads to a threshold policy equilibrium if it solves this functional fixed point equation.

Existence and uniqueness are then obtained under sufficiently noisy sharing. Theorem 3 gives the sufficient condition

$A=\int_0^1 a_i\,di.$0

under which there exists a continuous threshold function $A=\int_0^1 a_i\,di.$1 leading to equilibrium; the equilibrium is unique in the relevant function class, and the threshold can be approximated by iterating

$A=\int_0^1 a_i\,di.$2

The contraction logic is central: if sharing is too accurate, higher-order belief feedback may be too strong; if sharing is noisy enough, the operator becomes contractive. This directly parallels, at a different formal level, the idea that information pollution can damp strategic amplification rather than merely degrade accuracy.

4. Competing senders, effective information, and equilibrium on the square

"A game of information" (Brody, 14 Jun 2026) does not present a formal equilibrium concept named Polluted Information Equilibrium, but it studies a closely related mechanism in which information reaching the receiver is filtered through noisy, strategically chosen channels. The hidden state is binary, $A=\int_0^1 a_i\,di.$3, with prior $A=\int_0^1 a_i\,di.$4, and two players broadcast

$A=\int_0^1 a_i\,di.$5

where $A=\int_0^1 a_i\,di.$6 are chosen signal-to-noise ratios and the Brownian noises have correlation $A=\int_0^1 a_i\,di.$7.

Because both streams concern the same $A=\int_0^1 a_i\,di.$8, the receiver fuses them into one effective process

$A=\int_0^1 a_i\,di.$9

with

$A>\theta$0

The receiver’s posterior is

$A>\theta$1

At decision date $A>\theta$2, the players oppose one another over

$A>\theta$3

and the paper shows that this probability can be written as

$A>\theta$4

with

$A>\theta$5

The strategic problem therefore reduces to an elementary infinite game on $A>\theta$6. Since $A>\theta$7 is convex in each control, and since the objective is monotone in $A>\theta$8, the equilibrium classification depends on $A>\theta$9. For $\theta$0, Proposition 1 gives a complete solution: if $\theta$1, then $\theta$2; if $\theta$3, then $\theta$4; if $\theta$5, player A uses $\theta$6 and player B randomizes between $\theta$7 and $\theta$8 with probabilities $\theta$9 and $1-c$0. If $1-c$1, the equilibrium strategies are flipped.

The paper states that the effective-signal formula is the key “polluted information equilibrium” mechanism in the baseline model. In this usage, pollution is generated not by passive measurement error alone but by strategic interaction through correlated noisy channels. The disinformation extension,

$1-c$2

adds a bias term that shifts the fused signal and breaks the clean symmetry of the baseline game. The paper leaves a general Nash existence and characterization theorem open in that asymmetric setting.

5. Polluted information equilibrium in AI information markets

"The Economics of Information Pollution in the Age of AI: A General Equilibrium Approach to Welfare, Measurement, and Policy" (Zhang et al., 17 Sep 2025) defines Polluted Information Equilibrium explicitly as a decentralized outcome in an AI-transformed information market characterized by excessive low-quality content, diluted signal quality, and under-provision of verification. Formally, it is a Subgame Perfect Nash Equilibrium of a three-stage game,

$1-c$3

in which the platform chooses moderation and amplification, producers choose supply, and consumers choose verification.

The platform is monopolistic and chooses moderation intensity $1-c$4 and amplification weights $1-c$5 to maximize

$1-c$6

Given platform policy, producer $1-c$7 obtains per-unit profit

$1-c$8

and aggregate supply functions satisfy

$1-c$9

Consumers observe a noisy signal $h(y_i)$00, decide whether to verify, and the precision of the signal depends on pollution and collective verification: $h(y_i)$01

The core technological assumption is a CES production function,

$h(y_i)$02

with

$h(y_i)$03

This formalizes that AI and labor are gross substitutes in low-quality production and gross complements in high-quality production. The paper proves the cost asymmetry

$h(y_i)$04

with unit cost

$h(y_i)$05

On existence, the body states a theorem proving that a Polluted Information Equilibrium exists under standard regularity conditions, via backward induction using Brouwer’s fixed-point theorem for the consumer stage, the Theorem of the Maximum for producer continuity, and the Weierstrass theorem for the platform stage. The abstract further claims existence of a unique Polluted Information Equilibrium, while the displayed theorem in the excerpt explicitly proves existence.

The equilibrium is Pareto inefficient because of three market failures: a production externality, a platform governance failure, and an information commons externality. The platform may choose

$h(y_i)$06

and verification is underprovided because it is a public good. A key comparative-static result, labeled the “paradox of AI progress,” is

$h(y_i)$07

Interpreting $h(y_i)$08 as the AI capital cost, a fall in $h(y_i)$09 increases low-quality supply, raises effective pollution density, and lowers decentralized welfare.

The measurement device proposed in the paper is the welfare-weighted Information Pollution Index,

$h(y_i)$10

with endogenous welfare weights

$h(y_i)$11

Its four dimensions are Effective Pollution Density, Social Welfare Deadweight Loss, Decay of the Trust Commons, and Asymmetric Technology Risk. Policy analysis then argues that first-best requires a portfolio consisting of a Pigouvian tax on low-quality output, mandatory content provenance standards, and information fiduciary duties. The paper further advocates adaptive governance using real-time IPI readings so that policy tightens when pollution rises above target and relaxes when pollution falls below target.

6. Cross-cutting mechanisms, misconceptions, and research significance

Across these papers, polluted information equilibrium is best understood as an equilibrium in which informational contamination changes the mapping from signals to actions and, in some cases, the composition of the informational environment itself. The equilibrium objects differ—a fixed point in aggregate attacks, a fixed point in threshold functions, a Nash equilibrium over signal intensities, and an SPNE in a three-stage market game—but each model places strategic weight on information that is not cleanly exogenous (Grafenhofer et al., 2021, Mahdavifar et al., 2015, Brody, 14 Jun 2026, Zhang et al., 17 Sep 2025).

Several recurring mechanisms emerge. First, pollution alters higher-order beliefs. In noisy information-sharing games, the failure of the intuitive posterior-threshold rule shows that equilibrium depends on how agents think others interpret contaminated signals. Second, pollution can either damp or amplify coordination. Noisy sharing in the threshold-function framework supports uniqueness through contraction, whereas precise observation of others’ actions in coordination games generates multiple fixed points. Third, pollution need not mean low precision in a narrow statistical sense. In the two-sender game, strategic channel choice and correlation determine the effective information flow; in the AI market model, pollution refers to an equilibrium overproduction of low-quality content rather than a single noisy signal.

A common misconception is that more information necessarily restores the classical global-games selection logic. The coordination model with action observation shows the opposite: sufficiently precise observation of others’ actions can reintroduce multiplicity. Another misconception is that information pollution is reducible to sender deception. The AI market model instead treats pollution as an equilibrium consequence of cost asymmetries, platform incentives, and under-provided verification, even absent a single deceptive sender. Conversely, the two-sender disinformation extension shows that explicit bias can be added to an already strategically polluted information structure.

Taken together, these results suggest that polluted information equilibrium is a unifying label for settings in which the informational substrate is itself strategic, contaminated, or endogenously distorted. Its analytical significance lies in showing that equilibrium selection, uniqueness, multiplicity, welfare, and policy all depend not only on the amount of information available, but on the manner in which that information is generated, transmitted, shared, observed, and amplified.