6.4.2 Informed vs Uninformed Traders

Basic Prerequisites: Bayesian Statistics and 01 · From Zero.

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1. Intuition & Practical Objective

Every trade is an unresolved mystery: is this buyer informed (they know the price is going up) or uninformed (they just need to sell for cash)? The two types are observationally tangled - both hit your ask. This page is about type detection: how to use the pattern of order flow to update your belief that you are trading against someone who knows something.

The objective: build the inference machinery that the market maker uses every round. It has three ingredients:

1. The informed trader is directionally committed. They trade only in the direction of their (private) information. Uninformed traders’ directions are coin flips. So a concentrated run of one-sided buys is statistically much more likely to be informed-driven than a balanced mix.
2. The prior matters. If you believe insiders are rare ($\pi$ small), a single buy is weak evidence; a streak of buys is strong. The Bayesian posterior compresses all of this into one number: ${\theta }_t=\mathbb{P}(V=V_H\mid \text{orders seen so far})$.
3. Not every buy is toxic. $\mathbb{P}(\text{informed}\mid \text{buy})$ is generally larger than $\pi$ - a buy is a mild signal - but it is only overwhelming for large $\pi$ or long streaks. This quantification is the core of “toxicity.”

The tester’s habit. Whenever you see flow, first ask what would posterior $\theta$ be if this trader were uninformed? - the answer is “noisy around ${\theta }_{t-1}$.” Informed traders push $\theta$ in a consistent direction. Deviation from the noise benchmark is the signal.

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2. Mathematical Ground Truth & Derivations

In the GM economy, informed traders buy with probability $1$ when $V=V_H$ and sell with probability $1$ when $V=V_L$; uninformed buy or sell with probability $\frac{1}{2}$. The condition matters: an informed trader buys only in the high state, so

$$\mathbb{P}(B\mid I)=\mathbb{P}(V_H\mid I)={\theta }_{t-1},\mathbb{P}(B\mid U)=\frac{1}{2},\mathbb{P}(B)=\pi {\theta }_{t-1}+(1-\pi )\frac{1}{2}.$$

(Equivalently, conditioned on the state, $\mathbb{P}(B\mid V_H)=\pi +(1-\pi )\frac{1}{2}=\frac{1+\pi }{2}$ - the arrival law used on 03 · The Glosten–Milgrom Model. Conditioning on $I$ instead would give $\mathbb{P}(B\mid I)=1$, which is only correct in the degenerate $\theta =1$ case.)

Probability that a buy came from an informed trader (Bayes):

$$\mathbb{P}(I\mid B)=\frac{\mathbb{P}(B\mid I)\pi }{\mathbb{P}(B)}=\frac{\pi {\theta }_{t-1}}{\pi {\theta }_{t-1}+\frac{1-\pi }{2}}=\boxed{\frac{2\pi {\theta }_{t-1}}{2\pi {\theta }_{t-1}+1-\pi }},$$

which equals $\pi$ at the symmetric prior ${\theta }_{t-1}=\frac{1}{2}$ and exceeds it only when ${\theta }_{t-1}>\frac{1}{2}$. That is the right reading of the model: at a symmetric prior a buy tells you about $V$, not about whether the trader was informed ($\mathbb{P}(V_H\mid B)>{\theta }_{t-1}$ while $\mathbb{P}(I\mid B)=\pi$). The informed-ness becomes visible in the streak: after a run of buys the conditional $\theta$ rises, and a further buy is then genuinely more likely to be informed flow. At $\pi =0.1$ and $\theta =\frac{1}{2}$: $\mathbb{P}(I\mid B)=0.100$; at $\theta =0.9$ it is $0.167$.

Posterior belief that value is high after observing a buy at prior ${\theta }_{t-1}$ (the GM update, repeated here explicitly):

$${\theta }_t=\mathbb{P}(V_H\mid B_t)=\frac{\frac{1+\pi }{2}{\theta }_{t-1}}{\frac{1+\pi }{2}{\theta }_{t-1}+\frac{1-\pi }{2}(1-{\theta }_{t-1})}.$$

For a run of $n$ buys the update compounds. A sequence of buys moves $\theta \to 1$ (value-revealed as high), a sequence of sells moves it to $0$. The interesting, non-obvious content: the speed of learning is driven by $\pi$. Larger $\pi$ ⇒ each buy is more likely informed ⇒ posterior moves faster (Foucault Ch 3; Hasbrouck Ch 5).

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3. Computational Implementation - type detection and streak diagnostics (stdlib only)

Read the numbers as a toxicity screen: at $\pi =0.10$ a single buy barely moves the posterior ($0.50\to 0.55$), but ten consecutive buys push it to $0.88$ - you should now be quoting a heavily skewed book, not symmetric. Conversely a buy followed by five sells over-shoots the posterior below 0.5 ($\to 0.31$) - classic back-and-forth noise. Directional persistence, not any single order, is what identifies information. This is the seed of order-flow-imbalance (OFI) and Volume-Synchronized probability of toxicity measures.

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4. Failure Modes & First-Principles Breakdowns

1. Treating each order as independent noise. The toxic agent trades in one direction persistently. Averaging over “order flow is balanced” hides the local one-sidedness that is exactly the signal. Use run-lengths / imbalance windows, not per-order classification.
2. Ignoring the base rate. If insiders are genuinely rare ($\pi =0.01$), even a buy streak is weak. Over-flagging noise as toxicity gets you killed on opportunity cost; under-flagging gets you killed by the informed. The posterior quantifies the trade.
3. Believing the market maker “can’t see” the signal. The maker’s belief $\pi$ is what matters - $\pi$ is the maker’s belief about informed probability, which updates with experience (Hasbrouck Ch 5: “the stock doesn’t know that you own it”). A stale $\pi$ makes detection lag reality. The GM dynamics describe beliefs, not the true composition of traders.

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5. References

• Glosten & Milgrom (1985), JFE 14 (sequential Bayes; belief dynamics; martingale transaction prices).
• Hasbrouck (2007), Empirical Market Microstructure
• Easley, Kiefer & O’Hara (1997), The information content of the trading process, JFE 44
• Foucault, Pagano & Röell (2013), Market Liquidity

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6. Connected Graph Bridges

• Back: 01 · From Zero · Index Hub
• Forward: 03 · The GM Model (the full quoting solver)
• Sibling: Toxic Order Flow & VPIN · Order Flow & OFI