2.10.5 Failure Modes and Desk Practice

Basic Prerequisites: 04 - Predatory Trading & Execution Games.

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1. Intuition & Practical Objective

Every model in this folder is an equilibrium of a game whose payoffs are assumed to be common knowledge. That is a much stronger demand than “the parameters are roughly right”, and it fails in a characteristically nasty way: when the game’s payoffs are mis-specified, the equilibrium you are playing is not the equilibrium you solved, and the error does not show up as noise - it shows up as a systematic transfer of money, every trade, in the same direction.

This page names the failures precisely, gives the first-principles reason for each, and quantifies the dollar cost of the two that matter most: mis-specifying the adverse-selection equilibrium, and mis-specifying the informed share that the equilibrium takes as given.

The six failures, one line each:

1. Adverse selection is estimated, not observed. The equilibrium spread is linear in $\pi$; a wrong $\pi$ is a first-order money error, not a variance term.
2. The equilibrium spread is state-dependent, and desks quote constants. $A(\theta )-B(\theta )$ collapses toward zero as the belief sharpens, so a constant spread is wrong on every trade even if it is right on average.
3. The single informed trader is a fiction. With $K$ informed participants the revelation rate scales with $K\beta$ and $\lambda$ falls; calibrating a single-insider $\lambda$ over-states impact (page 06).
4. The game is non-stationary. $\pi$, ${\Sigma }_0$, ${\sigma }_u^2$ and the impact coefficients all move with news, volatility and time of day. A static equilibrium is a snapshot of a moving object.
5. Toxicity proxies are contested. PIN and VPIN are the practical surrogates for $\pi$, and the empirical literature does not agree that they measure it (Easley–López de Prado–O’Hara 2012 vs Andersen & Bondarenko 2014).
6. Predation is real but rarely identified. The mechanism’s signatures are indistinguishable from momentum and news in most data.

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2. Mathematical Ground Truth & Derivations

2.1 The cost of a mis-specified informed share. The maker’s equilibrium quotes are $A(\theta ;\hat{\pi })$ and $B(\theta ;\hat{\pi })$ - but the world draws trades with ${\pi }_{\text{true}}$. The maker’s expected profit per trade is

$$\mathbb{E}[\text{P\&L}\mid \hat{\pi },\pi ]=\frac{1}{2}{n}_A(\theta ;\hat{\pi })[A(\theta ;\hat{\pi })-\mathbb{E}[V\mid B;\pi ]]+\frac{1}{2}{n}_B(\theta ;\hat{\pi })[\mathbb{E}[V\mid S;\pi ]-B(\theta ;\hat{\pi })],$$

where the expectations on the right use the true $\pi$. When $\hat{\pi }=\pi$ it is identically zero. When $\hat{\pi }\ne \pi$ it is not second-order:

• $\hat{\pi }<\pi$: quotes too tight. The maker is picked off on the informed flow, and loses at a rate proportional to the shortfall in the spread - the P&L is strictly negative and grows linearly with $\pi -\hat{\pi }$.
• $\hat{\pi }>\pi$: quotes too wide. The maker earns a positive spread, but on these numbers only a small fraction of the shortfall, because the conditional losses to the informed flow are already sunk. In a model with competing makers the wide quotes lose order flow and the profit vanishes.

Crucially, in this game the loss is not a sampling error that averages out over a day. It is the same signed amount on every trade until the maker updates $\hat{\pi }$. This is the practical meaning of “adverse selection is an equilibrium rather than an assumption”: get the equilibrium wrong and you are systematically on the losing side of a transfer.

2.2 Why a constant spread cannot be the equilibrium. With $n_A=(1+\pi )\theta +(1-\pi )(1-\theta )$ and $n_B=(1-\pi )\theta +(1+\pi )(1-\theta )$,

$$A(\theta )-B(\theta )=\frac{V_H(1+\pi )\theta +V_L(1-\pi )(1-\theta )}{n_A}-\frac{V_H(1-\pi )\theta +V_L(1+\pi )(1-\theta )}{n_B}.$$

At $\theta =\frac{1}{2}$ this is $\pi (V_H-V_L)$. At the extremes $n_A$ and $n_B$ diverge asymmetrically and the spread collapses: as $\theta \to 1$ both $n_A\to 1+\pi$ and $n_B\to 1-\pi$, and the two conditional expectations converge on $V_H$. Numerically (§3): the spread falls from $0.4000$ at $\theta =\frac{1}{2}$ to $0.0785$ at $\theta =0.95$. A desk quoting the average spread is quoting too wide when the market is already informed and too tight when it is not - and the too-tight end is where the money is lost.

2.3 The toxicity of a fill is state-dependent. We derived

$$\mathbb{P}(\text{informed}\mid B)=\frac{2\pi \theta }{n_A},\mathbb{P}(\text{informed}\mid B){|}_{\theta =1/2}=\pi .$$

So the answer to “what was the chance my last fill was informed?” is not a constant: it is $0.0909$ at $\theta =0.20$, $0.2000$ at $\theta =0.50$ and $0.2857$ at $\theta =0.80$ for $\pi =0.2$. Any execution logic that treats toxicity as a fixed property of the name is mis-specified; toxicity is a property of the state.

2.4 Non-stationarity: everything moves together. ${\Sigma }_0$ (prior uncertainty), ${\sigma }_u^2$ (noise volume) and $\pi$ (the informed share) are correlated in the worst possible direction. On a news day, ${\Sigma }_0\uparrow$ and $\pi \uparrow$ simultaneously, so $\lambda =\frac{1}{2}\sqrt{{\Sigma }_0/{\sigma }_u^2}$ and the spread $\pi (V_H-V_L)$ both jump - in the direction that makes a yesterday-calibrated model under-charge on both. And $\lambda =\sqrt{{\Sigma }_0/({\sigma }_u^{2}T)}$ in continuous time depends on the horizon, so the same flow implies different impact at different measurement horizons. A single $\lambda$ is a statement about a (name, regime, horizon) triple.

2.5 Estimation: PIN, VPIN and the identification problem. The practical surrogates for $\pi$ are:

• PIN (Easley–Kiefer–O’Hara–Paperman 1996): a three-parameter mixture ($\alpha$ = information-event probability, $\mu$ = informed arrival rate, $\epsilon$ = noise arrival rate) estimated by maximum likelihood, giving $\text{PIN}=\alpha \mu /(\alpha \mu +2\epsilon )$. This is an equilibrium-based estimator, but its identification rests on the assumption that information events occur once per day and that buy/sell classification is correct.
• VPIN (Easley–López de Prado–O’Hara 2012): the volume-bucketed order-imbalance $\mathbb{E}[|V_B-V_S|]/V$. Cheap, high-frequency, and contested - Andersen & Bondarenko (2014) show it has almost no power to predict toxicity and that its “flash crash” evidence is a construction artefact.

The lesson is not “use a better proxy”. It is that the $\pi$ the equilibrium needs is a latent payoff-relevant object, and every observable surrogate for it is a noisy, contested statistic. The discipline is to treat the spread as a range and to re-estimate $\pi$ continuously, not to fix it.

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3. Computational Implementation - the cost of getting the game wrong

Stdlib only. Part 1 tabulates the equilibrium spread against the maker’s belief for three different assumed $\pi$; Part 2 simulates the maker’s P&L per trade over many days when it assumes $\hat{\pi }$ but the world runs at ${\pi }_{\text{true}}=0.2$; Part 3 shows the toxicity of a fill is state-dependent.

What this measures:

1. The equilibrium is correct exactly where it should be. At $\hat{\pi }={\pi }_{\text{true}}=0.2$ the maker’s P&L per trade is $+$$0.00029 - zero to within Monte-Carlo noise on 480,000 trades. That is the validation of the whole framework: get the game right and the maker breaks even.
2. Getting it wrong is a first-order, signed cost. Quoting as if there were no informed flow ($\hat{\pi }=0$) costs the maker $-$ $0.19816 per trade, on 100% of trades. At a modest 10,000 shares/day that is -$1,982/day, or roughly -$0.5m/year - from a single scalar mis-estimate, with no variance reduction available.
3. The asymmetry is brutal and worth internalising. Under-estimating $\pi$ (too-tight quotes) costs $-0.198$. Over-estimating it ($\hat{\pi }=0.5$) “earns” only $+0.023$ - a tenth of the magnitude, and that is before accounting for the order flow a wide quoter loses. The payoff to a maker is convex in the wrong direction: being too tight is punished far harder than being too wide is rewarded.
4. The spread is a state-dependent object. The true equilibrium spread runs $0.4000$ at $\theta =0.5$ down to $0.0785$ at $\theta =0.95$, i.e. it falls by a factor of 5 as the belief sharpens. The two mis-specified columns show the cost of a constant-spread assumption: quoting the $\hat{\pi }=0.5$ spread at $\theta =0.95$ ($1.0000$) is more than 12x the true equilibrium ($0.0785$).
5. Toxicity moves with the state, not with the name. $0.0909\to 0.2000\to 0.2857$ as $\theta$ moves from $0.20$ to $0.80$, at a fixed $\pi =0.2$. Any flow-toxicity metric that is a property of the instrument rather than of the current belief is measuring the wrong thing.

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4. Failure Modes & First-Principles Breakdowns

1. Equilibrium mis-specification is a transfer, not a variance. Cost: $-$$0.198/trade for $\hat{\pi }=0$ against ${\pi }_{\text{true}}=0.2$; zero at $\hat{\pi }={\pi }_{\text{true}}$. First-principles cause: the spread is linear in $\pi$, so the error is linear in the error of $\pi$ - there is no second-order protection anywhere in this game.
2. The state-dependence of the equilibrium is ignored in practice. Spreads collapse by 5x as beliefs sharpen, and toxicity moves by 3x. First-principles cause: the equilibrium quotes are conditional expectations, and conditionals are functions of the belief, not constants. Fix: quote off your belief, and re-estimate the belief continuously.
3. Single-insider calibration over-states impact. With $K$ informed traders the information is impounded at rate $K\beta$: ${\Sigma }^{\prime }={\Sigma }_0{\sigma }_u^2/((K\beta {)}^2{\Sigma }_0+{\sigma }_u^2)$ falls to $50\%$, $80\%$, $90\%$, $96\%$ of the prior for $K=1,2,3,5$ (page 06). A $\lambda$ fitted assuming one insider is too large for a market with many. First-principles cause: competition among the informed accelerates revelation, which deepens the market.
4. Non-stationarity has a signature direction. On stress days ${\Sigma }_0$ and $\pi$ rise together, so both $\lambda$ and the spread increase - exactly when you most need your model to be right. First-principles cause: liquidity provision withdraws under uncertainty (the same adverse-selection logic as Liquidity Risk & Funding). Fix: state-dependent parameters, re-solved intraday.
5. Toxicity proxies are contested and fragile. PIN requires an MLE with an unobservable information-event structure; VPIN is cheap but Andersen & Bondarenko (2014) find it has negligible predictive power and that its flash-crash evidence is constructed. First-principles cause: these are proxies for a latent equilibrium object; the map from data to $\pi$ is not invertible without strong assumptions.
6. Predation and momentum are observationally close. Both produce elevated volume and a directional price move in stressed names. First-principles cause: the models differ in who knows what, which is unobserved; identifying predation requires an instrument for the victim’s forced-ness that data rarely provides.
7. Zero-profit quotes assume free entry of liquidity providers. With finite maker capital the equilibrium spread sits above zero-profit, and the “fair” spread computed here becomes a lower bound rather than the market’s spread. Confusing the two under-estimates the cost of provision.
8. The models are silent on venue. All of the machinery above is single-venue. Fragmentation, dark venues, and fee structures change the equilibrium by changing what a maker can condition on (see Smart Order Routing & Fragmentation).

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5. References

• Easley, David; Kiefer, Nicholas M.; O’Hara, Maureen; Paperman, Joseph B. - “Liquidity, information, and infrequently traded stocks,” Journal of Finance 51(4), 1405–1436 (1996). PIN: the MLE estimator of the informed share the games take as given.
• Easley, David; López de Prado, Marcos; O’Hara, Maureen - “Flow toxicity and liquidity in a high-frequency world,” Review of Financial Studies 25(5), 1457–1493 (2012). VPIN.
• Andersen, Torben G.; Bondarenko, Oleg - “VPIN and the flash crash,” Journal of Financial Markets 17, 1–46 (2014). The critique: VPIN’s predictive power and its flash-crash evidence both fail under scrutiny. Read it before using any toxicity number.
• Glosten, Lawrence R.; Harris, Lawrence E. - “Estimating the components of the bid/ask spread,” Journal of Financial Economics 21(1), 123–142 (1988). *How to decompose a measured spread into adverse-selection and other components
• Hasbrouck, Joel - Empirical Market Microstructure (2007)
• Almgren, Robert; Chriss, Neil - “Optimal execution of portfolio transactions,” Journal of Risk 3(2), 5-40 (2000)
• Brunnermeier, Markus K.; Pedersen, Lasse Heje - “Predatory trading,” Journal of Finance 60(4), 1825–1863 (2005). The theory that is hardest to identify empirically.
• Andersen, Torben G.; Bondarenko, Oleg

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6. Connected Graph Bridges

• Back: 04 - Predatory Trading & Execution Games · 03 - Glosten–Milgrom · Index Hub
• Forward: 06 - Advanced Extensions
• Measurement of the inputs: Toxic Order Flow & VPIN · Spread Decomposition & the Roll Model · Market Impact & Depth
• Model risk discipline: Model Risk & Validation
• The non-strategic failure list, for comparison: Almgren–Chriss: Failure Modes & Practice