2.7.2 The Latency Arms Race

Basic Prerequisites: 01 · From Zero and basic game theory (dominant strategy, Nash equilibrium).

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1. Intuition & Practical Objective

Why do HFT firms spend $300 M digging a straighter cable to shave 3 milliseconds off New York–Chicago round-trip time? Because the continuous limit order book is a first-past-the-post institution: it processes messages serially, so somebody is always first, and whoever is first captures the stale-quote arbitrage. That prize is the “rent” of the arms race, and the objective of this page is to show, mathematically and with a simulation, that competition for that rent destroys it.

The core argument, due to Budish, Cramton & Shim (2015, QJE):

1. At human time scales the S&P-instruments ES (Chicago) and SPY (New York) are near-perfectly correlated - the arbitrage “should” not exist.
2. Zoom to high frequency and the correlation breaks down: 0.0923 at 10 ms, 0.0073 at 1 ms (2011). Because each security trades on its own continuous book, and light needs ~4 ms to cross 1,180 km, their prices simply do not move in lock-step at the microsecond frontier.
3. That breakdown creates frequent purely technical arbitrage opportunities - ~1,000/day in ES–SPY, ≈$75 M/yr - available to whomever is fastest.
4. Racing for it is a prisoner’s dilemma: everyone would be better off not racing, but each individual firm deviates and invests anyway.

The one-sentence essence. “A continuous limit-order book hands permanent, renewable arbitrage rents to whoever is marginally fastest; competition for those rents turns the entire stream into deadweight speed investment - a socially wasteful arms race with the prize eventually paid for by fundamental investors via wider spreads.”

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2. Mathematical Ground Truth & Derivations

The rent (BCS §3–§4). From millisecond direct-feed data, ES–SPY arbitrage opportunities number ~1,000/day at a median profit of ~0.08 index points/unit; total ≈$75 M/yr, conservatively (BCS’s back-of-the-envelope extrapolation to the whole universe reaches “billions”). Crucially, the duration of an opportunity collapsed from a median of 97 ms (2005) to 7 ms (2011) - the race didn’t shrink the prize, it just raised the bar (needed speed) to grab it. A mechanically constant prize available only to speed is the definition of a rent.

Tullock contest model of the race. Let $N$ symmetric, risk-neutral, fast firms each invest $x_i$ in speed to win a rent $V$ per period with success probability

$$p_i=\frac{x_i}{{\sum }_{j=1}^N{x}_j}.$$

Firm $i$‘s objective: ${\pi }_i=p_{i}V-x_i$. First-order condition $\partial {\pi }_i/\partial x_i=0$ gives $V\frac{{\sum }_{j\ne i}{x}_j}{({\sum }_j{x}_j{)}^2}=1$. Symmetric equilibrium $x_i=x$ with $\sum x_j=Nx$:

$$x^{\ast }=\frac{N-1}{N^2}V,\text{total investment}=N{x}^{\ast }=\frac{N-1}{N}V,$$

and the fraction of the rent destroyed (spent, not earned) is

$$\boxed{\frac{\text{wasted}}{V}=\frac{N-1}{N}\stackrel{N\to \infty }{\to }1.}$$

Each firm’s retained profit collapses to $V/N^2\to 0$. Every firm invests, and (in the aggregate) the whole rent evaporates into computers, cables, and microwave towers.

The prisoner’s dilemma (BCS §5.7). The cooperative outcome “nobody invests” makes everyone better off; the equilibrium is “everyone invests” because deviation is individually rational. This is exactly a prisoner’s dilemma built into the market design - and it is why BCS call the prize more a “mechanical constant of the continuous limit-order book” than an inefficiency that competition removes.

Who pays? In BCS’s model the speed spend is funded by liquidity providers, who must price the risk of being “sniped” into the spread they charge. Net: fundamental investors pay for the whole race through wider spreads and thinner markets (and Hasbrouck EMM Ch 14 documents that trading costs land on the liquidity-demanding side).

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3. Computational Implementation - the rent-dissipation simulator

Runs on the standard library only. It computes the Tullock equilibrium for the BCS ES–SPY rent ($V=$$75M/yr) and shows the entry dynamics that turn the rent into waste.

Read the table top-to-bottom with $N$ (the number of seriously-fast firms) growing over 2005–2011: total investment heads toward the entire $75M, each firm’s retained profit heads to zero - yet no firm dares not invest. That is the arms race in numbers, and it is the economic engine behind the batch-auction remedy in 06 · Batch Auctions.

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4. Failure Modes & First-Principles Breakdowns

1. Treating speed spend as if it created value. The Tullock result proves (under symmetry) that aggregate speed investment is pure deadweight: $V(N-1)/N$ spent to capture a rent that could be earned once. First-principles check: if your latency project would not be funded absent the rival race, it is arms-race waste.
2. Believing “faster equities means better prices.” The race persistently shortens opportunity duration (97 ms → 7 ms) without shrinking the prize - faster ≠ more efficient, it just re-qualifies who’s fastest.
3. Ignoring the spread channel. The rent is not “found money”; it is transferred from fundamental investors. Hasbrouck EMM Ch 14 models execution cost as landing on the liquidity demander, so the arms race is a tax on the very investors the market is supposed to serve.
4. Pricing sniping risk at zero. If you quote in a continuous book, a stale quote is picked off for its entire size; the fear makes books thinner (BCS) - a first-principles driver of shallow markets, not just wide spreads.

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5. References

• Budish, Cramton & Shim (2015), QJE 130(4), 1547–1621
• Hasbrouck, Joel - Empirical Market Microstructure
• Menkveld, Albert J. (2013)
• Biais, Bruno; Foucault, Thierry; Moinas, Sophie (2015) - “Equilibrium Fast Trading”

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6. Connected Graph Bridges

• Back: 01 · From Zero · Index Hub
• Forward: 03 · Colocation & Networks (the specific physical spend) and 06 · Batch Auctions (the design fix)
• Spread/information economics: Market Microstructure & Order Types · Queue Position & Fill Probability
• Market-making cost: Market Impact & Depth · Hendershott et al. (2014) Price Pressures