Probabilistic Backrunning in Blockchain Timing Games

• Probabilistic backrunning is a blockchain timing game where participants commit actions before a randomly timed trigger, capturing MEV opportunities.
• The formal model uses timing-game dynamics and equilibrium conditions to show zero expected profits and systematic replication of transactions.
• Key implications include excessive transaction spam, welfare loss from redundant costs, and design cues for mitigating first-come-first-served inefficiencies.

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Probabilistic backrunning is a form of blockchain timing competition in which an MEV opportunity is created by an event whose exact occurrence time is uncertain within a known window and is not observed immediately. Searchers therefore do not merely react after observing a state change; they choose ex ante timing plans and submit transactions “optimistically” or “probabilistically,” hoping that one attempt lands immediately after the trigger. In the formal treatment of this problem, repeated costly actions correspond to spam, and the strategic question is how many actions to submit and when to place them when ordering is effectively first-come-first-served after a randomly timed opportunity [2602.22032].

1. Definition, scope, and motivating environments

The motivating environments are settings in which an opportunity may arise at some random time between two information updates and participants cannot condition on the realized event time when they act. The main examples are trying to land immediately behind a price-impacting trade on an exchange, reacting to an oracle update that creates an arbitrage or liquidation opportunity, and, more generally, competing to be the first transaction after a target event that may occur sometime between two information updates. The paper emphasizes private mempools and delayed state revelation: between two blocks, or between streamed state updates, an opportunity may arise at a random time, but if a searcher waits until the event is certainly known, the opportunity may already be lost [2602.22032].

The central distinction from deterministic backrunning is that the event time is not known when actions are chosen. This changes the problem from rapid post-observation reaction to strategic positioning over time. The operative uncertainty is therefore not merely latency; it is the combination of a random trigger time and delayed observation. In blockchain terms, each attempt has marginal cost (c) through gas, tips, or infrastructure expenditure, and because only one transaction ultimately matters for allocation, the rest are socially wasteful duplication.

A common misconception is to treat probabilistic backrunning as simply “faster backrunning.” The formal model rejects that reduction. The key difficulty is not instantaneous reaction to a known event, but ex ante scheduling under uncertainty about whether the opportunity has already materialized. This suggests that delayed observability is not an incidental fric­tion but a defining structural feature of the phenomenon.

2. Formal timing-game model

The game is denoted (\mathcal G(n,c,G)). There are (n) risk-neutral players, a single opportunity of value (1) appears at random time (T \in [0,1]) with absolutely continuous strictly increasing distribution (G) and density (g), and each action costs a fixed marginal amount (c \in (0,1)). Proposition 1 shows that (\mathcal G(n,c,G)) and (\mathcal G(n,c,\mathcal U[0,1])) are strategically equivalent, so the equilibrium analysis can be reduced without loss of generality to (T \sim \mathcal U[0,1]) [2602.22032].

A pure strategy for player (i) is any finite subset (N_i \subseteq [0,1]), interpreted as the set of action or transaction times. A mixed strategy is a probability measure over finite subsets. Since the prize is worth (1), any strategy with more than
[
\beta=\left\lfloor \frac{1}{c}\right\rfloor
]
actions is strictly dominated by doing nothing, so the effective strategy space can be represented by ordered vectors with (+\infty) denoting “do not send further transactions.”

Winning is defined through directed distance,
[
d_+(x,y) = |x-y|1{x\geq y}+\infty 1{x<y},
]
and, for a finite set (A),
[
d_+(A,t) = \min{x-t: x\in A, x\geq t},
]
with (\min \emptyset = +\infty). Hence (d_+(A,t)) is the waiting time from opportunity time (t) to the first action in (A) at or after (t). The winner is the player with the earliest action weakly after (t); ties are split uniformly.

Player (i)’s payoff under pure profile (N) is
[
u_i(N) = \int_0^1 \frac{1{i\in \mathcal{W}_t(N)}}{|\mathcal{W}_t(N)|}g(t)\,dt - c\cdot |N_i|,
]
with the convention that the fraction is (0) if (\mathcal W_t(N)=\emptyset). Under mixed profile (\sigma), payoffs are obtained by expectation. The model’s assumptions are a single unit-value opportunity, absolutely continuous strictly increasing arrival distribution, ex ante choice of finite action sets, homogeneous action cost, earliest-after-trigger ordering, uniform tie-breaking, symmetry, risk neutrality, and a focus on symmetric Nash equilibria.

A technically important object is the opponents’ void probability,
[
V_i^{\sigma}(t,z):=\Pr_{N_{-i}\sim\sigma_{-i}}!\bigl[N_{-i}\cap[t,z]=\emptyset\bigr],\qquad 0\le t\le z\le 1.
]
It summarizes the probability that no opponent action intervenes on the relevant interval and becomes the central state variable in the equilibrium analysis.

3. Equilibrium characterization

For (n\ge 2) and (c\in(0,1)), Theorem 2 establishes three facts: there is an almost surely unique symmetric Nash equilibrium; each player gets zero expected payoff in that equilibrium; and the equilibrium admits a recursive ((X,\psi))-representation. Specifically, there exist i.i.d. random variables
[
X_i\in [c,1]\cup{+\infty}
]
and a map
[
\psi:[c,1]\cup{+\infty}\rightarrow[c,1]\cup{+\infty}
]
that is strictly increasing on (\psi^{-1}([c,1])), such that player (i)’s random action set is
[
\sigma(X_i,\psi)={\psi^{(k)}(X_i):k\ge 0}\cap[0,1].
]
Thus the equilibrium randomizes only the initial action time, while later actions are generated deterministically by repeated application of the successor map (\psi) [2602.22032].

Several preliminary structural results sharpen this picture. Proposition 2 states that (\mathcal G(n,c,\mathcal U[0,1])) admits no pure Nash equilibrium. Lemma 1 shows that equilibrium intensity measures are atomless, eliminating systematic ties. Lemma 2 shows that the earliest support point of equilibrium play is exactly (c), that any two actions in a pure best reply must be separated by at least (c), and that the support of the intensity measure is all of ([c,1]). The support beginning at (c) encodes the economic statement that an action earlier than (c) cannot generate sufficient expected gross payoff to cover cost (c).

The winning probability of an ordered action vector (y=(y_1,\dots,y_k)), with (y_0:=0), decomposes as
[
W_i^{\sigma}(y)=\sum_{\ell=0}^{k-1}\int_{y_\ell}^{y_{\ell+1}}V_i(t,y_{\ell+1})\,dt.
]
Each action (y_{\ell+1}) covers opportunities arriving after the preceding action and before itself, provided that no opponent acts in the relevant interval. First-order conditions for interior best-reply points then link each action to its successor via the void probability. For (j=1,\dots,k-1),
[

\frac{\partial}{\partial x_j}W_i^{\sigma}(x)

1+\int_{x_{j-1}}^{x_j}\partial_2V_i(t,x_j)\,dt - V_i(x_j,x_{j+1})=0,
]
and for the last coordinate,
[

\frac{\partial}{\partial x_k}W_i^{\sigma}(x)

1+\int_{x_{k-1}}^{x_k}\partial_2V_i(t,x_k)\,dt=0.
]

A pivotal simplification is the single-point characterization. A symmetric profile is a Nash equilibrium if and only if every single-action-time deviation has zero payoff:
[
W_i^\sigma({x},\sigma_{-i}) = c
\qquad \forall x\in[c,1].
]
Equivalently,
[
\Pr_{N_{-i}\sim\sigma_{-i},T}\bigl[N_{-i}\cap[T,x]=\emptyset,\ T\le x\bigr]=c,\qquad \forall x\in[c,1].
]
Hence every isolated timing choice in the support exactly breaks even. The zero-profit result is therefore not an auxiliary observation but a pointwise equilibrium condition.

4. Recursive layers and uniqueness

The recursive structure is formalized through a continuation value
[
J(y) :=\max_{\substack{m\ge 0\ y=y_0<y_1<\cdots<y_m\le 1}}
\sum_{\ell=0}^{m-1}\Bigg(\int_{y_\ell}^{y_{\ell+1}} V_i(t,y_{\ell+1})\,dt - c\Bigg),
]
with (m=0) yielding (0). On that basis, the continuation objective
[
\mathcal H(x,y) :=
\Big(\int_{0}^{x} V_i(t,x)\,dt - c\Big)
+
\Big(\int_{x}^{y} V_i(t,y)\,dt - c\Big)1{y<+\infty}
+
J(y)
]
defines the minimal best continuation
[
\psi(x):=\inf\arg\max_{y\in[x,1]\cup{+\infty}} \mathcal H(x,y).
]
A key lemma shows that (\mathcal H) has increasing differences, implying that (\psi) is monotone nondecreasing. The equilibrium action set then takes the orbit form
[
N_i={\psi^{(k)}(x_1(N_i)):k\in\mathbb Z_{\ge 0}}\cap[0,1]
\quad \text{a.s.}
]
and (\psi) is almost everywhere strictly increasing on its finite part [2602.22032].

The first layer is pinned down by the distribution of the initial action time. If (F(x)=\Pr[X_i\le x]), then on the first layer (x\in[c_1,c_2]), with (c_1:=c),
[
\int_0^x \bigl[1 - F(x) + F(t)\bigr]^{\,n-1}\,dt = c.
]
Differentiation yields
[
F'(x) =
\frac{1}{(n-1)\displaystyle\int_0^x \bigl(1 - F(x) + F(t)\bigr)^{\,n-2}\,dt},
]
or, equivalently, the ODE system
[
F'(x)=\frac{1}{(n-1)\,I_{n-2}(x)},\qquad
I_k'(x)=1-k\,F'(x)\,I_{k-1}(x),\qquad
I_0(x)=x,
]
where
[
I_k(x)=\int_0^x \bigl(1 - F(x) + F(t)\bigr)^k dt.
]
By Picard–Lindelöf, this has a unique solution, so the law of the initial action is unique.

For (n=2), the first-layer solution reduces to the log-uniform formula
[
\sigma(x)=
\begin{cases}
\log(x/c), & x\ge c,\
0, & x\le c.
\end{cases}
]
Equivalently, (F_1(x)=\log(x/c)) on its support. Later layers are determined recursively through the inverse (g=\psi^{-1}). With layer endpoints (c_{k+1}:=\psi(c_k)) and (Y_{i,k}:=\psi^{(k-1)}(X_i)), if (F_k) is the CDF of (Y_{i,k}), then on layer (k+1),
[
\int_0^x \left[F_k(g(t))+(F_k(t)-F_k(g(x)))1_{{t>g(x)}}\right]^{n-1}dt = c.
]
For (n=2), this simplifies to
[
g'(x) = \frac{1}{F_k'(g(x))(x-g(x))}
]
and equivalently
[
\psi'(y) = F_k'(y)(\psi(y)-y).
]
The uniqueness proof first determines (F) on the first layer and then recursively determines (g=\psi^{-1}) layer by layer.

5. Spam, welfare, and inefficiency

The welfare interpretation is framed explicitly in terms of spam and cost dissipation. Total expected spam is defined as
[
\mathrm{Spam}(\sigma)=\mathbb E_{N\sim \sigma}\left[\sum_{i=1}^n |N_i|\right].
]
In blockchain terms, this is the expected total number of transactions broadcast in competition for a single opportunity. Since only one transaction is socially necessary to capture the opportunity, any additional attempts are wasteful overhead [2602.22032].

Let
[
A := \left{\left(\bigcup_{j=1}^n N_j\right)\cap[T,1]\neq\emptyset\right}
]
be the event that some player captures the opportunity. Because equilibrium payoffs are zero,
[
\sum_{i=1}^n u_i(\sigma)=\Pr[A]-c\cdot \mathbb E_\sigma[|N|]=0,
]
so
[
\Pr[A]=c\cdot \mathbb E_\sigma[|N|]=c\cdot \mathrm{Spam}(\sigma).
]
This identity gives the core accounting relation of the model: the total fee burn equals the total probability that the opportunity is captured.

Proposition 11 bounds equilibrium spam. If (\sigma^{(n)}) is the unique symmetric equilibrium with (n) players, then
[
\mathrm{Spam}(\sigma^{(n)})>\frac{1-c}{c},
]
and the source’s typesetting for the upper bound is garbled, but the intended upper bound, as stated in the proof, is
[
\mathrm{Spam}(\sigma^{(n)})\le \frac{1-c^{\,n/(n-1)}}{c}.
]
In particular,
[
\lim_{n\rightarrow+\infty}\mathrm{Spam}(\sigma^{(n)})=\frac{1-c}{c}.
]
Thus total spam is tightly pinned around ((1-c)/c), and as (n) grows the lower bound becomes tight.

The model therefore yields a stark welfare conclusion. Equilibrium is an all-pay competition in which the opportunity value is largely dissipated into transaction costs or sequencer revenue, while searchers earn zero expected profit. A plausible implication is that low submission cost relative to opportunity value does not merely intensify competition; it transforms competition into systematic duplication of economically redundant actions.

6. Comparative statics, design implications, and terminological boundaries

The strongest comparative static concerns action cost. If (c) is high, players may send at most one transaction in equilibrium; if (c) is low, they send multiple transactions; and spam scales on the order of (1/c). In the motivating two-player case, if (c\ge 1/e), equilibrium involves at most one transaction per player, while lower costs induce multiple transactions. The model also states that the maximal number of actions scales like (\lfloor 1/c\rfloor), so spam becomes large as (c\to 0) [2602.22032].

Dependence on the number of players is more subtle. The first-layer distribution changes with (n) through
[
\int_0^x [1-F(x)+F(t)]^{n-1}dt=c,
]
but total spam is not monotone increasing in (n). Proposition 11 states this explicitly. Numerically, for (c=0.05), total spam decreases with (n) and converges toward (19=(1-c)/c). The common intuition that more competitors must produce more aggregate spam is therefore incorrect in this model; more competition can instead reorganize timing so that aggregate cost converges to near-complete value dissipation.

The arrival distribution (G) is strategically irrelevant up to monotone transformation, because Proposition 1 reduces any absolutely continuous strictly increasing (G) to the uniform case. What matters economically is not the shape of (G) but the existence of a random opportunity time together with delayed observability and first-come-first-served ordering. This is why the model is especially relevant to chains or rollups that sequence by arrival time, or by priority fee with arrival-time tie-breaking.

The mechanism-design implications are direct but cautious. Lower transaction costs increase spam, delayed observability induces probabilistic probing, and first-come-first-served ordering creates timing races. The paper explicitly notes that
[
\mathrm{Rev}(\sigma)=c\cdot \mathrm{Spam}(\sigma),
]
so higher spam can increase sequencer or validator revenue, but this does not imply higher social welfare because spam consumes blockspace, raises processing burden, may crowd out useful user transactions, and worsens user experience. The suggested design lesson is therefore to reduce incentives for first-come-first-served timing competition through mechanisms such as batching, auctions for ordering rights, delayed reveals or synchronized execution, and designs that reduce the benefit of sending many redundant attempts.

The term should also be distinguished from the unrelated use of “running probabilistic programs backwards” in probabilistic programming, where “backrunning” refers to computing preimages of output sets under a measure-theoretic semantics for a first-order probabilistic language with recursion [1412.4053]. In that setting, backward execution means propagating output constraints to feasible input sets rather than competing over transaction ordering. The terminological overlap is therefore superficial: in blockchain economics, probabilistic backrunning is a timing game with spam; in probabilistic programming, backward execution is a semantics of constrained inference.