Timing Games: Probabilistic backrunning and spam

TL;DR

The main motivation is the study of ``probabilistic backrunning" on blockchains, where arbitrageurs want to place an order immediately after a trade that impacts the price on an exchange or after an oracle update.

Abstract

There are $n$ players who compete by timing their actions. An opportunity appears randomly on a time interval. Whoever takes an action the fastest after the opportunity has arisen wins. The occurrence of the opportunity is observed only with a delay. Taking actions is costly. We characterize the unique symmetric equilibrium of this game and study worst-case inefficiency of equilibria. Our main motivation is the study of ``probabilistic backrunning" on blockchains, where arbitrageurs want to place an order immediately after a trade that impacts the price on an exchange or after an oracle update. In this context, the number of actions taken can be interpreted as a measure of costly ``spam" generated to compete for the opportunity.

Figures (6)

• Figure 1: The equilibrium strategy for $c=1/e$.
• Figure 2: Equilibrium point process $\sigma(X,\psi):=\{\psi^{(k)}(X):k\geq0\}\cap[0,1]$. The red-hatched area is the support of the distribution of the initial point.
• Figure 3: Game with players $A$ and $B$. Player $A$’s transactions are shown in red, and player $B$’s transactions in blue. $T$ denotes the arrival time of the opportunity. In this arrangement, both players place $3$ transactions and player $B$ wins the opportunity.
• Figure 4: CDFs $F_k$ and densities of random variables $\psi^{(k)}(X_i)$ for $k=0,\ldots,4$ for $n=2$, $c=0.1$.
• Figure 5: Initial point distribution for $n=2,3,4,5$ players in the case $c=1/e$.

Theorems & Definitions (37)

• Remark 2.1
• Proposition 2.2
• Proposition 2.3
• Theorem 3.1: Main theorem
• Lemma 3.2
• proof
• Lemma 3.3
• Lemma 3.4: Properties of $V_i$
• proof : Proof of Theorem \ref{['theorem:main:payoff']}
• Lemma 3.5