Is Folk's theorem the Blockchain's nightmare

Posted by Bruno Mazorra on May 20, 2022

Is Folk’s theorem the Blockchain’s nightmare?


Editors Note:

Could not find these slides, so these are my own notes. There is more content than this in the presentation.

Economic rationality

Individual rationality

An agent is individually rational if they try to maximize its own revenue. Example: Construct block with max fees

A miner receives a set of transactions \(xx _{1}, \ldots, tx x _{n}\) with gas price \(b_{1}, \ldots, b_{n}$ and $g_{1}, \ldots, g_{n}\) units of gas. The miner can choose any subset of transactions $T X$ such that \(\sum_{t x \in T X } g_{ tx } \leq \max\) Gas.

script[type=’math/tex’; mode=display] \begin{equation} T X \text { such that } \sum{ tx \in T X } g{ tx } \leq \max \text { Gas. } \end{equation}

Example: Transaction inclusion

  • A “dummy” node orders transaction by timestamp, by transaction hash or randomly.
  • A rational node tries to solve the following optimization problem: \begin{equation} \begin{array}{ll} \max & \sum{i=1}^{n} x{i} b{i} g{i}
    \text { s.t. } & \sum
    {i=1}^{n} x{i} g{i} \leq \max G a s, \ & x{i} \in{0,1} \text { for } i=1, \ldots, n \end{array} \end{equation}

This is known as the Knapsack problem and is a NP-problem. In general, Ethereum nodes use a greedy approximation algorithm to obtain an approximation of the optimal solution. source,

Game Theory

The Stage Game

A game is a tuple \(G =(N, A, u)\) where:

  • $N={1, \ldots, n}$ is the set of players.
  • $A=\prod_{i=1}^{n} A_{i}$, where $A_{i}$ denotes the set of actions for a player $i$.
  • $u_{i}: A \rightarrow R$ is the utility function of a player $i$.
  • Players want to maximize $u_{i}$ and take actions simultaneously.


A pure strategy can be thought as a plan subject to the observations they make during the course of the game of play. A mixed strategy is an assignment of a probability to each pure strategy.

Nash equilibrium

A mixed strategy $s=\left(s_{1}, \ldots, s_{n}\right)$ is a Nash equilibrium if for every player \(i\), and any strategy \(\tilde{s}_{i}\), we have that

script[type=’math/tex’; mode=display] \begin{equation} u{i}\left(s{i}, s{-i}\right) \geq u{i}\left(\tilde{s}{i}, s{-i}\right) \end{equation}


Every game has a Nash equilibrium.

Example 2: L2 game


Assume \(N=\{1,2\}\) and \(t=2\).

  • $EV =$ The value that can be extracted if players know the content of txs per block.
  • $CR =$ Commit and Reveal. If possible, slash the other player.
  • RC $=$ Reveal and Commit. If possible, extract EV.
  • $R =$ Reward per Block.
  • $S =$ Slashing value s.t. $S » EV$.

Problems and difficulties to cooperate

  • Anonymous players.
  • Unable to commit to future strategies.
  • Economic incentives to deviate from commitment. Conclusion on stage game cooperation Hard to achieve consensus to cooperate.

What if games are played indefinitely?


Players are non-myopic if they are concerned for presents and future payoffs. Given an infinite sequence of payoffs \(r_{0}, r_{1}, r_{2}, \ldots\) for a player $i$ and a discount factor \(\delta\) with \(0 \leq \delta<1, i^{\prime}\) s future discounted reward is \begin{equation} \sum{i=0}^{\infty} \delta^{i} r{i} \end{equation} Intuition on discount factor:

  • The agent values about near term profits than future profits.
  • The discount factor models the players’ patience.

\begin{equation} \sum{i=0}^{\infty} \delta^{i} r{i} \end{equation}

Repeated game

Repeated games

The stage game is played indefinitely many times. Players can observe past actions. All player: share the same discount factor $\delta$. Player’s utility Let $x_{t}$ be the tuple of actions played at round $t$, then the utility of a player $i$ with discount factor $\delta$ is: \begin{equation} U{i}=\sum{t=0}^{\infty} \delta^{t} u{i}\left(x{t}\right) \end{equation}

Folk theorem with perfect monitoring

Folk Theorem

Let $G$ be any $n$-player game.

  • For all strictly pure-action individually rational action profiles ã, that is, $u_{i}(\tilde{a})>\operatorname{minmax}_{i}$ for all $i$, there is a $\bar{\delta} \in(0,1)$ such that for every $\delta \in(\bar{\delta}, 1)$, there exists a subgame-perfect equilibrium of the infinitely repeated game with discount factor $\delta$ in which $\tilde{a}$ is played in every period.
  • For all feasible tuple $v$, there is a $\bar{\delta} \in(0,1)$ such that for every $\delta \in(\bar{\delta}, 1)$, there exists a subgame-perfect equilibrium, with payoff $v$, of the infinitely repeated game with public correlation and discount factor $\delta$.

Example 1: Arbitrage competition and the Folk theorem

Since $(50 \,/\ 50 )$ is a feasible payoff, we have by Folk theorem that, if both players are enough patient (for $\delta \geq 2 / 3$ holds), there exists a Nash equilibrium \(\left(s_{1}, s_{2}\right)$ such that $u_{i}\left(s_{1}, s_{2}\right)=50 \\). In these setting, we have that collision among searchers induce:

  • $+$ profits for searchers,
  • profits for miners. compared to the stage game.

System performance

There exists Nash equilibrium that do not lead to egalitarian distribution of rewards.


Example 2: L2 with Threshold decryption scheme

Repeated game: All for nothing

If players are enough patient $(\delta \approx 1)$, then there exists a Nash Equilibrium where both players, play the Reveal-Commit strategy and extract the MEV. System performance Since miners extract MEV from users, the users’ revenue decreases compared to the myopic model.