用棋盘游戏理解共识问题

Mar 19, 2026

I have an early adulthood trauma from struggling to understand consensus amidst a myriad of poor explanations. I am overcompensating for that by adding my own attempts to the fray. Today, I want to draw a series of pictures which could be helpful. You can see this post as a set of missing illustrations for Notes on Paxos, or, alternatively, you can view that post as a more formal narrative counter-part for the present one.

The idea comes from my mathematics of consensus lecture, with versions in English and Russian.

The Preamble

I am going to aggressively hand wave the details away, please refer to Notes for filling in the blanks.

And, before we begin, I want to stress again that here I am focusing strictly on the mathematics behind the algorithm, on the logical structure of the universe that makes some things impossible, and others doable. Consensus is but a small part of the engineering behind real data management systems, and I might do something about pragmatics of consensus at some point, just not today ;)

The Problem

There’s a committee of five members that tries to choose a color for a bike shed, but the committee members are not entirely reliable. We want to arrive at a decision even if some members of the committee are absent.

The Vote

The fundamental idea underpinning consensus is simple majority vote. If R0, … R4 are the five committee members, we can use the following board to record the votes:

A successful vote looks like this:

Here, red collected 3 out of 5 votes and wins. Note that R4 hasn’t voted yet. It might, or might not do so eventually, but that won’t affect the outcome.

The problem with voting is that it can get stuck like this:

Here, we have two votes for red, two votes for blue, but the potential tie-breaker, R4, voted for green, the rascal!

To solve split vote, we are going to designate R0 as the committee’s leader, make it choose the color, and allow others only to approve. Note that meaningful voting still takes place, as someone might abstain from voting — you need at least 50% turnout for the vote to be complete:

Here, R0, the leader (marked with yellow napoleonic bicorne), choose red, R2 and R3 acquiesced, so the red “won”, even as R1 and R4 abstained (x signifies absence of a vote).

The problem with this is that our designated leader might be unavailable itself:

The Board

Which brings us to the central illustration that I wanted to share. What are we going to do now is to multiply our voting. Instead of conducting just one vote with a designated leader, the committee will conduct a series of concurrent votes, where the leaders rotate in round-robin pattern. This gives rise to the following half-infinite 2D board on which the game of consensus is played:

Each column plays independently. If you are a leader in a column, and your cell is blank, you can choose whatever color. If you are a follower, you need to wait until column’s leader decision, and then you can either fill the same color, or you can abstain. After several rounds the board might end up looking like this:

The benefit of our 2D setup is that, if any committee member is unavailable, their columns might get stuck, but, as long as the majority is available, some column somewhere might still complete. The drawback is that, while individual column’s decision is clear and unambiguous, the outcome of the board as whole is undefined. In the above example, there’s a column where red wins, and a column where blue wins.

So what we are going to do is to scrap the above board as invalid, and instead require that any two columns that achieved majorities must agree on the color. In other words, the outcome of the entire board is the outcome of any of its columns, whichever finishes first, and the safety condition is that no two colors can reach majorities in different columns.

Let’s take a few steps back when the board wasn’t yet hosed, and try to think about the choice of the next move from the perspective of R3:

As R3 and the leader for your column, you need to pick a color which won’t conflict with any past or future decisions in other columns. Given that there are some greens and blues already, it feels like maybe you shouldn’t pick red… But it could be the case that the three partially filled columns won’t move anywhere in the future, and the first column gets a solid red line! Tough choices! You need to worry about the future and the infinite number of columns to your right!

Luckily, the problem can be made much easier if we assume that everyone plays by the same rules, in which case it’s enough to only worry about the columns to your left. Suppose that you, and everyone else is carefully choosing their moves to not conflict with the columns to the left. Then, if you chose red, your column wins, and subsequently some buffoon on the right chooses green, it’s their problem, because you are to their left.

So let’s just focus on the left part of the board. Again, it seems like blue or green might be good bets, as they are already present on the board, but there’s a chance that the first column will eventually vote for red. To prevent that, what we are going to do is to collect a majority of participants (R0, R2, R3) and require them to commit to not voting in the first columns. Actually, for that matter, let’s prevent them from voting in any column to the left:

Here, you asked R0, R2 and R3 to abstain from casting further votes in the first three columns, signified by black x. With this picture, we can now be sure that red can not win in the first column — no color can win there, because only two out of the five votes are available there!

Still, we have the choice between green and blue, which one should we pick? The answer is the rightmost. R2, the participant that picked blue in the column to our immediate left, was executing the same algorithm. If they picked blue, they did it because they knew for certain that the second column can’t eventually vote for green. R2 got a different majority of participants to abstain from voting in the second column, and, while we, as R3, don’t know which majority that was, we know that it exists because we know that R2 did pick blue, and we assume fair play.

That’s all for today, that’s the trick that makes consensus click, in the abstract. In a full distributed system the situation is more complicated. Each participant only sees its own row, the board as a whole remains concealed. Participants can learn something about others’ state by communicating, but the knowledge isn’t strongly anchored at time. By the time a response is received the answer could as well be obsolete. And yet, the above birds-eye view can be implemented in a few exchanges of messages.

Please see the Notes for further details.