Ever had to go out with your colleagues for lunch and experienced how hard it can be to come to a conclusion on where to eat? Individual decision making is hard. Collective decision making is much harder (an understatement)! But it is extremely important, especially in decentralized autonomous organizations (DAOs). To be able to come to a conclusion using code as mediator is challenging. But before we dive into the complexities of governance whether on-chain or off-chain, let's understand some of the fundamentals from social choice theory and the decades of work that has been put into understanding decision making in a democratic structure.

Most of the DAOs that have implemented governance mechanism, essentially boils down their voting method into single choice "yes", "no" or "abstain" option. While this seems intuitively simple and completely eliminates the complex intricacies involves in ranked choice voting, it's not the best system for all scenarios. And most of the projects do not seem to highlight this:

Let's say for platform X (sarcasm intended), a decision has to be made about which logo, out of logoA, logoB and logoC, should be adopted. One can argue that instead of creating one proposal and tallying using ranked choice preference, we can split the proposal into 3 and carry-on with the usual voting methods adopted by DAOs:

- Should we go with logoA? Options: "yes", "no", "abstain"
- Should we go with logoB? Options: "yes", "no", "abstain"
- Should we go with logoC? Options: "yes", "no", "abstain"

This opens up a can of worms! What happens if logoA and logoB both have "yes" as the winner? Should we then create another proposal to resolve the tie? Are the same voters going to vote on that? What if some significant portion of the old voters does not show up? What if new voters show up? Would this not increase voter fatigue?....

While most DAOs try to avoid such proposals, it can still happen depending on the topic of discussion. There is a reason why ranked choice preference tallying is avoided but that does not mean it cannot be made practical. In this article, we will look into a few of the well-known theorems in social choice theory and to keep in mind when designing any governance mechanism.

## May's Theorem

The reason that most DAOs use single choice simple majority voting method is because of the May's theorem:

For two alternative voting, May's theorem states that simple majority is the only anonymous, neutral and positively responsive social choice function.

**Anonymous:** There is no distinction between the votes and each voter is identical.

**Neutral:** Reversing the choice of each voter, reverses the group outcome.

**Positively Responsive:** If some voters change their preference in favor of one option, while others remain the same, the proposals outcome does not change in opposite direction. If the previous outcome was a tie, the tie is broken in the direction of the change.

The theorem only applies if there are two options. In most DAO's ballot (set of options in proposals), "abstain" vote is not counted and hence the theorem applies.

## Arrow's Impossibility Theorem

Arrow's impossibility theorem applies only for *ranked choice voting*. A *voting rule* is a method of choosing winner from a set of options (ballot) on the basis of voter's ranking of those options. Before jumping into the theorem, lets examine a couple of different voting rules.

In **plurality rule**, the winning option is the option which was *ranked first* the most than any other option. E.g. for 3 options $X$, $Y$ & $Z$, if 40% voters liked $X$ best i.e. ranked it first, 35% liked $Y$ best and 25% liked $Z$ best, then $X$ wins, even though it is short of an over-all majority (greater than 50%).

40% | 35% | 25% |
---|---|---|

X | Y | Z |

In **majority rule**, the winning option is the option that is preferred by a majority to each other option. E.g. for 3 options $X$, $Y$ & $Z$, if 40% voters rank $X>Y>Z$, 35% rank $Y>Z>X$ and 25% rank $Z>Y>X$, the winner is $Y$ because majority of voters (35% + 25% = 60%) prefer $Y$ to $X$ and a majority (40% + 35% = 70%) prefer $Y$ to $Z$.

40% | 35% | 25% |
---|---|---|

X | Y | Z |

Y | Z | Y |

Z | X | X |

Note that plurality and majority rule leads to different outcome. This prompts the question: Which outcome is "right"? Or, which one is better to use? We can then ask a general question: Among *all* possible voting rules, which is the best?

Arrow proposed that we should first identify what we *want* out of the voting rule i.e. what properties should be satisfied. The best voting rule will then be the one that satisfy all of them. Those properties are:

#### 1. Decisive/Unrestricted domain

All preferences of all voters should be accounted for and there should always be a winner and there shouldn't be more than one winner.

#### 2. Pareto Principle

If all voters rank $X$ above $Y$ and $X$ is on the ballot, $Y$ should not be the outcome.

#### 3. Non-dictatorship

No single voter's choice should decide the outcome.

#### 4. Independence of irrelevant alternatives

Given a voting rule and voter's ranking, if $X$ is the winner, then if we remove $Y$ from the ballot as it was not the winning choice (*irrelevant*), $X$ should still win i.e. independent from an irrelevant choice. To give an example, in the plurality rule example above, if option $Z$ was removed and all those who chose $Z$ as their first choice now chooses $Y$, making $Y$ the winning choice with 60%. Politically speaking, $Z$ is a *spoiler*. Even though $Z$ was not going to win in either cases, it ended up determining the outcome. This happens in democratic elections several times. This property serves to rule out *spoilers*.

Plurality rule is vulnerable to *spoilers* and hence violates the *independence* property. Majority rule, satisfies the independence property i.e. if $X$ beats each of the other choices, it continues to do so if one of the choices is dropped. But majority rule does not satisfy the *decisiveness* property i.e it doesn't always produce a winner. E.g. in the following table of ranked choices, $Y$ beats $Z$ by a majority (68% to 32%), $X$ beats $Y$ by a majority (67% to 33%) and $Z$ beats $X$ by a majority (65% to 35%) - so there is no option which beats the other two. This is called **Condorcet's Paradox.**

35% | 33% | 32% |
---|---|---|

X | Y | Z |

Y | Z | X |

Z | X | Y |

Arrow tried to find a voting rule that satisfies all the properties but eventually led to the conclusion that there is **no** voting rule that satisfies all 4 properties!

The name of the theorem is itself a source of pessimism: if something is "impossible", its pretty hard to accomplish. This theorem prompts the question: *Given that no voting rule satisfies all properties, which rule satisfies them most often?* One plausible answer is that in majority voting, if one particular class of ranking (e.g. $Z>Y>X$) is removed with high probability of it not occurring, then majority rule will always have an outcome. In this case, majority rule does *not* violate the *decisive* property.

There is another theorem called the domination theorem. It states that for any voting rule that differs from majority rule, if it works well for a particular class of rankings, then majority rule must also work well for that class. Furthermore, there must be some other class of rankings for which majority rule works well and the voting method we started with does not. Whenever another voting rule works well, majority rule must work well too, and there will be cases where majority rule works well and the other voting rule does not.

This applies only if there is a possibility of identifying a class of ranking that is highly unlikely to occur. In case of DAOs, the question arises who is responsible for identifying and eliminating such a class of ranking for each proposal. Simply eliminating the least voted class of ranking results into utter neglect of minority.

## Gibbard–Satterthwaite Theorem

Gibbard-Satterthwaite's theorem is applicable on ranked choice voting that chooses a single winner. It follows from Arrow's impossibility theorem. For every voting rule, one of the 3 things must hold:

- There is a dictatorship i.e. one voter's choice determines the outcome
**OR** - There are only two choices (in ballot) and hence only two possible outcome
**OR** - Tactical voting is possible i.e. there exist some strategy where a voter's ranked choice does not show their sincere opinion but gives the outcome they want.

**Borda count:** For the ranked choice ballot of each voter with $n$ options, assign $n-1$ points to the top option, $n-2$ to the second option, ... and $0$ to the last option. Option with the most point is the winner.

To demonstrate the theorem, consider 3 voters $A$, $B$ & $C$ and 4 options $W$, $X$, $Y$ & $Z$ and their ranked preference is as follows:

Voter | Choice 1 | Choice 2 | Choice 3 | Choice 4 |
---|---|---|---|---|

Alice | W | X | Y | Z |

Bob | Y | X | Z | W |

Carol | Y | X | Z | W |

Based on Borda's count ($W$: 3, $X$: 6, $Y$: 7, $Z$: 2), $Y$ is the winner. But if Alice changes its ballot as follows:

Voter | Choice 1 | Choice 2 | Choice 3 | Choice 4 |
---|---|---|---|---|

Alice | X | W | Z | Y |

Bob | Y | X | Z | W |

Carol | Y | X | Z | W |

From Borda count ($W$: 2, $X$: 7, $Y$: 6, $Z$: 3), $X$ is the winner and Alice's preference of $X$ over $Y$ is still maintained. We can say that there exists a strategy where Borda count is manipulable.

## Conclusion

Before designing any governance mechanism for DAOs, it is extremely important to understand that on-chain voting is a great tool but it does not solve the inherent problems in social choice theory. We also want to experiment with new systems but first, base our work on decades of research that have already been proven to work or not work. This article gives an overview of why ranked choice voting is complex to implement and even though most DAOs are currently opting for single choice voting, it is also prone to manipulation.

Most DAOs allow the voters to *weigh* the preference by either using more tokens or time-locking tokens (conviction voting). This is limited to putting the weight towards one option in single choice voting. Ranked choice voting is complex to begin with and introducing weights can potentially add more complexities and result in unforeseen outcomes. As shown in Gibbard-Satterthwaite theorem, Borda count is manipulable. And adding weights will open up more possibilites to game the system. But nonetheless, a great domain to research and experiment!