Posts made by cfrank

@mosbrooker Mike, this forum is for structured discussions on voting theory, not personal blog posts. If you want to share your personal thoughts in a blog format, you’re welcome to start your own blog elsewhere.

@matija yeah, it seems that there were many factors that converged. I guess we never know what might have been or might be especially instrumental per se in the rise of anti-democratic regimes. As we’re seeing now in the USA, even supposedly “stable” vote-for-one systems are vulnerable to the most basic kind of fragmentation, namely splitting in half.

I’m not sure how stable these systems are then, considering the bipolar oscillations of government. The same gridlocks that make legislature impotent appear in a more blatant form. To me, that means vote-for-one is even less desirable than previously imagined, since even its claim to purported stability is moot.

@matija is that really how they elected the president? How did they vote?

Hi, as a fairly busy researcher, I still want to understand to whatever degree possible the context of the rise of the Third Reich and fascism in Germany. What factors enabled or failed to frustrate this rise in terms of the structure of the German government? For example, leading up to the rise, what was the parliamentary structure? How, if at all, was power separated between branches of government? What form of voting system was used—was it a proportional representation system? What do scholars speculate about in terms of what changes to the structure of government might have checked, frustrated or disabled the rise of fascism there?

Some answers to these are obvious, but I wanted to ask these questions openly to spur a conversation about the roles of these factors in the failure to keep fascism in check. For example, one speculation is that PR was a factor, leading to an unstable government as parties arose, formed and broke coalitions, and dissolved. This led to a weak legislative branch of government. With this in mind, how can we design or provide counterbalances to systems of proportional representation that don’t suffer from the same vulnerability to fragmentation and instability?

Another aspect was the capability of the executive to seize essentially unilateral control in “emergency” events.

If anybody has insight about this, I think this is an important discussion to have.

Hi! I’ve noticed an increase in spam posts, especially advertisements, which as a moderator I have been deleting at my discretion based on the clear policies on content for this forum. I think maybe our policies should require moderator approval for posts by new users, or some additional safeguards against spam, whatever those might be. Do you have any thoughts or suggestions?

@toby-pereira I have a moment now to take a look and I’m taking notes. I have a specific question for you though: Generally, do you think that registering for political parties and having that registration play a role in the voting process is something to be avoided?

@toby-pereira thank you!! I will definitely follow this as you continue. Due to other obligations it’s been difficult to find bandwidth to dive into your other content, but I intend to once I find an opening.

@Toby-Pereira I tried to fix this problem, and came up with some formulas to define the quota compliance multiplier using entropy. It’s kind of complicated, but it’s well-defined, and it encourages diverse compromises between parties. It probably now gives more bargaining power to larger parties, and “puppeteering” can still be strategic, but I think those things are in direct conflict unfortunately.

This is a markdown file that defines the formula.

# Proposed Compliance Multiplier Formula

This document presents a proposed formula for the compliance multiplier in the proposed system, which compares the observed distribution of ambassador seats with the ideal distribution based solely on voter shares.

## 1. Partition Functions

The ideal partition function for a given party \(X\) is defined as:

[
Z_P(X) = P(X \sim X) + \sum_{i \neq X} P(X \sim i) + \sum_{j \neq X} P(j \sim X)
]

The observed partition function is defined as:

[
Z_Q(X) = P(X \sim X) + \sum_{i \neq X} Q(X \sim i) + \sum_{j \neq X} Q(j \sim X)
]

*Note: For party consistency, we impose that \(Q(X \sim X) = P(X \sim X)\) for every party \(X\).*

## 2. Entropy–like Quantities

The ideal (maximum) entropy is given by:

[
E_P(X) = \frac{-P(X \sim X)\ln P(X \sim X) - \sum_{i \neq X} P(X \sim i)\ln P(X \sim i) - \sum_{j \neq X} P(j \sim X)\ln P(j \sim X)}{Z_P(X)} + \ln Z_P(X)
]

The observed entropy is given by:

[
E_Q(X) = \frac{-P(X \sim X)\ln P(X \sim X) - \sum_{i \neq X} Q(X \sim i)\ln Q(X \sim i) - \sum_{j \neq X} Q(j \sim X)\ln Q(j \sim X)}{Z_Q(X)} + \ln Z_Q(X)
]

## 3. Compliance Multiplier

The compliance multiplier for party \(X\) is then defined as:

[
\text{Multiplier}(X) = \frac{E_Q(X)}{E_P(X)}
]

Since for every off-diagonal entry we have \(Q(X \sim Y) \leq P(X \sim Y)\) (with equality on the diagonal), the normalized observed entropy \(E_Q(X)\) is less than or equal to the ideal entropy \(E_P(X)\). Therefore, it follows that:

[
\text{Multiplier}(X) \leq 1.
]

This guarantees that a party's compliance multiplier never exceeds 1, reflecting that the observed (normalized) diversity of ambassador nominations cannot surpass the ideal (maximally spread) distribution.

And here are some examples

# Four-Party Examples of Compliance Multipliers

This document presents computed examples for a 4-party system under different scenarios. We consider four parties—A, B, C, and D—with the following voter shares:

- **Party A:** 0.4  
- **Party B:** 0.3  
- **Party C:** 0.2  
- **Party D:** 0.1

The ideal ambassador seat allocation is given by:

[
P(X \sim Y)=P(X) \times P(Y)
]

Thus, the **ideal matrix** \(P\) is:

| From \(\backslash\) To | A    | B    | C    | D    |
|------------------------|------|------|------|------|
| **A**                | 0.16 | 0.12 | 0.08 | 0.04 |
| **B**                | 0.12 | 0.09 | 0.06 | 0.03 |
| **C**                | 0.08 | 0.06 | 0.04 | 0.02 |
| **D**                | 0.04 | 0.03 | 0.02 | 0.01 |

For each party \(X\), we define the **partition functions** as follows:

- **Ideal:**
  [
  Z_P(X)=P(X\sim X)+\sum_{Y\neq X}P(X\sim Y)+\sum_{Y\neq X}P(Y\sim X)
  ]
- **Observed:**
  [
  Z_Q(X)=P(X\sim X)+\sum_{Y\neq X}Q(X\sim Y)+\sum_{Y\neq X}Q(Y\sim X)
  ]
  
with the assumption that for all parties, \(Q(X \sim X)=P(X \sim X)\).

We then define the **entropy–like quantities**:
[
E_P(X)=\frac{-\sum_Y P(X\sim Y)\ln P(X\sim Y)}{Z_P(X)}+\ln Z_P(X)
]
[
E_Q(X)=\frac{-\sum_Y Q(X\sim Y)\ln Q(X\sim Y)}{Z_Q(X)}+\ln Z_Q(X)
]
and the **compliance multiplier** is given by:
[
\text{Multiplier}(X)=\frac{E_Q(X)}{E_P(X)}.
]

Under the condition that for every off-diagonal entry \(Q(X\sim Y)\leq P(X\sim Y)\), the normalized observed entropy cannot exceed the ideal one—so \(\text{Multiplier}(X)\leq 1\).

---

## Scenario 1: Full Compliance

**Situation:**  
Every party fills its ambassador seats exactly as in the ideal, so for all \(X, Y\):
[
Q(X\sim Y)=P(X\sim Y).
]

**Results:**  
- **Party A:** Multiplier = 1.000  
- **Party B:** Multiplier = 1.000  
- **Party C:** Multiplier = 1.000  
- **Party D:** Multiplier = 1.000  

---

## Scenario 2: Sabotage by Party B Against Party A

**Modification:**  
Party B refuses to elect any ambassadors from A. In our observed matrix, we set:
[
Q(B\sim A)=0 \quad \text{(instead of the ideal }0.12\text{)}.
]
All other entries remain ideal.

### Computed Values

**For Party A:**

- **Ideal Partition Function:**  
  \(Z_P(A)=0.16+ (0.12+0.08+0.04)+(0.12+0.08+0.04)=0.16+0.24+0.24=0.64.\)

- **Observed Partition Function:**  
  - Row A remains: \(0.16+0.12+0.08+0.04=0.40.\)  
  - Column A: Instead of \(0.12+0.08+0.04=0.24,\) we have \(0+0.08+0.04=0.12.\)  
  So, \(Z_Q(A)=0.16+0.24+0.12=0.40.\)

- **Entropy–like Quantities (Approximate):**  
  - \(E_P(A) \approx 1.840.\)  
  - \(E_Q(A) \approx 1.472.\)

- **Compliance Multiplier:**  
  [
  \text{Multiplier}(A)\approx \frac{1.472}{1.840}\approx 0.800.
  ]

**For Party B:**

- **Ideal Partition Function:**  
  \(Z_P(B)\approx 0.51.\)

- **Observed Partition Function:**  
  \(Z_Q(B)\approx 0.27.\)

- **Entropy–like Quantities (Approximate):**  
  - \(E_P(B) \approx 1.824.\)  
  - \(E_Q(B) \approx 1.524.\)

- **Compliance Multiplier:**  
  [
  \text{Multiplier}(B)\approx \frac{1.524}{1.824}\approx 0.834.
  ]

**For Parties C and D:**  
No sabotage occurs, so:  
- **Multiplier(C) = 1.000.**  
- **Multiplier(D) = 1.000.**

---

## Scenario 3: Puppet Scenario – Party D as a Puppet for Party A

**Modification:**  
Party D acts as a puppet for Party A to harm Party B. We set:
[
Q(D\sim B)=0 \quad \text{(instead of the ideal }0.03\text{)}.
]
All other entries remain ideal.

### Computed Values

**For Party A:**  
- \(Z_Q(A)\) remains nearly ideal, so  
  **Multiplier(A) \(\approx 1.000\).**

**For Party B:**  
- Losing support from D reduces its observed diversity, so  
  **Multiplier(B) \(\approx 0.910\).**

**For Party C:**  
- Fully compliant, so  
  **Multiplier(C) = 1.000.**

**For Party D (the puppet):**  
- Due to its refusal to elect from B, its observed diversity is reduced:  
  **Multiplier(D) \(\approx 0.940\).**

### Effective Representation for the A Coalition

If effective main-platform representation is given by the product of voter share and multiplier, then:
- **Party A:** \(0.4 \times 1.000 = 0.400.\)
- **Party D:** \(0.1 \times 0.940 \approx 0.094.\)

Thus, the total effective representation for the A coalition is:
[
R(A \text{ coalition}) = 0.400 + 0.094 \approx 0.494.
]
This is slightly less than the 0.5 (50%) of the vote they would control if D were fully compliant, reflecting the cost of splitting support.

---

## Summary of Computed Multipliers

- **Scenario 1 (Full Compliance):**
  - A: 1.000  
  - B: 1.000  
  - C: 1.000  
  - D: 1.000

- **Scenario 2 (Sabotage by B Against A):**
  - A: ≈ 0.800  
  - B: ≈ 0.834  
  - C: 1.000  
  - D: 1.000

- **Scenario 3 (Puppet – D as Puppet for A):**
  - A: ≈ 1.000  
  - B: ≈ 0.910  
  - C: 1.000  
  - D: ≈ 0.940

---

## Interpretation

- **Full Compliance:** All parties achieve ideal diversity, so their compliance multipliers are 1.
- **Sabotage:** When Party B refuses to elect ambassadors from Party A, both A and B see their normalized diversity reduced (multipliers drop to ≈0.800 and ≈0.834, respectively).
- **Puppet Scenario:** Using Party D as a puppet to harm Party B, while A remains nearly ideal, both B and D are penalized—the puppet (D) suffers a lower multiplier (≈0.940) and B’s multiplier drops to about 0.910. Moreover, the A coalition's effective representation becomes slightly diluted (≈0.494 instead of 0.500).

These examples illustrate that while parties might attempt sabotage or use puppets to undermine competitors, the system’s reliance on normalized diversity ensures that such strategies come at a cost to all parties involved.

I think with careful consideration, it might be possible to adjust the multiplier so that “puppeteering” actually harms a party in proportion to the harm it could cause to its largest rival. For example, maybe raising the multiplier to some power or passing it through some other function would disincentivize puppeteering more.

Hi @toby-pereira! This is a lot of information. Do you have resources you would recommend to learn more about PR methods and the different specific methods and modules that you mention here? I think you did link a lot of things, but is there a “crash course” of sorts that could make it easier to digest?

@toby-pereira you were on the right track about something: in this system, small parties can effectively sacrifice themselves to take down larger parties. This could lead to the creation of “puppet parties” that are essentially used as weapons to take down representation of another party, which is kind of a virtual form of MAD warfare.

@toby-pereira yes I agree, thanks for taking a look! I’m curious about your point about parties ganging up, do you have a generic example in mind?

Each party’s main platform representation is limited by the minimum of compliance rates with all ambassador quotas involving that party. So for example, if we have three parties A, B, C, and the compliance rates are

(A~B): 80% of ambassador seats filled
(A~C): 90% of ambassador seats filled
(B~A): 100% of ambassador seats filled
(B~C): 85% of ambassador seats filled
(C~A): 100% of ambassador seats filled

then A will only be allowed to fill 80% of its main platform seats, B will only be allowed to fill 80% of its main platform seats, and C will only be allowed to fill 85% of its main platform seats. So the lack of compromise cuts both ways.

I imagined voting being approval based but with each voter registered under a single party. I figure parties would nominate any candidates they like, and the party affiliations of voters and the party nominations of candidates would determine the filling of “quotas” in some way. For example, if a voter registers under party A, and party B nominates a candidate, I figured a vote cast by that voter to that candidate would contribute to the (A~B) ambassador quota.

I’m now thinking about whether a candidate might be able to acquire multiple seats as ambassadors across many parties. I’m not necessarily thinking of this as a formal decision procedure for how to allocate the seats, but as an apparatus or framework within which seats might be allocated by an additional, more formal mechanism. Does that make sense?

A kind of basic “support quota” for an (A~B) ambassador would be 50%, meaning at least half of the voters registered for party A approved of the B-nominated candidate. B-nominated candidates with less than the support quota from A cannot be given an (A~B) ambassador seat.

@toby-pereira this seems like a tricky thing to deal with. I appreciate your posts here, PR methods seem to be pretty complex. The whole process seems to be a “cake cutting” problem, and a while back, I was trying to think about how to pose the situation in a way that enables the optimal cake cutting protocol to work. It might not be clear, fully hashed out or easy to follow, but I wonder what your thoughts about the concept are:

https://www.votingtheory.org/forum/topic/299/pr-with-ambassador-quotas-and-cake-cutting-incentives?_=1739756465750

@toby-pereira failure of monotonicity is interesting and makes sense, since voters are effectively “forced to buy” in an algorithmically greedy fashion, whereas one could imagine them waiting strategically and hoping others put their money up first so they can freeload and increase their representation. And in practice this would be done by choosing not to approve of a desired candidate, betting on others bearing the cost.

By graph based I was thinking of some matching or coloring scheme, but in effect that is what a Phragmén method is, although it’s equivalent to an iterative greedy process. I speculate that computing a globally optimal solution to most “good” objectives is not polynomial, but sometimes there are guaranteed bounds of divergence from the global optimum. Just musing.

I still haven’t dived into this due to other time obligations. But definitely want to learn more.

@toby-pereira I really like the concept of the Phragmén method. You’re also right that the proportionality constant doesn’t change things, I was crossing wires about the ratio.

Lots of interesting things to think about. Do you know of any graph-based PR algorithms?

@toby-pereira I didn’t get to check this in detail but I find it interesting. It also makes sense, although I wonder if it should be stated a bit more generally, saying maybe that there is some fixed positive constant C such that for all epsilon>0, there is some k such that for all k’>=k, C*Var(l)/k’<epsilon.

Is there a reason for choosing the normalized variance Var(l)/k’ rather than the normalized standard deviation sqrt(Var(l))/k’? Or even expressing in terms of sqrt(Var(l))/E(l)?

@jack-waugh I think anything except the minimum for unmarked candidates makes it too easy to mark bullet burials. But I don’t know.

@masiarek this may be slightly tangential, but another consideration for the future is making sure any encryption in voting systems is also quantum secure.

Youtube Video

@wolftune IRV wasn’t largely in the public consciousness of the USA until groups like FairVote started promoting it in the 1990s. It took a decade for acknowledgment of the system to grow, and by then it had subsumed the name “ranked choice.” So it wasn’t as if people made a concerted effort to change the widely accepted name in that case. In fact, I would say more people try to reverse the name change, because it’s a presumptuous moniker that obscures other ranked choice systems.

Anyway, maybe there are examples. But I doubt whether they were efforts not in line with the mainstream or status quo.