@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.

@cfrank Good question. I'm not sure I can think of one place that gives a good intro to all the stuff. Anything I know I've picked up piecemeal over the years, and there isn't really that much about proportional cardinal methods in general I'd say. The stuff on the Electowiki could probably do with being massively overhauled.

Obviously the Thiele and Phragmén methods have been around for over a century but academic research into this type of method only really picked up again relatively recently, and dry academic papers aren't the best place for a beginner to learn about them anyway.

Edit - Maybe I could start a thread with some basic info. Others can add to it and can also take issue with anything they might disagree with!

Also I've been doing some work on my COWPEA paper to see if I can get it to publication standard. So hopefully that will happen at some point.

@gregw COWPEA doesn't pass consistency. Fairness is subjective. But COWPEA itself is just a proportional weighting thing rather than an election method itself. The lottery version could be used for elections but being non-deterministic would likely be a difficult sell.