It’s not need to cover the generic concepts and theorems with a lot of specific terminology and even more support theorems that consider partial cases of the general theorem. The concepts as “deterministic” or “probabilistic” should be used by professionals in calculation models to get a specific result. In reality, most voting processes are deterministic, while when calculating the probabilities of something in science or in game choice - should be used the probabilistic model mainly.
Do you think that Goonswarm voting order for CSM as Aryth>Merckelchen>Innominate>Xenuria>…Vily>? is probabilistic?
Do you think that Legacy voting order for CSM as Vily>Cornak>Dunk Dinkle>? is probabilistic?
Do you think that republican’s vote for Trump is probabilistic? There weren’t scenarios, in the republican camp, like this: Ah, today is a voting day! I should vote for somebody. Let me think. IDK, I’ll vote for Trump because I like his tie.
Theory: Deterministic formal models assume that voters make decisions with certainty while probabilistic models assume that voters’ intentions are uncertain. The classic deterministic model assumes that individuals choose the candidates whose positions are nearest their ideal points with certainty. In contrast, probabilistic models posit that the likelihood of voting for a particular candidate increases as distance decreases, but to vote against the nearest candidate remains possible.
There is another, more practical, definition to notice when the Arrow’s theorem criteria can happen - ordinal utility. We must rely on ordinal utility not on deterministic concept to define the usage of Arrow’s theorem areas of application.
In economics, an ordinal utility function is a function representing the preferences of an agent on an ordinal scale. The ordinal utility theory claims that it is only meaningful to ask which option is better than the other, but it is meaningless to ask how much better it is or how good it is (as CSM voting happens).
The ordinal approach is based on the fact that the preferences of an individual regarding the alternatives proposed for selection cannot be measured quantitatively, but only qualitatively, that is, one alternative is worse or better than the other.
Ordinal utility contrasts with cardinal utility theory: the latter assumes that the differences between preferences are also important.
Within the framework of the cardinal approach, which implies the quantitative measurability of preferences, Arrow’s theorem in general does not work.
Or, in more public terms
Arrow has said “Most systems are not going to work badly all of the time. All I proved is that all can work badly at times.”
You mentioned the Nakamura number. Its calculation is like a quantum mechanics in social statistics and mainly optimised for games. In real elections we should consider the human factor. CSM elections, as many real elections, uses the STV voting model. STV voting model uses the Quota concept, as I mentioned, which “adds” the cardinal utility and Nakamura number ideas to the voting model indirectly. It’s a rough definition of course. Anyway, the STV voting model is vulnerable as I demonstrated for elections between Goonswarm and NC few posts above.
In general, all these things as CSM elections looks “fishy” as I mentioned in the OP. Just a rhetorical question. Do you think that Exookiz got more direct votes than Sort Dragon? No! He got the same amount of votes. But he got more shared votes from Bloc lists also: from Innominate, Xenuria, Dunk Dinkle, Matterall?
My expectation for CSM14, a month ago was:
- Goownswarm 2 + [3]: Aryth, Innominate, [Merkelchen], [Xenuria], [The Judge];
- NC 1 + [1]: Killah Bee, [Matterall];
- TEST 1 + [1]: Vily, [Cornak Firefist];
- Horde 1 + [1]: Gobbins, [Arqui Nurbs];
- BRAVE 1: Dunk Dinkle;
- DARKNESS 1: Sort Dragon;
- ProviBloc 1: Carbon Alabel;
- H-Sec 1 + [2]: Steve Ronuken, [Mike Azariah], [Manic Velocity];
- L- J-Space 1 + [?]: Olmeca Gold, Mathew Dust, Exookiz, Blood Ruin and e.t.c.
Official data:
Summary
Round 1 beginning - 44 candidates remain
32994 votes, 3000 quota
Initial talley:
7200 “Aryth”
5089 “Gobbins”
3232 “Olmeca Gold”
3011 “Vily”
2196 “Killah Bee”
1500 “ExookiZ”
1500 “Sort Dragon”
1389 “Dunk Dinkle”
937 “Stitch Kaneland”
920 “Steve Ronuken”
717 “Cornak Firefist”
682 “Matterall”
476 “Mike Azariah”
466 “The Judge”
446 “Merkelchen”
324 “Innominate”
…
Actions:
Elected: “Aryth”
Elected: “Gobbins”
Elected: “Olmeca Gold”
Elected: “Vily”
Transfer from “Aryth”:
Votes: 7200.000000, Factor: 0.583333, Excess: 4200.000000
3857.000000 votes to “Merkelchen”
233.916667 votes to “Innominate”
24.500000 votes to “Xenuria”
22.166667 votes to “The Judge”
15.166667 votes to “Killah Bee”
13.416667 votes to “Steve Ronuken”
5.833333 votes to “Dunk Dinkle”
5.250000 votes to “Stitch Kaneland”
4.083333 votes to “Matterall”
4.083333 votes to “ExookiZ”
3.500000 votes to “Sort Dragon”
…
The “cycle” exists in this scenario.
Voter1: a > b > c;
Voter2: b > c > a;
Voter3: c > a > b.
Imagine this scenario in real situation: 3M voters prefer a>b>c and 3M voters prefer b>c>a and 3M voters prefer c>a>b from overall 9M voters.
Here’s another example - 8 voters and 3 candidates:
- 3 voters: A > B > C;
- 2 voters: C > A > B;
- 2 voters: B > C > A;
- 1 voter: C > B > A.
A > B: 5 voters prefer A to B, B > A:3, A > C:3, C > A:5, B > C:5, C > B:3. Here’s a “cycle” between A > B, C > A and B > C.
Now lets suppose that 8 voters voted for 3 candidates differently:
- 3 voters: A > B > C;
- 2 voters: B > A > C;
- 1 voter: B > C (we are allowed to vote for different amount of candidates in any order);
- 2 voters: C > B > A.
A > B:3, B > A:4, A > C:5, C > A:2, B > C:6, C > B:2; B > C, B > A and A > C => B > A > C. Candidate B won, because candidate A didn’t won the quota.
So, “voting order” matters and it’s determined by ordinal utility mainly, because votes are deterministic by default in STV systems. Due to Arrow’s theorem and ordinal utility, Sort Dragon with 1500 initial votes lost while Merkelchen with 446, Innominate with 324 votes and Steve Ronuken with 920 initial votes won. An example how the N-Sec Blocs voting. On the other side, Exookiz got a big support from N-Sec Blocs while Olmeca Gold didn’t get a single vote from them. So, entire 0.0 Coalition of 33k + 17k +17k +8k + … players didn’t like Olmeca Gold, but he won! He won just due to the quota and direct votes, while Exookiz won due to significan support from transferred excess votes. Now enjoy an elite speech: "IDK what you are talking about! Look! A WH candidate won! Don’t whine!
Therefore, the Condorcet Paradox - it is a stick with two ends.