# Electoral systems math is so cool!

It’s a good day when a student comes to office hours wanting a refresher on the various formulas. Even better when at the end he says it is “so cool” how the math explains politics.

## 8 thoughts on “Electoral systems math is so cool!”

1. Nathalie S. says:

Honestly, the math for D’Hondt and Hare-LR is so satisfying, I completely understand where that student is coming from.

• Tom Round (aka Gaudiatrix) says:

I must say I always picture STV operating like one of those machines that sorts coins neatly into different tubes according to their respective sizes/ denominations.
Kaimipongo Wengerer, a US law prof who teaches corporate governance law, distributes M&Ms among his students to explain the concept of cumulative voting.
I once mocked up a combat board game that was something like “Risk” with a “Senate”, or a more abstract “Kingmaker”. For every two provinces your armies controlled, you got five popular votes, which you could trade with allied players. You could cash in twelve popular votes for three Senate seats, or five PVs for one seat,when the “elections” were held. Players gained or lost Senate seats as their armies conquered or ceded provinces. Never marketed it.

• Tom Round (aka Gaudiatrix) says:

(Once you controlled, I think, 25 Senate seats you could declare yourself “President for Life” and win the game.)

• Alan says:

When people still played board games (I believe this period ended sometime around 1500) there was a game called Roman Republic where the object was to gain control of the senate. And then there was Kingmaker which required players to master bicameralism in the Wars of the Roses.

• Tom Round says:

Board games still have a niche as an excuse to meet people in realspace and share pizza, but they have to be simple. Rivalry from MMPORGs means that manufacturers can no longer expect a profit from 15,000-piece historically accurate re-enactments of the siege of Stalingrad.

My one attempt at playing KINGMAKER suggests that it, too, will by now have been overtaken by something like AGE OF EMPIRES where a computer calculates all the dice-rolls for you.

Anecdotal perusal of a few toy stores recently shows that 90% of inventory has hardly changed since I was a kid in the Seventies, apart from franchise-based variations (“House of Cards” RISK, “Fear the Walking Dead” MONOPOLY, etc).

Ie, the board games that survived are very simple — in some ways too simple. RISK and MONOPOLY (and POLECONOMY — an Australian variant) are a bit too dice-based for my liking, with very little room for alliances, strategizing, and Pontoon-type tactics. I would like to see a “city council” added for MONOPOLY. Election for one of five “seats” in rotation every time someone passes go — players pick up cards worth 10 to 50 votes each, “bid” these, highest combined bid wins, and hope like Joe Kennedy Senior that you didn’t pay for a landslide. Player(s) controlling at least three of the five seats can expropriate other player’s houses and hotels, or selectively waive the five-building limit.

My combat game was to be very simple and abstract, and set in a stylised, made-up country. Players could, revocably, delegate command of their forces to a fellow player. This would make your alliance much more militarily efficient — if Red delegates to Green, Green can then move any of the Red or Green pieces on both Red’s and Green’s turns — but it also creates an agent/ principal (or a Hobbesian sovereignty) problem, eg, Red suspects Green has deliberately used up the Red troops first as cannon fodder, so secedes from the alliance.

I got as far as a rough attempt with some friends using Lego blocks on an A-3 sheet back in 1995, I think, but then my PhD-writing took precedence.

I have tried to come up with some way to model election results in a game (other than POLECONOMY-style dice rolls), and have come to the reluctant conclusion that manoeuvring among the players is the best proxy for real voters.

(* Slight correction to my first post above: “You could cash in twelve popular votes for three Senate seats, or seven [not five] PVs for one seat…” ie the electoral system was semi-proportional, so as to encourage both electoral pacts before voting and coalition deals in the “Senate”).

2. Alan says:

There are some complex boardgames still being published and played, but the market must be tiny compared with computer games. There are also some fairly sophisticated computer games like the Democracy series. I feel terrible hijacking yet another thread but at east we are not talking about a certain electoral system.

• Tom Round (a.k.a Gaudiatrix) says:

I’m not sure the original thread had a destination airport, so “time for some GAME THEORY!!!” is as good a non-hijack as any.
I do have a preference for divisors over remainders (outside the context of STV elections, of course) because (a) they are more monotonic, ie avoid the “Alabama paradox” (where that great State got, I think, 10 seats in a US House of 300 but only 9 in a House of 299, and (b) can be used to “prioritize”, in an iterative or chronological order that allows “queuing”: eg, “Okay, so far 9 of the first 10 seats allocated have gone to male candidates, so the next goes to the female candidate with the highest comparison number…” rather than “Oh, blow, we’ve accidentally elected 10 male candidates on whole quotas, so we’ll have to retrospectively unseat one of them… sorry buddy”.
Having said that, almost all student council and trade union cons & regs I’ve come across seem to use “one Councillor per whole Hare quota or highest remainder” so perhaps divisors are too hard to explain. The closest _simple_ approximation to D’Hondt would be something like: “Initially the quota is ‘divide by seats plus 2’, but if that allocates too many seats, re-calculate with ‘divide by seats + 1’, and if that allocates too few seats, round up the largest remainders”. Remainders are not quite as bad with the Droop quota as with Hare since fewer seats are allocated to sub-quotas under the former.