WitA: Victory Condition?

[NoPunIn10Did]

The victory condition for War in the Americas was originally 31. Playtesting has shown that this is a bit too high of a bar to cross. For PlayDiplomacy, we need to set a different victory condition.

NanookTheEskimo has posted on this topic as well, and he has recommended 24 centers.

31 SC is too high a total for a solo. 24-28 is the sweet spot, and my recommendation would be for 24. There’s math behind this (courtesy of nopun) that I can get into later. 31 is too hard for most countries to get to playing with standard build rules.

What happens if two people reach 24 at the same time? Luckily, we already have a precedent on site for what to do, in 1900. I would propose the same rules apply to WitA.

For this workshop post, I want to share a little of the "math behind this" that Nanook refers to, and I want to get feedback on where players think the sweet spot might be for WitA.

[NoPunIn10Did]

Why not just use the majority?
Some players tend to assume that a proper Diplomacy variant should always require the majority of SCs for a solo. They may take this as a matter of principal. If you find yourself in this camp, you may find the rest of what I describe to be a whole lot of kabalism and fuzzy math. But hear me out.

We've seen that non-majority SC goals can still make for engaging gameplay in Diplomacy variants; our case in point currently present on the site is Baron VonPowell's 1900, which requires 18 out of a possible 39 to win. What we also know is that with a lower threshold, achieving a solo is much easier (that is, not necessarily easier for a specific player, but rather that a game is much more likely to end in a solo rather than a draw). There's also an inherent difference in difficulty between playing a 7-player game and achieving an outright majority of all SCs versus playing a 5-player or 10-player game with the same goal, especially if you're playing toward the solo (rather than toward a particular draw size).

I wanted to find a way to generalize and quantify the difficulty of achieving a solo as a metric that would reflect SC counts and player counts. I ended up using an existing metric: Sum-of-Squares.

Some Background on SOS
Sum-of-Squares (SOS) is a scoring model used in many face-to-face Diplomacy tournaments as well as on webDiplomacy. SOS rewards players for not only having more SCs, but also having more SCs in relation to one's opponents. Games that end in evenly-split draws tend to have smaller SOS scores for their board-toppers, while games that end with one board-topper and a bunch of small, divided opponents will tend to have larger SOS scores.

We're not going to use this thread to debate the pros and cons of playing SOS games (where it is a normative metric), but one thing that it is very good at is modeling is a player's relative proximity to victory based on the board state (where it is an observational metric).

That is to say, a player who tops the board with an SOS score of 55 had a greater likelihood of being closer to a solo than a player who topped the board with an SOS of 44.

What does this have to do with the victory condition?
I ran a bit of a numerical experiment using SOS and several different variants. For this exercise, I looked at the SOS score range for hypothetical soloists (i.e. I ignored the rule in SOS scoring that converts a soloist's score to 100).

My simulations assume the following

1. All SCs on the board have been captured by somebody (no neutrals left).
2. The soloist has exactly the right number of SCs to win.
3. No other player has that many SCs.
4. Each map will have a high scenario and a low scenario.
5. In the high scenario, no player has been eliminated.
6. In the low scenario, the fewest number of players are on the map (where conditions 1-3 still apply).

I call the high point of this range the Rigidity of a variant. Because the low point of the range tends to change primarily based on SC victory conditions rather than player count, I measured it but didn't find it as useful.

Example Rigidity: Classic Diplomacy
For example, for Classic Dip, my high scenario was 18-3-3-3-3-2-2, which yields an SOS for the winner of 88.043. The low scenario was 18-16-0-0-0-0-0, which yields an SOS of 55.862.




Other Examples
I have an imgur album of examples of several other variants, including 1900, Ancient Med, Hundred, and several more.

Diplomacy Variant Rigidity on imgur

You'll also see four difference simulations for victory conditions in War in the Americas: 31, 25, 24, and 21.

So What is the Right Rigidity?
Well, there isn't a "right" number. What it comes down to is more a question of what other variants you want a given map to be most similar to in terms of ease-to-solo. In 1936, for example, the proportion of SCs to win is a very low 18/50 for a player count of 7; that was done in part because the designer (Charles Feaux de la Croix) wanted almost every game of 1936 to end in a solo. 1936 has a rigidity of 65.323.

So What is the Right Rigidity for WitA?
Also not a clear answer here, but if you want War in the Americas to more closely resemble 1900 in terms of ease of soloing, and you don't want to make the game harder to solo than Classic (which is already very hard to solo), it makes sense to aim for a rigidity close to that of 1900 (81.203).

Setting a victory condition for WitA of either 24 or 25 SCs will achieve that.

Disclaimer
Rigidity is a model for measuring difficulty to solo. It doesn't take into account stalemate lines or other features of a map that might make it more-or-less likely to end in a solo as opposed to a draw. Those factors are far harder to quantify, so we generally have to rely on playtesting to help us understand them.

[Enriador]

Excellent analysis! I would prefer a 21 out of 61 SC criteria. It's more than a third of all SCs and will make the map more dynamic.

On tiebreaking: I am not a big fan of 1900's approach. A victory criteria is supposed to define when the game ends and a winner exists. Continuing the game after the criteria has been fulfilled kinda runs against what it was created for.

I prefer vDiplomacy.com's approach: the game always ends when someone reaches the victory criteria. If two or more players got the criteria, the game checks who had more SCs the previous year. If it's a tie again, the game checks the year before, and so on.

Another alternative is 1812 Overture's "double victory" scenario. If two players grab the victory criteria at the same time (fat chance!) they are both awarded a victory.

If coding limitations are on the way, then 1900's tiebreaking is an acceptable option.

[Enriador]

And out of curiousity, Canton's SOS results would be:

Canton, 7p, Raw SCs, High Scenario = 19, 3, 3, 3, 3, 3, 2.
Canton, 7p, SOS, High Scenario = 88.048

Canton, 7p, Raw SCs, Low Scenario = 19,16, 0, 0, 0, 0, 0.
Canton, 7p, SOS, Low Scenario = 58.508

So basically identical to Classic.

[NoPunIn10Did]

But if there’s a tie for first, then neither player has actually met the criteria.

The nice thing about 1900’s victory condition is that it can apply to the vast majority of variants.

1. The player must have more centers than every other individual player AND
2. The player must have at least X centers (18 in 1900’s case)

[Nanook]

I would advocate pretty strongly for a 24 SC victory condition, with 1900 tiebreak rules. 31 is just too high on this map, without build anywhere rules there's only two, maybe three countries with home centers located in a way that would give them a reasonable chance to solo on a board against decent/competent players.

21 is what we used in our last PbF run, and while it's better than 31, I think it encourages a bit too much over-paranoia that stifles the mid and end game.