a song of ice and fire - How big of a deal is Westeros?

Of course, Westeros is the main stage for A Song of Ice and Fire, but how important is it at a planetary scope?

There is a lot of civilization beyond the Narrow Sea and, especially around the Narrow Sea, most of it is very evolved in terms of society and economy. Also the Dothraki may live a very wildling-ish life, but they appear neither afraid of nor foreign to higher cultures. There are also the Wildlings north of the wall, which also seems to be a large civilization.

So how important is Westeros to the entire planet? Are they a few among many or are they the most sophisticated society on that planet and take a rather leading role?

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More information I found

I re-watched the TV show and did a little army counting on the side. The numbers don't feel mathematically sound, but maybe the Westerosi are very good at accounting for losses in most recent battles, while I'm not. Give or take a few garrisons, all of Westeros has (had) about 300K soldiers. If I had to guess I'd say that's a little more than the Wildlings and all of Essos put together, but I can't find solid numbers on those. So, measuring in military size and assuming there aren't any other big inhabited continents we don't know of, Westeros is about half the world. I'm still having trouble gauging the size of Essos.

In Yunkai alone, there are about 200K slaves (according to TV show S3E7). This allows for speculation that Yunkai alone has a population somewhere between 200K and 1M people. If Yunkai does not greatly outsize all other Free Cities, Essos probably has a much larger population than Westeros. Kings Landing has a population of 500k.

Even more information I found

Just reading A Storm of Swords and it would appear Mereen, Yunkai and Astapor are not of the nine Free Cities. IIRC, I got this impression from the Daenerys chapter where she talks to Kraznys mo Nakloz for the first time. Comparing impressions from reading the books vs. the TV series, according to the book there are way more Dothraki and other tribes/people living on that side of the narrow sea. So in terms of population I would reduce my guess, making Westeros about one third of the known world.

In terms of society and technology I also wouldn't call Westeros leading. Their beliefs and knowledge seem to have devolved over time. There are the new gods and people somehow accept there are old gods as well. Everything else is little more than folly, if at all. The Maesters can choose to study magic but few do and none can actually perform magic. The alchemists seem a pathetic guild of engineers. Nobody believes there are things like mammoths, giants, skinchangers and such. The remaining acknowledged supernatural thing are the Children of the Forest, but even here everybody knows there aren't any left anywhere.

In Essos, many beliefs are practiced, but people are hardly judged by that. Maegi are more than tales and everything that may seem impossible/unrealistic can be done by at least one person from Asshai. If not, you will still find at least one person who will claim they once met someone from Asshai who could. The Warlocks in Qarth live more ore less openly among other people. In general, the Free Cities appear bigger not only in population and size but also the architecture, than any dwelling in Westeros. All of Essos seems well arranged, even settled, while all of Westeros goes to war whenever a highborn firstborn hits puberty.

Answer

I think it's important to understand the planet, and how much is known. Currently there are 4 known continents of the known world: Westeros, Essos, Sothoryos, and Ulthos. We should remember that civilisation in the known world is currently at a state where there are many parts of the world still uncharted. It is entirely possible that there are more continents that are not known. Consider the late discovery of the Americas, and the even later discovery of Australia, by Eurasian civilisation.

Further to this, we don't even know a significant amount about the four continents. Little is known about Sothoryos, and almost nothing is known about Ulthos. Even the two continents that are known, Westeros and Essos, are not fully charted. No complete map of Essos exists, and the polar regions to the far north of Westeros, way beyond the Wall are still unexplored. However, these lands are unlikely to contain any great civilisation, if life at all, so we can consider the knowledge to be sufficient.

So this means that any answer can only truly be based on a comparison between Westeros and Essos.

The main difference between Westeros and Essos is that the former is united as The Seven Kingdoms (clearly the civil wars in A Song of Ice and Fire throw this into question, but lets look at this from the start of the series). With the exception of a few raids from the wildlings, a few pirates along the narrow sea, and bandits roaming the kingdom, Westeros has enjoyed relative peace. This has allowed their civilisation to prosper. Lords are free to devote resources to keeping roads safe, and clearing bandit raids, allowing smaller villages and farmsteads to enjoy a safer life.

Essos consists of many different peoples. A few of the most significant are the Dothraki, the Free Cities, the Lhazareen, Qarth, and many other regions. One of the main territories was the Valyrian Freehold. Once a great, powerful and knowledgeable territory, left in ruins and all but extinct since the Doom of Valyria.

As Essos is not united, the internal fighting, and the threat of the fierce Dothraki, has shaped the continent into a very different civilisation to Westeros. Rather than many villages and farms surrounding a Lord's castle, as is the case with Westeros, much of the Essos civilisation consists of huge, fruitful cities, with high walls and little outside. An obvious example of this is Qarth. An impressive port city, booming on trade, but surrounded by nothing but wasteland. The reason for this sort of growth, is that smaller villages struggle to withstand the Dothraki. The Lhazareen are constantly under attack from the Dothraki, and the Kingdom of Sarnor has been left in ruins from Dothraki attack. Therefore the best option for civilisation in Essos is a concentrated city, near the sea, allowing them to amass a large, single army to defend them, surrounded by high city walls.

It's hard to compare different civilisations in this world, as it is not a global world such as our own. Westeros is quite insular, but is clearly a significant land mass and civilisation within the known world. I'll try and find canon examples of mentions of Westeros' significance, but hopefully this will provide you with enough information.