Okay, so, the Paraguayan leadership made two big assumptions that would allow victory in this scenario.

The first was the mobilization assumption. Paraguay had the largest standing army of the region, precisely so that the prospects of its neighbors needing to mobilize their population, a very expensive and unpopular thing to do, would not be appealing at all and instead the other countries would try to appease Paraguay’s interest as much as possible. That didn’t work, because of Paraguay’s previous decades-long isolationism. The country played so little a role in previous conflicts in the region that, when it tried to gain influence through military strength, Brazil and Argentina’s foreign policy towards it was literally just to ignore it, that it might talk a big game but would do nothing. The biggest example of this is Paraguayan President Solano Lopes’ ultimatum he sent to Brazil before invading, an ultimatum that never made it to the Brazilian Emperor because Paraguay was considered an irrelevant entity that didn’t warrant urgency in case of hostilities, a mindset that changed drastically when the war began.

The second assumption that turned out to be false was a Hitler-like belief by the Paraguayan President that Brazil and Argentina were rotting countries, with no real national unity or loyalty to patriotic beliefs. Argentina had just finished a bloody civil war, and the government in power was resented by a large sum of the population. Brazil was seen as an artificial construct created by the Portuguese royal family to remain in power in the continent, with huge amounts of land to its west very sparsely populated and isolated, i.e. for easy conquest, colonization and assimilation for a foreign power coming from the center of the continent. Most of those ideas fell flat in practice, as although a lot of Argentinian citizens despised the current government, it wasn’t enough to support a foreign invasion of their country. In Brazil, the news of the invasion caused general outrage and a wave of patriotic support that would fuel the war effort in its first year, although support did indeed waver once it became an extended conflict. Nonetheless, the emperor took the war very personally and would refuse all attempts at a white peace or a compromise, only considering acceptable proposals were the Paraguayan President would turn himself in to the Brazilian army to be judged for his crimes.

So, the TL;DR is that that South America was an unstable region in the 19th century, with new countries and rapidly changing borders, that allowed for a flawed understanding by the Paraguayan leadership of their prospects for war with their neighbors. Paraguay was one of the most stable countries in South America thanks to their president for life dictatorships, leading to the assumptions that it could capitalize on the regional instability.

Sources: Maldita Guerra, by Francisco Doratioto

As Barbas do Imperador, by Lilia Schwarcz

1889, by Laurentino Gomes