Splitting defensive alliance into chaining alliance + isolating alliance.

The classic defensive alliance

Call the countries Ashland, Bogland, Cragland, Dryland. Ashland and Bogland form a defensive alliance because they fear Dryland. (perhaps Bogland fears that Dryland will invade to steal water, whatever). Now the people of Ashland and Bogland feel twice as safe, because their united armed forces are twice the size that each mustered individually. The way this usually works out is that Ashland and Bogland square root it

Square rooting it

It is common for an illustrative example to talk about things getting halved or doubled. Often talking of things doubling is useful. Increasing fourfold is a large increase and may raise other issues. Often a 10% increase makes little sense in context. If two equal countries form a defensive alliance, their military strength is greatly increased, not just by 10%. If Lanchester's Square Law applies, then their military strength is indeed increased four fold. Let us skip that complication, and just go with doubling.

There are many other calls on the public purse beside defense spending. If Ashland and Bogland feel much safer due to their defensive alliance they will likely feel free to transfer some money from guns to butter. We just want familiar numbers for an illustrative example. Root two is approximately 1.4. One over root two is approximately 0.7. Rather than invent detailled budgets for Ahsland and Bogland, let us assume that each cuts defense spending by 30%. So each has a military 70% of the strength they had when they stood alone and afraid. But combined, in is 40% stronger. That factor of two increase has been split into 40% stronger and 30% cheaper, because 1.4 times 0.7 is two, near enough. We call spliting a factor of two change into two factor of 1.4 changes square rooting it and it is convenience when we need to be realistic about political trade-offs

Extending guarantees

Consider now Bogland giving security guarantees to Cragland. Perhaps Dryland attacks Cragland anyway. Bogland goes to war with Dryland to honour their guarantee to Cragland. And now for the tricky bit: does Ashland go to war with Dryland to honour their alliance with Bogland? When the time comes the answer might be "no" because Ashland never agreed to extend security guarantees to Cragland. Or the answer might be "yes" becuase honouring alliances is what you do. A differen way to think about this is to imagine that there is clarity before the event.

A chaining alliance is one in which Ashland allies with Bogland. Bogland guarantees the security of Cragland. Dryland attacks Cragland. This starts a war between Cragland and Bogland, and Ashland is dragged into fighting Dryland to defend Bogland by the chains of the alliance.

An isolating alliance is one in which Ashland allies with Bogland. Bogland wants to extend security guarantees to Cragland, but pauses for thought. The treaty with Ashland is explicit. Ashland isn't guaranteeing the security of Cragland and if Bogland wants to comes to Cragland's aid, it will fight Dryland alone. Ashland isn't chained to Bogland and will not be dragged in to an expanding war.

But why call this an "isolating alliance"? Who is being isolated from whom? This goes back to square rooting it. Bogland was happy to ally with Ashland. It wasn't just about being more secure. It was about saving money on defence leaving more to spend on schools, pensions, and hospitals. That is all very nice until one faces the prospect of fighting alone. Before the wretched so called alliance with Ashland, the politicians of Bogland had a strong military and standing in the world. If they wanted to offer Cragland security guarantees, they had the troops to back that up. But the alliance has lead to defense cuts. If the alliance doesn't cover the war that might follow from protecting Cragland from Dryland, it leaves Bogland in a bind. The ruling class of Bogland want to guarantee Cragland against Dryland, but now find that they lack the military strength. They have been tricked and manoevered into isolationist policies by an isolating alliance.