Gameplay Issue

Currently, when a player sieges an AI city, the AI will almost always sortie immediately. Once in a sortie, the AI is forced to be aggressive, since it will lose the city if the turn timer runs out and it has not outright won. Taken together, this becomes a frustrating behavior where the AI will start a fight it probably can't win, then run its units out of its fortifications to die ineffectively. Players feel this makes sieges against the AI feel too easy and exploitable.

I don't know much about AI programming so I don't know why the AI does this. However, I think at least part of the cause is that the current siege mechanics reinforce this behavior and can even make it strategically sound. Currently, siege engine production and militia drain mean that as a siege is maintained, the attacker gets stronger and the defender gets weaker. As a result, the defender should generally sortie as soon as possible. This can be true even if the defender will probably lose the sortie, since their prospects of keeping the city will only get worse if they wait, and they will do more attrition damage to the attacker if they attack with their maximum free militia rather than wait for them to starve. 

This feels like it is counter to player expectations of how sieges should play out. Historically, a protracted siege was often bad for an attacking army, and a fortified defender could hope to wait out a besieging force while it burned through supplies, manpower, and morale. In some cases, a sieging army could be exhausted to the point of having to retreat - or even end the war - without the defender mounting a committed counteroffensive. 

Proposed Changes

Other genre games like Europa Universalis simulate attacker attrition by having a passive penalty for a sieging army. This puts the attacker on a timer, so they have to decide how much they want to invest in a siege before they are forced to assault or back off. The longer the siege lasts, the more cost-inefficient the victory becomes, and the greater the risk of a successful counterattack.

-Each turn the attacker maintains a siege, they must pay an "attrition cost." This could either be HP (e.g. sieging army loses 5 HP per turn, representing attrition losses) or War Support (e.g. the attacker loses 5 WS per turn, representing supplies and morale running out). 

-The "maintain" button in the siege panel has the attrition cost directly on it (with a tooltip breakdown) to make it obviously a decision the attacker is making to invest in the siege.

-The defender's option to sortie is removed, and only the attacker can launch a battle to take the city. (The defender can counterattack the sieging army with one of their armies, resulting in a normal battle.)

-Some defensive infrastructures increase an attacker's attrition penalty (e.g. Stone walls and Bastions each deal +5HP damage per turn to sieging units). 

-Some economic infrastructures like granaries and grain silos reduce the rate at which the defender's miltia is drained during a siege (in addition to their current effects), so the defender can make sieges last longer and force the attacker to suffer greater attrition. 

Analysis

With current siege mechanics, it seems like the pressure the attacker is intended to face is the risk of defensive reinforcements. In practice, the AI almost always has defensive reinforcements immediately available, and reinforcements can join a battle in progress anyway - so it is still generally strategically better to sortie immediately with maximum defender's advantage, then bring in reinforcements as they arrive. Regardless, the AI seems to struggle with the complex logic to determine when it should sacrifice its defender's advantages to wait for reinforcements (which can be hard to determine even for human players).

Adding a more direct cost/reward calculation for the attacker would shift the complex choice of when to start battle onto the attacker, and potentially make sieges play out more like players want them to. Theoretically, an attrition penalty means the attacker wants to attack as soon as they can reasonably secure the city, so they need to make a tough choice about whether siege engines and militia drain will be worth the HP/war support they invest, plus the heightened risk of reinforcements arriving while they're weak. Meanwhile, all the defender needs to do is buy time, either to wear down the attacker to prepare for a counterattack they can actually win, or just make the city more costly to take (without needing to commit to a suicidal sortie).

HP loss is the more tried and tested penalty, but I think War Support might work a bit better for HK mechanically, as it would not have to determine which units are "involved" in a siege (players could not exploit the reinforcements system to avoid attrition damage). A WS penalty would mean that if an attacker brings overwhelming force to a siege (or the defender doesn't have good defenses), they could assault right away and earn full WS for winning the city. If the forces are more balanced, the attacker could decide to sacrifice some WS to prolong the siege and reduce their risk of defeat - but must be careful not to overextend themselves, or they will end up losing more WS than they gain from victory (similar to how a normal battle can result in a net WS loss for the victor if they lose more units). Since occupying a city passively generates War Support over time, this could lead to situations where it is worth "spending" banked WS in the short term to take a city that you can hold for a long war.  

While the attrition penalty isn't mutually exclusive with sorties, removing them as an option would prevent the AI from instantly sortieing, without having to change its logic to understand that an instant sortie is usually no longer the best option. The AI might still be overeager to counterattack a sieging army with its non-militia forces, but it would be able to use the same logic it does for engaging in normal battles (which has gotten pretty good, and much better than its sortie behaviour). More importantly, if the AI loses a counterattack rather than a sortie, it doesn't instantly lose the city and all its militia; the attacker must still attack the city or continue the siege. This way, militia can only appear in defensive combat, where they can safely hide behind walls instead of playing suicidal capture the flag.