Today, I'll focus mainly on the role of the infantry on the battlefield in Lion Rampant, the medieval wargame rules by Dan Mersey.
The
 Middle Ages features a pretty static infantry: except fiece foot, who 
were mainly found at the fringes of European territories, the standard 
infantry unit deployed in a static defensive spearmen armed formation. 
The main aim was to resist the assualts of enemy's cavalry: superb 
knights, sergeants or arabs mamelukes. Occasionally they could face some
 barbaric infantry, classed as fierce foot (let apart for a second the 
possibility to grade as fierce foot halberdier or sword and buckler, we 
will come back to this later).
During Dark Age, the 
situation was similar: the Roman infantry lacked the might of the past, 
and the Barbarian Infantry, often of low quality (levy, better 
represented as foot yeoman, IMHO) was used as a rally point for noble 
cavalry, according to Procopius. Even in the East, against Sassanid, the
 battles were mainly fought between  opponent cavalry.
Things
 become very different when talking about Classical Antiquity, between 
the classic Greece and the Roman Empire (principate at last), the 
infantry was the key of the battles, with cavalry often considerated an 
auxillary corps, at least until 3rd century AD. With the rules, as they 
are written, you cannot simulate this in any realistic way. So we must 
deal with Classical Antiquity separately (i.e. in a other post!)
So, back to Infantry in Middle Ages. In my humble opinion, there is one main issue. Two infantry units (segeants or yeomen) facing 
each other, causing a situation in which no one wants to attack, because
 the attacker rolls 5+ and defender 4+. I can see the logic of this: we 
just said that during this period sergeants formed a defensive formation
 and all that stuff. But, in some particular context, if you're fielding
 pretty specular armies (for example two Communals 
Italian armies) I'd advide to allow a sort of  incentive the clash of 
infantries
 that happened frequently during the period. It's true (and a bit 
hollywodian as the author likes) that infantry stood in front of enemies
 knights, but infantry lines often clashed.

 
The author 
probably decided that the variety and interplay of arms would avoid such
 situations, and it wanted us to focus on different arms, so adding a 
cavalry unit, or a foot man at arms or a fierce foot unit (not very 
common during the feudal age), could break the balance, and in most 
games, this will work. But mono-dimensional armies exist, even if the 
rules allow you to field max 12 points or 4 units of any determined kind
 of unit, but nothing could prevent you to field 3 sergeants units 
and/or foot yeoman units.
During our game, however, sometimes happened (maybe the knight ran away or were deployed at the other side 
of the battlefield, or whatever) that two units of foot sergeants stood 
watching each other for an entire game, none of them (rightly) wanting 
to lose their advantage charging the enemy.
According to rules can 
give an offensive edge to your sergeants (and yeomen), making them "offensive" : spending 2 points more, the unit's attack become 4+ 
instead that 5+. Anyway, I think tthat 2 points for +1 attack is too expensive. 
What should we do?
- nothing, the troops variety or at least the difference in armour would break the balance.
 
- add a scenario optional rule, "Guelphs and Ghibellines":
 foot sergeants and yeomen gain +1 to dice rolled when attacking enemy's
 foot sergeants and yeomen for no cost. It could break the empasse. Use 
sparingly only in symmetric games (and under supervision of your 
parents).
 
Of course the latter option would diminishing the cost-effectiveness of purchasing offensive sergeants (still attacking 4+ on every
 enemy unit), but I already consider 2 points a bit too much for +1 when
 attacking. It's not even useful to break the balance, since the attack and defence are equal in sergeants or yeomen, but you can 
field 3 units vs 2. If the players with three static sergeants sits 
down, the offensive minded player will soon finds his men are too few to
 disrupt 3 enemy's units.
And this is the first issue 
of Medieval Warfare that could need a fix. Bear in mind I just proposed 
one of the possible solutions.
Second issue: where are the halbardiers and (later in period) sword and buckler men?
Well, that's an interesting topic: in the rules there are two ways to portray them:
- use offensive foot sergeants or yeomen (the unit costs +2 points) 
attack and defend on 4+ and lose schiltron, armour as the main unit (3 
or 2).
 
- use fierce foot (4 points) attack on 3+, defend on 6, wild charge! armour 2.
 
I see none of these option is very used by our local club, why?
Offensive
 foot (either sergeants or yeomen) are the best way to represent 
halberdiers, in our opinion, because they retain their defensive 
capabilities, but they cost too much. Fierce foot aren't really loved 
because they're wild charger, and stay almost halplessy if they fail to 
countercharge.
Possible solutions:
- offensive foot cost reduced to +1 point
 
- add a new upgrade for foot sergeants and yeomen (I really had sword 
and buckler in mind when I wrote this) (I cannot think about an 
appropriate name now, let's call them AlternativeOffensive), leaving the hot minded troops as fierce foot.
 
AlternativeOffensive@0points
The unit switch Attack with 
Move and Attack Value with Defence value. It lose Schiltron special 
rule, and gain Countercharge Infantry special rule.
So an Alternatively Offensive Foot Sergeants unit (Sword and Buckler for example):
Attack 5+  Attack Value 4+
Move 6+   Defence Value 5+
Shoot - Shoot Value-
Courage 4+ Armour 3 (2 if foot yeomen)
Max movement 6"
By
 the way, this is exactly how I would represent Early Migration (i.e. 
Early Dark Age) infantry in  those people still using massive infantry 
charges, instead that relying on cavalry for their attacks (the 
latter are often referred as Sarmatized Germans). Of course I would also
 represent in the same manner Celts infantry during Classical Antiquity.
 But I think this post is already long enough and I shouldtalk about them an other
 time.