Bidding
Due to multiple inquiries as to how to bid I've decided to make this post.  Not because I think that people should have figured this out but because the bidding system can be very confusing due to all the options players have.  Okay, I take that back.  It's not that it can be, it simply is confusing.  I'm going to explain, as simply as possible, how everything figures in; followed by a couple of examples.  If you have any questions please please please please please ask!  With the aid of Hoffman this matter on Essences has been cleared up.  It was both items that were posted.

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Values / Use
Unless otherwise noted, whenever you use something for a bid it is spent for the entire night; winning or losing the bid does not matter.

Base Bid - Always add this in.  For players it's going to be +3 or +4, depending on your Rank.  You never spend for this, you always have it.(This is most commonly forgotten)

Elements - Each dot adds +1 to your bid.  You can use as many as you'd like but only from the Physical category or Spiritual; not both.  A good rule to determine if you can use an element to add to your bid is to see if you can use it in a simple sentence to describe your action.  For example, you cannot be Commanding (a Physical Element) enough to dodge a bullet; it simply doesn't care about what you have to say.

Expertise - Each dot adds a total equal to the max number of dots you have in that expertise.  For example, if you had 4 dots for driving, each use of a dot would grant you +4 to a bid.  Note: you can only expend one dot of your expertise per bid.

Role - Each dot adds a total equal to the max number of dots that you have in that role.  This works the same as Expertise.

Essences - You automatically gain +1 per sphere known for related bids.  This bonus is automatic and does not require any additional expenditure.  When you activate spheres you gain +1 for general use, or +2 for specific uses.  This requires the expenditure of Forces to activate the Essence.  These bonuses can stack.

Forces - Each Force spend adds +3 to your bid.  Only a maximum of 3 Forces may be spent this way per bid.  Depending on the level of your Id/Ego/Will, it will affect all of your bids.  Please check this link.  This affect requires no expenditure and will always be applied to your bid.  Forces are the only item that is spent for the entire month versus a single night.

Equipment - Always add this in.  Your item card will notate the exact bonus applied to your bid.  Unless the item has a limited number of uses, you always never "spend" this bonus, it is always available when using the item.

Circumstantial - A Director will inform you of how much your bid is affected and the reasons for this.  (Unless we are evil and secretive for some reason)  Obviously this requires no expenditure on your part.

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Example
For this example we are going to use a VERY basic character sheet.  Also, these examples are for educational purposes, do not expect the same effects for your characters.

Dead Beat Poet
Guardsman - Base Bid +3
Physical Elements - Aggressive ooo, Graceful ooo, Quick ooo
Spiritual Elements - Aware ooo, Compassionate ooo, Eloquent ooo ooo
Expertise - (Class 1) Second Rate Poetry oooo
Role - Self-Assigned Prophet oo
Forces - Id (Narcissist) ooo, Ego (Bard) ooooo, Will (Judge) o
Mutare oo

Let's say that our Dead Beat Poet gets stoned and is suddenly inspired.  To create his work he automatically bids a minimum of 4 without any effort: +3 from his base, and +1 because his Ego (which has at least 3 dots) is applicable.  The maximum that he could bid for this would be 31: +3 Base, +1 Ego, +12 (spending ALL spiritual essences), +4 Second Rate Poetry, +2 Self-Assigned Prophet, +9 (spending 3 forces of ego).  On the flip side, this weed could have been fertilized with sphinx manure, giving him some serious hallucinations, imposing a penalty on his bid. 

Seeing how fantastic this (31 bid) poem is he opts to write one about how awesome he is.  This time for a minimum bid he'll get +5, +3 Base Bid, +1 Ego, and +1 Id (like his Ego, it has at least 3 dots.)  Now, even though he used three dots of Ego, he is still treated as having 5 Ego for the purposes of the automatic bonus, not 2.  Since he has already spent all of his Spiritual Elements, the only Elements he might be able to use are his Physical, but in no way are any applicable.  As before he can use a dot of Second Rate Poetry to get +4 as well as Self-Assigned Prophet for an additional +2, coupled with a different three Forces for +9; this brings his maximum bid to 20.   Since the result was a 20 it is not quite as great as what he produced a moment ago.  It is, by mundane standards, still a profound piece of work.

Now with both of these works made he opts to walk around like Shel Silverstein (don't ask why).  He's pretty exhausted from everything he's done.  However he can still get +5 to his disguise without any real effort: +3 Base Bid, +2 for his two spheres in Mutare.  Let's say he has enough Forces left over from earlier, if he opted to activate Mutare 2 to look like Shel his bid would be +9: +3 Base Bid, +2 for his two spheres in Mutare, +4 for activating Mutare 2 for the specific purpose of looking like Shel.

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Okay, so I lied to you.  Essences are not always guaranteed to work, they just usually work.  The only time you need to determine if your Essences work is when they are competing with the activated Essences of another Awakened.  It's fairly easy to figure out, whomever has activated more spheres is the one to automatically win.  If both people activated an equal number of spheres they partake in a contested bid.  There is a bit of an exception to this rule, if there is strict numerical effects then you treat it normally; I'll give an example later.  However, when you have numerical effects combined with special effects things can get a tiny bit complicated.

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Example
No character sheets, just straight up comparisons and what the desired results are.  As before these are only examples.

Obscurity 3 VS. Wind 4.  The desired effect with Obscurity is to simply vanish.  The competing Knight uses the winds in the area to find where they are forced to move around an unseen object.  Since the Wind activated is greater than the Obscurity, the Knight using Wind automatically wins.

Stone 4 VS. Vigor 4.  The Stone 4 is meant to encase the other Knight and drag him into the ground, their response is to immediately destroy it before it can fully form.  Since both are at equal levels, they need to make contested bids.

Mutare 3 VS. Vigor 2.  The Mutare provides two to soak and ignores armor piercing while the Vigor grants +1 damage to their blows.  These both produce numerical effects, so in this instance the Vigor does not automatically lose, you add it to any damage that would normally be done.  This means you make normal bids to land the blow, if the Vigor Knight wins his damage of 3 (2 + 1) is reduced down to 1 from the soaking effect on the Mutare Knight.

Mutare 3/Resilience 2 VS. Fire 3/Lightning 3.  The first combination grants two soak and ignoring armor piercing while regenerating one health per round.  The second combination deals two damage from fire, while ignoring armor, with an additional two damage from lightning plus getting stunned for a round.  Now this ends up being treated as a 5 VS. 6 because you take the sum of the spheres activated.  Because the second combo states that it ignores armor it bypasses the effect of "ignoring armor piercing" because the second combo is a greater sphere.  This means that the first Knight is dealt four damage and is stunned for a round, but still automatically heals one health next round.