Value of a +X to d20 1/encounter vs static +1.

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What is +X 1/encounter worth?  (such as wizard's wand impliment, or shield).  This is for offense and defense.

NOT ALL ATTACKS ARE CREATED EQUAL.


Bonses for big/early attacks makes dynamic values more valueble.
Actions cost also needs to be taken into consideration (mainly interupts), making static more valueble. 



X = Number of attacks per encounter (or day, if you want to figure out how much a daily counts for)
Y = Bonus
*Note that this assumes that you can use the full bonus.  If you hit/miss on a Z, then Y=Z-1.
(i.e.  if you hit on a 6, then a +8 bonus will only be worth 5).

Static:
Y * .05 * X

Dynamic (1/encounter):
1-((1-(Y *.05))^X)                 1 - Chance of it being NOT used.

Next Attack:  (not really interesting, so i'm not gunna chart it)
Y * .05

Dynamic vs Static = number of attacks to be equal  (using a graph method).
+2 vs +1 = 16.47

+3 vs +1 = 19.1
+3 vs +2 = 6.55

+4 vs +1 = 19.76
+4 vs +2 = 8.5
+4 vs +3 = 3.83

+5 vs +2 = 9.31
+5 vs +3 = 5.1
+5 vs +4 = 2.7

You do between 5-15 attacks per encounter, depending on build (multi/off action attack / AOE do more)
You are attacked ....???  per encounter.

guides
List of no-action attacks.
Dynamic vs Static Bonuses
Phalanx tactics and builds
Crivens! A Pictsies Guide Good
Power
s to intentionally miss with
Mr. Cellophane: How to be unnoticed
Way's to fire around corners
Crits: what their really worth
Retroactive bonus vs Static bonus.
Runepriest handbook & discussion thread
Holy Symbols to hang around your neck
Ways to Gain or Downgrade Actions
List of bonuses to saving throws
The Ghost with the Most (revenant handbook)
my builds
F-111 Interdictor Long (200+ squares) distance ally teleporter. With some warlord stuff. Broken in a plot way, not a power way.

Thought Switch Higher level build that grants upto 14 attacks on turn 1. If your allies play along, it's broken.

Elven Critters Crit op with crit generation. 5 of these will end anything. Broken.

King Fisher Optimized net user.  Moderate.

Boominator Fun catch-22 booming blade build with either strong or completely broken damage depending on your reading.

Very Distracting Warlock Lot's of dazing and major penalties to hit. Overpowered.

Pocket Protector Pixie Stealth Knight. Maximizing the defender's aura by being in an ally's/enemy's square.

Yakuza NinjIntimiAdin: Perma-stealth Striker that offers a little protection for ally's, and can intimidate bloodied enemies. Very Strong.

Chargeburgler with cheese Ranged attacks at the end of a charge along with perma-stealth. Solid, could be overpowered if tweaked.

Void Defender Defends giving a penalty to hit anyone but him, then removing himself from play. Can get somewhat broken in epic.

Scry and Die Attacking from around corners, while staying hidden. Moderate to broken, depending on the situation.

Skimisher Fly in, attack, and fly away. Also prevents enemies from coming close. Moderate to Broken depending on the enemy, but shouldn't make the game un-fun, as the rest of your team is at risk, and you have enough weaknesses.

Indestructible Simply won't die, even if you sleep though combat.  One of THE most abusive character in 4e.

Sir Robin (Bravely Charge Away) He automatically slows and pushes an enemy (5 squares), while charging away. Hard to rate it's power level, since it's terrain dependent.

Death's Gatekeeper A fun twist on a healic, making your party "unkillable". Overpowered to Broken, but shouldn't actually make the game un-fun, just TPK proof.

Death's Gatekeeper mk2, (Stealth Edition) Make your party "unkillable", and you hidden, while doing solid damage. Stronger then the above, but also easier for a DM to shut down. Broken, until your DM get's enough of it.

Domination and Death Dominate everything then kill them quickly. Only works @ 30, but is broken multiple ways.

Battlemind Mc Prone-Daze Protecting your allies by keeping enemies away. Quite powerful.

The Retaliator Getting hit deals more damage to the enemy then you receive yourself, and you can take plenty of hits. Heavy item dependency, Broken.

Dead Kobold Transit Teleports 98 squares a turn, and can bring someone along for the ride. Not fully built, so i can't judge the power.

Psilent Guardian Protect your allies, while being invisible. Overpowered, possibly broken.

Rune of Vengance Do lot's of damage while boosting your teams. Strong to slightly overpowered.

Charedent BarrageA charging ardent. Fine in a normal team, overpowered if there are 2 together, and easily broken in teams of 5.

Super Knight A tough, sticky, high damage knight. Strong.

Super Duper Knight Basically the same as super knight with items, making it far more broken.

Mora, the unkillable avenger Solid damage, while being neigh indestuctable. Overpowered, but not broken.

Swordburst Maximus At-Will Close Burst 3 that slide and prones. Protects allies with off actions. Strong, possibly over powered with the right party.

I think it somewhat depends on build.  For instance,  a twin striking ranger or any avenger, probably values the +1 more.  On the other side of the coin, a wizard trying to land a daily probably values the +x more.

Some attacks simply matter more, and some classes by design make a lot more rolls than others.  I think generalizing here might actually be the wrong approach.  Rather than trying to sort out how many attacks per encounter, etc, it might be better to identify when it's better to take a penalty to hit on at-wills in order to maximize an encounter/daily.

This is a very good question.  I have markings of the victor on my goliath and its roughly the equivalent of a +5 to one attack roll/encounter.  It think it would vary a lot by the number of attacks, which varies significatly for classes.  But starting at 10 is a good start. 

And there are next "attack roll" bonuses and "next attack" variations.
Is it for a turn or for a single attack? This is important.
Mountain Cleave Rule: You can have any sort of fun, including broken, silly fun, so long as I get to have that fun too (e. g., if you can warp reality with your spells, I can cleave mountains with my blade).
Is it for a turn or for a single attack? This is important.

Everything's calculated per attack.

guides
List of no-action attacks.
Dynamic vs Static Bonuses
Phalanx tactics and builds
Crivens! A Pictsies Guide Good
Power
s to intentionally miss with
Mr. Cellophane: How to be unnoticed
Way's to fire around corners
Crits: what their really worth
Retroactive bonus vs Static bonus.
Runepriest handbook & discussion thread
Holy Symbols to hang around your neck
Ways to Gain or Downgrade Actions
List of bonuses to saving throws
The Ghost with the Most (revenant handbook)
my builds
F-111 Interdictor Long (200+ squares) distance ally teleporter. With some warlord stuff. Broken in a plot way, not a power way.

Thought Switch Higher level build that grants upto 14 attacks on turn 1. If your allies play along, it's broken.

Elven Critters Crit op with crit generation. 5 of these will end anything. Broken.

King Fisher Optimized net user.  Moderate.

Boominator Fun catch-22 booming blade build with either strong or completely broken damage depending on your reading.

Very Distracting Warlock Lot's of dazing and major penalties to hit. Overpowered.

Pocket Protector Pixie Stealth Knight. Maximizing the defender's aura by being in an ally's/enemy's square.

Yakuza NinjIntimiAdin: Perma-stealth Striker that offers a little protection for ally's, and can intimidate bloodied enemies. Very Strong.

Chargeburgler with cheese Ranged attacks at the end of a charge along with perma-stealth. Solid, could be overpowered if tweaked.

Void Defender Defends giving a penalty to hit anyone but him, then removing himself from play. Can get somewhat broken in epic.

Scry and Die Attacking from around corners, while staying hidden. Moderate to broken, depending on the situation.

Skimisher Fly in, attack, and fly away. Also prevents enemies from coming close. Moderate to Broken depending on the enemy, but shouldn't make the game un-fun, as the rest of your team is at risk, and you have enough weaknesses.

Indestructible Simply won't die, even if you sleep though combat.  One of THE most abusive character in 4e.

Sir Robin (Bravely Charge Away) He automatically slows and pushes an enemy (5 squares), while charging away. Hard to rate it's power level, since it's terrain dependent.

Death's Gatekeeper A fun twist on a healic, making your party "unkillable". Overpowered to Broken, but shouldn't actually make the game un-fun, just TPK proof.

Death's Gatekeeper mk2, (Stealth Edition) Make your party "unkillable", and you hidden, while doing solid damage. Stronger then the above, but also easier for a DM to shut down. Broken, until your DM get's enough of it.

Domination and Death Dominate everything then kill them quickly. Only works @ 30, but is broken multiple ways.

Battlemind Mc Prone-Daze Protecting your allies by keeping enemies away. Quite powerful.

The Retaliator Getting hit deals more damage to the enemy then you receive yourself, and you can take plenty of hits. Heavy item dependency, Broken.

Dead Kobold Transit Teleports 98 squares a turn, and can bring someone along for the ride. Not fully built, so i can't judge the power.

Psilent Guardian Protect your allies, while being invisible. Overpowered, possibly broken.

Rune of Vengance Do lot's of damage while boosting your teams. Strong to slightly overpowered.

Charedent BarrageA charging ardent. Fine in a normal team, overpowered if there are 2 together, and easily broken in teams of 5.

Super Knight A tough, sticky, high damage knight. Strong.

Super Duper Knight Basically the same as super knight with items, making it far more broken.

Mora, the unkillable avenger Solid damage, while being neigh indestuctable. Overpowered, but not broken.

Swordburst Maximus At-Will Close Burst 3 that slide and prones. Protects allies with off actions. Strong, possibly over powered with the right party.

+1 Static is better then. Per-attack doesn't help you on a nova turn but +1 Static will. +X/Turn is pretty much always worth it though.
Mountain Cleave Rule: You can have any sort of fun, including broken, silly fun, so long as I get to have that fun too (e. g., if you can warp reality with your spells, I can cleave mountains with my blade).
+1 Static is better then. Per-attack doesn't help you on a nova turn but +1 Static will. +X/Turn is pretty much always worth it though.

Nope.

Getting +2 to one attack is gunna increase your hits more then a static +1.

Unless you can pull off more then 10 attacks (probably closer to 15 or so) and less then 20, (+1 always wins at 20).

guides
List of no-action attacks.
Dynamic vs Static Bonuses
Phalanx tactics and builds
Crivens! A Pictsies Guide Good
Power
s to intentionally miss with
Mr. Cellophane: How to be unnoticed
Way's to fire around corners
Crits: what their really worth
Retroactive bonus vs Static bonus.
Runepriest handbook & discussion thread
Holy Symbols to hang around your neck
Ways to Gain or Downgrade Actions
List of bonuses to saving throws
The Ghost with the Most (revenant handbook)
my builds
F-111 Interdictor Long (200+ squares) distance ally teleporter. With some warlord stuff. Broken in a plot way, not a power way.

Thought Switch Higher level build that grants upto 14 attacks on turn 1. If your allies play along, it's broken.

Elven Critters Crit op with crit generation. 5 of these will end anything. Broken.

King Fisher Optimized net user.  Moderate.

Boominator Fun catch-22 booming blade build with either strong or completely broken damage depending on your reading.

Very Distracting Warlock Lot's of dazing and major penalties to hit. Overpowered.

Pocket Protector Pixie Stealth Knight. Maximizing the defender's aura by being in an ally's/enemy's square.

Yakuza NinjIntimiAdin: Perma-stealth Striker that offers a little protection for ally's, and can intimidate bloodied enemies. Very Strong.

Chargeburgler with cheese Ranged attacks at the end of a charge along with perma-stealth. Solid, could be overpowered if tweaked.

Void Defender Defends giving a penalty to hit anyone but him, then removing himself from play. Can get somewhat broken in epic.

Scry and Die Attacking from around corners, while staying hidden. Moderate to broken, depending on the situation.

Skimisher Fly in, attack, and fly away. Also prevents enemies from coming close. Moderate to Broken depending on the enemy, but shouldn't make the game un-fun, as the rest of your team is at risk, and you have enough weaknesses.

Indestructible Simply won't die, even if you sleep though combat.  One of THE most abusive character in 4e.

Sir Robin (Bravely Charge Away) He automatically slows and pushes an enemy (5 squares), while charging away. Hard to rate it's power level, since it's terrain dependent.

Death's Gatekeeper A fun twist on a healic, making your party "unkillable". Overpowered to Broken, but shouldn't actually make the game un-fun, just TPK proof.

Death's Gatekeeper mk2, (Stealth Edition) Make your party "unkillable", and you hidden, while doing solid damage. Stronger then the above, but also easier for a DM to shut down. Broken, until your DM get's enough of it.

Domination and Death Dominate everything then kill them quickly. Only works @ 30, but is broken multiple ways.

Battlemind Mc Prone-Daze Protecting your allies by keeping enemies away. Quite powerful.

The Retaliator Getting hit deals more damage to the enemy then you receive yourself, and you can take plenty of hits. Heavy item dependency, Broken.

Dead Kobold Transit Teleports 98 squares a turn, and can bring someone along for the ride. Not fully built, so i can't judge the power.

Psilent Guardian Protect your allies, while being invisible. Overpowered, possibly broken.

Rune of Vengance Do lot's of damage while boosting your teams. Strong to slightly overpowered.

Charedent BarrageA charging ardent. Fine in a normal team, overpowered if there are 2 together, and easily broken in teams of 5.

Super Knight A tough, sticky, high damage knight. Strong.

Super Duper Knight Basically the same as super knight with items, making it far more broken.

Mora, the unkillable avenger Solid damage, while being neigh indestuctable. Overpowered, but not broken.

Swordburst Maximus At-Will Close Burst 3 that slide and prones. Protects allies with off actions. Strong, possibly over powered with the right party.

Ah, but you're forgetting a key fact mello:

Reality doesn't play by the maths. You're talking about dynamic boosters. I'm talking about getting a boost to your next roll. I trust you can see where this is going?
Mountain Cleave Rule: You can have any sort of fun, including broken, silly fun, so long as I get to have that fun too (e. g., if you can warp reality with your spells, I can cleave mountains with my blade).
I think it's worth adding in variables for your Attack Bonus and the Defense to hit.  If you have an attack bonus of +10 and you want to hit Defense of 16, that means only 30% of your rolls would be subject to either the static or dynamic bonus and the dynamic bonus runs a higher chance of wasted bonus since a roll of 5 would "waste" 5% or more to-hit chance.

I think the forumla is more complex than you've created, but I'm not enough of an expert to apply all of the numbers needed to make it more accurate. 
Ah, but you're forgetting a key fact mello:

Reality doesn't play by the maths. You're talking about dynamic boosters. I'm talking about getting a boost to your next roll. I trust you can see where this is going?

Yes i am talking about dynamic boosters.

Next attack is pretty limited.  If you do 5 attacks, you need a +5 to match +1 to hit.  And it runs into a ceiling.  So if you hit on a 4, it's really only worth +3.

guides
List of no-action attacks.
Dynamic vs Static Bonuses
Phalanx tactics and builds
Crivens! A Pictsies Guide Good
Power
s to intentionally miss with
Mr. Cellophane: How to be unnoticed
Way's to fire around corners
Crits: what their really worth
Retroactive bonus vs Static bonus.
Runepriest handbook & discussion thread
Holy Symbols to hang around your neck
Ways to Gain or Downgrade Actions
List of bonuses to saving throws
The Ghost with the Most (revenant handbook)
my builds
F-111 Interdictor Long (200+ squares) distance ally teleporter. With some warlord stuff. Broken in a plot way, not a power way.

Thought Switch Higher level build that grants upto 14 attacks on turn 1. If your allies play along, it's broken.

Elven Critters Crit op with crit generation. 5 of these will end anything. Broken.

King Fisher Optimized net user.  Moderate.

Boominator Fun catch-22 booming blade build with either strong or completely broken damage depending on your reading.

Very Distracting Warlock Lot's of dazing and major penalties to hit. Overpowered.

Pocket Protector Pixie Stealth Knight. Maximizing the defender's aura by being in an ally's/enemy's square.

Yakuza NinjIntimiAdin: Perma-stealth Striker that offers a little protection for ally's, and can intimidate bloodied enemies. Very Strong.

Chargeburgler with cheese Ranged attacks at the end of a charge along with perma-stealth. Solid, could be overpowered if tweaked.

Void Defender Defends giving a penalty to hit anyone but him, then removing himself from play. Can get somewhat broken in epic.

Scry and Die Attacking from around corners, while staying hidden. Moderate to broken, depending on the situation.

Skimisher Fly in, attack, and fly away. Also prevents enemies from coming close. Moderate to Broken depending on the enemy, but shouldn't make the game un-fun, as the rest of your team is at risk, and you have enough weaknesses.

Indestructible Simply won't die, even if you sleep though combat.  One of THE most abusive character in 4e.

Sir Robin (Bravely Charge Away) He automatically slows and pushes an enemy (5 squares), while charging away. Hard to rate it's power level, since it's terrain dependent.

Death's Gatekeeper A fun twist on a healic, making your party "unkillable". Overpowered to Broken, but shouldn't actually make the game un-fun, just TPK proof.

Death's Gatekeeper mk2, (Stealth Edition) Make your party "unkillable", and you hidden, while doing solid damage. Stronger then the above, but also easier for a DM to shut down. Broken, until your DM get's enough of it.

Domination and Death Dominate everything then kill them quickly. Only works @ 30, but is broken multiple ways.

Battlemind Mc Prone-Daze Protecting your allies by keeping enemies away. Quite powerful.

The Retaliator Getting hit deals more damage to the enemy then you receive yourself, and you can take plenty of hits. Heavy item dependency, Broken.

Dead Kobold Transit Teleports 98 squares a turn, and can bring someone along for the ride. Not fully built, so i can't judge the power.

Psilent Guardian Protect your allies, while being invisible. Overpowered, possibly broken.

Rune of Vengance Do lot's of damage while boosting your teams. Strong to slightly overpowered.

Charedent BarrageA charging ardent. Fine in a normal team, overpowered if there are 2 together, and easily broken in teams of 5.

Super Knight A tough, sticky, high damage knight. Strong.

Super Duper Knight Basically the same as super knight with items, making it far more broken.

Mora, the unkillable avenger Solid damage, while being neigh indestuctable. Overpowered, but not broken.

Swordburst Maximus At-Will Close Burst 3 that slide and prones. Protects allies with off actions. Strong, possibly over powered with the right party.

I think it's worth adding in variables for your Attack Bonus and the Defense to hit.  If you have an attack bonus of +10 and you want to hit Defense of 16, that means only 30% of your rolls would be subject to either the static or dynamic bonus and the dynamic bonus runs a higher chance of wasted bonus since a roll of 5 would "waste" 5% or more to-hit chance.

I think the forumla is more complex than you've created, but I'm not enough of an expert to apply all of the numbers needed to make it more accurate. 

That's included.

Insightful Riposte has a .15 chance of changing a hit, and will give you (approximatly) .8 more hit's per encounter.
+1 is a .05 chance of changing it, and will give you (approximatly) .5 more hit's per encounter.

guides
List of no-action attacks.
Dynamic vs Static Bonuses
Phalanx tactics and builds
Crivens! A Pictsies Guide Good
Power
s to intentionally miss with
Mr. Cellophane: How to be unnoticed
Way's to fire around corners
Crits: what their really worth
Retroactive bonus vs Static bonus.
Runepriest handbook & discussion thread
Holy Symbols to hang around your neck
Ways to Gain or Downgrade Actions
List of bonuses to saving throws
The Ghost with the Most (revenant handbook)
my builds
F-111 Interdictor Long (200+ squares) distance ally teleporter. With some warlord stuff. Broken in a plot way, not a power way.

Thought Switch Higher level build that grants upto 14 attacks on turn 1. If your allies play along, it's broken.

Elven Critters Crit op with crit generation. 5 of these will end anything. Broken.

King Fisher Optimized net user.  Moderate.

Boominator Fun catch-22 booming blade build with either strong or completely broken damage depending on your reading.

Very Distracting Warlock Lot's of dazing and major penalties to hit. Overpowered.

Pocket Protector Pixie Stealth Knight. Maximizing the defender's aura by being in an ally's/enemy's square.

Yakuza NinjIntimiAdin: Perma-stealth Striker that offers a little protection for ally's, and can intimidate bloodied enemies. Very Strong.

Chargeburgler with cheese Ranged attacks at the end of a charge along with perma-stealth. Solid, could be overpowered if tweaked.

Void Defender Defends giving a penalty to hit anyone but him, then removing himself from play. Can get somewhat broken in epic.

Scry and Die Attacking from around corners, while staying hidden. Moderate to broken, depending on the situation.

Skimisher Fly in, attack, and fly away. Also prevents enemies from coming close. Moderate to Broken depending on the enemy, but shouldn't make the game un-fun, as the rest of your team is at risk, and you have enough weaknesses.

Indestructible Simply won't die, even if you sleep though combat.  One of THE most abusive character in 4e.

Sir Robin (Bravely Charge Away) He automatically slows and pushes an enemy (5 squares), while charging away. Hard to rate it's power level, since it's terrain dependent.

Death's Gatekeeper A fun twist on a healic, making your party "unkillable". Overpowered to Broken, but shouldn't actually make the game un-fun, just TPK proof.

Death's Gatekeeper mk2, (Stealth Edition) Make your party "unkillable", and you hidden, while doing solid damage. Stronger then the above, but also easier for a DM to shut down. Broken, until your DM get's enough of it.

Domination and Death Dominate everything then kill them quickly. Only works @ 30, but is broken multiple ways.

Battlemind Mc Prone-Daze Protecting your allies by keeping enemies away. Quite powerful.

The Retaliator Getting hit deals more damage to the enemy then you receive yourself, and you can take plenty of hits. Heavy item dependency, Broken.

Dead Kobold Transit Teleports 98 squares a turn, and can bring someone along for the ride. Not fully built, so i can't judge the power.

Psilent Guardian Protect your allies, while being invisible. Overpowered, possibly broken.

Rune of Vengance Do lot's of damage while boosting your teams. Strong to slightly overpowered.

Charedent BarrageA charging ardent. Fine in a normal team, overpowered if there are 2 together, and easily broken in teams of 5.

Super Knight A tough, sticky, high damage knight. Strong.

Super Duper Knight Basically the same as super knight with items, making it far more broken.

Mora, the unkillable avenger Solid damage, while being neigh indestuctable. Overpowered, but not broken.

Swordburst Maximus At-Will Close Burst 3 that slide and prones. Protects allies with off actions. Strong, possibly over powered with the right party.

If it's after you see the result (ala Heroic Effort), then I like the +X once per encounter better, provided X is sufficiently large (around 4ish would make me happy). If it's before you attack, then I'll take the permanent +1 every time.

Same goes for defenses. I'd rather have +4 to AC once per encounter when I know I can negate a hit, than +1 to AC all the time.
I just want to justify my bias toward always on boosts...

As you get more powerful as a party, the response of most DMs will to throw more powerfull stuff at you.  You'll increasingly hit higher-than-CR challenges.  Those combats will run longer, because you have to hit more times to drop a monster.  The longer the combat runs, the more always-on outperforms one-time.

(Also: AoE blasters really crank up the attack count.  So blasters will want always-on boosts more than single-target strikers will.  Controllers really crank up the difference between attacks, so controllers want situational larger boosts more than always-on boosts)

"Nice assumptions. Completely wrong assumptions, but by jove if being incorrect stopped people from making idiotic statements, we wouldn't have modern internet subculture." Kerrus
Practical gameplay runs by neither RAW or RAI, but rather "A Compromise Between The Gist Of The Rule As I Recall Getting The Impression Of It That One Time I Read It And What Jerry Says He Remembers, Whatever, We'll Look It Up Later If Any Of Us Still Give A Damn." Erachima

I didn't know how to write the formula, but I figured a simple excel simulation might help.  I ran 100,000 encounters each encounter assuming 10 attacks, with two characters.  One character at +11 attack vs 16 defense.  and the other at +10 attack, with an encounter bonus of +2 to hit.

Encounters: 100,000
Attacks: 1,000,000
Hits: 724,268  (72.43%)

Character 1 (Static +1)
Static bonus hits: 49,823 (4.98%)

Character 2 (Dynamic +2, once per encounter)
Dynamic bonus hits: 64,983 (6.50%)


So Mellored is correct that a dynamic +2 bonus to hit is better than a static +1 to hit over 10 attacks per encounter.


Interestingly the break even point is 11 attacks.  They are exactly equal at 11 attacks, and the +1 pulls ahead begining at 12 attacks per encounter.


Edit.  Good call on the percentage Mellored, I didn't count the encounters correctly. Fixed the mistake and I get the same result.

Code used
Show

encounters = 100000
attacks = 10
hits = 0
dbonusAttacksMax = 1
dbonusAttacks = dbonusAttacksMax
attack = 10
defense = 16
Dbonus = 2


For x = 1 To encounters
    For i = 1 To attacks
        roll = Rnd * 20
        If roll + attack >= defense Then
            hits = hits + 1
        End If
        If roll + attack = defense - 1 Then
            StaticHits = StaticHits + 1 'Static bonus hits
        End If
        If roll + attack < defense Then
            If dbonusAttacks > 0 Then
                If roll + attack + Dbonus >= defense Then
                    dbonusAttacks = dbonusAttacks - 1
                    dhits = dhits + 1 'Dynamic bonus hits
                End If
            End If
        End If
        roll = 0
    Next
    dbonusAttacks = dbonusAttacksMax
Next




I didn't know how to write the formula, but I figured a simple excel simulation might help.  I ran 100,000 encounters each encounter assuming 10 attacks, with two characters.  One character at +11 attack vs 16 defense.  and the other at +10 attack, with an encounter bonus of +2 to hit.

Encounters: 100,000
Attacks: 1,000,000
Hits: 674,823  (67.5%)

Character 1 (Static +1)
Static bonus hits: 49,989 (5.00%)

Character 2 (Dynamic +2, once per encounter)
Dynamic bonus hits: 52,824 (5.28%)


So Mellored is correct that a dynamic +2 bonus to hit is better than a static +1 to hit over 10 attacks per encounter.


Interestingly the break even point is 11 attacks.  They are exactly equal at 11 attacks, and the +1 pulls ahead begining at 12 attacks per encounter.

Hmm...

I got ,65132156, not .528

Something wrong with my math?

guides
List of no-action attacks.
Dynamic vs Static Bonuses
Phalanx tactics and builds
Crivens! A Pictsies Guide Good
Power
s to intentionally miss with
Mr. Cellophane: How to be unnoticed
Way's to fire around corners
Crits: what their really worth
Retroactive bonus vs Static bonus.
Runepriest handbook & discussion thread
Holy Symbols to hang around your neck
Ways to Gain or Downgrade Actions
List of bonuses to saving throws
The Ghost with the Most (revenant handbook)
my builds
F-111 Interdictor Long (200+ squares) distance ally teleporter. With some warlord stuff. Broken in a plot way, not a power way.

Thought Switch Higher level build that grants upto 14 attacks on turn 1. If your allies play along, it's broken.

Elven Critters Crit op with crit generation. 5 of these will end anything. Broken.

King Fisher Optimized net user.  Moderate.

Boominator Fun catch-22 booming blade build with either strong or completely broken damage depending on your reading.

Very Distracting Warlock Lot's of dazing and major penalties to hit. Overpowered.

Pocket Protector Pixie Stealth Knight. Maximizing the defender's aura by being in an ally's/enemy's square.

Yakuza NinjIntimiAdin: Perma-stealth Striker that offers a little protection for ally's, and can intimidate bloodied enemies. Very Strong.

Chargeburgler with cheese Ranged attacks at the end of a charge along with perma-stealth. Solid, could be overpowered if tweaked.

Void Defender Defends giving a penalty to hit anyone but him, then removing himself from play. Can get somewhat broken in epic.

Scry and Die Attacking from around corners, while staying hidden. Moderate to broken, depending on the situation.

Skimisher Fly in, attack, and fly away. Also prevents enemies from coming close. Moderate to Broken depending on the enemy, but shouldn't make the game un-fun, as the rest of your team is at risk, and you have enough weaknesses.

Indestructible Simply won't die, even if you sleep though combat.  One of THE most abusive character in 4e.

Sir Robin (Bravely Charge Away) He automatically slows and pushes an enemy (5 squares), while charging away. Hard to rate it's power level, since it's terrain dependent.

Death's Gatekeeper A fun twist on a healic, making your party "unkillable". Overpowered to Broken, but shouldn't actually make the game un-fun, just TPK proof.

Death's Gatekeeper mk2, (Stealth Edition) Make your party "unkillable", and you hidden, while doing solid damage. Stronger then the above, but also easier for a DM to shut down. Broken, until your DM get's enough of it.

Domination and Death Dominate everything then kill them quickly. Only works @ 30, but is broken multiple ways.

Battlemind Mc Prone-Daze Protecting your allies by keeping enemies away. Quite powerful.

The Retaliator Getting hit deals more damage to the enemy then you receive yourself, and you can take plenty of hits. Heavy item dependency, Broken.

Dead Kobold Transit Teleports 98 squares a turn, and can bring someone along for the ride. Not fully built, so i can't judge the power.

Psilent Guardian Protect your allies, while being invisible. Overpowered, possibly broken.

Rune of Vengance Do lot's of damage while boosting your teams. Strong to slightly overpowered.

Charedent BarrageA charging ardent. Fine in a normal team, overpowered if there are 2 together, and easily broken in teams of 5.

Super Knight A tough, sticky, high damage knight. Strong.

Super Duper Knight Basically the same as super knight with items, making it far more broken.

Mora, the unkillable avenger Solid damage, while being neigh indestuctable. Overpowered, but not broken.

Swordburst Maximus At-Will Close Burst 3 that slide and prones. Protects allies with off actions. Strong, possibly over powered with the right party.

I didn't know how to write the formula, but I figured a simple excel simulation might help.  I ran 100,000 encounters each encounter assuming 10 attacks, with two characters.  One character at +11 attack vs 16 defense.  and the other at +10 attack, with an encounter bonus of +2 to hit.

Encounters: 100,000
Attacks: 1,000,000
Hits: 674,823  (67.5%)

Character 1 (Static +1)
Static bonus hits: 49,989 (5.00%)

Character 2 (Dynamic +2, once per encounter)
Dynamic bonus hits: 52,824 (5.28%)


So Mellored is correct that a dynamic +2 bonus to hit is better than a static +1 to hit over 10 attacks per encounter.


Interestingly the break even point is 11 attacks.  They are exactly equal at 11 attacks, and the +1 pulls ahead begining at 12 attacks per encounter.





Maybe I'm missing something, but those results seem wrong to me. I'm seeing Case 1 as a base 80% to hit (miss on 1-4), and Case 2 as a base 75% with a one time 85% hit percentage. Your hit results should be around 750,00-800,000.


Maybe I'm missing something, but those results seem wrong to me. I'm seeing Case 1 as a base 80% to hit (miss on 1-4), and Case 2 as a base 75% with a one time 85% hit percentage. Your hit results should be around 750,00-800,000.

Cool Not me then.

Mabey he used a d12 instead of d20 for attack rolls...

guides
List of no-action attacks.
Dynamic vs Static Bonuses
Phalanx tactics and builds
Crivens! A Pictsies Guide Good
Power
s to intentionally miss with
Mr. Cellophane: How to be unnoticed
Way's to fire around corners
Crits: what their really worth
Retroactive bonus vs Static bonus.
Runepriest handbook & discussion thread
Holy Symbols to hang around your neck
Ways to Gain or Downgrade Actions
List of bonuses to saving throws
The Ghost with the Most (revenant handbook)
my builds
F-111 Interdictor Long (200+ squares) distance ally teleporter. With some warlord stuff. Broken in a plot way, not a power way.

Thought Switch Higher level build that grants upto 14 attacks on turn 1. If your allies play along, it's broken.

Elven Critters Crit op with crit generation. 5 of these will end anything. Broken.

King Fisher Optimized net user.  Moderate.

Boominator Fun catch-22 booming blade build with either strong or completely broken damage depending on your reading.

Very Distracting Warlock Lot's of dazing and major penalties to hit. Overpowered.

Pocket Protector Pixie Stealth Knight. Maximizing the defender's aura by being in an ally's/enemy's square.

Yakuza NinjIntimiAdin: Perma-stealth Striker that offers a little protection for ally's, and can intimidate bloodied enemies. Very Strong.

Chargeburgler with cheese Ranged attacks at the end of a charge along with perma-stealth. Solid, could be overpowered if tweaked.

Void Defender Defends giving a penalty to hit anyone but him, then removing himself from play. Can get somewhat broken in epic.

Scry and Die Attacking from around corners, while staying hidden. Moderate to broken, depending on the situation.

Skimisher Fly in, attack, and fly away. Also prevents enemies from coming close. Moderate to Broken depending on the enemy, but shouldn't make the game un-fun, as the rest of your team is at risk, and you have enough weaknesses.

Indestructible Simply won't die, even if you sleep though combat.  One of THE most abusive character in 4e.

Sir Robin (Bravely Charge Away) He automatically slows and pushes an enemy (5 squares), while charging away. Hard to rate it's power level, since it's terrain dependent.

Death's Gatekeeper A fun twist on a healic, making your party "unkillable". Overpowered to Broken, but shouldn't actually make the game un-fun, just TPK proof.

Death's Gatekeeper mk2, (Stealth Edition) Make your party "unkillable", and you hidden, while doing solid damage. Stronger then the above, but also easier for a DM to shut down. Broken, until your DM get's enough of it.

Domination and Death Dominate everything then kill them quickly. Only works @ 30, but is broken multiple ways.

Battlemind Mc Prone-Daze Protecting your allies by keeping enemies away. Quite powerful.

The Retaliator Getting hit deals more damage to the enemy then you receive yourself, and you can take plenty of hits. Heavy item dependency, Broken.

Dead Kobold Transit Teleports 98 squares a turn, and can bring someone along for the ride. Not fully built, so i can't judge the power.

Psilent Guardian Protect your allies, while being invisible. Overpowered, possibly broken.

Rune of Vengance Do lot's of damage while boosting your teams. Strong to slightly overpowered.

Charedent BarrageA charging ardent. Fine in a normal team, overpowered if there are 2 together, and easily broken in teams of 5.

Super Knight A tough, sticky, high damage knight. Strong.

Super Duper Knight Basically the same as super knight with items, making it far more broken.

Mora, the unkillable avenger Solid damage, while being neigh indestuctable. Overpowered, but not broken.

Swordburst Maximus At-Will Close Burst 3 that slide and prones. Protects allies with off actions. Strong, possibly over powered with the right party.

It's also worth considering that sometimes landing that big single attack is what sets up following attacks.  If you hit with something like like Lead the Attack or Knockout, the next hit (or hits) is pretty much guaranteed.  Even things like Righteous Brand have consequences that might favor the one-time bonuses when you're setting up your teammate for a nova.

Didn't this come up a while back when people were arguing Elf Acc vs. Heroic Effort? It wasn't the main topic, but I thought it was mentioned.
Just to avoid confusion, probably good to explain what your dynamic bonus to hit means. Based on your formula, it means you get to CHOOSE when to apply that +X bonus, so the power of when to use it is in your hands. If you get a +X bonus on your next attack BEFORE you roll it, then this kind of bonus is much less helpful. If instead, you get a "floating" +X bonus you can apply to your attack AFTER you roll it, which means you only use it when it matters (makes you hit), then this is much more powerful. This may help some who are trying to run excel files on this 

Since you only have 2 variables (X, Y) you could plot this on a 3D graph with the X axis number of attacks and the Y axis bonus to hit and the Z axis being the result. In case someone wants to plot this in 3D that is...

Larry 
I'll have to agree, a 1/encounter +2 to-hit will usually outperform a static +1 to-hit. Assuming you can apply it after the roll, and hear by how much you've missed. It's easy to see why:


  • Assumption: You need to roll a 10 to hit a monster without the bonus (it holds for any number between 4 and 20, though).

  • A static +1 to-hit has an exact chance of 5% to turn any missed attack roll into a hit, since you have to roll exactly 9 for the bonus to turn a miss into a hit.

  • An activated +2 to-hit has an exact chance of 10% chance to turn a missed attack roll into a hit, since you can turn a miss into a hit on the roll of 8 or 9. However, only one missed attack per encounter can be turned into a hit.


So, as long as we roll exactly one 9 during an encounter, the two are equal in terms of how often you hit (regardless of how often we roll 8). If we roll an 8, but not a 9, the 1/enc bonus wins out. Rolling two nines puts the static bonus ahead.

But there's a really big selling point for the +2 once per encounter: It can turn an 8 into a 10. This matters when you activate a powerful encounter or daily power early in a fight. Average chance to hit means squat for nova rounds - you need your best attacks to hit, and you need it now, not on the at-will you use in round four. Same goes for those insane daily control powers you use to lock down that nasty elite.

And even if you're using only MBAs, the once/encounter bonus has a selling point: Its effect is frontloaded. Your first ten attacks have a higher chance to hit, meaning your first opponent will drop faster, on average.

The same idea goes for defensive bonuses, by the way. My late heroic avenger/swordmage may be sitting on a passive 26 AC, it takes a total attack roll of 32 to hit her outright (courtesy of Shield and Cloak of Displacement, with a Githzerai racial and some proactive defense bonuses for backup). Which is a nice and fuzzy safe feeling (were it not for the DM and Murphy conspiring - so far about half his d20s against me turned up 17+).
It doesn't really need simulation. If move away from mellored's initial assumption of perfect knowledge and opportunity then it's fairly straightforward.

N: Number of attacks
P: Base to hit chance
E: Encounter bonus/20
S: Static bonus/20

Expected Value for Hits/encounter with Static bonus: N(P+S)
Expected Value for Hits/encounter with Enc bounus: (P +E) + (N-1)P

Difference: (P +E) + (N-1)P  - N(P+S)
               : E - NS

which is the straightfoward result that one would expect, that is, +4 once is the same as +1 4 times, assuming P+E < 1
 
 
Lets move away here from Mello's assumption that all attacks are created equal, because that is not normally true and Charop deals with what 'normally' happens.

The best choice depends on your role. A defender will want the steady +1 to hit in order to keep the bad guy focused on him. Strikers have been already mathmatically shown by the others to get more damage from a static bonus.

Now things get interesting. Most Controllers have VERY powerful dailies, with the wizard leading the pack. The Controller with few exceptions would care much more about landing his big daily then getting a static bonus to At-Wills. The Leader OTOH has his bread and butter in at-wills (Commanders Strike, Misdirected Mark, Magic Weapon, the list goes on) and so would favor the static bonus even more than some strikers.

This is all generalized and a lot of the tine it comes down to what the build has/does. But in general most would like the static bonus better mathematically and in Actual Play.
It's not a matter of Right vs. Left, it's a matter of right vs. wrong My Charop Works Who Am I Really? The Theme Handbook I'm an
D&D Home Page - What Class Are You? - Build A Character - D&D Compendium
I would say let's look at your assumption of 10 attacks.

Lets assume a 5 round encounter; lets next assume you use an action point every other encounter.  Lets assume your DM uses lots of mixes of monsters, from solos (who are solo, and sufficiently customized to still mean something) to massive numbers of minions - the point of this assumption is I'm just going to make a follow-up assumption on AoE's.

Lets take 5 characters:
Paladin
Non-Tempest Fighter
Ranger
Wizard
Artificer|Lazylord

I'm taking these characters as being fairly commonly played, and fairly close to average compared to similar characters, purely based on number of attacks per round. An avenger would be a fighter (or paladin if no extra attacks and no good MBA for lazy leader), a sorcerer would be a wizard, a non lazy leader would be a paladin or fighter depending on immediate action access, etc.

Paladin:  Probably gets 1 attack per round, plus 1 attack from an action point. 5.5 attacks.  Dynamic boost is FAR better than +1 static to hit, simply since he won't be attacking all that many times.
Fighter: Probably gets 1 attack per round, plus 1 attack from action point, plus 3 attacks from either multi-attack powers, immediate actions, or free action powers, 2 extra attacks from lazy leader. 10.5 attacks.  Could go for the dynamic OR the static.
Ranger: 2 attacks per round, plus 2 from action point, plus 4 from (larger) multi-attack powers, free action powers, minors, and immediates. 15 attacks, definitely wants the static boost.
Wizard: 3 targets one round (either first or second depending on set up), 2 targets for 3 rounds, 1 target on the last round. 3 targets on an action point, 4 extra targets from dailies, minors, slides, all kinds of fancy things.  Total 15.5, again loves the static boost.
Artificer|Lazylord: grants attacks on 2 rounds, attacks on 3 rounds with 1 target, action point for 1 target, 1 immediate action. Total: 4.5 attacks. Dynamic bonus is so amazing it's not even a race.  Would probably take a +5 dynamic boost over a +3 static (but possibly only a +2 static).


With most of these cases what we see is that picking dynamic or static has very little to do with how good the two compare with each other.  Instead the important thing is what class and what kind of attacks you are using. 

Multi-attackers and AoE:  Take static boosts.
Most leaders or defenders who don't get extra attacks like Paladin or Shielding Swordmage: Take dynamic boosts.

The only people that have to decide are people who only make 1 attack per standard action, but get lots of non standard action attacks, like fighters or rogues or avengers.  However, please note that most of these classes already have inherently high accuracy.  A rogue for instance will likely be hitting on a 2, 3 or 4+  The dynamic bonus will usually be overkill for him, so a low static boost is better, even if in a vacuum it wouldn't be.  The avenger on the other hand usually rolls at least one dice high and doesn't care what bonus he's getting; but because he will sacrifice +hit, when he does roll double low, it really hurts - the dynamic boost is better for him.  The fighter: Well, the fighter will probably hit on 5+ish.  He could go either way.

So in the end what we see is that defenders (or maybe aggressive leaders) who get lots of non-standard attacks are the only characters who really need to choose.  In these cases, the options will often be chosen for them; a defender usually gets stance like things that are static for the whole fight.  So he gets to use static bonuses.  A leader often gets one shot boosts.  So he'll use those.  But in reality, either one would work, and provide roughly equal bonuses; so at that point, you might as well go with player preference.


Note: This is all conjecture, which is of course against my own nature as a mathemetician, and against the nature of this thread.  But I am trying to show that Mellored should acknowledge that most optimized characters will have their choice chosen far before they get to the stage of picking between 2 powers, one static, and one dynamic.  However, given that, I think it's a very cool mathematical excercise.

Also note: Yes this is almost completely contrary to what Is612 said.

Edit:  Actually I realize a secondary point to this thread.  The point really isn't to compare static vs dynamic at +1 vs +2 is it Mellored?  The point is to find out what bonus, a ranger for instance would choose the dynamic boost over the static one.  I would say.. you'd need a +4 dynamic boost for the multi-attacker to choose it over a +1 static.  For a wizard/aoe blaster, +5 due to those times you really need minion clearing (and most super dailies have awesome effects even on a miss, or don't rely on hitting at all, or have orbs that make you auto hit for the part that counts).

For the in-between guys (fighter/aggressive leader), a +3 dynamic boost would be better than a +1 static boost.  For a Rogue (or other super accurate character, like a seeker who always has CA, or a psychic obsessed psion), overkill on accuracy means the static boost is still almost invariably better (you'd probably want like a +8 dynamic boost to choose it over a +1 static, and only for that 'I kill the monster 10 levels higher than me!' moment)
As long as you are making less than 20 attacks per round, a dynamic (post hit/miss) +2 to hit is better than a static +1 bonus for all characters that have to-hit rolls.  Obviously it leans even more in favor of dynamic when using encounters/daily powers with hit efects. 

For those humans with less than 95% hit rates, Heroic Effort is amazing and one of the better scaling racial powers that exists.


You did mean 20 attacks per encounter right?  Also, as pointed out on the second page, 20 is far too high, because you have to account for all the times a person will roll exactly one lower than needed (before either bonus) more than once in an encounter, since the static bonus will every one of those attacks to hit.  (At fantastically low odds, you could in fact hit all 20 times because of the +1 static; but it's not particularly bad odds that you will do it twice in an encounter, which the dynamic bonus can never do no matter how good it is.)  Thus why we end up with 12 attacks being the tipping point if you compare +1 static and +2 dynamic.
Interesting question.  If I've figured it correctly, a +2 dynamic modifier is generally better than a +1 static modifier, up to about 17 attacks per encounter, depending on how accurate you are before either modifier is added.  But the difference in outcomes isn't large.

This might seem counterintuitive, because you'd think the static modifier would win out as the number of attacks increases.  After all, it applies to every attack!  The dynamic modifier can be applied only once.

I don't get to apply my dynamic modifier to any miss. I have to miss by 2 or by 1.  But when I do get to use it, I get a 100% increase in hit rate for that one attack.  As the number of attacks increase, a single-use bonus may seem less useful, but the odds that I'll get to use it go up until I'm almost certain to get an extra attack out of it. By the time we get to 17 attacks, it's more than 80% likely that I got to use it, and after that the static modifier starts winning because I continue to benefit from it for any number of attacks.

Here's how I set up the math:

b = Base To-Hit = 7 : the number I must roll to hit, prior to adding any modifier
s = Static Modifier = +1
d = Dynamic Modifier = +2
n = Number of Attacks per encounter = 10 in the example calculations below

Without any modifier, my base hits per encounter will be:
n x (20-b+1)/20 = 10 x (20-7+1)/20  = 7 hits per encounter.
This is a pretty obvious result.  I only miss on 1-6, or 6/20=30% of the time.  So, 70% of my ten attacks will hit, on average.

Adding the static modifier just makes me 5% more likely to hit, because now I only miss on 1 through 5.
n x (20-b+1+s)/20 = 10 x (20-7+1+1)/20  = 7.5 hits per encounter.

To understand how the dynamic modifier will perform, we calculate how likely it is that I will get one extra hit in addition to the hits I am likely to score without it.  


  • Number of expected misses = number of attacks per encounter - number of expected hits per encounter, or 10-7 = 3 expected misses.

  • Odds that any single miss could benefit from my dynamic bonus = 2 / 6 = 33.3%  (I can miss on a 1-6, but I can only make that a hit if I roll a 5 or 6).

  • Odds that I will be able to use the dyanamic bonus successfuly = 1-(1-1/3)^3 = 70%. In english, each time I miss there is a 2/3 chance that I will not be able to use the dynamic bonus because I missed by more than 2.  But if am going to miss three times, the odds that I will by more than 2 all three times are actually quite small:  2/3 x 2/3 x 2/3, or 30%, which means I have a 70% chance of using it.

  • If I have a 70% chance of getting one extra hit in the encounter, my expected number of hits goes up to 7 + 1 x 70% = 7.7.


So in summary:
No modifier:  7 hits per encounter
+1 static: 7.5 hits per encounter
+2 dynamic: 7.7 hits per encounter

There is a nominal difference, but in this case it really isn't signifcant and a player could choose whichever was most attainable or thematically desirable and not feel disadvantaged.

Base accuracy also influences the outcome, but not as much as one might expect.  If I'm already highly accurate, the dynamic modifier pays off more reliably when I miss because individual misses are more likely to qualify for it, but I don't miss very often.  If I'm less accurate, I have more chances to use it, but it is less likely to pay off when I miss.  Eiter way, dynamic is nominally better up until around 17 +/- 3 attacks.

Here are the calculations in more detail:
Show















































































































































































































































































Base To-Hit Target             7
Static Bonus Modifier             1
Dynamic Bonus Modifier             2
Attacks per Encounter           10           11           12           13           14           15           16           17           18           19           20
Base Case
Hits  per Encounter         7.0         7.7         8.4         9.1         9.8       10.5       11.2       11.9       12.6       13.3       14.0
Static Modifier
Hits  per Encounter         7.5         8.3         9.0         9.8       10.5       11.3       12.0       12.8       13.5       14.3       15.0
Dynamic Modifier
Expected Misses         3.0         3.3         3.6         3.9         4.2         4.5         4.8         5.1         5.4         5.7         6.0
Miss Interval Size             6
Odds a single miss could benefit?         0.3
Odds of getting one extra hit?         0.7         0.7         0.8         0.8         0.8         0.8         0.9         0.9         0.9         0.9         0.9
Hits per Encounter         7.7         8.4         9.2         9.9       10.6       11.3       12.1       12.8       13.5       14.2       14.9
Dynamic Hits - Static Hits         0.2         0.2         0.2         0.1         0.1         0.1         0.1         0.0       (0.0)       (0.0)       (0.1)




Grondag: I think you've overcomplicating this. Your actual hit/miss chance has nothing to do with it. There are only two cases that matter:


  • You would miss by exactly 1.

  • You would miss by exactly 2.


Each of these cases correspond to exactly one side of your d20. You need a 5 on your d20 to hit? The first case is rolling a 4, the second case a 3. In the first case, both the static modifier and the dynamic one turn a single hit into a miss. In the second case, the dynamic one turns a hit into a miss. To throw math at it, we'll need to compare the expected number of extra hits from the static modifier, with the chance that you get to apply your dynamic modifier during an encounter.

For each die roll, the chance of an extra hit from static mods is exactly 0.05 (corresponding to exactly one side of your d20). In mathy stuff:

E(extra hits) = Our expected number of additional hits due to the bonus during the encounter.
N = The number of attack rolls we make during the encounter.
P(static) = The chance that our static modifier turns a miss into a hit.

E(extra hits)static = P(static) * N = 0.05 * N

The dynamic modifier is slightly more complicated, as it applies at most once. If we get to apply it during an encounter, our number of hits increases by 1. Otherwise, it does nothing. The easiest way to calculate this is to consider the chance we can not apply the modifier. For a single attack, we can apply the bonus with chance 0.1 (corresponding to two sides of a d20), thus the chance that we do not apply it is 0.9 (the other 18 sides).

P(not enc) = The chance of never being able to apply the bonus during the entire encounter.
P(not attack) = The chance of not being able to apply the bonus to a given attack, assuming it's still unused.

P(not enc) = P(not attack) N = 0.9 N
E(extra hits)dynamic = 1 - P(not enc) = 1 - 0.9 N

Plugging in N=10:

E(extra hits)static = 0.05 * 10 = 0.5
E(extra hits)dynamic = 1 - 0.9 10 = 1 - 0.348... = 0.651...
And we can conclude that mellored's math is correct ;)

Considering that not all attacks are equal, and that you usually use your best attacks first, the actual game advantage of the dynamic bonus is even greater. It does a lot more than the static bonus when you absolutely need to hit that daily power. Even for multiattackers: Blade Cascade and similar powers stop on a miss, so the dynamic bonus does an awful lot for those powers (miss by 2 -> you don't get any more attacks -> static bonus does nothing for the power). Be sure to frontload your big guns to get the most out of your bonus.
Next question...  Since not everyone makes the same number of attacks...

How many attacks per encounter will make static +1 equal to dynamic +2?

So...
1-((1-(2 *.05))^X) = 1 * .05 * X
1-(.9^X) = .05X
 (any idea how to solve this??)


Binary Search:
X=10
0.6513215599 = .5

X=15
0.794108867905351 = 0.75

X=16
0.8146979811148159 = .8

X = 17
0.83322818300333431 = .85



17 attacks to make static better.

guides
List of no-action attacks.
Dynamic vs Static Bonuses
Phalanx tactics and builds
Crivens! A Pictsies Guide Good
Power
s to intentionally miss with
Mr. Cellophane: How to be unnoticed
Way's to fire around corners
Crits: what their really worth
Retroactive bonus vs Static bonus.
Runepriest handbook & discussion thread
Holy Symbols to hang around your neck
Ways to Gain or Downgrade Actions
List of bonuses to saving throws
The Ghost with the Most (revenant handbook)
my builds
F-111 Interdictor Long (200+ squares) distance ally teleporter. With some warlord stuff. Broken in a plot way, not a power way.

Thought Switch Higher level build that grants upto 14 attacks on turn 1. If your allies play along, it's broken.

Elven Critters Crit op with crit generation. 5 of these will end anything. Broken.

King Fisher Optimized net user.  Moderate.

Boominator Fun catch-22 booming blade build with either strong or completely broken damage depending on your reading.

Very Distracting Warlock Lot's of dazing and major penalties to hit. Overpowered.

Pocket Protector Pixie Stealth Knight. Maximizing the defender's aura by being in an ally's/enemy's square.

Yakuza NinjIntimiAdin: Perma-stealth Striker that offers a little protection for ally's, and can intimidate bloodied enemies. Very Strong.

Chargeburgler with cheese Ranged attacks at the end of a charge along with perma-stealth. Solid, could be overpowered if tweaked.

Void Defender Defends giving a penalty to hit anyone but him, then removing himself from play. Can get somewhat broken in epic.

Scry and Die Attacking from around corners, while staying hidden. Moderate to broken, depending on the situation.

Skimisher Fly in, attack, and fly away. Also prevents enemies from coming close. Moderate to Broken depending on the enemy, but shouldn't make the game un-fun, as the rest of your team is at risk, and you have enough weaknesses.

Indestructible Simply won't die, even if you sleep though combat.  One of THE most abusive character in 4e.

Sir Robin (Bravely Charge Away) He automatically slows and pushes an enemy (5 squares), while charging away. Hard to rate it's power level, since it's terrain dependent.

Death's Gatekeeper A fun twist on a healic, making your party "unkillable". Overpowered to Broken, but shouldn't actually make the game un-fun, just TPK proof.

Death's Gatekeeper mk2, (Stealth Edition) Make your party "unkillable", and you hidden, while doing solid damage. Stronger then the above, but also easier for a DM to shut down. Broken, until your DM get's enough of it.

Domination and Death Dominate everything then kill them quickly. Only works @ 30, but is broken multiple ways.

Battlemind Mc Prone-Daze Protecting your allies by keeping enemies away. Quite powerful.

The Retaliator Getting hit deals more damage to the enemy then you receive yourself, and you can take plenty of hits. Heavy item dependency, Broken.

Dead Kobold Transit Teleports 98 squares a turn, and can bring someone along for the ride. Not fully built, so i can't judge the power.

Psilent Guardian Protect your allies, while being invisible. Overpowered, possibly broken.

Rune of Vengance Do lot's of damage while boosting your teams. Strong to slightly overpowered.

Charedent BarrageA charging ardent. Fine in a normal team, overpowered if there are 2 together, and easily broken in teams of 5.

Super Knight A tough, sticky, high damage knight. Strong.

Super Duper Knight Basically the same as super knight with items, making it far more broken.

Mora, the unkillable avenger Solid damage, while being neigh indestuctable. Overpowered, but not broken.

Swordburst Maximus At-Will Close Burst 3 that slide and prones. Protects allies with off actions. Strong, possibly over powered with the right party.

I was messing around with that myself, but my math is a bit rusty: I really can't remember how to work out these equations.

Fortunately, you cannot make partial attacks, so you can just plug in some integer values and see what works. I checked your math and got the same results using my own calculations. So it seems 17 is indeed the correct threshold when you try to maximize the expected number of hits in an encounter.
I was messing around with that myself, but my math is a bit rusty: I really can't remember how to work out these equations.

Fortunately, you cannot make partial attacks, so you can just plug in some integer values and see what works. I checked your math and got the same results using my own calculations. So it seems 17 is indeed the correct threshold when you try to maximize the expected number of hits in an encounter.

Thanks for the double check.

But yea...  Even for multi-attackers and AOE-ers, it seems dynamic +2 is one of the best investments.  

Skill Power (Insightful Riposte) is a win.
Past Soul (heroic effort) is better then expertise (in heroic).
Wand of Accuracy DOESN'T suck, even if you only have 14 Dex.

Goes for defense boosts, and skills rolls as well.

guides
List of no-action attacks.
Dynamic vs Static Bonuses
Phalanx tactics and builds
Crivens! A Pictsies Guide Good
Power
s to intentionally miss with
Mr. Cellophane: How to be unnoticed
Way's to fire around corners
Crits: what their really worth
Retroactive bonus vs Static bonus.
Runepriest handbook & discussion thread
Holy Symbols to hang around your neck
Ways to Gain or Downgrade Actions
List of bonuses to saving throws
The Ghost with the Most (revenant handbook)
my builds
F-111 Interdictor Long (200+ squares) distance ally teleporter. With some warlord stuff. Broken in a plot way, not a power way.

Thought Switch Higher level build that grants upto 14 attacks on turn 1. If your allies play along, it's broken.

Elven Critters Crit op with crit generation. 5 of these will end anything. Broken.

King Fisher Optimized net user.  Moderate.

Boominator Fun catch-22 booming blade build with either strong or completely broken damage depending on your reading.

Very Distracting Warlock Lot's of dazing and major penalties to hit. Overpowered.

Pocket Protector Pixie Stealth Knight. Maximizing the defender's aura by being in an ally's/enemy's square.

Yakuza NinjIntimiAdin: Perma-stealth Striker that offers a little protection for ally's, and can intimidate bloodied enemies. Very Strong.

Chargeburgler with cheese Ranged attacks at the end of a charge along with perma-stealth. Solid, could be overpowered if tweaked.

Void Defender Defends giving a penalty to hit anyone but him, then removing himself from play. Can get somewhat broken in epic.

Scry and Die Attacking from around corners, while staying hidden. Moderate to broken, depending on the situation.

Skimisher Fly in, attack, and fly away. Also prevents enemies from coming close. Moderate to Broken depending on the enemy, but shouldn't make the game un-fun, as the rest of your team is at risk, and you have enough weaknesses.

Indestructible Simply won't die, even if you sleep though combat.  One of THE most abusive character in 4e.

Sir Robin (Bravely Charge Away) He automatically slows and pushes an enemy (5 squares), while charging away. Hard to rate it's power level, since it's terrain dependent.

Death's Gatekeeper A fun twist on a healic, making your party "unkillable". Overpowered to Broken, but shouldn't actually make the game un-fun, just TPK proof.

Death's Gatekeeper mk2, (Stealth Edition) Make your party "unkillable", and you hidden, while doing solid damage. Stronger then the above, but also easier for a DM to shut down. Broken, until your DM get's enough of it.

Domination and Death Dominate everything then kill them quickly. Only works @ 30, but is broken multiple ways.

Battlemind Mc Prone-Daze Protecting your allies by keeping enemies away. Quite powerful.

The Retaliator Getting hit deals more damage to the enemy then you receive yourself, and you can take plenty of hits. Heavy item dependency, Broken.

Dead Kobold Transit Teleports 98 squares a turn, and can bring someone along for the ride. Not fully built, so i can't judge the power.

Psilent Guardian Protect your allies, while being invisible. Overpowered, possibly broken.

Rune of Vengance Do lot's of damage while boosting your teams. Strong to slightly overpowered.

Charedent BarrageA charging ardent. Fine in a normal team, overpowered if there are 2 together, and easily broken in teams of 5.

Super Knight A tough, sticky, high damage knight. Strong.

Super Duper Knight Basically the same as super knight with items, making it far more broken.

Mora, the unkillable avenger Solid damage, while being neigh indestuctable. Overpowered, but not broken.

Swordburst Maximus At-Will Close Burst 3 that slide and prones. Protects allies with off actions. Strong, possibly over powered with the right party.


Wand of Accuracy DOESN'T suck, even if you only have 14 Dex.


Well, having to wield a wand can suck. But yeah, you need 13 dex for DIS anyway, so 14 dex is a reasonable investment, and the power is pretty nice.

Skill boosts are especially powerful compared to static skill bonuses, since you'll rarely need to make more than a handful of consecutive checks.
Paladin:  Probably gets 1 attack per round, plus 1 attack from an action point. 5.5 attacks.  Dynamic boost is FAR better than +1 static to hit, simply since he won't be attacking all that many times.
Fighter: Probably gets 1 attack per round, plus 1 attack from action point, plus 3 attacks from either multi-attack powers, immediate actions, or free action powers, 2 extra attacks from lazy leader. 10.5 attacks.  Could go for the dynamic OR the static.
Ranger: 2 attacks per round, plus 2 from action point, plus 4 from (larger) multi-attack powers, free action powers, minors, and immediates. 15 attacks, definitely wants the static boost.
Wizard: 3 targets one round (either first or second depending on set up), 2 targets for 3 rounds, 1 target on the last round. 3 targets on an action point, 4 extra targets from dailies, minors, slides, all kinds of fancy things.  Total 15.5, again loves the static boost.
Artificer|Lazylord: grants attacks on 2 rounds, attacks on 3 rounds with 1 target, action point for 1 target, 1 immediate action. Total: 4.5 attacks. Dynamic bonus is so amazing it's not even a race.  Would probably take a +5 dynamic boost over a +3 static (but possibly only a +2 static).

Not bad estimates.  Still at the break point is 17.

Anyways....  More math

Dynamic vs Static = number of attacks to be equal  (using a graph method)
+2 vs +1 = 16.47

+3 vs +1 = 19.1
+3 vs +2 = 6.55

+4 vs +1 = 19.76
+4 vs +2 = 8.5
+4 vs +3 = 3.83

+5 vs +2 = 9.31
+5 vs +3 = 5.1
+5 vs +4 = 2.7 

guides
List of no-action attacks.
Dynamic vs Static Bonuses
Phalanx tactics and builds
Crivens! A Pictsies Guide Good
Power
s to intentionally miss with
Mr. Cellophane: How to be unnoticed
Way's to fire around corners
Crits: what their really worth
Retroactive bonus vs Static bonus.
Runepriest handbook & discussion thread
Holy Symbols to hang around your neck
Ways to Gain or Downgrade Actions
List of bonuses to saving throws
The Ghost with the Most (revenant handbook)
my builds
F-111 Interdictor Long (200+ squares) distance ally teleporter. With some warlord stuff. Broken in a plot way, not a power way.

Thought Switch Higher level build that grants upto 14 attacks on turn 1. If your allies play along, it's broken.

Elven Critters Crit op with crit generation. 5 of these will end anything. Broken.

King Fisher Optimized net user.  Moderate.

Boominator Fun catch-22 booming blade build with either strong or completely broken damage depending on your reading.

Very Distracting Warlock Lot's of dazing and major penalties to hit. Overpowered.

Pocket Protector Pixie Stealth Knight. Maximizing the defender's aura by being in an ally's/enemy's square.

Yakuza NinjIntimiAdin: Perma-stealth Striker that offers a little protection for ally's, and can intimidate bloodied enemies. Very Strong.

Chargeburgler with cheese Ranged attacks at the end of a charge along with perma-stealth. Solid, could be overpowered if tweaked.

Void Defender Defends giving a penalty to hit anyone but him, then removing himself from play. Can get somewhat broken in epic.

Scry and Die Attacking from around corners, while staying hidden. Moderate to broken, depending on the situation.

Skimisher Fly in, attack, and fly away. Also prevents enemies from coming close. Moderate to Broken depending on the enemy, but shouldn't make the game un-fun, as the rest of your team is at risk, and you have enough weaknesses.

Indestructible Simply won't die, even if you sleep though combat.  One of THE most abusive character in 4e.

Sir Robin (Bravely Charge Away) He automatically slows and pushes an enemy (5 squares), while charging away. Hard to rate it's power level, since it's terrain dependent.

Death's Gatekeeper A fun twist on a healic, making your party "unkillable". Overpowered to Broken, but shouldn't actually make the game un-fun, just TPK proof.

Death's Gatekeeper mk2, (Stealth Edition) Make your party "unkillable", and you hidden, while doing solid damage. Stronger then the above, but also easier for a DM to shut down. Broken, until your DM get's enough of it.

Domination and Death Dominate everything then kill them quickly. Only works @ 30, but is broken multiple ways.

Battlemind Mc Prone-Daze Protecting your allies by keeping enemies away. Quite powerful.

The Retaliator Getting hit deals more damage to the enemy then you receive yourself, and you can take plenty of hits. Heavy item dependency, Broken.

Dead Kobold Transit Teleports 98 squares a turn, and can bring someone along for the ride. Not fully built, so i can't judge the power.

Psilent Guardian Protect your allies, while being invisible. Overpowered, possibly broken.

Rune of Vengance Do lot's of damage while boosting your teams. Strong to slightly overpowered.

Charedent BarrageA charging ardent. Fine in a normal team, overpowered if there are 2 together, and easily broken in teams of 5.

Super Knight A tough, sticky, high damage knight. Strong.

Super Duper Knight Basically the same as super knight with items, making it far more broken.

Mora, the unkillable avenger Solid damage, while being neigh indestuctable. Overpowered, but not broken.

Swordburst Maximus At-Will Close Burst 3 that slide and prones. Protects allies with off actions. Strong, possibly over powered with the right party.

Great thread, mellored! I think these calculations can be a great tool for evaluating many character options.

My blog about 4e rules and news: Square Fireballs The Magic Item Reset: A standalone set of items for 4E


Grondag: I think you've overcomplicating this. Your actual hit/miss chance has nothing to do with it.




You're right!  The way I set it up would apply if the dynamic modifier were a separate random event contingent on missing, like the Memory of a Thousand Lifetimes racial power for Devas.

In this case, the outcome is completely determined by the initial die roll, and the way you set it up is a more accurate and simpler representation of what's happening.

Running it through both ways, the differences are barely significant ( like 0.2 extra attacks) when accuracy is very high (90%+) already, and immaterial otherwise.

I believe my other comments still apply, and the break-even point is indeed around 17, plus or minus depending on your accuracy.

The key point is that a +2 dynamic modifier is as good or better than a +1 static modifier until you get above 17 or so attacks per encounter.  If you can't afford a static bonus due to a feat shortage, then you shouldn't feel to bad about settling for an encounter power +2 instead.  Obviously, both is better when you can get both. Smile

Wand of Accuracy DOESN'T suck, even if you only have 14 Dex.

Well, having to wield a wand can suck. But yeah, you need 13 dex for DIS anyway, so 14 dex is a reasonable investment, and the power is pretty nice.

Doesn't seem like you need to cast though the wand.  Just hold one off hand (which you'd do for the DIS bonus).

Make it a master's wand of magic missile, auto damage & push 1 has it's occasional uses, and you don't care if it falls behind.

guides
List of no-action attacks.
Dynamic vs Static Bonuses
Phalanx tactics and builds
Crivens! A Pictsies Guide Good
Power
s to intentionally miss with
Mr. Cellophane: How to be unnoticed
Way's to fire around corners
Crits: what their really worth
Retroactive bonus vs Static bonus.
Runepriest handbook & discussion thread
Holy Symbols to hang around your neck
Ways to Gain or Downgrade Actions
List of bonuses to saving throws
The Ghost with the Most (revenant handbook)
my builds
F-111 Interdictor Long (200+ squares) distance ally teleporter. With some warlord stuff. Broken in a plot way, not a power way.

Thought Switch Higher level build that grants upto 14 attacks on turn 1. If your allies play along, it's broken.

Elven Critters Crit op with crit generation. 5 of these will end anything. Broken.

King Fisher Optimized net user.  Moderate.

Boominator Fun catch-22 booming blade build with either strong or completely broken damage depending on your reading.

Very Distracting Warlock Lot's of dazing and major penalties to hit. Overpowered.

Pocket Protector Pixie Stealth Knight. Maximizing the defender's aura by being in an ally's/enemy's square.

Yakuza NinjIntimiAdin: Perma-stealth Striker that offers a little protection for ally's, and can intimidate bloodied enemies. Very Strong.

Chargeburgler with cheese Ranged attacks at the end of a charge along with perma-stealth. Solid, could be overpowered if tweaked.

Void Defender Defends giving a penalty to hit anyone but him, then removing himself from play. Can get somewhat broken in epic.

Scry and Die Attacking from around corners, while staying hidden. Moderate to broken, depending on the situation.

Skimisher Fly in, attack, and fly away. Also prevents enemies from coming close. Moderate to Broken depending on the enemy, but shouldn't make the game un-fun, as the rest of your team is at risk, and you have enough weaknesses.

Indestructible Simply won't die, even if you sleep though combat.  One of THE most abusive character in 4e.

Sir Robin (Bravely Charge Away) He automatically slows and pushes an enemy (5 squares), while charging away. Hard to rate it's power level, since it's terrain dependent.

Death's Gatekeeper A fun twist on a healic, making your party "unkillable". Overpowered to Broken, but shouldn't actually make the game un-fun, just TPK proof.

Death's Gatekeeper mk2, (Stealth Edition) Make your party "unkillable", and you hidden, while doing solid damage. Stronger then the above, but also easier for a DM to shut down. Broken, until your DM get's enough of it.

Domination and Death Dominate everything then kill them quickly. Only works @ 30, but is broken multiple ways.

Battlemind Mc Prone-Daze Protecting your allies by keeping enemies away. Quite powerful.

The Retaliator Getting hit deals more damage to the enemy then you receive yourself, and you can take plenty of hits. Heavy item dependency, Broken.

Dead Kobold Transit Teleports 98 squares a turn, and can bring someone along for the ride. Not fully built, so i can't judge the power.

Psilent Guardian Protect your allies, while being invisible. Overpowered, possibly broken.

Rune of Vengance Do lot's of damage while boosting your teams. Strong to slightly overpowered.

Charedent BarrageA charging ardent. Fine in a normal team, overpowered if there are 2 together, and easily broken in teams of 5.

Super Knight A tough, sticky, high damage knight. Strong.

Super Duper Knight Basically the same as super knight with items, making it far more broken.

Mora, the unkillable avenger Solid damage, while being neigh indestuctable. Overpowered, but not broken.

Swordburst Maximus At-Will Close Burst 3 that slide and prones. Protects allies with off actions. Strong, possibly over powered with the right party.

What kind of difference does it make if the dynamic bonus must be proactive (for instance, a minor action "+2 to your next attack" vs. an immediate/free "+2 to the triggering attack")? Surely that changes the math a lot, but I don't have nearly enough math skills to figure out how.
What kind of difference does it make if the dynamic bonus must be proactive (for instance, a minor action "+2 to your next attack" vs. an immediate/free "+2 to the triggering attack")? Surely that changes the math a lot, but I don't have nearly enough math skills to figure out how.



It's huge. All of the value of a dynamic bonus is being able to apply it on a known miss. If you have to apply a dynamic bonus before the roll, then the number of extra hits is just 
(DynBonus - (#Attacks*StaticBonus)) / 20,
where unless the Dynamic bonus is significantly larger than the static bonus, your usually worse off. 

The scenario that mellored laid out where it's more like Memory of a Thousand lives or the Accurate Wand feature, where you can chose to apply the Dynamic bonus when you miss, makes it much more valuable, as the numbers in the previous posts show. There is no closed form solution however, because you end up solving an equation involving #Attacks and the natural log of #Attacks.  
Doesn't seem like you need to cast though the wand.  Just hold one off hand (which you'd do for the DIS bonus).

Make it a master's wand of magic missile, auto damage & push 1 has it's occasional uses, and you don't care if it falls behind.


Indeed, you only have to wield a wand for Wand of Accuracy (offhanding counts as wielding, iirc). Still, I'm not sold on using wands. I think Wand of Shield is kinda decent. Master's Wand of Magic Missle also has its uses, particularly if you don't already have Magic Missile (having a range 20 option is nice). I'd much rather be carrying an Orb of Nimble Thoughts in my offhand, though: Int mod to initiative is awesome, as is shifting lots as encounter power.

As for proactive bonuses to attack rolls: The "your next attack roll" variety mostly sucks. Too much chance to do nothing when you use it. For comparison, unless you miss on an 11, it is literally impossible to gain 0.5 hits per encounter off the proactive bonus (and the bonus would have to be +10 to-hit or higher to achieve that). For a more likely scenario - miss on a 6 - you can at most gain 0.25 attacks per encounter off a +5 (or higher) proactive bonus. Only bother if it's cheap to get, or you have a signature ability that absolutely must connect.

An attack bonus for a turn is better. You can apply that to entire multiattacks, nova rounds and area attacks.
Nimble Thoughts was erratad some time ago.
Nimble Thoughts was erratad some time ago.

Yes, it went from being OMFGBBQ crazy good to merely quite nice.  You can get up to a +6 item bonus in the initative round and then swap it out for some other implement when needed, and swap it back in if you want to use the shift power.

There are other ways to get item bonuses but they take up a slot permanently. (Timeless Locket, for example.)

My Wiz wears a battle harness so that he can juggle orbs to fit the circumstance, but a familiar (disembodied hand, rakshasa claw, etc.) would work just as well.  

No reason you can't use both an orb and a wand to good effect though to save gold perhaps only one or two will have a top tier enhancement bonus.  Hold that one while you attack with your main implement for the Dual Implement Spellcasting benefit.
Hmm. Now I'm picturing a character based around retroactively adding bonuses to their allies' attacks (and/or defenses). Prescient Bard is the obvious choice. Maybe go deva and MC Warlord for Commander's Memory, take Guided Shot as one of your utilities . . . I can see it working. We can call it the Durkon Bard.
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