## Decline & Fall IV: I Hate Math

One of my take home lessons from reading http://gamebalanceconcepts.wordpress.com/ was that I really needed to do the math when designing games.  Because I’m pretty happy with combat at the moment my focus is on the length of the game, and how players score Glory (victory points).

Running out the clock.

Current levers for ending the game:
(1) A player moves from 0 to 100+ Glory (most likely option).
(2) The Galactic Empire moves from 25 to 0 sectors.
(3) The number of leaders drops from 100 to 0.
(4) No remaining atomic power (the least likely option).

Ways for leaders to be eliminated:
(1) Death in battle (if one leader committed chance of death is 1/24 or 4.2%, if two leaders committed, chance of one death is about 8.2%, chance of two deaths 4.3%)
(2) Participate in Civil War (1-P, where P is number of players with claimants, 100% end up dying, as the Emperor will die at start of next Civil War)
(3) Card Play (# of cards/deck size, probability of card, chance of success,  options to remove one or many leaders?) Possible card: Purge: play when Emperor, target leader, eliminate on 13+, reduce Con by 1, continue until failure.

Estimating battle mortality for (1), assuming Civil War duration of four turns, and each player starting one battle per turn with both sides committing leaders, with an expected loss of 0.172 leaders per battle:
– Four players, 16x 0.172 =  2.752
– Five players, 20x 0.172 = 3.44
– Six players, 24x 0.172 = 4.128
– Seven players, 28 x 0.172 = 4.816

For (2), key variable is the number of players in the game.  Assuming 100% rate of claimant nomination by players:
– Four players = 100/4 or 25 Civil Wars
– Five players = 100/5 or 20 Civil Wars
– Six players = 100/6 or 17 Civil Wars
– Seven players = 100/7 or 14 Civil Wars

Combining the two sets for conflict loss:
– Four players, a civil war costs 6.752 leaders
– Five players, a civil war costs 8.44 leaders
– Six players, a civil war costs 10.128 leaders
– Seven players, a civil war costs 11.816 leaders

Number of Civil Wars required to eliminate 100 leaders:
– Four players = 14.8 Civil Wars
– Five players = 11.8 Civil Wars
– Six players = 9.7 Civil Wars
– Seven players = 8.5 Civil Wars

I’m not sure what the optimal number of Civil Wars/game would be.  One possibility is to reduce the pool of leaders in games with less than seven players.  If we think 8-9 Civil Wars is a good point to aim for, then we can reverse engineer an optimal leader mix:

Possible 100, 85, 75, 60 to keep things at nice round numbers.

Empire Zero

There are 25 sectors in the game.  Current levers for shrinking the Galactic Empire:
(1) -1 Sector per Civil War (assuming 100% chance of a crisis on the map)
(2) Cardplay (# of cards/deck size, probability of card, one Sector or many Sectors)

24 Civil Wars will end the Galactic Empire through sector decay, but based on leaders, only 8-9 Civil Wars will occur, suggesting scope for a number of other mechanics to accelerate sector loss.  Option: if any player Glory is 50+, then lose 2 sectors after Civil War, if any player Glory is 75+, lose 3 sectors after Civil War (where possible).  This might mean a loss of 4+4+8 sectors, or 16/25 sectors through Civil Wars, so perhaps 4-8 card events are needed to collapse the remaining sectors.

This mechanic means that if leader commitment in Civil War  is low, the clock still runs out.  Having a card/action based mechanic for additional sector loss means the front-runner must sacrifice some actions to run the clock out, a negative feedback loop.

Glory 100

This clock really depends on the rate at which Glory can be scored.  Feedback loops are important here (negative feedback, glory can be reduced and being in front can be a disadvantage, positive feedback, scoring gets faster and faster so an initial frontrunner is hard to overtake).

Glory sources:
(1) Resolve crisis (+1-6, risky as action can fail)
(2) Auction (1-4, +2 for highest bid) [cardplay or turn sequence?]
(3) Battle (% x 1-6)
(4) Civil War Glory Pool (1-P6)
(5) Card play (numerous flavours, could be Penny Trades of +/-1 Glory, or more substantial 1d6 Glory gains)

(1) Resolve Crisis:
– not a guaranteed success (but with good leader and power, can be close to 95%)
– average +3.5 glory (possibly slightly more than this, as attempt must be successful, which is more likely with higher rolls)
– incentives: if it removes a crisis and restores confidence then non-Emperors may not want to do (unless already a strong contender)
– assumption: one-two players per turn will do this (so more players means a lower average score) [Might need to make it so that in a 4 player game, crisis resolution does not boost Confidence]

Glory gained will depend on length of game.  Assume 25 full turns,

Four players = +44 Glory each?  Five players = +35 Glory each? Six players = 30 Glory each? Seven players = +25 Glory.  Feels about right for seven players, may need a way of discounting score/increasing risk of failure in games with fewer players.  Could increase the target number required for success by +1 per player missing.

(2) Auctions
– Option A: card initiated auction winner, if sufficient power, can get +1-4 glory, + winner bonus (1-6 Glory)
– Option B: fixed auction every turn with a winner bonus, fix at +1, increase by +1 per player with more glory than you (a catch up mechanic), +1 for each other player that bids, -1 for lowest bidder (so no bid = -1, lowest bid = 0-1, other bids 1, winning bid 1+)
– incentives: if player initiated in option A, only likely when initiating player can win, if it happens once/emperor turn as with option B then its a steady source of Glory.  High spending also requires a player to have atomic power to burn, which must be balanced against the anticipated costs of participating in civil wars and future auction bids.
– assumption for option A: if initiated by card, assume one play per card per game, worth average +4 glory/player, so six cards = +24 Glory, with an outlier of +60 Glory (about a 1 in a million chance)
– assumption for option B: if per turn, then more likely to be 1+0.5 Glory per player (because there will be well more than six turns played, scoring must be lower) and 25 turns per game is possible.

Not sure how the winner bonus might work out for option B, but possibly:
4 players = 3 Glory per turn to winner, assuming even spread of players ahead/behind
5 players = 3.5 Glory per turn
6 players = 4 Glory per turn
7 players = 5 Glory per turn (this is fine because turns will take longer to complete with more players)

(3) Battle
– winner has 13.3% chance to score Glory, loser has 20% chance to score Glory
– most competent military leaders have low Glory values, +1
– incentives, player will not always choose Glory (may prefer other options), wash with possible higher Glory values?
– expectation of a glory proc is 26.64%
– if expected score is +1, then in each four turn CW a player will gain +1G from battle (not much, but could be much higher with one of the rare competent high Glory leaders)
– so +8-9G per game

(4) Glory Pool
– depends on freq of war, assume 8.5 CW per game
– depends on player negotiation, assume half of players benefit each war (rounded down)
– depends on #leaders committed, and Glory values
– incentives, confident of winning = commit high G leader, share with fewer players, reverse if less confident, but, if losing commit high value leader and hope to get lucky, reverse if winning to reduce chances of people catching up/overtaking
– assuming 100% commitment and 2 Glory leader values

4 player = 8 Glory pool, 4 Glory share, entire game gain = 0-34 Glory, avg +17 Glory
5 player = 10 Glory pool, 5 Glory share, entire game gain = 0-42 Glory, avg +21 Glory (if > player share, more equivalent)
6 player = 12 Glory pool, 4 Glory share, entire game gain = 0-34 Glory, avg =17 Glory
7 player = 14 Glory pool, 4.7 Glory share, entire game gain = 0-40 Glory, avg +20 Glory

Combined Civil War glory expectations = 25-30 Glory over entire game. But extreme possibility is that one player could pick up that much Glory from one war (assuming some skillful play, luck, and the cooperation/poor play by other players).

Balance: each source of scoring worth around 25% of final score?  With (5) card plays to even off the edges?

Duration of Game

This section is a bit rough.  Number of rounds of play required to score Glory (G) in peace, anticipated frequency of Civil War ( CW).  CW frequency depends on high fast the different Confidence (Con)  tracks can be run to zero.

Triggers for CW:
(1) 1/36 chance each player turn
(2) Any Con track hits zero (most likely cause)
(3) Card play (automatic or chance based)
(4) No power in bank (unlikely )

Three Con tracks.  Value 1-12 with each new Emperor.  Likely that one value will be lower than others due to the way leader values were generated (40 point buy, four attributes, attribute cost is points equal to square of attribute value, with some fudging for a few über leaders and a few woeful leaders).

Crisis chart: 27.7% -1 MilCon, 19.4% -1PopCon, 19.4% -1EliteCon [So cards should reduce Pop/EliteCon more often than MilCon]
Exploitation of Crisis: -1 Con of player choice (subject to crisis availability), incentive is to reduce the lowestCon to trigger the next CW and chance to become Emperor (done instead of trying to resolve a crisis, so incumbent Emperor is unlikely to do this)

Will take perhaps 4-5 player turns in typical play for lowest Con to get a crisis, and then that Con can then drop by -1 per player turn.

Assume Con value of 7 (likely in early game), 10-11 player turns to proc CW (and 10/36 chance of RNG CW), 2-3 full rounds of play between CW.
Midgame Con value of 4, 7-8 player turns to proc CW, 1-2 rounds between CW
Endgame Con value of 1, 4-5 turns to proc CW, 0-1 round between CW.

Rough estimate that the first half of the game requires 8-12 player turns.  Second half of the game perhaps 6-10 turns.  So feels like Glory accumulation will end most games before leaders/sectors are exhausted, but outlier games would be possible.

Still a bit of work left to do to fully explore the synergy between game duration, methods of scoring Glory, and levers for running out the clock, but I think this back of the envelope doodling has been useful for tuning several game mechanics to a more balanced point.