Happiness

Every town has a "Happiness" level which indicates the satisfaction of citizens. A new town's happiness count will start out at a medium level, 196, yielding a population growth rate of 3.12 per hour. As the town's number of citizens increases, the happiness is diluted and the population growth rate gradually decreases. Without happiness, your population may remain stagnant and can even decrease. Population growth rate is governed by happiness, with the rate being a function of your happiness score. __NOWYSIWYG__

Buildings that affect happiness
The Tavern and the Museum play a major role of the growth of your happiness.


 * The first building that you get that affects your happiness is the Tavern. The Tavern gives +12 happiness per level. Using the Tavern, you can provide wine to your citizens. The more wine you give, the more happiness you provide, with +60 happiness per extra load of provided.


 * The second building you can build to increase happiness is the Museum. The Museum gives +20 happiness per level. Using the museum, you can make a Cultural Asset Treaty with another player to exchange exhibits, with each exchange to give +50 happiness.
 * You can have Museum in each town.
 * You can not have more treaties than the total number of levels of every Museum combined.
 * The Governor's Residence reduces the corruption in colonies. Corruption affects happiness, by reducing the effect of wine and Cultural Asset Treaties on happiness, so your first aim when you make a new colony is to get rid of any corruption that it will gain when built. __NOWYSIWYG__

Researches that affect happiness

 * Researches that are needed to build buildings that increase happiness:
 * In order to build Taverns, you have to research Wine Culture.
 * In order to build Museums, you have to research Cultural Exchange.


 * Researches that directly improve happiness are:
 * Well Digging increases happiness by 50 in your capital only.
 * Holiday increases happiness by 25 in all of your towns.
 * Utopia increases happiness by 200 in your capital only.
 * Economic Futures increase happiness by 10 per level in all of your towns.

Notes:

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Governments that affect happiness
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Governments that can be good for your happiness

 * Democracy - Increases happiness in all cities by +75. __NOWYSIWYG__
 * Theocracy - Increases happiness in all cities with a temple by max +150.

Governments that can be bad for your happiness

 * Aristocracy - Reduces happiness in your colonies only by 3% because of corruption
 * This corruption can be removed if you raise your Governor's Residence up to a level that is higher than what is required for the number of colonies that you own.
 * Dictatorship - Reduces happiness in all cities by -75.
 * Oligarchy - Reduces happiness in your cities by 3% because of corruption.
 * This corruption can be removed if you raise your Palace and/or Building:Governor's Residence up to a level that is higher than what is required for the number of colonies that you own.
 * Theocracy - Reduces happiness in every city, that does not have a Temple, by 20. __NOWYSIWYG__

Happiness Levels
There are five levels of satisfaction in your town. The higher the satisfaction, the more population you gain per hour. The levels are:

Formulas
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Happiness (Satisfaction)
As you play the game you will encounter a number of things that change this value. These events, good or bad effect the growth rate of the town.


 * The formula for happiness is:

Happiness = Basic Bonuses ( 196 ) +/- ( Government Bonuses/Penalties ) + ( Research Bonuses ) + ( Wine = Tavern Base + Bonus ) + ( Culture = Museum Base +  Bonus ) - ( Population ) - ( Corruption = Corruption Rate * Bonuses )


 * Or, in short:

Happiness = Bonuses - ( Population + Corruption Rate * Bonuses )


 * or

Happiness = ( 1 - Corruption Rate ) * Bonuses - Population

The formula showing happiness over time, if bonuses and corruption remain constant during this time and the population keeps growing, is the following:

$${ h(t) = h_0 \times e^{ - \frac { t } { 50 } } }$$, where


 * $${ h_0 }$$ is the happiness at an arbitrary starting point of time
 * $${ h(t) }$$ is the happiness after $${ t }$$ hours
 * $${ e }$$ is.

Notes:

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Population growth
The formula that shows the connection between happiness and population growth generally is:


 * Growth Rate = Happiness * 0.02 __NOWYSIWYG__

Additionally, the formula showing growth rate over time, if bonuses and corruption remain constant during this time and the population keeps growing, is the following:

$${ g(t) = g_0 \times e^{ - \frac { t } { 50 } } }$$, where


 * $${ g_0 }$$ is the growth rate at an arbitrary starting point of time
 * $${ g(t) }$$ is the growth rate after $${ t }$$ hours
 * $${ e }$$ is Euler's constant. __NOWYSIWYG__

Population after certain time
The formula that shows what the population will be after that amount of time if the Happiness doesn't change (ie., Museum or Tavern are not improved, population doesn't reach the maximum, etc) is:

$${ p(t) = p_0 + h_0( 1-e^{ - \frac{ t }{ 50 } } ) }$$, where


 * $${ p_0 }$$ is the population at an arbitrary starting point of time
 * $${ p(t) }$$ is the population after $${ t }$$ hours
 * $${ h_0 }$$ is initial happiness at the same starting point of time as $${ p_0 }$$
 * $${ e }$$ is Euler's constant. __NOWYSIWYG__

Time before the Town hall gets filled
You can calculate the amount of hours prior to your Town hall to get full as:

$${ t=50( \ln ( h_i ) - \ln ( h_f ) ) }$$
 * or

$${ t=50 \ln \left( \frac{ h_i }{ h_f } \right ) }$$, where


 * $${ \ln }$$ is the natural logarithm function
 * $${ h_i }$$ is the initial happiness
 * $${ h_f }$$ is the final happiness - which can additionally be calculated as (current happiness + current population - town capacity)

If the town capacity is greater than or equal to $${ B_c = Bonuses * ( 1 - Corruption Rate ) }$$, the Town hall will never be full, and the formula will result in an error.

This is because generally the town's population can approach up (but never be exactly equal to) the marginal number $${ B_c }$$. Thus it will never fill it's town capacity(TC), if the last is greater than (or equal to) this marginal number $${ B_c }$$. i.e.: $${ TC \ge B_c \Rightarrow \lim_ { t \to \infty } p(t) = B_c \And p(t) < B_c \le TC, \forall t \ge 0 }$$

Note:

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Time until happiness halves
If there are no changes on happiness, it halves every 34 hours, 39 minutes and 26 seconds (roughly, 1 day and 11 hours), while the population grows in the same amount in which happiness diminishes. This is a very good guide to know which of your towns needs your attention first. __NOWYSIWYG__