The Reinforcement Card can be played to gain additional armies.
The reinforcement card supports two modes: fixed and progressive.
Fixed Reinforcement Cards are simple: play one to be given a specific number of extra armies to deploy that turn. Each card is always worth the same amount.
When the Reinforcement Card is set to the progressive setting, it gives a varying number of armies. Typically, you want your Reinforcement Cards to give fewer armies at the start of a game to keep them from being too powerful, and more armies as players accumulate larger incomes to keep the cards from becoming worthless.
It's important to note that the value of the card is determined when you receive it (or when it is made whole) rather than when the card is played.
The card increases either by turn number or by territories held:
 Progressive By Turn Number
A Reinforcement Card that progresses by turn number will simply multiply the number of turns that have passed in the game (seen in the upper-left corner of the game screen) by the defined percentage.
The advantages to using this method are that players cannot learn anything about a game that they otherwise could not know and that it is easy to calculate ahead of time how much your cards will be worth.
The disadvantage to using this method is that cards are unbounded; they will continue growing in value forever. In long games, this tends to decrease the fun as cards become the key component to winning and bonuses grow obsolete.
 Progressive By Territories Held
A Reinforcement Card that progresses by territories held will multiply the total number of territories held by all players on the entire map by the defined percentage. This is designed for games that start mostly filled with neutrals as it will allow the cards to grow along with the game.
The advantages of using this method are that the cards will stop growing in value once the world is fully taken. This allows the card significance to, more often than not, stay in sync with the pace of the game and keep from getting out of control.
The disadvantages of using this method are that clever players can use the value to calculate how much of the world has been taken over in a fog game. This typically is not very valuable information, so this does not usually harm the game.
In the event of a remainder in these calculations, the value will be rounded to the nearest whole number. A remainder of 0.5 will round up.