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 Posted: 31 May 2015, 11:17 pm
 Modified: 28 Oct 2021, 08:37 pm

Post #1
The essence of a winning move and a bit of math.
This is an attempt to show a bit of how I think when I play an increasing card game.
To do so I will look at two alternatives setups of the same game.
But first...
Some important facts:
The statistics:
Below is a picture of the statistics of some game. We are the blue player and receive 6 troops this turn. The card turnin value is 15.
Now here's the plan:
If we kill the orange player we will lose about 17 troops (37 + 6  17 = 26 troops left) and get 4 cards. We already have 3, so that makes 7 cards.
With these 7 cards, we can almost be sure to have a double turn in: which gives us 15 + 20 = 35 troops.
Notice that we now have 1 card left and something like 26 + 35 = 61 troops.
With our remaining troops kill red: we will lose approximately 27 troops, and get 4 cards. Together with our remaining 1 card, we now have 5 cards, so we can turn in a set and receive 25 new troops.
The balance now is 61  27 + 25 = 59 troops.
With these 59 troops, we can kill the 41 troops from the cyan player and we won the game  yay!
But...
The board:
If we look at the board however, we cannot use all our troops  only the troops in Clwyd can move.
And 10 + 6 = 16 is not going to be enough to kill orange, because you leave one troops behind on each territory
(you have about 16.5% chance to kill orange  you should never do that).
So we can't win  aaaw.
A different setup:
What would happen if we didn't spend so much troops defending our region?
Now we suddenly have 21 + 6 = 27 troops available, which is more than enough to kill orange  yay!
This shows you that, even though regions are useful in the early game, in the end of the game they don't matter  killing your opponents and getting their cards is the important thing.
Some odds:
For those who are interested in the actual battle odds:
16 vs 1,3,1,6,2,5 = 16.5% chance of winning
27 vs 1,3,1,6,2,5 = 84.8% chance of winning
These odds are calculated using an odds calculator (Note the (One less than in territory) for the attacker).
A somewhat more simple to use calculator is this one: http://riskodds.com/
This is an attempt to show a bit of how I think when I play an increasing card game.
To do so I will look at two alternatives setups of the same game.
But first...
Some important facts:
 With increasing cards every time you or anyone else turns in a set of cards, the value of the next set increases.
These are the values: 4, 6, 8, 10, 12, 15, 20, 25, 30, 35, ... (+5 each set).  With 7 cards you almost always turn in two card sets (the probability is about 94%). This is called a double turn in.
 If you kill a player you lose troops, but you get his cards. If these cards are worth more troops than it costs to kill him, than you make a profit.
 If this profit is big enough to kill the next player and get his cards, you can get a chain reaction and kill everyone. This is generally how you win an increasing card game.
The statistics:
Below is a picture of the statistics of some game. We are the blue player and receive 6 troops this turn. The card turnin value is 15.
The statistics (click to show)
Now here's the plan:
If we kill the orange player we will lose about 17 troops (37 + 6  17 = 26 troops left) and get 4 cards. We already have 3, so that makes 7 cards.
With these 7 cards, we can almost be sure to have a double turn in: which gives us 15 + 20 = 35 troops.
Notice that we now have 1 card left and something like 26 + 35 = 61 troops.
With our remaining troops kill red: we will lose approximately 27 troops, and get 4 cards. Together with our remaining 1 card, we now have 5 cards, so we can turn in a set and receive 25 new troops.
The balance now is 61  27 + 25 = 59 troops.
With these 59 troops, we can kill the 41 troops from the cyan player and we won the game  yay!
But...
The board:
The original board (click to show)
If we look at the board however, we cannot use all our troops  only the troops in Clwyd can move.
And 10 + 6 = 16 is not going to be enough to kill orange, because you leave one troops behind on each territory
(you have about 16.5% chance to kill orange  you should never do that).
So we can't win  aaaw.
A different setup:
What would happen if we didn't spend so much troops defending our region?
An alternative board (click to show)
Now we suddenly have 21 + 6 = 27 troops available, which is more than enough to kill orange  yay!
This shows you that, even though regions are useful in the early game, in the end of the game they don't matter  killing your opponents and getting their cards is the important thing.
Some odds:
For those who are interested in the actual battle odds:
16 vs 1,3,1,6,2,5 = 16.5% chance of winning
27 vs 1,3,1,6,2,5 = 84.8% chance of winning
These odds are calculated using an odds calculator (Note the (One less than in territory) for the attacker).
A somewhat more simple to use calculator is this one: http://riskodds.com/
"Strength doesn't lie in numbers, strength doesn't lie in wealth. Strength lies in nights of peaceful slumbers." ~Maria