As long as you always cash at 5, and as long as you always take the best cash (ie if you have 3 reds, a blue and a green, you cash in the mixed instead of the red) then the math is simple.

You have 21 possibilities for the types of cards you have. Beside each number is the likelihood of that occuring.

rbg

500 1

410 5

401 5

320 10

302 10

050 1

140 5

041 5

032 10

230 10

005 1

104 5

014 5

203 10

023 10

311 20

131 20

113 20

221 30

212 30

122 30

31/81*6 + 50/81*10 = 8.469

Each card is worth a third of that, or 2.823!!!

It is not as important to take cards as it would seem unless you can get it for 1 or 2 troops.

Except each card value also has a potential +2. So, on classic map 1/42*2 for each territory you own.

+.05 if you own only one territory. If you own 20 territories, the value of a single card is increased by 1 - to 3.82.

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While I am here, I figured I would analyze 4 and 3 card setups as well.

rbg

400

310

3019

040

130

0319

004

103

0139

211

121

11236

220

202

02218

So, 18/81*0 can be disregarded.

1/3*6 + 4/9*10 = 6.444

Divided by 3 means that the spoil is only worth 2.148!

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rbg

300 1

030 1

003 1

111 6

210

201

021

012

120

10218

Using the same math as before, the value of a spoil is 0.96.

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Conclusion: when an opponent has 3 spoils, each is worth 0.96, for a total estimated value of 2.9 troops.

When he gets a 4th spoil, that one is worth 2.15, but all the other ones also increase in value to 2.15 - the total value is 8.6 (except he can only cash 3 of them, so there is an effective max of 6.5).

The 5th spoil causes another increase in value to 14.115. Obviously, that value is impossible to cash, but the effective value is just under 8.5.

The mod for territory +2 stays the same through all the numbers. It needs to be calculated different from every map though.

As you can see here (and what most players probably realized already) is that the 4th spoil is the most important. First to 4 spoils in a flat rate gets a definite advantage, and if you can hold an opponent at 3, you are doing good.

As always, these numbers are derived using some assumptions. Keep in mind that in reality, your opponent either has a mixed set or doesn`t. There are no probabilites. Rather, there is a probability of either 1 or 0 - you just don`t know which

Another thing this brings up: how idiotic the RISK rules actually are. Why is the most common set to get the one that gives the highest reward?

I have often heard him say that he would not begin life again if he had to pay for it by his years at school. There is, he is accustomed to say, only one crime which is beyond pardon, the crime which poisons the pleasures and kills the smile of a child