Expected value for spoil troops in Flat Rate per turn

[Eddygp]

Just a mathematical divertimento.
I calculated the expected value in the following way:
Let P be the average number of troops that you can get when you cash in, namely (4+6+8+10)/4.
There is 1/3 probability to play your spoils in 3 turns, which accounts for 1/3*1/3*P troops. (probability/turns*troops)
There is 2/3 probability to play your spoils in 4 turns, which accounts for 2/3*1/4*P troops.
There is 3/3 probability to play your spoils in 5 turns, which accounts for 3/3*1/5*P troops.

The expected value, assuming that the player conquers a territory every turn, is 3.34444444444 troops per turn (3+31/90 troops per turn).

[ManBungalow]

Not a bad estimate, but in some places you're quite off the mark.

For instance, it doesn't account for choices...if you have RRR you can wait 2 turns to see if you get RRRGB...and similarly with RRRGB you can choose to cash RRR or RGB.

And I'm not sure your probabilities are quite right...I could have a look at the numbers later and see if I can figure it out.

[DoomYoshi]

http://www.conquerclub.com/forum/viewtopic.php?f=6&t=197650#p4326386

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

rbg
400
310
301
9
040
130
031
9


004
103
013
9

211
121
112
36

220
202
022
18

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!

--------------------------------------------------------------

rbg
300 1

030 1

003 1

111 6

210
201
021
012
120
102
18

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

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?

[nippersean]

Can't be 3.33(4) / card - that assumes a mixed set every time.

2.8 must be about right........,mmmm it's very high tho?? Average cash 8.4.
That bit where you get a value for 3 cards (0.96 for cards 1-3) looks dodgy. You're assuming something, so it is diff to OP question. I think this (0.96) assumes you have to cash after 3 or throw them away if you haven't a set. Maybe you said that.

More likely i'm talking cock. There's a thread somewhere..................

[timogl]

flat rate sucks.

[DoomYoshi]

Just a mathematical divertimento.
I calculated the expected value in the following way:
Let P be the average number of troops that you can get when you cash in, namely (4+6+8+10)/4.
There is 1/3 probability to play your spoils in 3 turns, which accounts for 1/3*1/3*P troops. (probability/turns*troops)
There is 2/3 probability to play your spoils in 4 turns, which accounts for 2/3*1/4*P troops.
There is 3/3 probability to play your spoils in 5 turns, which accounts for 3/3*1/5*P troops.

The expected value, assuming that the player conquers a territory every turn, is 3.34444444444 troops per turn (3+31/90 troops per turn).

After reading this again, your math is way off.