Can someone help me with my math

[FamBonnet]

[Q&A]
I like to play flat rate spoils. Sometimes in a long game I have to decide if I should attack just to get a card...no strategic importance. I always wait until I have 5 cards to cash in a set. I will usually attack one or two but not three troops. Don't factor in the number of terts I have...I can do that math. Thanks for the help.


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[/Q&A]

[betiko]

Well sometimes it s not about the troop math.. Maybe not attacking can be the best solution to make other opponents mad at each other and go berserk.. Or not attacking can bother one of your opponent as you give him no card spot and he makes an unreasonable attack on you. More than arithmetics, i d say it depends on who is where and how they play.

[Jippd]

I go by the general rule of thumb that every card is worth 3.33 troops on average for flat rate. I assume I will cash 10 sets which doesn't always happen. I am also not taking into account bonus troops for having a bold card which certainly does happen. On the average I say it balances out and for the most part a set cashed at 5 cards in flat rate would net you close to an average of 10 troops.

So if the card costs you more than 3.33 troops to get you have a net loss.

[betiko]

I go by the general rule of thumb that every card is worth 3.33 troops on average for flat rate. I assume I will cash 10 sets which doesn't always happen. I am also not taking into account bonus troops for having a bold card which certainly does happen. On the average I say it balances out and for the most part a set cashed at 5 cards in flat rate would net you close to an average of 10 troops.

So if the card costs you more than 3.33 troops to get you have a net loss.

Still, not all about troop count in multisingles. Depends a lot on the profile of oponents and configuration. i don t like being an early troop leader either.

[FamBonnet]

Thanks guys but I'm pretty sure the average is less than 3.3. Can't expect a rainbow every time. I understand that there are other factors but sometimes it's just math.

[AyeTrain]

You get a maximum of 10 for a set, but a minimum of 4: so all that work to cash in, and you may only get 4 from it. If it makes sense to take a 1-troop territory, I'll do it for spoils. But often it makes sense to wait patiently: if attacking might antagonize another player; if taking the territory exposes your border to an enemy stack (I love using someone's 1s to keep them away from me); or if this card will be #5. In a long game, I sometimes get 4 cards and then wait---then when I do attack, I'll have extra troops on the next turn to help against the counter-attack.

[DoomYoshi]

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?

[DoomYoshi]

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

rbg
400
310
301

040
130
031

004
103
013

211
121
112

220
202
022

So, 3/15*0 can be disregarded.
9/15*6 + 3/15*10 = 5.6

Divided by 3 means that the spoil is only worth 1.87!

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

rbg
300

030

003

111

210
201
021
012
120
102

Using the same math as before, each spoil is only worth 0.9333

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

Conclusion: when an opponent has 3 spoils, each is worth 0.9333, for a total estimated value of 2.8 troops.

When he gets a 4th spoil, that one is worth 1.87, but all the other ones also increase in value to 1.87 - the total value is 7.48 (except he can only cash 3 of them, so there is an effective max of 5.6).

The 5th spoil causes another increase in value to 11.9. Obviously, that value is impossible to cash, but the effective value is just over 7.

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

[drunkmunky]

You get a maximum of 10 for a set, but a minimum of 4: so all that work to cash in, and you may only get 4 from it.

Not entirely correct; minimum you can get for a set is 4, but the most is 16. For all other answers, see DY above (^_^)

[FamBonnet]

Ha ha DY. The math doesn't seem simple to me but I'll take your word for it as your result is about what I expected. As for the extra bonus troops, I figure - % of map I control x 2 for each card.
Thanks everyone

[DoomYoshi]

Ha ha DY. The math doesn't seem simple to me but I'll take your word for it as your result is about what I expected. As for the extra bonus troops, I figure - % of map I control x 2 for each card.
Thanks everyone

That`s exactly right!

[zimmah]

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!!!

I am pretty sure you need to multiply the 31 by 3 and the 81 would then become 143.

The end result would then be 7.399/3 = 2.466

Also, I don't think you have factored in the card you have left over for a second cash in. If you have for example RRRGB And you turn in RGB then the likelihood of you next sets being RRR is increased by a lot, since you already have 2 red cards.

In either case, the value of a single card is somewhere between 2 and 3. + the bonus for number of regions compared to the total regions you own.

Ha ha DY. The math doesn't seem simple to me but I'll take your word for it as your result is about what I expected. As for the extra bonus troops, I figure - % of map I control x 2 for each card.
Thanks everyone

That`s exactly approximately right!

there's a small margin for error here considering the cards you already have (and the unknown cards the opponents have), but it's close enough.

[Kaskavel]

Good math work has been done here and I want to add a bit myself.
A card is worth
X+Y+Z+A
X= the expected value of the card, calculated by some people with some disagreement, but I also calculated 2.823 as Doom
Y=card value because of probability of a +2 bonus. This equals to Ter. possesed plus one (the one you gonna get) divided to total territories, multiplied by 2. A small adjustment can be made for the cards you already see in your hand.
Those are already mentioned, but I think we need to add 2 more variables
Z equals the attacking advantage. Just because a card may finally worth of 2.9 troops for example, this does not mean that it is wrong to attack a 3. You are not going to lose 3 troops to kill the 3 enemy troops. You are expected to lose something like 2.63 troops for example (random number). In other words, we must not compare to enemy troops, but to expected troops we will lose in order to kill those enemy troops. I have no time to analyze it by maths now, but I think this factor may be necessary to move to the other side of the equation to work properly (we need to have decided the value of the card by the other 3 factors and multiply the attacking advantage to the best accesible region we can attack)
A is a factor that depends by the mumber of players involved. Let's say that a card deserves 3 troops and you have acess only to an enemy 4. You are expected to lose lets say 3.75 troops to kill those 4 (random number). Is it wrong to attack? Not necessarily. You lose 0.75 troops indeed, but your OPPONENTS (I think it is better to work considering all opponents together), all together, also lose 4 troops. If you have 4 opponents, it is the same as you killed 1 troop from each one of them, so it is cool. If they are 7, it is the same as you killed 4/7=0.57 from each of them, so it is not cool!
There is also another variable, a more difficult one to calculate, the one that Zimmah mentions. You need to take into consideration the colors you already have in hand. If you have 2 reds, odds are much worse compared to having a green and a blue. Not only you theoriticaly need to calculate the odds again for your next set, but you should also calculate the expected odds for your next set as well. If you have 3 reds, it is not just that you are going to have a bad set after 2 turns, but you are also statisticaly probable to be left with a bad hand for your next set as well.
I will work on the maths again later after work, this is interesting, unless someone does them for me in the meantime.

[Kaskavel]

there's a small margin for error here considering the cards you already have (and the unknown cards the opponents have), but it's close enough.[/quote from zimmac]

Yes (and no). We can adjust B because of the cards we see in our hand but no, the cards the other players have are unknown to us and for calculating purposes, it is irrelevant if they are picked up to their hands or if they still in the deck.
For example in comparison, if you throw a coin without seeing the result, you should still make the maths considering it is a 50-50 probability. The fact that the coin is already flipped is irrelevant to you. It is the same like it has not yet been flipped.

[Kaskavel]

OK, I will try to provide some kind of answer.

1. The estimated trade value of a single card equals 2.823 troops, as calculated by Doom Yoshi, a number I have confirmed myself as well by using his board as starters. If his board is correct (which looks like being correct at a casual glance), I confirm his number.
2. Second parameter is the +2 possibility. Trying to create a reasonably simple and useful result, I will make the following assumption. All players control similar amounts of territories, as it useally happens in multi-flats-stalemate games. With this assumption, each player controls about 1/N of the board, N being the number of players. This assumption makes the result attractive, because we do not need to include the total number of regions in the equation, just the number of players (which is necessary anyway in step 3). A spoil now values 2.823+2/N troops
3. X is the number of defenders in territory of question (typicaly 3 in practice). You will kill X troops, which needs to be divided by N, in order to see what average damage you ve done to opponents. This is in your favor and can counterbalance it with the loss of troops in order to take the region. This is X/N
4. Finaly, we need to know how many troops you are estimated to lose while conquering a region. This is more tricky, because there is always a reasonable chance that you will fail to win, and the equation collapses. It is also a difficult calculation because I am not aware of any bot telling us the exact number of estimated average damage in order to win a battle. Calculations are different for different defending numbers. In the example of 3s, I will assume randomly that it is something like a 11 vs 3 attack or 11 vs 4 attack (the results should not be much different for a 9 vs 3 or 15 vs 3 attack). You are about to lose 2.35 troops against 3 troops and 3.25 against 4 (this is a very complex calculation, taking into account the senarios you will lose the battle and thus the 2.823 plus 2/N gain. It is completely pointless to try and reproduce it here)

So we have, for a case of a 3, in order that the attack is favourable. 2.823+2/N+3/N>2.35 (always favourable) and in a case of a 4, 2.823+2/N+4/N>3.25 (again always favourable).
In case of a 5, is starts to make sence. Seems like the attack is favourable in a game of 3.4 or 5 players, but not in a game of 6,7 or 8 players (something like N<5.25 occurs as a result)
In case of a 6, it seems like you should attack in a 3 player game only

I absolutely do not guarantee that my conclusions are correct. I will be happy to listen to objections and corrections, at least as far as my thoughts and calculations are included in the post, many are not, they are just in my mind lol...
Also keep in mind that there are far too many assumptions in the process.
1. All possible strategic points of the game are ignored. You are not to break any enemy bonuses, you are not attacking a region that also blocks another player, you do not open a big enemy stack's way to break you, you do not conquer an objective, you do not clear a path that allows you to reinforce useless troops into your bonuses etc etc
2. Deciding to refrain from the battle does not influence next round's situation. That means, you are not trapped and you are going to remain trapped again next round if you refrain from attacking. Obviously, if you are permanently trapped by a 6 or something, you need to kill it anyway
3. You do not have an especialy good or bad hand. The 2.823 calculation of Doom Yoshi assumes that you either have no spoils or you have 3 spoils with a rainbow (which are going to be cashed in the next 1-2 rounds, meaning that technicaly you have an empty hand). If you have 2 reds, odds are gravely lower. If you have a blue and a green, odds are higher
4. We ignore the possible +2 bonuses already in hand. If your current spoils posess +2 bonuses, your chance to gain new ones decreases and vice versa
5. We assume all neutrals have been killed
6. We assume that the board is more or less equally divided. If you are a leader, the equations tend to favor more attacking than no attacking. If you are remarkably weak, you are closer to non attacking
7. The game is static and close to stalemating situations, it is not that you are desperate for a cash next turn or something like that
8. You have very good chances of winning the battle in question. If the question is about making a 7vs 4 or a 4 vs 2 attack, things get different
9. We assume that you insist attacking if 3 vs 1, 2 vs 1, 3 vs 2 and M vs N (N>M) situations occur, while in practice you may stop if the battle begins with bad results
10. Gaining a region is not going to increase next round's income or decrease an opponent's income(this is an important one since in one third of the case you are going to win a whole troop if you conquer the region)
etc
etc

I have made too many assumptions in the calculations, many of them are not even included in the post, but I have reached the following conclusion
IT IS ALWAYS FAVOURABLE TO ATTACK A 3 OR A 4
IT IS FAVOURABLE TO ATTACK A 5 IN A 3-5 PLAYERS GAME, BUT NOT IN A 6-8 PLAYERS GAME
YOU SHOULD NEVER ATTACK A STACK OF 6 IN AN AT-LEAST-4 PLAYER GAME
IN A 3 PLAYER GAME, YOU ARE ALLOWED TO ATTACK EVEN BIGGER STACKS (I cannot calculate the exact top number, but I guess you should not provoke the opponent too much and getting trapped in a 3 player game is quite rare anyway)

[DoomYoshi]

Just so that it is all together, here are my expected worth calculations so you can compare expected troops lost vs expected troops gained.

Let's say it's a 20 round limit no spoils game on Feudal Epic (I don't really like the other Feudal map). This is a really simple example, and helps me illustrate a concept.

First, assume the game will go to round limit. How many spaces do you want to attack?

So, calculate how many rounds of deploy you will get for a bonus. On round 1, its 19. Then calculate expected lost troops.

Spoiler

0.8534144 N - 0.2213413 (1 - (-0.525359)^N)
N=# of defenders

Or, more a more simple approximation is 5/6 # of defenders in all territories + 7/9 # of territories + 1/18 # of territories that have exactly 2 defenders - 1/9 # of terts with 1 army.

This simple approximation includes troops lost due to territory spread. Those troops still count for victory at round limit, so my model is overly conservative.

A bonus of 1 requires killing 4 troops spread over 2 territories. So, it takes 5 troops. It is only worth it in the first 15 rounds. There is no excuse to not take your entire 1-6 of the kingdom.

The map is symmetrical in that everyone has the same 10-1-1-1-5 to get through for the village bonus. That would lose an estimated 18.5 troops. It only takes 7 rounds of village bonus to win that back.

The real question is: if everyone goes for village bonus, how do you deal with it?

I would say that if the first person to arrive at the village takes it for 8-10 rounds and then pulls out to let the other guy get the bonus for a bit, that would be the optimal strategy. What does game theory say about the likelihood of this?

Do you use expected worth calculations in your day-to-day games?

In non-round limit games, its pretty simple except you can only calculate relative worth, not absolute.

I haven't read Kaskavel's yet, I am planning to read it tonight.

One thing I was thinking of, is if you don't take a card and your opponent does, his cards are worth twice as much. (since he gains 2.8 and you lose 2..

[MoB Deadly]

Spoiler