Month: March 2018

Aside: Those hot new Game Plans

Game plans have just dropped, and I’m here for your mathematical analysis. It’s hard to analyze a lot of the effects on the card, they are situational and often hard to quantify. I will probably take a look at ones I think the factions I plan would like to have at some point, but for now, I have some stats:

Initiative
Number of Instances
Influence
Number of Instances
12-15
2209
3314
44
53
62
72

That has two tables. The first (on the left) is the possible initiative score bonuses and the number of cards that have that bonus. I draw your attention to the fact that the distribution is closer to a bell curve than a flat distribution (like rolling one die would be). This means that more results tend towards the middle, in this case 4. You are way more likely to see a card near 4 than you are to see the cards on the ends. This is useful for planning going first, more on this in a second. On the right, is the possible influence bonuses, or penalties, and how many of the cards have that value. Most of the cards (half) have not bonus or penalty. Out of the other half of the cards more than half of those have a penalty. So getting a bonus to influence is a big deal since most cards don’t give that.

So let’s circle back around to the initiative scores. These add to the momentum you have left from the previous turn to see who goes first. The following table shows the chance of going first with different levels of momentum and different cards. The momentum goes down the side (this is how much you are ahead of your opponent) and across the top is the values of the card you play. So if you want your chance of going first if you are up two momentum and have an initiative 4 on the card you would look up 2 on the left and 4 across the top and see that you have a 78 percent chance to go first. (Note: That means that 78 percent of the possible cards they could play would allow you to go first.)

If you are down momentum, you can take the card initiative value across the top and the momentum across the left that you are down to find the value that your opponent goes first. Then subtract that from 100 to find your chance of going first.

Momemtum Advantage
Chance w/ 1 Initiative
2
3
4
5
6
7
03142847678192
1112544647889100
22544647889100100
344647889100100100
4647889100100100100
57889100100100100100
689100100100100100100
7100100100100100100100

Now, that table doesn’t take into account cards you have seen that can’t possibly be in their hand. Cards like the ones you have discarded, the ones in your hands, and the ones your opponent has already played. If you want to take those into account, you’ll have to do that on the fly during the game. I have devised a system to work that out, but I’m not sure if it’s even worth doing. It requires the numbers above and for one to record all the initiative values they’ve seen. You can take those values and sort them quickly (hopefully in your head) of values lower and higher than the card you are playing + momentum and reduce those groups down, one to one, until only one group has anything. That is how many cards either favor you (the ones higher that are left would make it less likely they have a card that beats yours) or they would make it more likely the opponent has a card that beats yours (because the cards that are left are lower). Each of those numbers left would add or subtract 5.5% (1 out of 18 cards) to the percents in the table above. This is the same sort of thing someone “counting cards” in blackjack would do (but more work, and more accurate).

Example: You have the following initiative value cards in your hand at the start of turn three: 1,3,4, and 6 and played a 4 turn two. At the beginning of the game you chose to discard a 1 and a 7. Your opponent played a 2 at the start of turn two. That is the known information. All the cards you have possibly seen (you have not seen the rest of your opponent’s hand, the cards they discarded, or the cards left in the deck that no one saw, unless you are a cheater).

You are up 1 momentum and are thinking about choosing the 4 card in your hand. Let’s calculate the chance of going first. We start by sorting the numbers into groups that are higher or lower than 5, the initiative + momentum. Lower: 1,3,4,4,1,2 and higher: 6,7. There are 4 more cards you’ve seen in the lower category than the higher. That makes it more likely that they can play a card that beats you. Take that number of cards, 4, and multiply by 5.5, which is 22%. So it’s 22% more likely that you LOSE with this card than if you didn’t know any cards. So let’s take that into account. From the table, we can see that having a 4 + 1 momentum gives a base chance of 64% to go first. Subtract 22% because we are less likely to win with all those lower cards out there and we see that our chance of winning by playing the 6 goes down to 42%. We went from better than a coin flip to worse after we account for the fact that they can’t have a lot of cards that lose to use. We haven’t seen a lot that win (6, 7 and 5’s half the time), so it’s more likely that your opponent has and will play those cards.

As I said when I started this, I’m not sure all that is worth the precious clock time to calculate in a tournament game. It doesn’t give you enough extra information for the investment. Having a general sense of “have I seen more cards that are higher or lower” can give you a good idea of which way the math leans. In the example just knowing that I’ve seen more cards below my value of 5 means I know I am less likely to win the roll and I don’t think it matter too much by how much I’m less likely.

Now, the other area on the cards that we can mathematically analyze for what is “good” would be the influence values in the bottom right. I have the distribution above, but also calculated the chances of a few things happening. Having to go down influence is probably not pleasant, so I calculated the chances of getting a hand that, after discarding, you didn’t have any cards that have an influence penalty. You have 73% chance of having no negative influence cards in hand after discarding 2 of your starting 7. They chance of not having any negative influence cards in your hand of 7 is much lower (5.4%). The complete flip of that would be getting stuck with all 5 negative influence cards, meaning you would have to keep at least 3 of them after discard. That’s only going to happen 1.25% of the time. Somewhere in the middle is the chance of having to keep at least one negative card (or leaving your opponent with at least one negative card). That’s a 27% chance. So in about 1/4 games you’ll have to keep at least one negative influence card even if you don’t like those cards.

That wraps up my analysis of the game plan cards for today.