Two New Concepts
- Castles or factories which produce armies, gained by Risk card turn-in
- Always having allied armies, usable by winning poker games
Factories vs. Immediate Armies from Cards
Cards work differently in the New Rules. There is no "perpetual growth" of values of turn-ins, for one. You can either turn in a set of cards at the start of your turn for some number of armies determined by die roll, or at the end of your turn to get a factory on one of your territories. Factories produce two armies on the territory where they are located, forever, but they may be conquered by other players, who then get the armies. For immediate armies from a card turn-in, roll two dice.
Playing Poker for Control
All armies are on the board at all times. If there are only two players, then four of these armies will be allied armies, able to be used by any and all "live" players. Each allied army is assigned at least one poker suit, i.e., hearts, diamonds, clubs, spades, or ♥ ♦ ♣ ♠.
NOTE: The queen shall be the high card in the hearts and diamonds, ♥ ♦ the red suits.
The right to use an allied army is accomplished by winning a hand of poker (or being unchallenged) against the other players. At the end of every successful turn, in addition to a Risk card, a player draws a card from a poker deck.
The way a poker game works is that at the end of your turn, if you have three or more poker cards, you can declare the intent to use one of the allied armies. You must declare which army you wish to use, and if you win the hand, you must win with a card in the suit of that army in your winning hand. If you lose a hand, you may play again only if you have three more cards, which will only happen if you have six at the end of your turn.
Any player with at least three poker cards may elect to join the poker game. If he has more than three cards, he must choose three and set aside any extras.
At the end of your turn, if you have six poker cards, you have to play, and you have the option of doubling down with two hands. You have to ante for each hand, however.
Also known as Eastern United States Hold'em.
To join the hand, a player must have at least three poker cards accumulated, and choose three if holding more than that number, setting the other one or two aside for the duration of the hand. Each player joining the hand of poker must compute the number of armies he would receive for territory, continent bonuses, and factories and place that in his betting pile.
Each player must ante up one army for each hand he is playing (which will almost always be one). The dealer then turns up one card (the "flop"). Betting begins with the player whose turn it is, going around giving all players the chance to check, call, raise, or fold. The dealer lays out the next face-up card (the "river") and betting resumes. A player may go all-in, but this does not result in a side pot. The best 5-card hand made from a player's hand and the two "hole cards" wins the pot. If hands are tied, the pot is split, but if the player whose turn it is wins, and his winning hand contains the suit of the allied army he selected, he gets to use that allied army. Winning players convert all armies in the pot into their own color(s).
Playing an Allied Army
An allied army is played just like your own army. You accumulate Risk cards and poker cards, and may play them as normal to gain armies for the allied army. Only players that win the right to play an allied army may look at that army's cards, or, of course, anything else. Allied armies collect territory and continent bonuses as well as two armies for any factory on its territory. A player using an allied army can initiate a poker hand using the cards of the allied army he controls, though of course any winnings would go to the allied army to be used by the next player getting control.
At the end of a turn, a player is able to transfer as many armies from one territory to another as the largest army he has on any single territory.
For instance, if he has four armies on one territory, and this is the largest army, then he has three troop transfers he can make, regardless of any other armies on other territories. Now, this transfer may be in the form of moving all three armies of the main force to a neighboring territory, or, on the other extreme, moving one individual army on three different places to an adjacent territory. It's also possible to move two armies in one place and one army in another place, so long as the total is less than or equal to three, in this example. Those are what we call "adjacent transfers."
Alternatively, the player may choose to make one single move of a single body of troops, again, numbering equal to or less than the size of the largest army, from one territory to another that may be reached along a contiguous path through territories controlled by the player.
As something of a summary, at the beginning of his turn, a player receives the total count of territories divided by three plus continent bonuses, as normal, unless his troop count was determined by a previous poker game, in which case he deploys however many of those troops left in his "betting pile" after any poker losses or gains. Next, he receives two armies for every factory on any territory he controls. He can play any Risk cards in exchange for armies (by roll of two dice).
At the end of his turn, a player may play cards to buy factories (which start to pay off the following turn).
NOTE: If a player has already counted up his betting pile for a poker game prior to his turn, he does not count up his territory and continent bonuses again upon the start of his turn. He simply places all armies (if any) left in his betting pile. He does, however, get factory armies and may turn in Risk cards for armies, regardless of previous poker participation.
If a player fails to get territory and continent bonuses on his turn, he will get them at the start of his next turn or at the next poker hand he joins, whichever comes first.
QUESTION: Should allied armies that control factories get two armies on that spot even if no player plays them. If so, when should these armies be placed?
Values of Factories and Turn-in Armies
The relative value of a factory versus a set of armies for a card turn-in are both computed based on the average number of turns required to get another set of cards to turn in. In other words, if I turn in for a factory and get two armies per turn, I will get another set of cards to turn in for armies at the soonest after three turns and at the latest by five turns (guaranteed set of three). In that period of time I will get between six and ten armies from a factory. Now, if I roll two dice, I have the chance of 2 to twelve armies. Are the averages of both outcomes eight armies? Maybe. It would work well if so.
I think this works. The probability of getting more or fewer turn-in armies averages seven. So it's seven for all three of 3, 4, and 5 cards, but it also has a greater range...6 to 36.
The value of factory armies is proportional to 6 x chance of getting three-card match, plus 8 x chance of four-card match, plus 10 x chance of 5-card match (100%?).
The chance of getting three cards is three-of-a-kind 1 x 13/41 x 12/40 = 0.095
The chance of one of each 1 x 28/41 x 14/40 = 0.239
So the chance of either happening should be 0.095 + 0.239 = 0.334
And getting three-of-a-kind after 4 draws is the number of permutations on getting two cards of two different types (since three would be a one-of-a-kind) taken from the total number of possible 4-card draws. Probably combinations are okay, since I don't care the order.
one-of-each after 4 should be the same, since you need to draw two cards of two different suits, since any third suit would complete the set, or give three-of-a-kind.
At five cards it's 100% that you'll get one or the other combination.
The results here may be trivial, but I haven't worked it out yet. The crux is that the return on a factory is certain to be 6 at three cards, 8 at four, and 10 at five, and differences in those probabilities of getting a card set could change the average, and then there is also the greater range with turn-in armies by dice.
How to chose four cards, two pairs of two suits, how many different ways to pick them? There are three possible suits, so picking the second card