Coup is an excellent new card game involving bluffing and deception. Here are the rules from boardgamegeek.com:
In Coup, you want to be the last player with influence in the game, with influence being represented by face-down character cards in your playing area.
Each player starts the game with two coins and two influence – i.e., two face-down character cards; the fifteen card deck consists of three copies of five different characters, each with a unique set of powers:
- Duke: Take three coins from the treasury. Block someone from take foreign aid.
- Assassin: Pay three coins and try to assassinate another player’s character.
- Contessa: Block an assassination attempt.
- Captain: Take two coins from another player, or block someone from stealing coins from you.
- Ambassador: Draw two character cards from the Court (the deck), choose which (if any) to exchange with your face-down characters, then return two. Block someone from stealing coins from you.
On your turn, you can take any of the actions listed above, regardless of which characters you actually have in front of you, or you can take one of three other actions:
- Income: Take one coin from the treasury.
- Foreign aid: Take two coins from the treasury.
- Coup: Pay seven coins and launch a coup against an opponent, forcing that player to lose an influence. (If you have ten coins, you must take this action.)
When you take one of the character actions – whether actively on your turn, or defensively in response to someone else’s action – that character’s action automatically succeeds unless an opponent challenges you. In this case, if you can’t reveal the appropriate character, you lose an influence, turning one of your characters face-up. Face-up characters cannot be used, and if both of your characters are face-up, you’re out of the game.
If you do have the character in question, you reveal it, the opponent loses an influence, then you shuffle that character into the deck and draw a new one, perhaps getting the same character again and perhaps not.
The last player to still have influence – that is, a face-down character – wins the game!
Apparently the game has sold out, but it’s easy to re-create with a standard deck of playing cards. Play with a fifteen-card deck of three of each broadway card. Ace = assassin, king = duke, queen = contessa, jack = captain, ten = ambassador. Use loose change for the treasury.
Before reading further I recommend you play a few games of Coup, both because it’s a lot of fun and because what follows won’t make sense otherwise.
Okay, ready? Pretty fun, right? It becomes clear rather quickly that there is some strategy, (I lost my first five games, definitely not just from being unlucky) but it’s not obvious how to find the best plays. One reviewer on boardgamegeek wrote, “I love ‘hidden roles’ games in general, but here it seems you don’t really have much information to go on, so it becomes a game of guessing blindly and hoping for the best. What am I missing?!” While it’s true that we don’t have much information to go on, we can still do better than “blindly guessing.” To do so we can use some familiar concepts from poker such as combinations, balance, and blockers.
I’m going to try to design a balanced strategy for the player who goes first in a heads-up game of Coup. By the end of this post, I’ll recommend an action for every possible starting hand. The goal is to craft a strategy that gives us a good chance of winning and cannot be thwarted by any counter-strategy our opponent could use.
If we consider each of the 15 cards to be unique there are 105 possible starting hands. There are 15 * 14 ways of being dealt two cards (15 possibilities for the first card, 14 for the second), but this double-counts each starting hand because it counts ace-king as a different hand from king-ace. We can divide by two to get rid of the duplicates, which leads to 15*14/2 = 105.
Those 105 combinations are comprised of 15 distinct hands: three combinations each of the “pairs” (KK) and nine each of the unpaired hands (KQ). The number of combinations will become important as we try to create a balanced strategy. This is a chart that represents all our starting hands:
There are five possible actions we can take on our first turn: take three, take two, take one, steal two, or exchange cards. We don’t necessarily have to use all of those actions. In fact, there’s only one action it’s obvious we should use at all: take three. Taking three coins is one of the most powerful moves because it moves us toward a coup very quickly and it can’t be blocked. Let’s say we take three with all of our Kx hands. As we’ll see later, it may be desirable to use some of our Kx hands for other purposes, but as a starting point taking three whenever we have a king makes a lot of sense. This also allows us to bluff a king fairly often and we’ll have to choose which hands to use to bluff.
Can we get away with taking three with every hand? Of the 105 hand combinations, 39 contain a king and 66 do not. So if we tried to take three every time, our opponent could always challenge us and we’d be caught bluffing too often. Our legitimate Kx hands can carry some bluffs, but they’re not numerous enough to let us bluff with all our non-king hands. Exactly how often we should bluff a king is an interesting question. One of our goals is to be resistant to whatever counter-strategy our opponent chooses, so we have to bluff often enough that our opponent can’t exploit us either by challenging or by passing. We could aim for a 50/50 split of Kx hands and bluffs, but that allows our opponent to effectively make the game into a coin flip by always challenging. Since going first gives us an advantage, we shouldn’t allow him to make the game into a coin flip. We should bluff less often, but perhaps not by much. As a starting point, aiming for a 60/40 split of kings and bluffs seems reasonable. That means we’re looking for 26 combos to use as king bluffs. Let’s leave aside the question of which hands to use as king bluffs for now and consider which other moves we might want to use.
The most powerful action we can take — even more powerful than taking three — is stealing two from our opponent. If we can steal two every turn, we’ll accumulate coins rapidly while stunting the opponent’s development. This strategy is so powerful, in fact, that if our opponent wants to have any chance of winning the game he can’t let us do it. He has to either challenge whether we have a jack, or block with a jack or ten (whether or not he actually has it). Fortunately for him, he will be able to legitimately block our steal attempt rather often. The game designer clearly recognized the power of this action, because it’s the only one that can be blocked by two different cards. If we want to go for a steal, blockers become very important. A blocker is a card in our hand that figures in our opponent’s relevant hands. If we’re going for a steal, we have to be cognizant of how many jacks and/or tens we hold. If we have no jacks or tens, the chance that our opponent has a jack or ten is 73.1%. If we have one jack or ten, the chance is 64.1%. If we have two jacks or tens, the chance is 46.2%. So we can immediately see that stealing without two blockers isn’t going to work out very well. We can expect the opponent to block every time, and he’ll actually have the jack/ten often enough that there’s not much we can do about it. If we challenge, we’ll lose more often than not, but if we don’t challenge we’ll just have wasted our turn.
Stealing with two blockers is more promising. If our opponent blocks every time (whether he has it or not) we can challenge and win a little more than half the time. Then again, if he knows we’re auto-challenging, there’s no point in him trying to bluff. His best strategy in that case would be to always block if he has the jack or ten (knowing that we’ll challenge and lose), but if he doesn’t have a J/T, challenge whether we do. If we try to steal with all our double-blocker hands (JJ, JT, TT) we’ll show up with a jack 80% of the time (3 combos of JJ, 9 combos of JT, and 3 combos of TT adds up to 12/15 total combos that have a jack). He only wins 20% of the time, which doesn’t sound great, but is actually pretty good if the alternative is a sure loss. In fact, this set of strategies works out better for him than for us. When he has a jack or ten, he gets us every time; when he doesn’t, he gets us 20% of the time. Since he has a jack or ten 45.1% of the time, the overall chance that he gets one of our cards is .451(1) + .549(.2) = .56. He eliminates one of our cards 56% of the time, whereas we get one of his cards only 44%. That’s not the end of the story though, because if we know he’s not bluffing, we no longer have to challenge. When he blocks with a jack or ten, we can just let it pass. We’ve wasted our first turn, but we’re still very much in the game.
In a situation like this, where each side can switch between exploitative strategies, it’s often necessary to resort to a mixed strategy wherein you perform different actions with different frequencies. In other words, he should challenge sometimes and block sometimes; and when he blocks, we should challenge sometimes and allow it sometimes. Without trying to figure out the optimal frequencies for all of those actions, a good starting point for us would be to challenge with blockers: if he blocks with a jack, we challenge if we have JJ or JT, but not if we have TT. Overall, it seems reasonable to expect we can win around 50% of the time by stealing with JJ, JT, TT. That’s not great, because we expect to win more often than not when going first, but we should expect most of our advantage to be concentrated in our Kx hands because the king is by far the strongest card. If we can break even with our other cards, we can count on the king hands to pull up our overall win percentage. Let’s leave aside JJ, JT, and TT for now, content that we’ve found a reasonable way to play some of our non-king hands.
Recall that of the 105 total combinations, 39 include a king and 66 don’t. JJ, JT, and TT comprise 15 combos, so with those out we’re down to 51 combos. That’s still too many to use them all to bluff kings, so we’re going to have to adopt at least one more action into our strategy. The remaining actions are take one, take two, and exchange cards.
As a first attempt, I’m not going to include exchanging cards in my strategy. It may turn out to be part of an optimal strategy, but it’s very hard to say without more empirical evidence. Ideally I’d like to have access to results of games between top players. In effect, exchanging cards allows us to trade a bad starting hand going first for a better starting hand going second. Without more experience it’s hard to say how big of an advantage going first is, which the worst hands are, or how much separates them from the best hands. It’s possible we should be exchanging sometimes, but I won’t be able to figure it out without more familiarity with the game.
At first glance take-two is more attractive than take-one because, well, two is more than one. But taking two is extremely problematic, especially if we’re using all our Kx hands to take three. If that’s the case, we’ll never have a king blocker when we take two. The chance of our opponent having a particular card if we have no blockers is 42.3%. If he blocks with all his kings, and bluffs with a frequency such that he has a king twice for every one time he bluffs (a conservative strategy), he’ll still block us 63.5% of the time without allowing us to challenge profitably. In that case, when we try to take two, we’ll actually get two less than half the time, which averages out to less than one. It seems that if we want to create a take-two range that’s more effective than take-one, we’ll have to borrow some Kx hands from our take-three range. This does have a certain, sneaky appeal. For example, it could work out nicely to take two with KK, hoping to induce a king block, which we’ll then challenge, knowing there’s only one king left in the deck. But it seems dubious to cannibalize Kx hands from our take-three range, voluntarily taking two — and allowing our opponent the opportunity to block — when we could take three without the possibility of being blocked. Why take a sub-optimal action with some of our best starting hands, just to buff part of our strategy that isn’t that great anyway? As a starting point, I think it makes the most sense to never take two, but it’s possible that we could create an effective and deceptive take-two strategy that includes some Kx hands.
That leaves take-one. Tactically this works out well, because the most attractive hands to take one with are Ax hands, as taking one puts us at exactly the three we need for an assassination; and Qx hands, because they block our opponent’s queens, which can block our assassination. Ax and Qx are exactly the unassigned hands we have left. The best hands to take one with are AQ and QQ. AQ has an ace and a queen blocker, while QQ has two queen blockers.
At this point we have 39 combos still unassigned. We want to use 26 of those to bluff kings, and the other 13 will be used to take one. Since we generally want an ace in our hand when we take one, the best hands to use as king bluffs (take three) seem to be QJ and QT. We’ve arrived at these hands by process of elimination: they’re not especially useful for anything else. As in poker, it makes sense to use our weakest hands as bluffs. That gives us 18 combos of king bluffs, so we still need eight more. Of the remaining Ax hands, I don’t see any factors that make them better suited to take one or take three, so we may as well randomize our strategy to make it harder for our opponent to put us on a hand. The remaining hands are AA, AJ, and AT, which make up 26 combos. We can achieve our desired frequency by using those hands to take three if we hold the ace of spades, and to take one otherwise. Two of the three AA combos contain the ace of spades, while three of the nine combos of AJ and AT do (2 + 3 + 3 = 8). It doesn’t have to be the ace of spades, you can pick any ace, just make sure the card you’re looking for is actually in the deck, because you’re only playing with three of the four aces.
This then is our complete strategy:
Red = take three Blue = steal two Yellow = take one Purple = take three if you have the ace of spades; otherwise, take one
I’m sure there are problems with this strategy, but it’s a reasonable starting point. A good next step would be to put ourselves in our opponent’s shoes and think about how we would respond to this strategy. As we learn more about the game it may turn out that we want to include taking two or exchanging cards in our first turn strategy. It may also be a good idea to be more deceptive. This is a fairly straightforward strategy (that is, we usually have the cards we’re representing), but there is definitely some value in mis-representing your hand early on in order to spring a surprise later. As we go beyond turn one, the game becomes more complex because all of the previous actions provide information about what each player holds.
Bonus question: Suppose you’re playing Coup and you have made it through this whole post and believe it to be more-or-less sound. The same is true of your opponent. What strategy should you employ going first?