How Risk Aversion Promotes Lying in Coup

Coup is one of my favorite card games because it is relatively simple yet incredibly devious. And anyone who has played a game with me knows I love games with a certain amount of scheming. What particularly attracts me to Coup (and its not-so-distant cousin, The Resistance) is the idea of hiding your identity and motives.

Awhile back I wrote a review over at thegeekticket.com. I wanted to revisit Coup with a focus on game theory, but first, a quick recap on how Coup works.

coup-header

The Game

You can read my review – if you’re not familiar with the game, I highly recommend that – but here is the basic description:

The plot behind Coup is simple. As the Resistance is doing its thing, tearing apart (or failing to tear apart) the government, the people in power are playing their own game: Survival. Each player has influence over two members of the ruling class, and the goal is to wipe out everyone else’s influence — basically by staging a coup d’etat. This is done through, as the tagline states, “secret identities, deduction, deception.”

In the beginning, you’re given at random two influence cards representing members of the royal court and an income of two coins. On your turn, you can either take your standard income of one coin, take two coins from foreign aid, or pay seven coins to initiate a coup against another player. Complicating matters is that each member of the royal court has a special position: Duke, Assassin, Ambassador, Captain, or Contessa. The positions have special actions or counteractions. Example, a Duke can levy a tax and take three coins from the bank or block another player’s foreign aid.

Here’s where the game gets fun: Your influence cards are a secret, so you can totally lie about which members of the court you represent. On my turn, for example, I can say my Duke is blocking another player’s foreign aid, but the cards in my hand are actually a Captain and a Contessa. The other players have to decide whether I’m lying or not.

Liar, Liar

So let’s say I’m Player 1 and I have made the claim above: My Duke is going to block Player 2’s foreign aid. Player 2 has to make a decision whether to call my bluff or not. The payoff matrix is below. Player 2’s options represent the rows and my actions are the columns.

Player 2’s Options
Has Card Doesn’t Have Card
Don’t Call Bluff Opponent Gains Opponent Gains
Call Bluff Lose a Card Opponent Loses a Card

Or to give you a better idea of the outcomes…

Player 2’s Options, Outcomes Ranked
Has Card Doesn’t Have Card
Don’t Call Bluff Not Good Not Good
Call Bluff Worst Best

Player 2’s best strategy is to not call my bluff because it won’t result in an immediate loss. It’s basic risk aversion. If he/she doesn’t call my bluff, the worst thing that happens is I prevent him/her from taking in income. In the long run, that’s bad. But in the short run, nobody loses a card. And in a game where you only have two cards, losing one is a big deal.

Let’s look at it from my point of view.

My Options, Outcomes Ranked
Doesn’t Call Bluff Calls Bluff
Have Card Gain Best
Don’t Have Card Gain Worst

I have a dominant strategy to only play cards that I actually hold. However, knowing that Player 2 has the most to lose from calling my bluff, my payoff matrix looks more like this.

My Options, Outcomes Ranked
Doesn’t Call Bluff
Have Card Gain
Don’t Have Card Gain

In other words, lying is a great strategy.

This doesn’t mean you can spend the entire game lying. As the game progresses and the other players start forming theories about which cards you hold, the believability of your bluffs decreases, like in the plot below.

coup-bluffing-fig1

Your best bet in Coup is to take advantage of a couple of lies at the beginning. Certain cards, like the Ambassador, allow you to exchange cards with the deck, altering players’ perceptions of the cards you hold. But by the end of the game, it becomes a little more obvious who is bluffing.