Graphic by Ivan Vuong

The card game BS is known for its notoriously risky decisions and rage-inducing gameplay. The objective is to get rid of all of your cards through deceptive means — the person who does this first is the winner. 

The deck is first divided equally between a group of players, and the person who is dealt the ace of spades begins the game by placing the ace in the center. Gameplay proceeds with the cards being placed face down in sequential order — the next player places a two card face down on the pile, the next puts down a three and so on. Players must announce the value of the card that they are placing down, as well as how many of that card they are playing. 

The game continues to cycle through the cards sequentially, restarting the sequence if necessary, but there is a catch — the players can lie about the card they have placed onto the pile, despite whether or not it’s in their hand, and they can be called out for it. If another person suspects that the card a player put down is not what they have stated it to be, they can call their bluff by saying “BS” during that round. If the caller is correct, the player who lied must pick up all of the cards that have been placed in the pile so far and add them to their hand. If the caller is incorrect, they must pick up the entire pile instead. For a player with their mind set on winning, a well-known game theory concept can assist in helping them secure a victory: the prisoner’s dilemma.

The prisoner’s dilemma originates from a thought experiment developed in 1950 by mathematicians Merrill Flood and Melvin Dresher at RAND Corporation, a United States think tank. The prisoner’s dilemma represents a scenario in which two convicts have a choice to either “confess” or “deny.” If both prisoners were to confess to their crimes, they would both serve five years. However, if one confessed and one denied, the latter would have to serve 20 years, and the former would serve zero. If both prisoners were to deny, they would both serve one year.

In the case of the prisoner’s dilemma, where the prisoners are in an isolated situation and are not aware of the other’s choice, the optimal decision is to confess. This is an example of Nash equilibrium: a solution to a game in which neither player can benefit from changing their strategy, assuming the other player’s strategy is fixed. If one prisoner were to confess, the other prisoner’s best choice is to also confess. Both serve five years, instead of one serving zero and the other serving 20. Therefore, when one is unaware of their fellow prisoner’s choice, the optimal choice is to confess. On the other hand, if the conspirators had had time to discuss what their decision would be, and were aware of the sentences tied to their choices, their best decision as a pair would be to both deny and serve one year each. This, however, would not be an example of Nash equilibrium, as one prisoner could benefit from changing their strategy, assuming that the other prisoner’s strategy remains fixed. 

Something interesting to note about the Nash equilibrium in the prisoner’s dilemma is that it is not Pareto-efficient. For a given set of choices to be considered Pareto-efficient, all the alternatives in which one player benefits must make the other player worse off. In the case of the prisoner’s dilemma, the Nash equilibrium does not constitute a state of Pareto efficiency, as there exists a strategy in which both prisoners are strictly better off — namely when both deny.

Much like the prisoner’s dilemma, the game of BS is based on lies and truths, and what incentives lead the player to choose one or the other.

To demonstrate this scenario, a payoff matrix — a diagram that describes a player’s possible choices in a game  — representing a round of BS can be created. Let D be the number of cards in the pile at the time, and N be the number of cards put down by the player. “Points” in the payoff matrix correspond to the change in the number of cards in the respective players’ hands through the turn based on the choices they make. For example, in the first Lie/Call BS quadrant, one can see that, if the player chooses to lie and is caught by their opponent who calls BS on the play, they must pick up +D cards while their opponent picks up no cards. –N + N signifies their initial action of putting down N cards, but then having to pick that same number of cards, N, up again after being caught lying, as well as the rest of the pile, +D.

If the player tells the truth rather than a lie, they are guaranteed to get rid of the cards they placed down, in addition to forcing the other player to pick up D + N cards if they incorrectly call “BS.” If the player were to lie, and if another player were to correctly call “BS,” the initial player would have to pick up the D + N cards, which would bring them further away from the goal of getting rid of all their cards.

In the larger context of the entire game, the best strategy that a player can implement is to tell the truth unless they are absolutely forced to do otherwise. According to a computer simulation of the game with over 120,000 runs, this particular method of play has a 51% win rate — the greatest out of any of the other strategies one may employ. Furthermore, calling “BS” is a rather risky play, therefore, players who refrained from doing so had a similarly high win rate. Naturally, this is due to the fact that calling “BS” incorrectly results in the significant penalty of picking up the entire pile of cards that have been played.

This highlights a great similarity between the prisoner’s dilemma and BS: the player’s optimal strategy, when unaware of the strategy of their opponent, is to tell the truth. There are many principles of the prisoner’s dilemma that can be applied to BS, and learning how to play card games through the lens of game theory can be instrumental in helping players achieve the success they desire.

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