# Identify this game with 4 players, each have 2 choices

The game is played with any deck of playing cards, or just using hand gestures without any items at all.

How I've seen it usually played - There are 4 players. Every round, each player will choose between a red card or black card. All players then reveal the card they chose and points are scored based on what cards each team chose:

No. of Red Cards No. of Black Cards Reds Score Blacks Score
4 0 -4
3 1 +1 -3
2 2 +2 -2
1 3 +3 -1
0 4 +2

Example: if 3 players play a red card, 1 player plays a black card - each player that played red gets 1 point, the player that played black loses 3 points. If all players play red, all players lose 4 points; if all players play black, all players get 2 points.

(*Summary - the only way to get points is if everyone plays black, or you play red and get a point from every black; but if everyone plays red, everyone loses points. This will not be told to players, but they can deduce it easily from the table.)

The goal is to get the highest score from adding all 4 players' scores together.

I remember playing this game in school, clubs, or camps. It was mainly used to teach players to look at the big picture/goal and trust each other, instead of being greedy/competing with each other. Before teaching what to do and the scoring system, the goal of the game is explained at the very beginning in hopes of players later forgetting to work together and instead compete for the highest score by themselves. Understandably after playing the first time, all players will know to just always play black as anyone playing red will not increase the overall total score, as such this game is only meant to be played once just to teach this lesson.

(I am not quite sure if this originally used playing cards, so this may not be the correct stackexchange to ask this. So please let me know where I can ask this.)

• This is the sort of Prisoner's Dilemma game that is studied by economists. I think it is Stag Hunt (en.wikipedia.org/wiki/Stag_hunt) but I am not sure. Commented Aug 9, 2021 at 6:34
• Thanks @Flounderer, I managed to Google it on my own and posted an answer, also thanks for the different version Stag Hunt Commented Aug 9, 2021 at 9:06
• @Flounderer The Stag Hunt and Prisoner's Dilemma are different games. This is a really important distinction. Notably, the Stag Hunt has a global utility maximizing Nash Equilibrium, where the Prisoner's Dilemma does not. They are both symmetric 2x2 game theory games; the Prisoner's Dilemma is just one game in that category (albeit the most famous) and is not the name for the category itself.
– Zags
Commented Aug 9, 2021 at 18:12
• @Zags thanks for clarifying it the game theory. I'm not very familiar with the actual terms. Commented Aug 10, 2021 at 0:44

This is what game theory would term a 4-player symmetric game. Based on the payoff matrix, it seems most similar to being a 4-player version of Chicken (A.K.A. Hawk-Dove). The key to Chicken is that pushing a little can gain you some advantage but if everyone tries to gain an advantage, it's catastrophic.

One reduction of your payoff matrix to two players is the following, which is a textbook game of Chicken:

Black Red
Black +2, +2 -1, +3
Red +3, -1 -4 , -4

It's not a Prisoner's Dilemma because the Nash Equilibrium is not for all players to defect (i.e. play red). If everyone plays red, any player would have incentive to play black.

It's also not a Stag Hunt because all players cooperating (i.e. playing black) isn't a Nash Equilibrium. If all players play black, any single player could do better by playing red.

It's worth mentioning that the equilibrium analysis of all of these games changes when it becomes a repeated game (you play more than once with the same players). For example, while the Nash Equilibrium for the Prisoner's Dilemma is for both players to defect, conditional cooperation can be a dominant strategy in the repeated game. Assuming rational actors, you would likely see a similar effect in the game you describe, where conditional cooperation (playing black as long as everyone else is playing black) would emerge as a stable strategy in the repeated game.

As a final note, your statement "The goal is to get the highest score from adding all 4 players' scores together" is quite at odds with tracking individual scores per player. The game is only interesting if each player is competing to get the highest individual score.

• In the version I played, it was meant to teach a lesson about players to work together and achieve the common goal, but were purposely given the option to gain more for themselves while not increasing the 4-team-total, not about making the game interesting/competitive. It's also not meant to be played with the same people more than once as the strategy is obvious - everyone always choose black. Commented Aug 10, 2021 at 0:54
• the 2 player reduction also doesn't seem to match the score table I gave. In this version, as long as there's at least 1 red and 1 black, the overall total gain will be 0. Is there a version of a symmetric game more closely resembling mine, or is mine just an unofficial modified version of one of the game examples provided? Commented Aug 10, 2021 at 1:00
• @NickLeeJy Playing it as a repeated game does add some analysis; I've added a paragraph on repeated games.
– Zags
Commented Aug 10, 2021 at 13:50
• @NickLeeJy My two-player reduction is literally the first, fourth, and fifth rows of your score chart. I took those rows rather than rows 1, 3, and 5 because this isn't a zero-sum game; it's much more relevant that there is incentive for any one player to defect (i.e. playing red when all players play black increases your score) than it is that defection causes no net-change in total score.
– Zags
Commented Aug 10, 2021 at 13:59
• @NickLeeJy Any 2-player game with the property W > T > L > X is a game of chicken. The 2-player reduction I presented here has the values +3, +2, -1, -4. The values +2, +1, -2, -4 would get you the property you care about (when players play different numbers, there is no net-change in total score), but that is not a strict subset of your 4-player payout matrix, while +3, +2, -1, -4 is.
– Zags
Commented Aug 10, 2021 at 14:00

After Googling through various game theory rabbit holes, I've found one type of this game: The Prisoner's Dilemma. (and also thanks Flounderer for commenting a different version of the game - Stag Hunt)

General summary of a symmetric game (this type of game)

• If all players cooperate, everyone can win, gain something, or lose less.
• If some try to get ahead on their own or betray others, they will gain more and others will lose more.
• If everyone betrays each other, everyone loses more.

(edited the correct term for the game theory from Zags's comment)

• It's not a Prisoner's Dilemma. See my answer for an explanation
– Zags
Commented Aug 9, 2021 at 18:29