Netflix “Squid Game” and Game Theory
Spoiler alert! Don’t read if you want to watch the show. The article also contains spoilers in the last section “The Experience”, so I did not read it as well.
The new Korean show “Squid Game” on Netflix is a game where a mysterious host invites people from all over South Korea who are desperate for money and makes them play games from their childhood. The games are rather simple and easy, but the cost of losing is death. Each participant’s life is worth 100 million Korean Won, which is about 100k in US Dollars. There are a total of 456 participants. I have not watched the entire show, so I am not sure about the logistics of winning: whether multiple people can win and they split the prize money or if they compete until there is only one man standing. If the latter is the case, the winner gets a total of 45.5 million dollars, which is what the dead 455 participants’ lives would be worth. The writer and director of this series Hwang Dong-hyuk points out that we “mostly look at how winners struggle, but in ‘Squid Game,’ we focus on the losers. Because without losers, will there ever really be winners?” He is referring to the concept of zero-sum. In this game, one man’s death is another man’s chance at life with prize money.
These players do not have a choice but to join the game, even if the cost of losing is death. The host of this game invited only those who are in large debts and have no way to come back at life unless they play the game and win the prize money. Without joining the game, the participants have no way to pay off their debt, and life would just be hell on earth, which is not any better than death. Therefore, the participants deemed playing the game is the dominant strategy.
The participants are not given any hints of what games they are playing. Instead, they find out at the start of each game. Some games are played individually, but others require the participants to group up or partner up. Who to team up with largely depends on whether the next game has an advantage for men or women. Some games were more frequently played by girls, so the women participants have a better understanding of them, and vice versa for games more frequently played by boys. The two matrixes below show whether picking men or women is the better choice.
A game where having more men on team has a better chance at winning.
| Team 2 | Men | Women | |
| Team 1 | |||
| Men | 0,0 | +1,-1 | |
| Women | -1,+1 | 0,0 | |
When two teams are competing, and both teams largely consist of men, neither team has an advantage over the other; same goes for when both teams mostly consist of women. However, whenever one team has more men and the other doesn’t, the team with more men has an advantage to win by +1, which puts the other at an immediate disadvantage by -1.
Now, a game where having more women on team has a better chance at winning.
| Team 2 | Men | Women | |
| Team 1 | |||
| Men | 0,0 | -1,+1 | |
| Women | +1, -1 | 0,0 | |
Because the participants have no idea what the next game is, they cannot decide which of the two matrixes they are dealing with, so they team up with a good balance of both men and women. However, one of the participants is an insider who knows that the next game is tug of war. This participant has a clear advantage over the others because he knows his dominant strategy is to recruit strong men.
Another game the participants play is guessing whether the opponent has an even or odd number of beads in his hand. If he guesses right, he gets the number of beads he bet, and if he guesses wrong, he loses the number of beads his opponent had in his hand.
| Player 2 bets b | Has Odd | Has Even | |
| Player 1 bets a | |||
| Guesses Odd | +a, -a | -b, +b | |
| Guesses Even | -b, +b | +a, -a | |
In this matrix, one has no clear dominant strategy, so we need to figure out the mixed strategy equilibrium. Let’s say the probability of guessing odd is p, and the probability of guessing even is 1-p. The probability of has odd is q, and the probability of has even is 1-q.
The payoff for player 2 to play “has odd” is -a*p + b*(1-p).
The payoff for player 2 to play “has even” is b*p – a*(1-p).
To achieve mixed strategy equilibrium, these payoffs should be equal.
-a*p + b*(1-p) = b*p – a*(1-p)
a + b = 2p(a+b)
p = 1/2
The payoff for player 1 to play “guess odd” is a*q -b*(1-q).
The payoff for player 1 to play “guess even” is -b*q + a*(1-q).
a*q -b*(1-q) = -b*q + a*(1-q)
a + b = 2q(a+b)
q = 1/2
The mixed strategy equilibrium for playing this game is p=1/2, q=1/2.
The show “Squid Game” not only uses the concept of zero-sum, but also the games the participants play can be analyzed by Nash Equilibrium and dominant strategy.
Link: https://cnnphilippines.com/life/entertainment/television/2021/9/20/squid-game-series.html
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