Liar Game, Minority Rule Game Analyzed
One great anime (if there are those looking for recommendations) that features lots of game theory is Liar Game. The premise of the second round of the competition in this game is the minority rule game. This is a variation of El Farol Bar problem, further simplifying the decision so that the players want an unequal outcome in which they are in the minority. In the anime players were given the opportunity to vote on questions yes or no. If they answered in the majority, they were eliminated from the game. Thus the protagonist tries to find a way to “win” at this game. Interestingly, another Cornell blog post from 2018 talked about the phenomenon I wished to comment on: “The true genius of this round is at the end when there are only 4 contestants left, and the seconds lead of the manga, Akiyama displays to everyone that he is voting yes regardless of the question and does it immediately after the question is asked.” The blog post then concludes “Liar Game puts a practical view on Game Theory for people in desperate situations and is still able to show how the goodness of humanity sometimes is able to trump the deceitfulness present in all of us.” Indeed, this is one of the highlights of this anime, and one of the reasons it is enjoyable to watch. However I want to look through another lens at this situation.
When the contestant, Akiyama, says that he is voting yes, he breaks the game. The Minority Rule game operates on independent same time choices made by the players in the problem. By declaring his intention Akiyama takes his odds (in a four person challenge – which would also not be the format of the Minority Rule as it leads to obscure conclusions) from net 0 (-1 * .5 + 1 * .5) to net positive as he is practically guaranteed to not lose. As the blog post from a few years ago aptly explains : “Since there are only 3 contestants left to vote, the next player to vote must vote No if he or she wants to be the minority, however if all 3 remaining contestants vote No in order to be the minority automatically Akiyama will be the minority and win the game.” Thus this game can be broken, drastically upping your odds if you violate independent decisions made. It is also worthy to note that though this game falls through when someone pulls a move much like Akiyama did, it can be easily recovered. Another contestant can declare the opposing intention (saying no) securing themself equal odds to Akiyama to winning. A different contestant can then declare their vote arbitrarily at no harm to themself, as it is as if they are doing what Akiyama did but with 2 fewer people. If there is an even number of people it leads to a stalemate. If it is an odd number of people, it will eventually depend on the last person who will vote either way arbitrarily as they are guaranteed a loss. (Or perhaps they are in the best spot as they can be swayed with other incentives as they are guaranteed a loss). In either way, after two people (in this case Akiyama and one other) declare their intentions we can go back to the Minority Rules problem operating on independent decisions again. In either way, this episode may not be the goodness in humanity and spite of rationality for cooperation it was portrayed as if one other person threw down their opposing vote in the same way Akiyama did. Regardless, it was an entertaining episode and I highly encourage this show!
Source: 🙂