Game Theory in the anime Death Note
Death Note is centered on Light, a high school boy who possesses a mysterious notebook that can kill anyone whose name is written in it, and L, the best detective in the world who is chasing the culprit behind all the mysterious deaths happening in the world.
One of the most perplexing scenes in the anime is when L reveals his identity to Light (Episode 9, 14:40); L was anonymous throughout the story till now and because Light didn’t know L’s identity, he couldn’t kill him. Light wanted to kill L, because L was trying to find out who the murderer really was. L had a suspicion that Light was the murderer, so to confirm his suspicion, he revealed his identity to Light which was a really big gamble. L had deducted by now that the murderer needed a name and a face to kill someone, but he wasn’t hundred percent sure of this fact. Nevertheless, he decided to take a gamble and reveal himself to Light using a fake name. Using the payoff matrix below, we can figure out if this was actually the best strategy or not.
1 -> Winning (Winning for L means catching Light and winning for Light means killing L and thus not getting caught)
0 -> Losing (Losing for L means getting killed and losing for Light means getting caught). We can assume safely from L’s personality that if he is alive, he is always going to search for the murderer.
L
Reveal Not Reveal
Light Kills 0, 1 1, 0
Not Kills 0.5, 0.5 0.1, 0.1
(Not Kills, Not Reveal) / (0.1, 0.1): Before L revealed himself, there was an equal probability of L eventually winning by finding out who Light was and Light eventually finding out who L was and killing him. The probability of either happening was really low (0.1), as both characters were basically firing shots in the dark; they had no solid leads.
(Kills, Not Reveal) / (1, 0): If Light discovered the identity of L and killed L, he would have won.
Both character were at the (Not Kills, Not Reveal) case currently in the story i.e. either of the characters could win later in the story. The story was at a deadlock at this point, since L’s suspicion of the murderer being Light was disproved just in the previous episode, but L still had a lingering doubt and wanted to make sure his suspicions about Light were false. The flow of the plot would totally change in the following ways if L was to reveal his face (but not his name) to Light:
(Kills, Reveal) / (0, 1): If L revealed himself, and if Light tried to kill him, there would be two possibilities based on the accuracy of L’s assumption that the murderer needed a name and a face to kill someone:
- L’s assumption was false, and thus Light would have killed L. Since Light was the only person who knew L’s identity, he would have been arrested and would have lost.
- L’s assumption was true, and thus Light would have killed the person whose name L was using (a famous pop idol: since it was a famous pop idol, Light already knew his face and thus the death note would have worked to kill the idol) instead of L. This would have alerted L, and Light would have been caught immediately.
In either of the two cases, Light would have been caught, and L would have won the game.
(Not Kills, Reveal) / (0.5, 0.5): If Light didn’t try to kill L, both L and Light would have gotten closer to a more definite conclusion, as now they would have a much better platform to investigate each other. They would no longer be at a stalemate as before; each would have a concrete lead to focus on.
In either of the Reveal cases, L would have gotten closer to winning and would have an equal or higher chance of winning than Light. Thus, L should definitely reveal himself, and in response to this, Light should not kill L. The Nash equilibrium is (Not Kills, Reveal) here. This is exactly what happened in the story!