Network Interactions in Fate/Zero
One of the most popular anime/manga in last decade (according to the list that the community recognizes as the standard for rankings–myanimelist.net) is Fate/Stay Night and the prequels and sequels in the same universe.
In the story, seven people, known as Masters, complete for a magical artifact, the Holy Grail, that grants the victor any wish. These seven Masters summon Heoric Spirits from history to fight on their behalf and so on and so on.
One episode in particular caught my attention because it provided such an interesting dilemma. In the episode, one of the seven breaks the rules of the Holy Grail War and an addendum is made to the rules as follows: All Masters are to target and kill Caster, the one accused of breaking the rules, and the one to get the kill will receive an additional command spell–a valuable posession that would place that would give that person a formidable advatage for the remainder of the war, as all the other
Let us write out the payoffs that come from each strategy that the players can take.
1. Go after caster :
+ (1 command spell) * (1/number of people also pursuing Caster)
– (Risk of getting killed) * (number of people also pursuing Caster)
2. Remain hidden : 0
Another option that one character in the show presents is to go after the people going after the Caster. Thus, we can add in a third option.
3. Go after people going after Caster :
+ (Chance of killing someone) * (number of people also pursuing Caster)
– (Risk of getting killed) * (number of people also pursuing people pursing people pursing Caster)
It’s hard to quantize the risk that one is taking by going on the offensive and also hard to quantize exactly how potentially useful the extra command spell can be. Without actual numbers, it’s hard to see if this actually has a nash equillibrium or not for the number of people that will go on the offensive for option 1 or 3.
Of course, it all depends on the individual and the skills they have / relationships with the other characters, but it does provide an interesting problem. Because the risks / rewards involved scale directly to the number of people adopting each strategy, there is no dominant strategy. If everyone stayed hidden, then Option 1 would increase greatly in value beyond the other two options. At that point, Option 3 would also increase in value, and so on, until equillibrium was found. I guess the only way to figure out is to continue watching the show through finals week instead of studying.