The seventh season finale of Game of Thrones, the famous HBO adaptation of George R.R. Martin’s fantasy novels, was released on August 27, 2017, promising an epic war between The Living and The Dead in season 8. The game theory was pioneered by mathematician John Nash, portrayed in the cinema by Russel Crowe in ‘A Beautiful Mind’, and is used by economists but also by political scientists or military tacticians. What would be the strategy of Game of Thrones’ characters if they were aware of this theory?
We discuss a Nash equilibrium applied to a case of collaboration/betrayal between Cersei Lannister and Deanerys Targaryen in the fight against the Night King taking into account that Jon Snow bent the knee to House Targaryen. We consider a game with simultaneous moves and a competitive strategy where the players will choose the course of action that benefits them the most, no matter what the other ones decide to do.
First let us evaluate the military strength. If we assume that each Kingdom of the Seven Kingdoms has the same weight, here is how we might grant each player at the moment of the parley in King’s Landing. We do not consider the Riverlands (Tully, Frey).
We suppose that Deanerys lost the Reach and Dorne in the sea battle against the Greyjoys and Cersei lost the Westlands in the battle between Jaime and the dragons.
The players also have forces from Essos. We granted 2 points to the Dothraki as they seem to largely outnumber the other armies 
The social interactions can be described into a grid that shows their choices and their potential consequences.
Case A (They both collaborate)
Deanerys has 8 armies and Cersei has 4.
Case B (Deanerys collaborates – Cersei is treacherous)
Cersei being in a weak position will seek help in Essos by hiring the Golden Company and might want to take down one of Deanerys’ forces (e.g. the dragons could be very harmful if they would be controlled by the Knight King). As the Vale did not really pledge allegiance to House Targaryan she might try to have them by her side. She would then gain 2 armies and reduce 2 by Deanerys.
Case C (Deanerys is treacherous – Cersei collaborates)
Deanerys could ask Jaime to help either get one of the nearest Kingdom back, e.g. the Stormlands or the Reach to ensure her strength in the North, or conquer Dorne to surround Cersei.
Case D (They both betrayed each other)
Same moves as in B for Cersei and same moves as in C for Deanerys. Deanerys will lose one army but gain another one. Cersei will eventually lose one army.
The dominant situation for Deanerys is case C, the dominant situation for Cersei is case B. Cersei and Deanerys’ goal is to dominate Westeros. Jon Snow’s goal is to gather the biggest army to defeat the White Walkers. We all know that Cersei could (or will) stab Deanerys in the back. If he then wants to secure Deanerys victory after the war against The Dead, he should advise her to betray Cersei and choose case D.
This seems to lead to a prisoner’s dilemma . Deanerys and Cersei should play again after each battle against the Night King. Experiments  have proven that the best way to win an iterated prisoner’s dilemma is the tit-for-tat strategy . Jon should advise Deanerys to first collaborate then replicate Cersei’s behaviour the next moves.
But what would be your strategy? How would you play?