Thursday, May 15, 2014

Varys' Riddle - the game theory of game of thrones

In George RR Martin’s book of political intrigue one character poses the following riddle:

In a room sit three great men, a king, a priest, and a rich man with his gold. Between them stands a sellsword, a little man of common birth and no great mind. Each of the great ones bids him slay the other two. 'Do it,' says the king, 'for I am your lawful ruler.' 'Do it,' says the priest, 'for I command you in the names of the gods.' 'Do it,' says the rich man, 'and all this gold shall be yours.' So tell me- who lives and who dies?

Above is a link to the scene from the HBO tv show.

Game Theory, it turns out, has an answer. Consider what some call the repeated game of daily life. In each period, every member of society is randomly matched with someone else whereupon they play a Prisoner’s Dilemma. In each period, each player has two choices, either to be violent and slay the other, or to be peaceful, and cooperate. It is individually rational to be violent in any given period, but mutual cooperation is better for both than mutual violence.
In analyses of two player versions of this game, we see a wide variety of strategies can maintain mutual cooperation. For example grim-trigger or famously tit-for-tat. The common take away, people play nice in any given round for fear of being punished in the future.

In the large N versions, the problem becomes more difficult because you interact with somebody new each period. Thus in order to maintain cooperation you need some kind of social order. If somebody defects and decides to slay, we need to rely on somebody else to carry out the punishment.

Thus the answer to Varys' riddle depends on which equilibrium we are in, or perhaps if we are out of equilibrium, which sub-game of the current equilibrium.
Calvert (1992) finds an equilibrium of this game where players maintain equilibrium by choosing one member (a king) and using the king to collect information on who has defected, and thus giving the king power to decide who should be killed. Like in Hobbes’ Leviathan, we give the king power because he maintains order by punishing anyone who defects, thereby keeping everybody in line. Assuming this equilibrium is in play, the sell sword will listen to the king or risk being punished by whoever he encounters next.

Milgrom, North and Weingast (1990), find that merchants actually can maintain order without a government or king. By studying the “law merchants” of the Champagne trade fairs of medieval Europe (if you ever visit Paris, take a train to the town of Provins, today it’s basically a medieval times ren faire for French tourists, but you can still find the animatronic law merchants). Here, contracts were enforced by judges who kept a board that recorded the names of anyone who breaks a contract. In equilibrium, all merchants had to defect against anyone who ever violated a contract, or have their name recoded on the board as well (making sure the equilibrium is sub-game perfect, the judges themselves also would be punished if they were dishonest, ensuring sub-game perfection in their play as well). So long as this equilibrium of contract enforcement is in place, the merchant can credibly offer future money to the sell sword in exchange for his life.

Kandori (1993) gives conditions on when we can sustain cooperation based on some sense of moral authority alone that has a Kantian flavor. Consider a moral norm based on a grim trigger strategy where I cooperate in every period, unless I am ever wronged. However, if I am ever wronged, I will bring violence to everyone I encounter from here on out. Even though I may never see the person who wronged me ever again, if I start bringing violence to others, that should trigger a chain reaction where they start bringing violence to others, which brings down the entire system. Thus, people cooperate, to avoid setting off this chain reaction that brings down the system as a whole. A priest could appeal to the sellsword’s rational morality, that every act of violence has a chance of bringing down civilization as a whole.

Of course, all of these above institutions only work when we are on the equilibrium path. But when we are in an era when Winter is Coming, then we very well may fall out of equilibrium. In such conditions, Acemoglu and Robinson (2001) model the process where Latin American countries transition between cooperative behavior and a state of mutual violence, what Hobbes’ referred to as the “nasty, brutish and short” state of nature. In such a world, all that is left is violence, and whoever is best at violence, the sellsword, prevails. 

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