Game Theory and the Kingdom of God (A Quirky Series Installment), Part 4: "An Interlude: The Risk e-mail"

I don't share much personal stuff on this blog. But today I'm going to share a little incident that has taken on near mythic-status among some friends of mine on the ACU campus. The incident has nothing to do with the Kingdom of God, but it does have to do with game theory.

Some faculty friends and I occasionally gather to play the board game Risk. One night, while playing, one of the players was growing very strong (Risk is a world conquest game). In that situation, the other players tend to agree to mutually attack the strongest player. We call it "policing a threat." Well, on this night one of our players (who shall remain nameless) refused to help police the growing threat. This bothered me since the strong player did go on to win the game. I tried to argue why participating in mutual policing is important in Risk. The nameless player, in his defense, argued that Risk isn’t about cooperation. It is every man for himself. I responded that, although Risk is broadly a zerosum game (me against you), there are moments of non-zerosum when our interests overlap a wee bit (the enemy of my enemy is my friend). No one knew what I was talking about and no one wanted to sit through a lecture from me on game theory. So, I went home and drafted the following e-mail. It basically recapitulates the last few posts and adds information on how you mathematically define the Prisoner's Dilemma (if you cared to know). Mainly I offer this interlude to ask a simple question:

What kind of nut would write such an e-mail to make a point about a board game?

Welcome to my pathology.


The actual e-mail:


Gentlemen,
At our last Risk game I declared that certain phases of Risk reminded me of the Prisoner's Dilemma in Game Theory. Some of you asked me to clarify, but I declined at that time. Now, having worked out a more detailed analysis I want to share it with you.

1. A BRIEF INTRODUCTION TO GAME THEORY. Game theory is a branch of applied mathematics which enables a mathematical analysis of decision-making scenarios. A "game" is simply a situation where two or more "players" make "choices" leading to various outcomes called "payoffs." If the payoffs can be given numerical values, then game theory allows one to evaluate the different strategies players might employ.

2. ZEROSUM GAMES. In games where your winnings equal my losses, the sum of the losses and gains will equal zero. These are called "zerosum" games. They are games of total conflict. Poker is such a game. I can only win a dollar if someone has lost a dollar. The payoffs sum to zero and the end of the night. Risk is generally like this. A country can only be taken if another player has lost it. In general, game theory suggests that zerosum games have optimal playing strategies. Yet, it is another thing entirely if the players can, in practice, discover the optimal strategy. However, my point the other night was that although Risk is broadly a zerosum game, there are phases at which it takes on a non-zerosum flavor.

3. NON-ZEROSUM GAMES. In games where the players' interests are not completely opposed, the payoffs do not sum to zero. These are called non-zerosum games. Since, in non-zerosum games, the players' interests are not completely opposed, there are opportunities for cooperative outcomes. Win-Win scenarios become possible. The exact nature of the solution of such games, however, depends largely on the pattern of payoffs. The most notorious pattern of payoffs has received a name: The Prisoner's Dilemma.

4. THE PRISONER'S DILEMMA (PD). The classic formulation of the PD is this. Imagine two co-criminals are caught and are being interrogated in separate rooms. Each criminal has one of two choices: Keep Quiet or Rat Out My Partner. Here are the payoffs for this "game": If both keep quiet they each go to jail for 1 year, if both rat on each other they go to jail for 3 years, if one rats and one stays quiet the person who rats gets off free and the person who got ratted on goes to jail for 5 years. Okay, imagine playing such a game. What will you do? Most everyone chooses to rat to protect oneself. However, if both criminals make this choice they each go to jail for 3 years where if they both would have kept quiet they would only be facing a 1 year sentence. Rational play recommends the choice of ratting, but this appears to lead to a sub-optimal outcome. Hence the dilemma.

5. DEFINING THE PRISONER'S DILEMMA. PD's are defined by the following payoff structure: T < C < D < SP (or T > C > D > SP). Where T equals the temptation to defect (getting off free), C equals the payoff for cooperation (the 1 year sentence for both keeping quiet), D equals the payoff for mutual defection (3 years for both ratting), and SP equals the "sucker's payoff" (getting 5 years for keeping quiet while your pal is ratting you out). In my comment the other night, I was suggesting that some phases of Risk have just this payoff structure. I went home and worked it out.

6. POLICING A GROWING THREAT AS A PRISONER'S DILEMMA IN RISK. Here is a non-zerosum moment in Risk: Player 3 is growing strong, so strong that if left unpoliced he will ultimately win the game. Players 1 and 2 are now facing choices that have a non-zerosum flavor. First, they can refuse to cooperate and act selfishly by forcing the other player to police Player 3 unaided. Clearly this is a good outcome if you can get other players to do the policing for you. You can place all your armies offensively and not waste any policing Player 3. Like the PD, this is the defection response (ratting on each other). The cooperative outcome would be if both players agreed to police Player 3 equally. Here are the payoffs I worked out for this game: If both players cooperate and police Player 3 the payoffs are neutral, 0 for each player. If both players defect on each other and no one polices Player 3, the payoff to each is minus 10 since Player 3, in this situation, is sure to win. If one of the players polices Player 3 while the other player refuses to cooperate, the policing player gets an outcome of minus 15. I gave this a minus 15 because this is the worst situation to be in: Policing Player 3 by yourself while the other player expands as well. If a player refuses to cooperate and leaves the other player to police Player 3 alone, their payoff is plus 5 since this is a great situation to be in: Someone is policing Player 3 for me, so I can place all my armies offensively. If you look at these payoffs they form a PD: T > C > D > SP.

7. CONCLUSIONS. In non-zerosum games, optimal solutions are hard to find. Rational play in Risk suggests we leave to others the policing chores while I expand unimpeded. However, if all players make this choice growing threats continue unchecked and all the defecting players are sure to lose. If everyone could cooperate, the threat might be reduced and one of the policing players might have a chance to win. Game theory suggests that in these situations three ideas come to the fore: Communication, move order, and threats. Cooperation can emerge in a PD if players can talk and reach enforceable agreements with threats of retaliation. Move order is also important in that if I see you cooperating on a previous move I will be more likely to go along with the agreement. Moves can display good will. In sum, I think if there is a growing threat in a Risk game the players should reach an agreement to cooperate to minimize that threat. If a part of that coalition refuses or goes back on their word the other members of the coalition should attack the defector unmercifully. I believe this is the only way to enforce cooperation is such a scenario.

Have a great finals week.
Richard