Math, Belief and Hope

I left some loose ends in my previous post, so before I start discussing negotiation let me start clarifying some issues regarding my previous post. Negotiation will be the next topic. I promise.

My post on conflict and hope may lead many to believe that my hope is based on some simplistic argument: some equilibrium points in Game Theory and based on an old paper, written even before I was born. I invite those thinking that way to read Nash’s paper and tell me if the argument is simplistic. Others may criticize me as a crude reductionist, trying to explain complex human behavior using a mathematical model. As I said in my post, there are concepts that need to be defined further for the model. The payoff is one and, as I also said, we would have to find ways to include non-quantifiable variables in conflict. Someone could ask how to put moral principles as part of the payoff. To that I say principles are not negotiable. Another issue is related to rationality, but rationality in this case does not refer to reason or to the quality of reasoning, but, as I also said, rationality is based on utility theory, i.e. maximizing a payoff. How good could then be the model? To answer to this question I will summarize two experiments.

I found them in a book I’m currently reading: A Beautiful Math (Siegfried, Tom. A Beautiful Math. Joseph Henry Press. Washington, D. C. 2006.). In the first experiment, the biologist David Harper at the University of Cambridge fed a flock of 33 ducks in two separate patches of a pond at the university’s botanical garden. The strategy was to toss precisely weighted pieces of white bread. In one of the patches the feeder would toss one piece of bread every five seconds. In the other patch the feeder would toss one piece every 10 seconds. According to the author, ducks took about a minute to organize in two groups, approximately two thirds of them where bread was being tossed every five seconds and the rest, where bread was being tossed every 10 seconds. What this shows is not that the ducks know math but that they naturally organize themselves to maximize their food intake, their payoff.

The second is based on the work of another heavy-weight scientist: John Maynard Smith. He invented a simple animal-fighting game between hawks and doves. In this theoretical experiment Maynard Smith showed that one single strategy for the two types of birds would not produce a stable population, hawks aggressively and doves passively. The two types of birds in this theoretical experiment need to combine their strategies, doves sometimes behaving like hawks and vice versa, to become an “evolutionary stable strategy”. If hawks always fight for food, they’ll kill each other to extinction while food is wasted away. On the other hand, if doves never compete they will starve to extinction running away from any bird, including other doves. Of course, this is a theoretical experiment but shows clearly why a mixed strategy is necessary. A similar example I give my students when speaking of distributed intelligent agents. In my example, two or more collaborative agents will never go through a door because their courtesy yields to giving way to each other. On the opposite end, two competing agents will never go through the door fighting to get through first. This example also shows that mixed strategies work better than single strategies. Thus, Maynard Smith postulates that in evolution theory fitness should be replaced by utility and selection by rationality in the sense of game theory.

These examples show that the concept of rationality in game theory does not relate to reasoning but to maximizing payoff. The examples also suggest that the competing parties do not necessarily carry out mathematical computations, but that rationality is imprinted in evolution. To alleviate partially the dilemma nature versus nurture that may arise from this last statement, let me just say that the plasticity of the nervous system is a result of evolution but how this plasticity develops, is related to nurture: environment, education, culture, etc. A similar statement could be said of other systems in living organisms.

The next issue that may arise is how one construes hope from such an abstract mathematical expression. I would reply that some people found their hope in abstract expressions like the pursuit of happiness, achieving enlightenment or perfection or loving each other, including one’s enemies. I mean not to compare the moral or practical valor of these statements but only how abstract they all are. My point here is that I can ground hope, at least partially, in that conflict may turn positive and there are solutions where everybody can win. The problem paradoxically lies on finding rationality, when human reasoning is involved. My response is through principled negotiation but, as I said, this will be the topic of my next post.

Conflict and Hope

One of my reading assignments was “Fights, Games and Debates” by Anatol Rapoport. Professor Harry Schwarzlander was teaching a course on General Systems Theory at Syracuse University. It was 1985 or 1986 and I was starting my doctoral studies. Not only the reading but the discussions that Professor Schwarzlander promoted in class led me to develop a keen interest on conflicts and peace, first from a mathematical point of view and later on from a negotiation and conflict resolution perspective. At first I was disconcerted by the prospects of the mathematical concepts of stable and unstable equilibrium in the international conflicts of that time, particularly the cold war. I, a pacifist, had to face that mathematics seem to indicate that unarmed nations in conflict, were in a state of unstable equilibrium. Basically, a tiny pebble could become the force that would drive competing nations into war. On another end, comparably armed nations in conflict could be in a state of stable equilibrium. My only explanation to that was fear. Sides would avoid provoking an armed conflict, and peace could be maintained out of fear. The USA-USSR state of affairs at that time seemed to confirm this theory. Yet, I was not satisfied and so I decided to start studying conflict on my own.

Let’s step back a little on my reading assignment and its provoking title. I have to resort to my long term memory, very fragile, and to a few summaries I found on the book, to rebuild Rapoport’s theory, since I left this and many other books in Bogotá when I moved to Mayagüez, Puerto Rico. In Rapoport’s view, there are three kinds of conflicts:

1. Fights, in which the purpose of any party is to inflict harm to the opponent;
2. Games, where the purpose of any party is to outwit the opponent; and
3. Debates, in which the purpose of any party is to convince the opponent or any bystander.

Evidently, this taxonomy provides interesting insight about conflict. Fights in the pure sense assume no calculations or strategies. Games are based on rationality and strategic decisions based on the analysis of alternatives. Debates involve argumentation to make the opponent see things the way we see them. In an ideal situation, logic would be the right tool in a debate, but humans are not purely logic and it is not uncommon to see people using techniques of fights, e.g. threats, or of games, e.g. stratagems. Not only debates used mixed techniques, but games and fights combine them too. If one classifies war as a fight, which seems reasonable, in addition to attacks, threats, and other pure fight techniques, war involves a great deal of strategy. Many sports, and sports could be classified as games, allow some degree of fight: ice hockey in the USA is a good example. Artificial as it may seem, Rapoport’s conflict taxonomy results very useful when analyzing conflict as a process.

Conflict is natural and some authors claim it is necessary to prevent stagnation, but this, though, should be qualified. Ideally, conflict should not lead to fight, but that is idealism. Also, parties should be willing to resolve the conflict, otherwise conflict stagnates itself. One example is a couple that, to protect their children or their public image, do not voice their quarrels. Both the cause for complain and the conflict are stalled and I would hypothesize that conflict will escalate till it explodes. In this example, what could have been kept as a constructive debate could become a destructive fight.

Someone could argue that most human conflicts are debates and this makes sense since, at least, conflicts could start as one party tries to convince the other about one or more issues. As suggested before, people in debate use a variety of techniques, i.e. some strategy planned ahead of time, and some responses that arise in the heat of the situation. Strategies are typical of games while those unplanned or apparently irrational responses could seem more typical of fights. But realistic and interesting games not only involve pure strategy. Players, even in Chess, respond with surprises, threats, traps, etc.

The argument for conflict as debate is strong, but when we look at a good debate we see parties have prepared their strategy for argumentation so that their expressions, including body language are all in tune. After the opening statement, nonetheless, a good debater should be prepared to apply all the skills to respond adequately to the surprises the opponent brings to the table. Strategy should consider contingencies, having contemplated as many “what ifs” as possible and have the best replies to those unexpected situations. A good debater should analyze the opponents as carefully as possible during preparation, just like a game player analyzes opponents. But that is not enough. Traps will be set, emotions will enter the scene and all that planning may seem useless. Isn’t that exactly like a game? But in game as in conflict preparation isn’t useless if done carefully.

It is here that borders in Rapoport’s taxonomy become quite artificial or my rusty memories of the book do not allow me to see clear borders between the three grand types of conflict. Additionally, I do not know of any other general conflict taxonomy and, in the literature, I have only found some conflict-specific taxonomies, e.g. political conflict.

Anyway, this is convenient for my argument because I now bring another perspective to human conflicts: conflicts as games. This highly controversial perspective for some humanistic schools of thought brings hope to conflict resolution and I will explain this further down.

For my own internal debate and any external debate with any reader, I have some heavy-weight supporters of this perspective: John Von Neumann and Oskar Morgenstern, John Nash, Harold W. Kuhn, Anatol Rapoport, and many others. For the reader familiar with negotiation tactics I am using one known as “association”. Some of my “associates” (pardon my modesty) may be immediately recognized but, if not, I invite the reader to look for any or all of them in Wikipedia.

Now, you may wonder why treating human conflict like games brings hope. I invoke now the non-cooperative games that gave Nash an excellent topic for his doctoral dissertation and became a seminal work in Economics for which he received a Nobel Prize in 1994. The name of this kind of games is quite eloquent: each player for him/herself. This is not the aspect that gives me hope. What gives me hope is that in this type of game there is an equilibrium point where, and I quote Nash, “each player’s mixed strategy maximizes his payoff if the strategies of the other players are held fixed. Thus each player’s strategy is optimal against those of the others.” [http://www.lsi.upc.edu/~ia/nash51.pdf].

One historic note. When I got interested in this topic I had to struggle to get a physical copy of Nash's paper; the World Wide Web and Google were not even a figment of the imagination. That gives you an idea of how old I am. Now, I googled it out and I could reread it with almost as much difficulty as the first time: that guy is a genius :-)… and I’m not :-(.

Back from my digression, my hope resides on the fact that, based on Nash’s theory, each party in a conflict can maximize its payoff; sort of a win-win situation, if all the parties play their best mixed strategy. This is an ideal situation, because how does a party know what its best strategy is. The equilibrium points assume total rationality of the players, but who defines rationality or how? The debate about rationality has been going on for millennia and I’m not sure anyone has agreed on a definition, except, maybe, in mathematics regarding utility theory where rationality refers to maximizing a payoff function. Then, the next question comes to mind: How do we define that payoff, when many variables in conflict are not quantifiable? That is certainly not trivial, but I have never argued this was going to be easy. The equilibrium points give us a target at which we should aim so that, when in conflict, we all get the best we can; we all can win. If I were to continue with my mathematical arguments, we could approximate equilibrium through an iterative process. The final question is how? My answer is through negotiation.

P.S. I have been writing this post for several days, motivated by the longest student strike I have seen at the University of Puerto Rico. I had never been caught in the middle of this kind of conflict before. writing it has been my personal therapy and I hope will serve someone else too. I will now start to write my next post on negotiation as a sequel to this.