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再谈 “人生如棋还是人生如牌” - Human Behavior, Free Will, and other Game Theoretic Issues.

Dr. 武夷山’s article on 人生如棋还是人生如牌, my comment of 2/19/08 on it, and other comments and responses including the one by dummer 先生 led me to write this blog in an attempt to clarify and make precise  what I have said.

Human behavior and thinking are the domains of philosophy, psychology, religion,  economics, and many other disciplines. We have pondered over the QUESTION “Determinism vs Free Will”  since ages immemorial. Starting from the Greeks who stress the influence of character and its inevitable flaws in determining the course of tragic events to modern psychologists who believe forces outside determines what we do (e.g., the insanity defense, and the 17^th century philosopher Spinoza who advocate that God determines everything down to the last detail), I am not philosophically inclined nor am I competent to engage in such a discussion.

On a more narrower front , my original comment was an informal statement of what mathematical  game theory has to say on the subject of human behavior and the topic of 人生如棋还是人生如牌.Here I run into three problems. First, mathematics often used words which has very different formal meaning from its colloquial use. Usually this is no problem except when a mathematician tries to explain things to the public. Readers will impart meaning to words used which the author did not intend since he is unconsciously using them in the mathematical sense. This is one reason as to why popularizing science and mathematics (科普) are  difficult. Second, in order to be precise, statement in science and particularly mathematics require assumptions and logical deduction to prove they are correct. In popular literature, it is often not feasible to do this lest the writing will become very dull and difficult to follow. Third, mathematics when applied to the real world involves inevitable approximations since the assumptions required by theorem  may not always be satisfied or only approximately satisfied. As such, statement are often qualitative and meant to convey insight only and not quantitative results. Consequently, scientific or mathematical comments stated in everyday language are naturally subject to misinterpretation and misunderstanding. My original comment on Dr 武夷山’s article, and for that matter any attempt to popularize science and mathematics, suffers from these difficulties.

Now let me try to clarify what I meant to say. In the process, perhaps I can explain in a limited way what mathematical game and decision theory attempts to do with respect to shedding light on human behavior and decision making.

Thus, because of the philosophical difficulties of “Determinism vs. Free will” Game theorists and theoretical economists start by making a big assumption of RATIONALITY, i.e, people behave rationally in their self interest. More narrowly, a decision maker will always attempt to optimize his self interest.. We know this to be not always true. But without such assumption, game theory and theoretical economics cannot develop. Thus in playing a game, this assumption amounts to the fact that people “play to win” to the exclusion of everything else.  You do not behave to deliberately lose even though you may want to do so under certain circumstance, e.g., playing against your boss or with your young daughter. Consider the simple game of tic-tac-toe we all know as a child. We learn very quickly, there is a winning strategy if you move first. And if your opponent makes one mistake, then you are guaranteed to win. And if your opponent also plays optimally, the game always ends in a draw.  Thus, under the assumption that you always play to win or not to lose, the outcome is known and you quickly lose interest in the game. In this sense and under this assumption, tic-tac-toe is completely deterministic. More complicatedly, consider the game of checkers. Adults play this game and there are good and bad player of different skill level. However, recently, (New York Times July 20, 2007 http://www.nytimes.com/2007/07/20/science/20checkers.html?st=cse&sq=strategy+for+checkers&scp=1) someone  has found the winning strategy for all games by using a series of computer for 18 years to play every possible combination of a checkers game. In fact you are welcomed to play against this super player http://www.cs.ualberta.ca/~chinook/play/ . No matter what, you cannot win and at most a draw. As a result, it is also an ultimately deterministic game under the play-to-win assumption of the players. Just like the game of tic-tac-toe, there is no more free will. Now of course, because of the complexity of chess, we cannot exhaustively play out all chess games. But in principle, it is no different from the simple game of tic-tac-toe and the more complex checkers. It is in that sense and under that assumption, I stated that chess is in principle a deterministic game provided you have infinite computing power. Once free will on the part of the player is removed by the assumption of play-to-win, there is nothing random about chess.  It is a game of complete information. Nothing is hidden. In reality of course as I stated above, the assumption of playing to win only may not hold always for all kinds of emotional or other reasons. But these considerations are not part of the mathematical theory of games.

This simplified discussion of three games from the simplest to the complex illustrates how difficult it is to apply game theory to real world situation even if there is complete information in the sense of no inherent randomness. Free will of an intelligent opponent has been assumed away by the assumption of “play to win” or more generally “rationality”. From this game theoretic viewpoint, chess while complex is not that interesting a game. Other qualitative but fundamental game theoretic concepts, such as “prisoner’s dilemma”, “the battle of the sexes”, “core”, “coalition”, “negotiation set”, ”pareto optimality” “Shapley value”, etc, etc. that arise because of the nonzero sum or multiple player nature of a game and that are typical in real world human behavior are  not covered by chess-type of games. For a primer on these, I recommend the book “Games and Decisions” by D. Luce and H. Raiffa (even though it is some 50 years old). And it is for these reasons, I say 人生如棋 is not a proper description.

Now as to the game of bridge, randomness is built in from the shuffling of cards at the start and the incompleteness of information during play (i.e., I do not know exactly what cards my partner and my opponents hold). I only know the probabilities. And if the initial distribution of cards are not in my favor, I cannot win regardless of how skillful a player I am. Thus I must play to win on expected value basis. I cannot guarantee any particular decision during the game will always win. Thus, bridge is a game of incomplete information even under the assumption of “play to win”. It is to this additional complication of inherent randomness or incomplete information we apply the adjective of “stochastic”. Optimization must be done on the expected value basis (behind even such an assertion . there is a whole body of literature on utility theory which one must acquire to really justify playing on expected value basis. Utility theory of course has its own set of assumptions which one may or may not agree. The process almost never ends. It is like studying  philosophy in order to understanding the meaning of life. But then you must first explain what you mean by “meaning”, and the infinite regress starts).

As if these difficulties are not enough, games of incomplete information creates another set of major conceptual problems.  “How do you know what I am guessing based on what I think your are guessing which in turn are basis on what you think I am guessing you are guessing and so on . . . “. There is also a very subtle but real computational difficulty of even trying to solve the SIMPLEST game of incomplete information involving two players (In the forthcoming 2008 IEEE Conference on Decision and Control, there will be a panel discussing this famous  40 years old so-called Witsenhausen problem of incomplete information. ). To summarize, real world human behavior are at best qualitatively and only partially explained by game theory  more than 60 years after Von Neuman first invented it to explain economic behavior. Yet three Nobel prizes on economics including the 2007 prize have been awarded in recent years base on research in games.

So what are we to do when faced with a real problem of decision making? This is where Decision Theory comes in by making another big assumption.

“Anything too complex to understand or compute , we consider it as random”

Thus, in game situation when there are other decision makers and incomplete information., we simply replace these unknowns by random variables. This makes solve the problem from my viewpoint a problem of “decision making under uncertainty”. All the game theoretic difficulties and concepts are thus assumed away and the problem becomes that of stochastic optimization – a still difficult but often manageable and, in many case, computationally solvable problems. For a good and easily understandable explanation of the intricacies of decision analysis, I recommend “Introduction to Decision Analysis” by H. Raiffa, Addison-Wesley 1968. Furthermore, as I mentioned earlier in my original comment, the replacement of things complex by randomness is not without sound reasoning. Kolmogorov’s work on the equivalence of complexity and randomness backs up the above assumption [Li and Vitani, 1997] .

For these reasons, I claim it is more proper to consider 人生如牌 in view of the uncertainties. I apologize for this explanation of what my original one paragraph comment to Dr. 武夷山’s article on 人生如棋还是人生如牌 intended.  It is now over  2000 word and still incomplete and uneven in my opinion. But it is the best I can do. To really do justice to this argument and properly debate this , one need to study at least one or two graduate course on game theory, theoretical economics, or operations research. Until a common langauge and understanding are arrived at, our discussions will be bogged down by semantics from the start.

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