What is Mathematical Game Theory? 精选

What is Mathematical Game Theory? – My attempt at popularizing science我的科普试作
Because of interests in my earlier article on http://www.sciencenet.cn/blog/user_content.aspx?id=16273
I shall attempt below to explain a bit further what is Mathematical Game Theory and in what sense it tries to model  qualitatively human behavior and interaction. I am well aware of the difficulties of explaining science to lay reader and my attempt will probably raise more questions. Thus, let me state at the beginning that I fully expect and accept that my attempt may fail. I do not feel it is my duty to explain things satisfactorily to EVERYONE. I’ll try and do my best but I don’t wish to answer endless number of questions on this and that. But if sufficient number readers request clarification on a particular point AND if I feel I can do a better job, then I’ll try to answer. Learning about science is not easy. Some effort, thinking , and reflection on the part of the readers are also required. There is no magic learning pill (学科学的仙藥丸).
May I assume that every reader is familiar with the idea of a “contour map” – a map of concentric contours that show the elevation of land given heights above sea level?  A drastically simplified map is shown in fig. 1 below where the coordinate axes θ1 and θ2 are latitude and longitude respectively.
Fig.1     A simplified contour map
Imagine this contour map illustrates an abstract mountain peak. Let us assume that I, call me the decision maker one (DM1), have a goal to reach the highest point on this mountain. If we wish to climb this peak we can start at any point on this two dimensional diagram and move to the peak at the center by following any upward path. This assumes that DM1, can control his/her movement BOTH along the longitude and the latitude axis. But suppose I can only control θ1 and another decision maker DM2 controls θ2. Now where I may locate myself partially depends on the wishes or free will of DM2. However, regardless what DM2 choose (i.e., a value on the longitude axis θ2 ) my best response or reaction is to any θ2 is along the ridge line (山脊) of this mountain as shown in Fig,2
Fig.2 Best response of DM1, i.e., value of θ1 to any choice of DM2, θ2.
This ridge line can be described technically as the locus of tangent points of a contour with a vertical line at the chosen value of DM2, θ2. Intuitively, imagine taking a cross section of this mountain in three dimensions by cutting the mountain at the position θ2; then DM1, who wishes to reach as high as possible, will choose the highest point on this cross section, i.e., the ridge. This is the point on the curve R1 (stands for the best response of DM1 to any choice of DM2).
Similarly, The decision maker 2 has his/her own desire (objective or goal to maximize or a mountain top to reach) which can be expressed as another contour map J2 that in general are different from that of J1. For DM2 s/he faces that same consideration as DM1, namely s/he can control θ2 but not θ1 and s/he can construct a similar response curve, R2, as his/her reaction to any choice by DM1. Except technically R2 is the locus of tangent points of a horizontal line at any value of θ1 with the various contours (see the more intuitive explanation for R1 above. Same reasoning applies here). We now illustrate this in fig.3
Fig. 3 Best response of DM2, i.e., value of θ2 to any choice of DM1, θ1.
Now let us superimpose Figs 2 and 3 together to see the interactions of DM1 and DM2. This is shown in fig.4.

Fig.4 The Nash Equilibrium Solution

Note the intersection point of the R1 and R2 curve which is called the Nash Equilibrium solution of a two person Nonzero-sum game. Here both DMs have their own goal or objective. They cannot each reach their own desired peak since their decisions interact. The Nash point represents a possible solution to their interactions. The game is NONZERO-SUM because the objectives of the two decision maker is different but not necessarily in direct opposition (i.e., you gain is not necessarily my loss); it is NONCOOPRATIVE since the two decision makers are only interested in their own goal; it is in equilibrium because each can unilaterally enforce his/her own choice Thus. if DM1 chooses the value of θ1 corresponding to the equilibrium point, then DM2 has no choice but to follow with the value of θ2 for the equilibrium point and vice versa for otherwise, s/he will only hurt him or herself . This is because the equilibrium point is located on both ridge lines of the two mountains. It is named after John Nash of the Nobel prize and the movie/biographical fame of “A Beautiful Mind” since he first discovered this concept. Such concept qualitatively explains how two self-interested human decision makers may behave and interact under certain but not all circumstances. Example in real life may be the pricing behavior two companies in a market that has both competitive and noncompetitive aspects, e.g., one company makes truck and the other makes passenger autos. Another example may be the economic policies of the US and China.  A third example could be the Arms Race by the US and USSR during the cold war era. Both the US and the USSR wishes to feel secure and be able to defend their country. The equilibrium outcome is that both must engage in a costly and continuing arms race. It one side unilaterally disarm, it is immediately at a disadvantage. Thus, the undesirable (from economic viewpoint) equilibrium must continue.

Let me stop here and ask for feedback from readers. IS THIS SORT OF EXPLANATION USEFUL FOR AVERAGE READERS OF SCIENCENET? If so, I will continue with this example and explanation of more game theoretic concepts. If not, then it is simply a failed attempt and I shall do no more.

http://blog.sciencenet.cn/blog-1565-17894.html

何毓琦

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