Game Introduction 2.4 Extended
Extended game before the game is a standard formula for our expansion, we will introduce the following aspects:
- Extended illustrated example
- The concept Extended information set
- Extended concept formulas game
Shows an example of a game shown in FIG.
Game begins with participants of a decision tree section 1, then 1 participant to make a choice, according to the choices made participants 1, the game will be to a different branch. The final game will reach the end of the section, and the participants to benefit corresponding to the end section.
The concept of information collection
As shown, a prisoner's dilemma game Extended dashed line represents the above is a set of information. In simple terms, the following people do not know the dotted line above the dotted line information.
After a conceptual expression and expansion of the game.
definition:
- Decision node starts a single set of information (i.e. no nodes associated with dashed line) n (not including the first node)
- Section contains all decisions and end the game under the tree section n
- Segmentation is not formed to any set of information (i.e., if n is below a node n2, and n2 is other decision node in the same set of information must be under n)
Two examples are given
Below, there are two subgame
On the third point here for example, for the FIG., Only a sub-game, since the two branches 2 in the following three nodes placed on both sides of a centralized information, the right side can not be branches 2 subgame.
Chapter III incomplete information static game
Before talking about the two chapters are a perfect information game, that is, the two sides know each other payoff function of the game situation, this chapter will discuss the beginning of the game incomplete information, also known as Bayesian game, this chapter focuses on static incomplete information Game, the next chapter discusses dynamic game of incomplete information.
In fact, these two chapters of the core of what is actually the case in the previous two chapters added something about probability and the like, so it can be understood with reference to the previous method. Of course, with these two chapters of the study, we will be in front of the concept of the game and improve and expand, but is not that difficult to understand things. In the study after two chapters, I will reduce the derivation of the formula and its more elaborate idea is, first, because the formula too much too difficult to play ......
The second is very easy to derive some formulas, and to me, the more partial engineering, so the learning objectives of game theory does not focus on the process of the formula, but more attention on their problem-solving ideas and practical effect.
The following first to summarize the contents of this chapter:
This chapter focuses on Bayesian game, there is a simple example - Cournot competition under asymmetric information.
然后我们会对混合战略的概念进行延申，并且证明贝叶斯纳什均衡当所有非对称信息趋于0的时候，就变成了第一章所讨论的静态完全信息的纳什均衡。
再到后面我们会举一个拍卖的例子（当然，你会看到在非完全信息的博弈中，拍卖会很经常出现）
到最后，我们会给出并解释显示原理，这个将用于博弈设计方面。
3.1.a静态贝叶斯博弈和贝叶斯纳什均衡
我们先考虑一个非对称信息下的古诺竞争。
与第一章的古诺竞争相同，不过这里我们改变企业2的成本：企业2的成本有θ的概率是cH，有1-θ的概率是cL，这里cH>cL。当然，企业2是知道自己的成本是cL还是cH的，这里θ是对于企业1对企业2的的预测而言的。也就是在这场博弈中，企业2是占信息优势的：他知道自己的成本和企业1的成本，而企业一只能对企业2的成本做预测，并且企业2是知道企业1预测自己高成本的概率θ的，并且以上都是共同知识：企业1知道企业2占有信息优势，企业2知道企业1 知道自己占有信息优势等等。
那么很容易得到，对于企业2来说当取从cH时，q2*（cH）取值为：
对于企业2来说当取从cH时，q2*（cL）取值为：
类似的，我们可以得到企业1的q1*：
解方程得：
这就是非对称信息下古诺竞争的贝叶斯纳什均衡。很容易理解，这里企业2会根据自己的成本高低选择产出，同时，也会考虑企业1的最优反应，本质其实和第一章中所说的思想是相同的。