Game Theory with Applications to Finance and Marketing I, Lecture1 Notes(Sep. 22)
Game Theory with Applications to Finance and Marketing I, Lecture1 Notes(Sep. 22)
Category of Game
| static(靜態賽局) | dynamic(動態賽局) | |
|---|---|---|
| asymmetric(資訊對稱) | NE | SPNE |
| non-asymmetric(資訊不對稱) | BE | PBE(The most complicated) |
These are:
- Nash equilibrium (NE)
- subgame perfect Nash equilibrium (SPNE)
- Bayesian equilibrium (BE)
- perfect Bayesian equilibrium (PBE)
Examples of the Games
Normal form of a game
描述赛局时，通常包含以下三个部分：
- i. the set of players(赛局参加的玩家/决策者)
- ii. the strategies available to each player(决策者可以作出的决定)
- iii. the payoff of each player as a function of the vector of all players’ strategic choices(决策者做出决定后得到的好处)
以上称为一个赛局的normal form。
Example of a Game
- Players? Player1 and 2
- Strategies available?
- For player1, is ${U, D}$;
- For player2, is ${L, R}$;
- Players get what?
- We shall write it as a function:
- i.e If player1 chooses U and player2 chooses L, it shall be written as:
- $u_1(U, L) = 0$
- $u_2(U, L) = 1$
- Hint: 函数$u_1(., .)$和$u_2(., .)$被称为两个player的效用函数(payoff functions)
Definition of NE(Nash equilibrium)
假设player1的策略集为X，player2的策略集为Y，如果对$x^* \in X$, $y^* \in Y$都有: \(u_1(x^*, y^*) \ge u_1(x, y^*)\)
\[u_2(x^*, y^*) \ge u_2(x, y)\]相当于：
- 从player1的角度：对手出招$y^$，我无论出招什么，结果都不比$x^$这一招来的好；
- 从player2的角度：对手出招$x^$，我无论出招什么，结果都不比$y^$这一招来的好；
- 其实是一种站对方立场考虑的方式
这样的一个策略对(strategy pair) ($x^, y^$)被称为一个赛局的Nash equilibrium。
Mixed Strategy
一个混合的策略是说，player不是单纯的出招x，而是在ta可行的策略集合中，以一个概率分布来出招。 例如：
- player1 可以设定自己出招U的概率为1/2，出招D的概率为1/2，这就是一个混合策略；
- player2可以设定自己出招L的概率为1/3，出招R的概率为2/3，这也是一个混合策略。
NE for Mixed Strategy
以下面这个例子为例，求解player1和player2的混合策略？
解: 我们可以设定player1出招U的概率为p，出招D的概率为1-p；player2出招L的概率为q，出招R的概率为1-q。 先站在player1的角度考虑，player1的混合策略为：无论player2出招是什么，他出U、D结果要一样好。这样可以解 1) 对player1的策略U，player2采取混合策略的情况下，player1获得的payoff为： \(0 \times q + (-1) \times (1-q)\) 2) 对player1的策略D，player2采取混合策略的情况下，player1获得的payoff为： \(2 \times (q) + (-2) \times (1-q)\) 二者相等，解得$q = \frac {1} {3}$ > 同理，可以解得$p= \frac {1} {2}$
A more complicated Game
Consider a two-player game, where the two must pick an integer from the set {1, 2, …, 100} at the same time. If they pick the same number, then they each get 1; or else, they each get zero.
- Find the pure strategy NE’s.
- Find the mixed strategy NE’s.
Reference
This note is mainly derived from Chyi-mei Chen’s Game Theory with Applications to Marketing and Finance, I Course @ National Taiwan University. Thanks him for giving this kind of excellent lecture.