大学博弈论讲义.docx

1、大学博弈论讲义Week 11: Game Theory Required Reading: Schotter pp. 229 - 260 Lecture Plan1. The Static Game Theory Normal Form Games Solution Techniques for Solving Static Games Dominant Strategy Nash Equilibrium2. Prisoners Dilemma 3. Decision Analysis Maximim Criteria Minimax Criteria4. Dynamic One-Off Ga

2、mes Extensive Form Games The Sub-Game Perfect Nash Equilibrium1. The static Game TheoryStatic games: the players make their move in isolation without knowing what other players have done 1.1 Normal Form GamesIn game theory there are two ways in which a game can be represented. 1st) The normal form g

3、ame or strategic form game 2nd) The extensive form game A normal form game is any game where we can identity the following three things:1. Players: 2. The strategies available to each player.3. The Payoffs. A payoff is what a player will receive at the end of the game contingent upon the actions of

4、all players in the game. Suppose that two people (A and B) are playing a simple game. A will write one of two words on a piece of paper, “Top” or “Bottom”. At the same time, B will independently write “left” or “right” on a piece of paper. After they do this, the papers will be examined and they wil

5、l get the payoff depicted in Table 1.Table 1BALeftRightTop2, 10, 0Bottom0, 01, 2If A says top and B says left, then we examine the top-left corner of the matrix. In this matrix, the payoff to A(B) is the first(Second) entry in the box. For example, if A writes “top” and B writes “left” payoff of A =

6、 1 payoff of B = 2.What is/are the equilibrium outcome(s) of this game? 1.2 Nash Equilibrium Approach to Solving Static GamesNash equilibrium is first defined by John Nash in 1951 based on the work of Cournot in 1893. A pair of strategy is Nash equilibrium if As choice is optimal given Bs choice, an

7、d Bs choice is optimal given As choice. When this equilibrium outcome is reached, neither individual wants to change his behaviour.Finding the Nash equilibrium for any game involves two stages.1) identify each optimal strategy in response to what the other players might do. Given B chooses left, the

8、 optimal strategy for A is Given B chooses right, the optimal strategy for A is Given A chooses top, the optimal strategy for B is Given A chooses bottom, the optimal strategy for B is We show this by underlying the payoff element for each case.2) a Nash equilibrium is identified when all players ar

9、e player their optimal strategies simultaneously In the case of Table 2, If A chooses top, then the best thing for B to do is to choose left since the payoff to B from choosing left is 1 and the payoff from choosing right is 0. If B chooses left, then the best thing for A to do is to choose top as A

10、 will get a payoff of 2 rather than 0.Thus if A chooses top B chooses left. If B chooses left, A chooses top. Therefore we have a Nash equilibrium: each person is making optimal choice, given the other persons choice. If the payoff matrix changes as:Table 2BALeftRightTop-6, -60, -9Bottom-9, 0-1, -1t

11、hen what is the Nash equilibrium? Table 3BALeftRightTop0, 00, -1Bottom1, 0-1, 3If the payoffs are changed as shown in Table 3 2. Prisoners dilemmaPareto Efficiency: An allocation is Pareto efficient if goods cannot be reallocated to make someone better off without making someone else worse off. Two

12、prisoners who were partners in a crime were being questioned in separate rooms. Each prisoner has a choice of confessing to the crime (implicating the other) or denying. If only one confesses, then he would go free and his partner will spend 6 months in prison. If both prisoners deny, then both woul

13、d be in the prison for 1 month. If both confess, they would both be held for three months. The payoff matrix for this game is depicted in Table 4. Table 4BAConfessDenyConfessDenyThe equilibrium outcome 3. Decision AnalysisLet N=1 to 4 a set of possible states of nature, and let S=1 to 4 be a set of

14、strategy decided by you. Now you have to decide which strategy you have to choose given the following payoff matrix. Table 5NatureS=123412201211113040041130S=YouN=OpponentIn this case you dont care the payoff of your opponent i.e. nature.3.1 The Maximin Decision Rule or Wald criterionWe look for the

15、 minimum pay-offs in each choice and then maximising the minimum pay-off Let us highlight the mimimum payoff for each strategy. NatureS=1234122012111130400411303.2 The Minimax Decision Rule or Savage criterionOn this rule we need to compute the losses or regret matrix from the payoff matrix. The los

16、ses are defined as the difference between the actual payoff and what that payoff would have been had the correct strategy been chosen.Regret/Loss = Max. payoff in each column actual payoff For example of N=1 occurs and S=1 is chosen, the actual gain = 2 from Table 3. However, the best action given N

17、=1 is also to choose S=1 which gives the best gain = 2. For (N=1, S=1) regret = 0.If N=1 occurs but S=2 is chosen, the actual gain = 1. However, the best action given N=1 is also to choose S=1 which gives the best gain = 2. For (N=1, S=2) regret = 2-1. Following the similar analysis, we can compute

18、the losses for each N and S and so can compute the regret matrix. NatureS=123412201211113040041130Table 6: Regret MatrixNatureMaximumS=1234Regret1234After computing the regret matrix, we look for the maximum regret of each strategy and then taking the minimum of these.Minimax is still very cautious

19、but less so than the maximin.4. Dynamic one-off GamesA game can be dynamic for two reasons. First, players may be able to observe the actions of other players before deciding upon their optimal response. Second, one-off game may be repeated a number of times. 4.1 Extensive Form GamesDynamic games ca

20、nnot be represented by payoff matrices we have to use a decision tree (extensive form) to represent a dynamic game.Start with the concept of dynamic one-off game the game can be played for only one time but players can condition their optimal actions on what other players have done in the past. Supp

21、ose that there are two firms (A and B) that are considering whether or not to enter a new market. If both firms enter the market, then they will make a loss of $10 mil. If only one firm enters the market, it will earn a profit of $50 mil. Suppose also that Firm B observes whether Firm A has entered

22、the market before it decides what to do. Any extensive form game has the following four elements in common:Nodes: This is a position where the players must take a decision. The first position is called the initial node, and each node is labelled so as to identify who is making the decision.Branches:

23、 These represent the alternative choices that the person faces and so correspond to available actions. Payoff Vectors: These represent the payoffs for each player, with the payoffs listed in the order of players. When we reach a payoff vector the game ends. In period 1, Firm A makes its decisions. T

24、his is observed by Firm B which decides to enter or stay out of the market in period 2. In this extensive form game, Firm Bs decision nodes are the sub-game. This means that firm B observes Firm As action before making its own decision. 4.2 Subgame Perfect Nash Equilibrium Subgame perfect Nash equil

25、ibrium is the predicted solution to a dynamic one-off game. From the extensive form of this game, we can observe that there are two subgames, one starting from each of Firm Bs decision nodes. How could we identify the equilibrium outcomes? In applying this principle to this dynamic game, we start wi

26、th the last period first and work backward through successive nodes until we reach the beginning of the game. Start with the last period of the game first, we have two nodes. At each node, Firm B decides whether or not entering the market based on what Firm A has already done. For example, at the no

27、de of “Firm A enters”, Firm B will either make a loss of $10mil (if it enters) or receive “0” payoff (if it stays out); these are shown by the payoff vectors (-10, -10) and (50, 0). If Firm B is rational, it will stays out The node “Firm A enters” can be replaced by the vector (50, 0). At the second

28、 node “Firm A stays out”, Firm A has not entered the market. Thus, Firm B will either make a profit of $50mil (if it enters) or receive “0” payoff (if it stays out); these are shown by the payoff vectors (0, 50) and (0, 0). If Firm B is rational, it will enter thus we could rule out the possibility

29、of both firms stay out We can now move back to the initial node. Here Firm A has to decide whether or not to enter. If Firm B is rational, it is known that the game will never reach the previously “crossed” vectors. Firm A also knows that if it enters, the game will eventually end at (A enters, B st

30、ays out) where A gets 50 and B gets 0. On the other hand, if Firm A stays out, the game will end at (A stays out, B enters) where A gets 0 and B gets 50 Firm A should enter the market at the first stage. The eventual outcome is (A enters, B stays out) How to find a subgame perfect equilibrium of a dynamic one-off game?1. Start with the last period of the game cross out the irrelevant payoff vectors.2. Replace the preceding nodes by the uncrossed payoff vectors until you reach the initial node.3. The only uncrossed payoff vector(s) is