POLSC 135 1st Edition Lecture 7Outline of Last Lecture: Game Theory (cont.) Outline of Current Lecture: Solving the Game Current Lecture:I. Solving the Game. -We typically solve extensive form games like this for a particular type of Nash equilibrium called a “subgame perfect Nash equilibrium,” or SPNE. - A “subgame” is the part of a game beginning at one choice node and including all succeeding choice nodes.- A “SPNE” is a set of strategies such that each player plays a Nash equilibrium in every subgame.-Players in the game care about the consequences of their choices and, therefore, think ahead.-They try to anticipate how the other player will respond to their choices.-We find SPNEs by using a method called “backward induction.”- What would the other player do if I chose X? What would the player do if I chose Y?II.Game Example (solving by working backwards)-Game Theory Example: - Say for example that a citizen lives within a state. The state decides to raise taxes. The following chart below is a representation of how the citizen may react. - Presumptions: The state starts with 1 (representing tax income). -Key: L = Loyal, E= Exit, V=Voice. 1 = taxes. C=Cost. (Citizen;State)These notes represent a detailed interpretation of the professor’s lecture. It is best used as a supplement to your own notes, not as a substitute. CitizenVoiceLoyal (L; 1+L)Exit (E; -1) [state loses]StateIgnoreRespond (1-C;L)CitizenLoyal (0-C; 1+L)Exit (E-C; 1)- The citizen is making a choice. If I stay loyal, I get 0 - C. If I exit, I get E - C.- Easy to see that the choice will depend on whether E > 0.- If E > 0, we say the citizen has a credible exit threat (cost to state).- The citizen will choose to exit. Due to the cost of remaining Loyal. Now we ask what the state will do at the previous decision node.- The state is making a choice: If I respond, I get L. If I ignore, I get 1.- The choice will depend on whether L > 1.- If L > 1, we say the state is dependent on the citizen.- If L < 1, we say the state is autonomous.- Assume the state is dependent. In this case, the state will choose to respond. Now we ask what the citizen will do at the initial node.- If the citizen exits, she gets E.- If the citizen remains loyal, she gets 0.- If the citizen uses voice, she gets 1 - C.- The citizen will use voice. Conclusion to Scenario: - Assumptions- E > 0.CitizenLoyal (0-C; 1+L)Exit (E-C; 1)StateIgnoreRespond (1-C;L)CitizenLoyal (0-C; 1+L)Exit (E-C; 1)CitizenVoiceLoyal (L; 1+L)Exit (E; -1) [state loses]StateIgnoreRespond (1-C;L)CitizenLoyal (0-C; 1+L)Exit (E-C; 1)- E < 1 – C; note here that C < 1, because E > 0.- L > 1.- Subgame Perfect Nash Equilibrium- (Voice, Exit; Respond.)- Outcome- The citizen uses voice, the state responds.- Payoffs- The citizen gets 1 - C, the state gets L.Equilibria: - How to write down the equilibria.- Equilibrium in scenario 1 is (Voice, Exit; Respond).- (The citizen’s 1st action, the citizen’s 2nd action; the state’s 1st action.)End