### Game Theory and Nash Equilibrium Analysis#### Graphical RepresentationThe diagram shown is a representation of a two-player decision-making scenario often analyzed in game theory. The structure is a decision tree, also known as an extensive form game, where each node represents a decision point for a player.- **Player 1** makes the first decision, choosing between "Up" and "Down."- If Player 1 chooses "Up," then **Player 2** makes a decision at the next node, again choosing between "Up" and "Down."- If Player 1 chooses "Down," Player 2 is also faced with a decision between "Up" and "Down."Each endpoint of the tree specifies the payoffs for Player 1 (P1) and Player 2 (P2):- If both players choose "Up" (Player 1 chooses "Up" and Player 2 follows up with "Up"), P1 gets $45 and P2 gets $155.- If Player 1 chooses "Up" followed by Player 2 choosing "Down," P1 gets $100 and P2 gets $10.- If Player 1 chooses "Down" followed by Player 2 choosing "Up," both players get the same payoff of $85 each.- If both players choose "Down" (Player 1 chooses "Down" and Player 2 follows up with "Down"), both receive $95 each.#### Determining Sub-game Perfect Nash EquilibriumThe problem at the bottom of the diagram asks: "In the game above, what is/are the sub-game perfect Nash equilibrium?"There are four options provided:- (up, up)- (up, down)- (down, up)- (down, down) - The provided image indicates that this checkbox is selected, suggesting that this is the correct answer.#### AnalysisA sub-game perfect Nash equilibrium (SPNE) is a refinement of the Nash equilibrium concept which requires that players' strategies constitute a Nash equilibrium in every sub-game of the original game.In this game:1. If Player 1 chooses "Up," Player 2 will choose between "Up" (payoff 155 for P2) and "Down" (payoff 10 for P2). Rationally, Player 2 would choose "Up," leading to payoffs of $45 for P1 and $155 for P2.2. If Player 1 chooses "Down

We have sequential move game, where Player 1 moves first and then player 2 moves.

Up Down Up Down Player 1 In the game above, what is/are the sub-game perfect Nash equilibrium? (up,up) (up,down) (down, up) (down, down) No equilibrium exists Up Down Player 2 P1 gets $45 P2 gets $155 P1 gets $100 P2 gets $10 P1 gets $85 P2 gets $85 P1 gets $95 P2 gets $95

Up Down Up Down Player 1 In the game above, what is/are the sub-game perfect Nash equilibrium? (up,up) (up,down) (down, up) (down, down) No equilibrium exists Up Down Player 2 P1 gets $45 P2 gets $155 P1 gets $100 P2 gets $10 P1 gets $85 P2 gets $85 P1 gets $95 P2 gets $95

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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2. Consider the following game: u 1 r 6,0 0,6 d 3,2 6,0 Are there any Nash equilibria in pure strategies for this game? Find the mixed strategy Nash equilibrium.
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8. Solving for dominant strategies and the Nash equilibrium Suppose Tim and Alyssa are playing a game in which both must simultaneously choose the action Left or Right. The payoff matrix that follows shows the payoff each person will earn as a function of both of their choices. For example, the lower-right cell shows that if Tim chooses Right and Alyssa chooses Right, Tim will receive a payoff of 3 and Alyssa will receive a payoff of 6. Tim Left Left 5, 6 Right 4, 2 Alyssa Right 5,5 3,6 The only dominant strategy in this game is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Tim chooses and Alyssa chooses