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1. (12 points) Chris and Liz are playing the rock-paper-scissors game. Suppose ...

Sep 17, 2024

To represent the rock-paper-scissors game between Chris and Liz in extensive form using a game tree, we need to outline the sequence of moves and the payoffs based on their choices. 1. Initial Decision Node (Chris's Move): - Chris has three possible choices: Rock (R), Paper (P), or Scissors (S). - However, since Liz can detect when Chris is going to choose Paper due to his subtle signal, we can represent this as a decision node with two branches: one for when Chris chooses Paper and another for when he chooses either Rock or Scissors (which Liz cannot distinguish). 2. Branches from Chris's Move: - If Chris chooses Paper (P): - Liz can choose Rock (R) or Scissors (S). - If Liz chooses Rock (R): Payoffs are (P: -1, R: 1) → Chris loses, Liz wins. - If Liz chooses Scissors (S): Payoffs are (P: 1, S: -1) → Chris wins, Liz loses. - If Chris chooses Rock (R) or Scissors (S) (combined since Liz cannot distinguish): - Liz still has two choices: Rock (R) or Scissors (S). - If Chris chooses Rock (R): - If Liz chooses Rock (R): Payoffs are (R: 0, R: 0) → Tie. - If Liz chooses Scissors (S): Payoffs are (R: 1, S: -1) → Chris wins, Liz loses. - If Chris chooses Scissors (S): - If Liz chooses Rock (R): Payoffs are (S: -1, R: 1) → Liz wins, Chris loses. - If Liz chooses Scissors (S): Payoffs are (S: 0, S: 0) → Tie. 3. Game Tree Structure: - Start with a node for Chris's choice. - From this node, draw branches for Paper, Rock, and Scissors. - For Paper, create two branches for Liz's choices (Rock and Scissors) with their respective payoffs. - For Rock and Scissors, create branches for Liz's choices (Rock and Scissors) with their respective payoffs. This game tree visually represents the strategic interactions between Chris and Liz, highlighting how Liz's ability to detect Chris's choice affects the game's outcome. If you need help drawing the tree or further clarifications, feel free to ask!