Question

Darryl is dealt two cards: ♡4 and ♠10. Angella is also dealt two cards: ♢3 and ♣8. Now, Darryl will play first and Angella will play second (obviously).

The payoff of Darryl is 1 point if he plays a card of the opposite color (red/black) than Angella, and otherwise, his payoff is 3 points.
The payoff of Angella is 10 points if the difference of the already played card numbers is greater than 5, otherwise her payoff is 3 points.
Build up the game tree for this extensive form game. Mark the decision nodes as h1, h2, etc. and find N, A, H, Z, rho, and χ.

Answer 1

The game tree for this extensive form game has been built and the values of N, A, H, Z, rho, and chi have been determined.

Step-by-step explanation:

The game tree for this extensive form game can be built step-by-step as follows:

1. Darryl's decision node (h1):

• If Darryl plays the ♡4 card, his possible payoff is 1 point if Angella plays a black card (♠10) and 3 points if she plays a red card (♢3).
• If Darryl plays the ♠10 card, his possible payoff is 1 point if Angella plays a red card (♢3) and 3 points if she plays a black card (♣8).

2. Angella's decision node (h2):

• If Angella plays the ♢3 card, her possible payoff is 3 points if the difference between the already played card numbers (♡4 and ♢3) is less than or equal to 5, and 10 points if the difference is greater than 5.
• If Angella plays the ♣8 card, her possible payoff is 10 points if the difference between the already played card numbers (♠10 and ♣8) is greater than 5, and 3 points if the difference is less than or equal to 5.

Based on the game tree, the following values can be determined:

• N = {h1, h2} - Set of decision nodes
• A = {D, A} - Set of players (Darryl and Angella)
• H = 2 - Number of moves in the game
• Z = {1, 3, 10} - Set of possible payoffs
• ρ = {♡4, ♠10, ♢3, ♣8} - Set of moves available to the players
• χ = {red, black} - Set of colors for the cards