Question

math questions and answers / consider a new... Question: Consider a new card game between 2 players: Darryl (player 1) and Phyllis (player 2) Darryl is de... Consider a new card game between 2 players: Darryl (player 1) and Phyllis (player 2) Darryl is dealt two cards : ♡6 and ♠5 . Phyllis is also dealt two cards: ♢3 and ♣9 . Now, each of the players will play 1 card both at the same time. The payoff of Darryl is 1 points if he plays a card of opposite color (red/black) than Phyllis, and otherwise his payoff is 6 points. The payoff of Phyllis is 9 points if the difference of the already played card numbers is greater than 5, otherwise her payoff is 1 points. 1.Find the action sets of each player and the action profile of the game. 2.Represent the game in the Normal form. 3.Find the Best Responses for Darryl. 4.Find the Best Responses for Phyllis. 5.Find all the Nash Equilibriums of the game (if any).

Answer 1

1.

action sets:

$(♡6, ♢3), (♡6, ♣9), (♠5, ♢3), (♠5, ♣9)$

action profile:

(card of the opposite colour, difference of the already played card numbers is greater than 5)

(card of the opposite colour, difference of the already played card numbers is less or = 5)

(card of the same colour, difference of the already played card numbers is greater than 5)

(card of the same colour, difference of the already played card numbers is less or = 5)

2.

3.

best response to $(D>5)$ is SC

best response to $(D\le5)$ is SC

4.

best response to OC is $(D>5)$

best response to SC is $(D >5)$

5.

a pair (a₁, a₂) of actions is a Nash equilibrium if and only if

• player 1's action a₁ is a best response to player 2's action a₂
• and player 2's action a₂ is a best response to player 1's action a₁.

Nash equilibriums for this game:

$(SC, D>5), (SC, D\le≤ 5), (SC, D>5)$

so, we have:

$(SC, D\le≤ 5) = (♡6, ♢3), (♠5, ♣9)$