Question

Consider the following strategic setting. There are three players, numbered 1,2, and 3 . Player 1 has two cards, labeled King and Ace. At the beginning of the game, player 1 deals one of the cards to player 2 and the other card to player 3 ; that is, player 1 either gives the Ace to player 3 and the King to player 2 (call this the action A) or the King to player 3 and the Ace to player 2 (action K). Player 2 observes the card dealt to him; player 3 does not get to see the card dealt to her. Player 2 then must decide between switching cards with player \(3(\mathrm{~S})\) or not \((\mathrm{N})\). Player 3 observes whether player 2 made the switch, but does not see her card. Finally, player 3 responds to the question "Is your card the Ace?" by saying either "yes" (Y) or "no"" (N). If player 3 correctly states whether her card is the Ace, then she obtains a payoff of 1 and the other players get 0 ; otherwise, players 1 and 2 both get a payoff of 1 and player 3 obtains 0 . Represent this game in the extensive form.

Short answer

The game is represented by Player 1's choice (A/K), Player 2's action (S/N), and Player 3's response (Y/N) with payoffs based on correctness.

Step-by-step solution

Define the Players and Their Initial Actions

The game consists of three players: 1, 2, and 3. Player 1 starts the game by dealing one of two cards: King (K) or Ace (A). Action A refers to giving the Ace to player 3 and the King to player 2, while action K gives the King to player 3 and the Ace to player 2.

Consider Player 2's Possible Actions

Player 2, upon receiving a card, can see which card he has and chooses between two actions: to switch his card with player 3's (S) or not to switch (N).

Consider Player 3's Observation and Response

Player 3 observes whether player 2 has made a switch but does not see her card. She must then answer whether her card is the Ace by saying either "yes" (Y) or "no" (N).

Determine the Payoffs for Correct and Incorrect Answers

If player 3 correctly identifies whether her card is the Ace, she receives a payoff of 1, and players 1 and 2 get 0. If incorrect, players 1 and 2 receive a payoff of 1 each, and player 3 gets 0.

Constructing the Extensive Form Diagram

The game is represented starting with Player 1's action node with two branches (A and K). Each branch leads to Player 2's decision node, which has two branches: switch (S) or not (N). Each branch from Player 2's decision leads to Player 3's move (Y or N), which considers Player 3's observation of switching. The terminals of Player 3's decisions determine the payoffs based on correctness.

Key concepts

Strategic Interaction

In game theory, strategic interaction refers to the decision-making processes where each player's choice depends on or influences the actions of other players. It is akin to a chess match where players anticipate their opponent's next move to strategize effectively.

When exploring the given card game, each player must consider the potential decisions of others to maximize their payoffs. In this scenario:

• Player 1 must anticipate how Player 2 might use the revealed information to influence Player 3's response.
• Player 2 makes a decision to switch or not based on the card received, while also thinking about how Player 3 might react to such information.
• Player 3 needs to respond "yes" or "no" after observing Player 2's decision. Though blind to her card, Player 3 weighs the likelihood of holding an Ace based on the swap action.

All players must navigate this intricate setup, predicting others' actions to ensure the outcome aligns with their own incentives. Strategic interaction is at the heart of each player's choices, making understanding their interdependencies crucial.

Card Game Scenario

The card game scenario set in this exercise is a structured environment where each play seamlessly leads to another player's turn. Initially, Player 1 is tasked with giving one of two cards (King or Ace) to each of the other players, setting up an intriguing decision point for Player 2.

In this sequence:

• **Player 1**'s actions create the foundation of the game by deciding who receives the King and who receives the Ace. This choice is critical and fuels the uncertainty faced by Players 2 and 3.
• **Player 2** must decide whether to switch cards with Player 3 based on the card dealt. Though Player 2 has more information about the card he holds, he doesn't know Player 3’s position in terms of knowledge, which is only whether she sees the switch.
• Finally, **Player 3** makes the conclusive move by answering the Ace question post the switch observation. Since Player 3 is unaware of the card she holds and relies on the actions of Player 2 for interpreting any hints, her response carries a mix of logic and luck.

This card game highlights how initial choices and subsequent responses create complex yet fascinating dynamics, mirroring many real-world strategic decisions.

Extensive Form Representation

The extensive form representation is a way to visually illustrate the card game's mechanics and sequential decisions. It provides a structured diagrammatical depiction that outlines each player's options at different nodes of the game.

Creating this representation involves:

• Drawing a tree where **Player 1's decision** is the root, branching into two paths for allocating Ace and King (actions A and K).
• Each branch leads to **Player 2's decision node**, showcasing his choices to switch (S) or not (N). These branches further mirror the uncertainty spread to Player 3.
• As the branches diverge, they move towards **Player 3's decision** point, where she chooses "yes" (Y) or "no" (N) based on prior observations.

At the endpoints of these decision branches are the payoffs, determined by Player 3’s final answer relative to the truth of her card. This model captures players' strategic interactions explicitly, highlighting the dependency of later actions on earlier moves. Overall, it simplifies the understanding of a complex strategic game, allowing easy analysis of potential outcomes and strategies.