Question

Consider the following strategic setting. There are three people: Amy, Bart, and Chris. Amy and Bart have hats. These three people are arranged in a room so that Bart can see everything that Amy does, Chris can see everything that Bart does, but the players can see nothing else. In particular, Chris cannot see what Amy does. First, Amy chooses either to put her hat on her head (abbreviated by \(\mathrm{H}\) ) or on the floor (F). After observing Amy's move, Bart chooses between putting his hat on his head or on the floor. If Bart puts his hat on his head, the game ends and everyone gets a payoff of 0. If Bart puts his hat on the floor, then Chris must guess whether Amy's hat is on her head by saying either "yes" or "no." This ends the game. If Chris guesses correctly, then he gets a payoff of 1 and Amy gets a payoff of \(-1 .\) If he guesses incorrectly, then these payoffs are reversed. Bart's payoff is 0 , regardless of what happens. Represent this game in the extensive form (draw the game tree).

Short answer

A game tree with sequential decisions starting from Amy, followed by Bart, then Chris, with associated payoffs.

Step-by-step solution

Identify Player Decisions

In this game, there are two decisions to be made by the players. Amy makes the first decision by choosing either to put her hat on her head (H) or on the floor (F). Based on Amy's decision, Bart will then make his decision of either placing his hat on his head or on the floor.

Consider Chris' Decision

If Bart chooses to place his hat on the floor after seeing Amy's move, Chris will then have to make a decision. Chris will decide whether Amy's hat is on her head (saying "yes") or not (saying "no"). His decision ends the game.

Draw the Game Tree with Amy's Decision Node

Start the game tree with Amy's decision node at the top. Draw two branches from this node: one labeled 'H' for Hat on her head, and one labeled 'F' for Hat on the floor.

Add Bart's Decision Nodes

From each of Amy's choices ('H' and 'F'), add a decision node for Bart. Bart can either put his hat on his head or on the floor. If Bart chooses Hat on his head, he ends the game with payoffs of (0, 0, 0) for Amy, Bart, and Chris respectively. If Bart chooses Floor, the game continues to Chris.

Add Chris' Decision Nodes

If Bart puts his hat on the floor, draw branches from Bart's decision nodes indicating Chris' guesses: 'yes' or 'no'. This leads to possible outcomes based on Chris’s guesses correlating with Amy’s initial action.

Assign Payoffs for Chris' Decisions

If Chris guesses correctly (matching Amy's actual action), he receives 1 and Amy receives -1. If Chris guesses incorrectly, Chris gets -1 and Amy receives 1. Bart's payoff remains 0 regardless of the outcome.

Key concepts

Game Tree Representation

In an extensive form game, the game tree is a visual aid that helps us understand the sequence of events and decisions taken by players. Imagine this game as a tree starting from a trunk and branching out into different outcomes, where each branch represents a player's decision:

• Start with Amy's decision. She decides whether to put her hat on her head ('H') or on the floor ('F'). This choice branches the tree into two directions.

From each of Amy’s choices, another set of branches emerges based on Bart’s action since he observes Amy’s decision:

• If Amy chooses 'H', Bart can place his hat on his head or on the floor.
• If Amy chooses 'F', Bart has the same options.

Next is Chris’s decision. His branches only appear if Bart puts his hat on the floor:

• Chris chooses 'yes' or 'no' based on Amy’s original decision, and this ends the game.

Each end of the branches represents the final outcome of the game with specific payoffs for each player.

Strategic Decision Making

Strategic decision making in games involves players considering not only their own actions but also anticipating the reactions of others. In this scenario:

• Amy makes a strategic decision by choosing how to position her hat, fully aware that Bart and Chris will observe and act accordingly.
• Bart's strategy involves reacting to Amy’s choice with the knowledge that his actions pave the way for Chris’ decision.

When Bart decides, he needs to think about how Chris might guess Amy's choice, as any incorrect guess could disadvantage Amy. While Bart's payoff is always zero, his decision impacts the payoff of both Amy and Chris, showing how in-depth consideration is crucial.

Payoff Structure

The payoff structure of a game illustrates how the outcomes affect the players' rewards or penalties.

• If Bart puts his hat on his head, regardless of Amy’s choice, everyone's payoff remains (0, 0, 0).
• If Bart puts his hat on the floor, Chris’s decision to guess 'yes' or 'no' affects the final payoffs.

For Chris, guessing correctly means a payoff of 1, and Amy receives -1. An incorrect guess flips these results, with Chris getting -1 and Amy +1. Bart’s payoff remains unaffected at 0 in all scenarios. This payoff structure creates incentives for strategic decisions, as each player seeks their optimal outcome.

Player Decisions

Player decisions are about choosing actions that align with personal goals and anticipated reactions of others. In our example:

• Amy starts by deciding whether to put her hat on her head or throw it on the floor. Her choice pre-determines Bart’s possible strategies.
• Bart needs to choose his action based on Amy’s visible choice, where he can either put his hat on his head to end the game or on the floor to trigger Chris’ guess.
• Chris’s decision to guess 'yes' or 'no' relies on Bart's choice, affecting the payoffs and showing the consequences of his decision.

Each decision is interconnected, highlighting the complexity of the game. Understanding the implications of every player’s action is essential to navigate through such strategic settings.