Economics: Game Theory

Introduction: You have been assigned to create the new game show to be called "Millionaire." The basic rules of the game are as follows: 1) The game has two contestants; 2) The game begins with both contestants receiving $1 million; and 3) The contestants play a game that can increase or decrease their $1 million.

In order to understand your new "Millionaire" game, you decide to start with a simpler version of your game: "I want $3." In this scaled down version of the game, there are two contestants (they do not know each other and cannot communicate with each other during the game). The rules of this game are below:

• Each player is given $3 at the start of the game.
• Independently and simultaneously, the players announce whether they want to add $0, $1, $2 or $3 to their initial stake of $3. Doing so reduces their opponent's award by $0, $2, $4, or $6, respectively.

Whatever one player gains the other player loses twice that amount. The equation for a player's final prize amount is:

(Initial $3) + (their choice of $0, $1, $2, or $3) - 2 x (opponent's choice of $0, $1, $2, or $3).

If a player ends up with a negative prize amount, they must pay that amount.

Questions:

Part A) Using the game theory concept, create the game (payoff) matrix for the "I want $3" game, showing players, actions, and payoffs. Identify the game's Nash Equilibrium strategies and payoffs for the players.

Part B) Now that you understand how the "I want $3" game works, you are ready to analyze "Millionaire." Here are the rules:

• Each player's is given $1,000,000.
• Each player, simultaneously and independently of the other, chooses to add to their award $0, $1, $2, $3, $4, ...$999,999, or $1,000,000. Doing so decreases the other player's award by twice that amount.

Part C) The game matrix for this expanded game has 1,000,001 rows, 1,000,001 columns, and 1,000,002,000,001 pairs of payoffs (no need to illustrate this). Building on what was learned in Part A, answer the following two questions:

1. What is the Nash Equilibrium of this game?
2. What are the Nash Equilibrium payoffs for the players in this game?