You have been assigned to create a new TV game show, and you have an interesting idea that you call, “I WANT TO BE A MILLIONAIRE.” The basics are: 1) two contestants; 2) the show begins with each contestant being given $1 million (!), and then 3) they begin playing a game that can increase or decrease that $1 million. You worry that the initial outlay of $2 million will stun your producers, so you decide to prepare them with a simpler version of your game that you call: “I WANT $3.” There are four steps in this simpler game: I. There are two contestants/opponents (who do not know each other and cannot communicate with each other during the game). II. Each player is given $3 at the start of the game. III. Independently and simultaneously, each player must choose 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. IV. Each player knows that their payoff at the end of the game is based on their initial $3, their choice of an additional amount, and the reduction due to their opponent’s choice. Make sure you’re clear that step III means that, whatever one player gains, the other loses twice that amount. Consequently, 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). Finally, If a player ends up with a negative prize, they understand that they must pay that amount. Below: a) Create the ”I WANT $3” game matrix, showing players, actions, & payoffs. b) Identify the game’s Nash equilibrium strategies and payoffs for the players. c) Identify any strictly dominant strategies for the players.
6) You have been assigned to create a new TV game show, and you have an interesting idea that you call, “I WANT TO BE A MILLIONAIRE.” The basics are: 1) two contestants; 2) the show begins with each contestant being given $1 million (!), and then 3) they begin playing a game that can increase or decrease that $1 million. You worry that the initial outlay of $2 million will stun your producers, so you decide to prepare them with a simpler version of your game that you call: “I WANT $3.” There are four steps in this simpler game:
I. There are two contestants/opponents (who do not know each other and cannot communicate with each other during the game).
II. Each player is given $3 at the start of the game.
III. Independently and simultaneously, each player must choose 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.
IV. Each player knows that their payoff at the end of the game is based on their initial $3, their choice of an additional amount, and the reduction due to their opponent’s choice.
Make sure you’re clear that step III means that, whatever one player gains, the other loses twice that amount. Consequently, 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).
Finally, If a player ends up with a negative prize, they understand that they must pay that amount.
Below:
a) Create the ”I WANT $3” game matrix, showing players, actions, & payoffs.
b) Identify the game’s Nash equilibrium strategies and payoffs for the players.
c) Identify any strictly dominant strategies for the players.
6. b) Once your producers understand the “I WANT $3” game, you will present the “I WANT TO BE A MILLIONAIRE” game. Its rules are:
I. There are two contestants/opponents (who do not know each other and cannot communicate with each other during the game).
II. Each player is given $1 million at the start of the game.
III. Independently and simultaneously, each player must choose 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.
IV. Each player ends the game with a payoff based on their initial one million, the additional amount that they announced, and the reduction due to the opponent’s announcement.
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. I STRONGLY RECOMMEND THAT YOU DO NOT DRAW IT! But building on what you learned in part (a), answer the following two questions:
i) What is the Nash equilibrium of this game?
ii) What are the Nash equilibrium payoffs for the players in this game?
iii) Is there a strictly dominant strategy for either player? If not, explain why not. If so, identify it.
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