Question: Consider a couple, Pat and Sam

26 Aug 2024,2:54 AM

 

Consider a couple, Pat and Sam. Each of them is choosing whether to attend a country music event at a local bar, or a classical music event at a club to which they belong. Sam likes country music better than the classical music. Pat likes the classical music better than country music. The game is called Togetherness because, above all, both Sam and Pat prefer to attend the event that the other attends. For either person, the best outcome is to attend the event that features their own favourite music with their friend, the second-best outcome is to attend the event that features their friends favourite music with their friend, the third best out-come is to attend the event that features their own favourite music without their friend, and the worst outcome is to attend the event that features their friends favourite music without their friend.

(a) Assume that choices are made simultaneously and independently. Construct a strategic form for this game. Hint: (i) Who are the players? (ii) What are their possible strategies? (iii) What are their payoffs? How the payoffs are related to their preferences?
(b) Assume that Pat makes her choice first and then, knowing what Pat chose, Sam makes his choice. Construct an extensive form for this game.
(c) Continue with the previous part. Construct a strategic form for this sequential choice game.
(d) In the describing this situation, we assumed that for Pat and Sam to- getherness (attending the same event as their friend) was more important than listening to ones own favourite music. Assume instead that togetherness is less important than listening to ones favourite music. Assume also that choices are made simultaneously and independently. Construct a strategic form for this game.

3. There are two players, PRO and RES, and 100 ounces of prime Montana huckleberries to be split between them. PRO proposes a split, and RES accepts or rejects the proposal. PRO has 3 options: she can propose a 25/75 split (25 ounces for PRO and 75 for RES), a 50/50 split (50 ounces for PRO and 50 for RES), or a 75/25 split (75 ounces for PRO and 25 for RES). RES has 2 options: she can accept (A) the proposed split, in which case the berries are split between the two players in the proposed manner; or she can reject (R) it, in which case the berries are destroyed and neither player gets any.

(a) Assume that play is simultaneous. Construct a strategic form for this game. Articulate the assumptions that you make concerning prefer- ences and payoffs.
(b) Assume that play is sequential: PRO makes the first choice; then knowing PRO choice, RES makes her choice. Construct an exten- sive form for this game. Articulate the assumptions that you make concerning preferences and payoffs.
(c) Continue with the previous part. Construct a strategic form for this sequential choice game.

4. Rock-Paper-Scissors. You probably played this game when you were a kid. There is even a society devoted to the game, the World Rock Paper Scissors Society, that sanctions tournaments. Who ever knew?The game involves two players. Let us call them ROW and COL. A players strategy set is composed of three hand signals, R, P, S. Rock R beats Scissors S, Scissors S beat Paper P, Paper P beats Rock R, and if both players choose the same signal, neither wins and neither loses. Strategies are cho- sen simultaneously. Construct a strategic form for the game. Articulate the assumptions that you make concerning preferences and payoffs.

5. Consider the game Togetherness introduced in the problem 2, when Sam and Pat made their decision simultaneously. Pat is the row player and Sam the column player and this is the payoff matrix:

Classical Country Classical 1.5,1 0.5,0.5
Country 0,0 1,1.5

Now assume that choices are made sequentially: Pat chooses first, and then knowing Pats choice, Sam makes her choice. Construct the strategic form of this sequential choice game.

 

 

DRAFT/STUDY TIPS

Introduction

The concept of strategic decision-making is a cornerstone of game theory, which explores how individuals or groups make choices in situations where the outcome depends not only on their own actions but also on the actions of others. This essay critically examines several scenarios that involve strategic decision-making, focusing on a couple, Pat and Sam, and their choices regarding attending different music events, as well as another scenario involving the division of huckleberries between two players, PRO and RES. By constructing strategic forms and extensive forms for these games, we will explore the implications of simultaneous and sequential choices, as well as the role of preferences and payoffs in determining optimal strategies.

The Togetherness Game

(a) Constructing a Strategic Form for Simultaneous Choices

Players:
The players in this game are Pat and Sam.

Strategies:
Each player has two possible strategies: attend the Classical music event or attend the Country music event.

Payoffs:
The payoffs are determined by the players' preferences, with the best outcome being attending their favorite event with their friend, the second-best outcome being attending their friend's favorite event with them, the third-best outcome being attending their favorite event alone, and the worst outcome being attending their friend's favorite event alone.

Let's denote the Classical music event by "C" and the Country music event by "R." The strategic form can be represented in a payoff matrix as follows:

	Sam: Classical (C)	Sam: Country (R)
Pat: Classical (C)	(1.5, 1)	(0.5, 0.5)
Pat: Country (R)	(0, 0)	(1, 1.5)

Explanation of Payoffs:

• If both Pat and Sam choose Classical (C, C), Pat gets a payoff of 1.5 (attending their favorite event with their friend), and Sam gets a payoff of 1 (attending their less preferred event but with their friend).
• If Pat chooses Classical and Sam chooses Country (C, R), Pat gets a payoff of 0.5 (attending their favorite event but without their friend), and Sam gets 0.5 (attending their favorite event but without their friend).
• If Pat chooses Country and Sam chooses Classical (R, C), Pat gets 0 (attending their less preferred event without their friend), and Sam gets 0 (attending their favorite event without their friend).
• If both choose Country (R, R), Pat gets 1 (attending their less preferred event with their friend), and Sam gets 1.5 (attending their favorite event with their friend).

(b) Constructing an Extensive Form for Sequential Choices

In this scenario, Pat makes her choice first, and then Sam, knowing Pat’s choice, makes his choice. The extensive form represents this sequential decision-making process using a game tree:

1. Node 1 (Pat's Decision):

   • Pat chooses Classical (C) or Country (R).

2. Node 2 (Sam's Decision):

   • If Pat chooses C, Sam chooses between C and R.
   • If Pat chooses R, Sam chooses between C and R.

The game tree will have four possible outcomes at the terminal nodes, corresponding to the strategies chosen by Pat and Sam. The payoffs are the same as in the strategic form, depending on the combination of choices.

(c) Constructing a Strategic Form for the Sequential Choice Game

The strategic form for the sequential choice game involves identifying the optimal strategies for Sam, given Pat’s choice:

	Sam: C after C	Sam: R after C	Sam: C after R	Sam: R after R
Pat: C	(1.5, 1)	(0.5, 0.5)	—	—
Pat: R	—	—	(0, 0)	(1, 1.5)

In this table, Sam’s strategies are contingent on Pat’s initial choice, and the payoffs reflect the outcomes as previously described.

(d) Strategic Form When Togetherness is Less Important

If togetherness is less important than listening to one’s favorite music, the payoffs will change, as individual preferences for the type of music will take precedence. This scenario assumes that choices are made simultaneously and independently.

	Sam: Classical (C)	Sam: Country (R)
Pat: Classical (C)	(2, 0.5)	(1, 1)
Pat: Country (R)	(0.5, 1)	(1.5, 2)

Explanation of Payoffs:

• The highest payoff now corresponds to attending one’s favorite event, regardless of the friend’s presence, reflecting the decreased importance of togetherness.

The Huckleberry Division Game

(a) Constructing a Strategic Form for Simultaneous Play

Players:
The players in this game are PRO and RES.

Strategies:
PRO has three possible strategies: propose a 25/75 split, a 50/50 split, or a 75/25 split. RES has two strategies: accept (A) or reject (R) the proposal.

Payoffs:
The payoffs depend on the split and whether RES accepts or rejects the proposal.

Assumptions:

• Both players prefer to receive more berries.
• If RES rejects the proposal, both receive 0.

	RES: Accept (A)	RES: Reject (R)
PRO: 25/75 Split	(25, 75)	(0, 0)
PRO: 50/50 Split	(50, 50)	(0, 0)
PRO: 75/25 Split	(75, 25)	(0, 0)

(b) Constructing an Extensive Form for Sequential Play

In this scenario, PRO makes the first choice, and RES, knowing PRO’s choice, decides whether to accept or reject the proposal. The extensive form is represented by a game tree:

1. Node 1 (PRO's Decision):

   • PRO chooses 25/75, 50/50, or 75/25 split.

2. Node 2 (RES's Decision):

   • For each split chosen by PRO, RES decides whether to accept (A) or reject (R).

The payoffs at the terminal nodes will reflect the outcomes based on RES’s acceptance or rejection.

(c) Constructing a Strategic Form for the Sequential Choice Game

The strategic form for the sequential choice game will have RES’s strategy contingent on PRO’s choice:

	RES: A after 25/75	RES: R after 25/75	RES: A after 50/50	RES: R after 50/50	RES: A after 75/25	RES: R after 75/25
PRO: 25/75 Split	(25, 75)	(0, 0)	—	—	—	—
PRO: 50/50 Split	—	—	(50, 50)	(0, 0)	—	—
PRO: 75/25 Split	—	—	—	—	(75, 25)	(0, 0)

The Rock-Paper-Scissors Game

Strategic Form and Assumptions

Players:
The players in this game are ROW and COL.

Strategies:
Each player has three possible strategies: Rock (R), Paper (P), or Scissors (S).

Payoffs:
The payoffs are based on the rules of the game: Rock beats Scissors, Scissors beat Paper, Paper beats Rock. If both players choose the same signal, the payoff is a tie.

	COL: R	COL: P	COL: S
ROW: R	(0, 0)	(−1, 1)	(1, −1)
ROW: P	(1, −1)	(0, 0)	(−1, 1)
ROW: S	(−1, 1)	(1, −1)	(0, 0)

Explanation of Payoffs:

• A win gives the player a payoff of 1, while a loss results in a payoff of −1. A tie results in a payoff of 0 for both players.

The Togetherness Game in Sequential Play

Returning to the Togetherness game where Sam and Pat made their decisions sequentially, the strategic form for this game can be represented as follows:

	Sam: C after C	Sam: R after C	Sam: C after R	Sam: R after R
Pat: C	(1.5, 1)	(0.5, 0.5)	—	—
Pat: R	—	—	(0, 0)	(1, 1.5)

In this strategic form, Sam's strategy is contingent on Pat's initial choice, and the payoffs reflect the outcomes as previously described. If togetherness is less important, the payoffs would adjust to reflect each player's preference for their favorite music over being together.

Conclusion

Through the exploration of these games, we see how strategic decision-making varies depending on whether choices are made simultaneously or sequentially. The payoffs, preferences, and strategies of each player influence the outcomes, highlighting the importance of understanding the strategic forms and extensive forms in game theory. These models provide valuable insights into human behavior and decision-making processes in competitive and cooperative environments.