Free Printable Worksheets for learning Game theory at the College level

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Game Theory Info Sheet

Key Concepts

  • Game theory is the study of decision-making in strategic situations where the outcome of one person's choice depends on the choices of others.
  • Players are the decision-makers in the game.
  • The payoff is what a player gets as a result of the game.
  • Nash equilibrium is a solution concept of a non-cooperative game where each player's strategy is optimal given the strategies of all other players.
  • Dominant strategy is a strategy that is the best response for a player regardless of what other players do.
  • Prisoner's Dilemma is a classic example of a game theory scenario in which players each have a dominant strategy that leads to a suboptimal outcome for both players.

Strategies and Games

  • A strategy is a complete plan of action for a player in a game.
  • A mixed strategy is a probabilistic plan of action where the player randomly chooses a strategy based on a predetermined probability distribution.
  • A game tree is a visual representation of a game and its possible outcomes.
  • One-shot games are games played once, while repeated games are games played multiple times.

Applications

  • Game theory is used in economics to model and analyze markets, auctions, and pricing strategies.
  • It is also used in political science to model international relations, voting systems, and strategic decision-making.
  • Game theory has applications in computer science, biology, and psychology.

Takeaways

  • Game theory is the study of decision-making in strategic situations.
  • Nash equilibrium and dominant strategies are important solution concepts in game theory.
  • Strategies can be complete or mixed, and games can be represented by a game tree.
  • Game theory has applications in various fields, including economics, political science, and computer science.

Here's some sample Game theory vocabulary lists Sign in to generate your own vocabulary list worksheet.

Word Definition
Strategy A plan, method, or series of actions designed to achieve a specific goal.
Decision tree A graphical tool used to map out all possible decision paths and outcomes of a given decision scenario.
Nash equilibrium A concept in game theory that describes a state of affairs, where all players involved in a game will have no incentive to change their strategy, as long as the other players keep theirs unchanged as well.
Prisoner's dilemma A common example of game theory, which involves two players who are faced with a situation where they must choose whether to cooperate or betray each other, without knowing what the other will choose. In this game, the best outcome for both players would be to cooperate, but each player has an incentive to betray the other, which leads to a suboptimal outcome for both players.
Dominant strategy A strategy that is always the best option for a player, no matter what the other player chooses to do.
Payoff matrix A grid used to represent the possible outcomes of a game, where each cell lists the payoffs received by both players for a given combination of strategies.
Game theory A branch of mathematics that studies decision-making in situations where two or more individuals or groups have conflicting interests. Game theory provides a framework for analyzing complex interactions between rational decision-makers, and it has applications in a wide range of fields, including economics, politics, and psychology.
Equilibrium A state of balance or stability in a system. In game theory, an equilibrium occurs when all players involved in a game have chosen their optimal strategies given the other players' strategies.
Mixed strategy A strategy that involves a player choosing among several possible actions randomly, with each action being chosen with a certain probability. A mixed strategy can be used to avoid becoming predictable to an opponent in a game, and it can make it harder for the opponent to exploit weaknesses in the player's strategy.
Zero-sum game A type of game wherein the payoff for one player is exactly equal to the penalty suffered by the other player. In other words, one player's gain is exactly equal to the other player's loss. This means that there is no potential gain for both players at the same time. Therefore, the interests of the players in a zero-sum game are always opposed to each other.
Sequential game A type of game in which players make their moves in a specific order, with each player observing the moves of the previous players before making their own move. This type of game is often represented using a decision tree, and it can be used to model a wide range of real-world situations, such as bargaining, auctions, and negotiations.
Coordination game A type of game where the players can benefit from coordinating their actions. In a coordination game, the players must choose a strategy that is compatible with the strategy of the other players, in order to achieve the desired outcome.
Tragedy of the commons A concept in game theory, where individuals, acting in their own self-interest, behave contrary to the common good of all users of a shared resource. This creates a negative outcome for the group or community as a whole.
Cooperation A strategy in a game where two players work together to achieve a common goal. Cooperation can be difficult to achieve in a non-zero-sum game, where the interests of the players are not diametrically opposed.
Iterated game A game in which a sequence of similar games are played repeatedly over time. In iterated games, the outcome of one game can affect the choices made by players in future games, as players have a chance to learn from their past experiences.
Subgame In game theory, a subgame is a smaller version of a game that is reached after a certain sequence of moves has been played. The concept of a subgame is often used to simplify the analysis of complex games.
Pareto efficiency A state of affairs in a game where no player can improve their outcome without making another player worse off. In other words, it is impossible to make one player better off without making another player worse off. A Pareto efficient outcome is considered the most optimal outcome, as it maximizes the overall welfare of the players involved.
Free rider A person who benefits from a group effort but does not contribute to the group effort. In game theory, free riding can be a problem in situations where individuals are expected to contribute a certain amount for the benefit of everyone, such as in common resource situations.
Rationality The concept of rationality in game theory refers to the assumption that all players involved in a game are rational decision-makers. Rationality means that each player chooses their strategy based on an evaluation of the impact that it will have, given the other players' strategies and the likely outcomes of the game.
Asymmetric game A type of game in which one player has a different or superior set of resources or abilities than the other player. This can create an uneven or unfair situation in which the player with fewer resources or abilities has little chance of winning. Asymmetric games require specialized analysis because the strategies used by players must be adjusted to account for the differences in resources or abilities.

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Study Guide: Game Theory

Game theory is the mathematical study of decision making, conflict resolution, and strategy in social situations. It is a fundamental tool used in economics, political science, psychology, and other social sciences. This study guide is designed to introduce you to the basic concepts and techniques of game theory.

Introduction

What is game theory?

  • Definition of game theory
  • Historical context
  • Application areas

Game theory concepts

  • Players
  • Strategies
  • Payoffs
  • Nash equilibrium

Games and Strategies

Normal-form Games

  • Definition of normal-form game
  • Matrix representation
  • Example games: Prisoner's dilemma, Matching pennies, Coordination game

Dominance

  • Dominant strategies
  • Iterated elimination of dominated strategies (IEDS)
  • Example

Mixed Strategies

  • Definition of mixed strategy
  • Calculation of expected payoffs
  • Example

Extensive-form Games

  • Definition of extensive-form game
  • Tree representation
  • Backward induction
  • Example: Centipede game

Applications in Social Science

Economics

  • Oligopoly and monopolistic competition
  • Auctions
  • Public goods and externalities
  • Mechanism design

Political Science

  • Voting theory
  • Bargaining theory

Psychology

  • Behavioral game theory
  • Evolutionary game theory

Conclusion

Limitations and criticism

  • Assumptions and simplifications
  • Empirical testing
  • Alternative theories

Future directions

  • Behavioral economics
  • Network theory
  • Agent-based modeling

Here's some sample Game theory practice sheets Sign in to generate your own practice sheet worksheet.

Game Theory Practice Sheet

Question 1

Consider the following normal form game:

L R
U 3,5 0,0
D 1,1 4,2

(a) Write down the strategy form game.
(b) Find all pure strategy Nash equilibria.
(c) Can you find a mixed stategy Nash equilibrium?

Question 2

Consider the following normal form game:

L M R
U 1,1 0,0 0,3
D 0,0 2,2 3,0

(a) Write down the strategy form game.
(b) Find all pure strategy Nash equilibria.
(c) Can you find a mixed stategy Nash equilibrium?

Question 3

Consider a Stackelberg game with two firms with identical constant marginal cost c > 0. Firm 1 is the leader and observes its marginal cost and chooses its output quantity first. Then, firm 2, the follower, observes the output choice of firm 1 and chooses its output quantity.

(a) Write down the payoff function of each firm.
(b) Find the Stackelberg equilibrium.

Question 4

Consider a Cournot game with quantity-setting firms. There are two firms in the market and they have constant and identical marginal cost c > 0. Their demand function is p = a − bQ, where Q is the total amount of output and a, b > 0.

(a) Write down the profit function of each firm.
(b) Find the Nash equilibrium.

Question 5

Players A and B play a game. Each chooses left (L) or right (R). If both players choose L, player A gets 2 units and player B gets 1 unit. If both players choose R, player A gets 1 unit and player B gets 2 units. If the players choose differently, both get 0. The players choose simultaneously and independently.

(a) Construct the normal form of the game.
(b) Find all pure strategy Nash equilibria.
(c) Can you find a mixed strategy Nash equilibrium?

Question 6

Consider the following game where two players play rock-paper-scissors by choosing rock (R), paper (P), or scissors (S):

R P S
R 0,0 0,1 1,0
P 1,0 0,0 0,1
S 0,1 1,0 0,0

(a) Construct the normal form of the game.
(b) Find all pure strategy Nash equilibria.
(c) Can you find a mixed strategy Nash equilibrium?

Question 7

Consider a coordination game with two players where the payoff function is as follows:

L R
L 2,1 0,0
R 0,0 1,2

(a) Are there any dominant strategies?
(b) Find all pure strategy Nash equilibria.
(c) Can you find a mixed strategy Nash equilibrium?

Question 8

Consider a prisoner's dilemma game. Two players, A and B, have the following payoff matrix:

C D
C 2,2 0,3
D 3,0 1,1

(a) Is there any dominant strategy for either player?
(b) Find all pure strategy Nash equilibria.
(c) Can you find a mixed strategy Nash equilibrium?

Question 9

Consider a Hotelling's game with two firms selling a homogeneous product that is located at the endpoints of a line of length 1. Each consumer is located uniformly at random on the unit interval and has transportation cost t > 0 per unit distance. If two firms locate at x and y, respectively, then consumers choose the closest store.

(a) Write down the profit function of each firm.
(b) Find the location of the firms in the Nash equilibrium.

Question 10

Consider a war of attrition game between two players. The value of the contest for each player is H/2 where H is the common value of the prize. The players choose to quit the contest at some random time. The player who quits first wins nothing, and the other player wins the prize minus the total elapsed time.

(a) Write down the payoff function of each player.
(b) Find the symmetric Nash equilibrium.

Practice Sheet for Game Theory

Problem 1

Consider a game with two players, A and B. Player A has two strategies, X and Y, and player B has two strategies, P and Q. The payoff matrix is given below:

P Q
X 2,1 0,3
Y 3,0 1,2

What is the Nash equilibrium of this game?

Problem 2

Consider a game with two players, A and B. Player A has two strategies, X and Y, and player B has two strategies, P and Q. The payoff matrix is given below:

P Q
X 3,2 0,4
Y 4,0 1,3

What is the Nash equilibrium of this game?

Problem 3

Consider a game with two players, A and B. Player A has two strategies, X and Y, and player B has two strategies, P and Q. The payoff matrix is given below:

P Q
X 4,3 0,5
Y 5,0 1,4

What is the Nash equilibrium of this game?

Problem 4

Consider a game with two players, A and B. Player A has two strategies, X and Y, and player B has two strategies, P and Q. The payoff matrix is given below:

P Q
X 5,4 0,6
Y 6,0 1,5

What is the Nash equilibrium of this game?

Problem 5

Consider a game with two players, A and B. Player A has two strategies, X and Y, and player B has two strategies, P and Q. The payoff matrix is given below:

P Q
X 6,5 0,7
Y 7,0 1,6

What is the Nash equilibrium of this game?

Game Theory Practice Sheet

Introduction

Game theory is a branch of mathematics that studies the strategic interactions between different players in a game. It is used to analyze a wide range of situations, from business decisions to political campaigns and even military conflicts. This practice sheet will help you understand the basic concepts of game theory and how they can be applied in various situations.

Questions

  1. What is the core concept of game theory?
  2. What are the two types of game theory?
  3. What is a Nash equilibrium?
  4. What is the difference between a cooperative and a non-cooperative game?
  5. How can game theory be used to analyze business decisions?
  6. What is the Prisoner's Dilemma?
  7. What is the difference between zero-sum and non-zero-sum games?
  8. What is the difference between a pure-strategy and a mixed-strategy?
  9. How can game theory be used to analyze political campaigns?
  10. What are the different types of game theory models?

Here's some sample Game theory quizzes Sign in to generate your own quiz worksheet.

Below is a quiz to help you test your mastery of Game theory. The quiz will test different aspects of knowledge and insight about Game theory. The quiz is tailored to your College level of understanding about Game theory. The quiz is formatted in a table with the problems on the left and answers on the right. Good luck!

Problem Answer
What is Game theory? Game theory is a branch of mathematics that studies decision-making in situations where two or more individuals or organizations have conflicting interests.
What are the key elements of a game? The key elements of a game are players, strategies, payoffs, and rules.
What are the differences between simultaneous and sequential games? Simultaneous games are those in which players move simultaneously, while in sequential games, players move sequentially.
What is the Nash equilibrium? The Nash equilibrium is a stable state of a game in which no player can improve their payoff by changing their strategy, assuming the other players' strategies remain unchanged.
What is a dominated strategy? A dominated strategy is a strategy that is always worse than another strategy, regardless of what the other players do.
What is the minimax theorem? The minimax theorem states that in a two-player zero-sum game, there exists a value called the minimax value, which represents the best worst-case outcome for each player.
What is a mixed strategy? A mixed strategy is a strategy in which a player chooses each possible pure strategy with a certain probability.
What is a subgame perfect equilibrium? A subgame perfect equilibrium is a strategy profile that constitutes a Nash equilibrium of every subgame of the original game.
What is the difference between a cooperative and a non-cooperative game? In a cooperative game, players can form coalitions and jointly decide on strategies, while in a non-cooperative game, players act independently and do not form coalitions.
Can a game have more than one Nash equilibrium? Yes, a game can have more than one Nash equilibrium.

Note: This quiz is not exhaustive and is meant to provide a glimpse of the types of questions and topics that may be covered in a Game theory course.

Question Answer
What is a game theory? Game theory is a branch of mathematics that studies strategic decision making in situations where multiple players are involved. It is used to analyze and predict the outcomes of competitive situations and to determine optimal strategies for each player.
What is a Nash equilibrium? A Nash equilibrium is a state of a game where no player has an incentive to change their strategy. This is because if a player were to change their strategy, they would not be able to improve their outcome.
What is a zero-sum game? A zero-sum game is a game where the total gains and losses of all players sum to zero. This means that for every gain, there must be an equal loss.
What is a cooperative game? A cooperative game is a game where players can form coalitions and make binding agreements. This allows players to cooperate and share the gains from the game.
What is a dominant strategy? A dominant strategy is a strategy that is always the best choice for a player, regardless of the strategies chosen by the other players. It is a strategy that is always the most beneficial for a player.
What is a pure strategy? A pure strategy is a strategy that is always chosen by a player, regardless of the strategies chosen by the other players. It is a strategy that is always chosen by a player without any consideration for the strategies chosen by the other players.
What is a mixed strategy? A mixed strategy is a strategy that is chosen by a player with some consideration for the strategies chosen by the other players. It is a strategy that is chosen by a player with the intention of maximizing their expected payoff.
What is a payoff matrix? A payoff matrix is a matrix that shows the payoffs for each player in a game. It is used to analyze the outcomes of different strategies and to determine the best strategy for each player.
What is a strategic form game? A strategic form game is a game where the strategies of the players are known to all players. It is a game where the strategies of the players are known before the game begins.
What is a normal form game? A normal form game is a game where the strategies of the players are not known to all players. It is a game where the strategies of the players are not known before the game begins.

Quiz on Game Theory at the College Level

Questions Answers
What is the definition of game theory? Game theory is the study of mathematical models of strategic interaction between rational decision-makers.
What is the Nash equilibrium? The Nash equilibrium is a solution concept of a game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.
What is the Prisoner's Dilemma? The Prisoner's Dilemma is a classic example of a game analyzed in game theory that shows why two individuals might not cooperate, even if it appears that it is in their best interests to do so.
What is the minimax theorem? The minimax theorem is a fundamental concept in game theory which states that in a two-player zero-sum game, the player who makes the first move can guarantee at least a draw by playing the strategy that minimizes the maximum loss.
What is a mixed strategy? A mixed strategy is a probability distribution over a set of pure strategies, which is used by a player when they are uncertain of which pure strategy to use.
What is a dominant strategy? A dominant strategy is a strategy that is always better than any other strategy, regardless of what strategy the other player chooses.
What is a dominant strategy equilibrium? A dominant strategy equilibrium is a Nash equilibrium in which all players have a dominant strategy.
What is the Stackelberg game? The Stackelberg game is a game in which one player, the leader, makes their move first and the other player, the follower, makes their move second.
What is the Pareto efficiency? The Pareto efficiency is a state of allocation of resources in which it is impossible to make any one individual better off without making at least one individual worse off.
What is the Shapley value? The Shapley value is a solution concept in cooperative game theory that assigns a value to each player in a cooperative game based on their contribution to the total payoff of the game.
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