Game theory - fundamental theory, maximin and minimax principles - Question Bank

1. Which principle is used by a player who is uncertain about the opponent's move and wants to ensure the best possible outcome even if the opponent acts in the most unfavorable way for them?
A) Minimax
B) Maximin
C) Nash Equilibrium
D) Pareto Optimality
2. In the context of game theory, 'fundamental theory' primarily deals with:
A) Advanced algorithms for solving complex games.
B) The basic concepts, assumptions, and principles of strategic interaction.
C) The historical development of game theory.
D) Applications of game theory in specific fields.
3. What is the relationship between a player's security level and the value of the game?
A) The security level is always greater than the value of the game.
B) The security level is always less than the value of the game.
C) For optimal strategies, the security level equals the value of the game.
D) They are unrelated concepts.
4. If a game has multiple saddle points, what can be said about their values?
A) They will have different values.
B) They must all have the same value.
C) Only one saddle point can exist.
D) The value is indeterminate.
5. What is the primary goal when analyzing a game using the Minimax principle?
A) To guarantee the best possible outcome.
B) To find the strategy that minimizes the maximum possible loss (or minimizes the opponent's maximum gain).
C) To maximize the minimum possible gain.
D) To ensure cooperation with the opponent.
6. What is the primary goal when analyzing a game using the Maximin principle?
A) To exploit the opponent's weaknesses.
B) To find the strategy that offers the best worst-case scenario.
C) To achieve the highest possible payoff.
D) To force the opponent into a disadvantageous position.
7. The concept of 'rationality' in game theory implies that players:
A) Always make the best possible decision.
B) Have perfect foresight.
C) Act in accordance with their preferences and available information to achieve their goals.
D) Will always cooperate.
8. What does it mean if the Maximin value is strictly less than the Minimax value in a zero-sum game?
A) A saddle point exists.
B) The game is strictly determined.
C) Mixed strategies are required to find the optimal solution.
D) Player 1 has a dominant strategy.
9. A payoff matrix is used to represent:
A) The sequence of moves in a game.
B) The payoffs for each player for every possible combination of strategies.
C) The probabilities used in mixed strategies.
D) The rules of the game.
10. What is the role of probability in a mixed strategy?
A) To determine the exact outcome of the game.
B) To represent the player's uncertainty about the opponent's moves.
C) To assign likelihoods to different pure strategies, making the choice unpredictable.
D) To calculate the total payoff.
11. If Player 1's strategy guarantees a payoff of at least $5, and Player 2's strategy guarantees Player 1's payoff is at most $5, what can be concluded?
A) Player 1 has a winning strategy.
B) Player 2 has a winning strategy.
C) The game has a value of $5, and both players are playing optimally.
D) The game requires mixed strategies.
12. What is the main difference between the Maximin and Minimax principles when applied by the same player?
A) Maximin focuses on minimizing losses, Minimax on maximizing gains.
B) Maximin assumes opponent plays optimally, Minimax assumes opponent plays to hurt you.
C) Maximin is for pure strategies, Minimax is for mixed strategies.
D) Maximin is about maximizing minimum gains, Minimax is about minimizing maximum losses.
13. In a game theory context, 'optimal strategy' refers to:
A) The strategy that guarantees the best possible outcome in all circumstances.
B) A strategy that maximizes a player's expected payoff, assuming other players also play optimally.
C) Any strategy that a player chooses to follow.
D) A strategy that leads to a win.
14. What does the term 'payoff' represent in a game theory context?
A) The cost incurred by a player.
B) The utility or value a player receives from an outcome.
C) The strategy chosen by a player.
D) The number of players involved.
15. Which of the following scenarios is best modeled by game theory?
A) A single student studying for an exam.
B) A company deciding its advertising budget independently.
C) Two competing companies setting prices.
D) A person choosing a restaurant.
16. What is the fundamental concept that connects the Maximin and Minimax principles in solving zero-sum games?
A) Nash Equilibrium
B) The Minimax Theorem
C) Game Simplification
D) Randomization
17. For the payoff matrix [[3, 1], [2, 4]], does the game have a saddle point?
A) Yes, at value 2
B) Yes, at value 3
C) No
D) Yes, at value 1
18. Consider a game with the following payoff matrix for Player 1: [[3, 1], [2, 4]]. What is the Minimax value for Player 2 (minimizing Player 1's payoff)?
A) 1
B) 2
C) 3
D) 4
19. Consider a game with the following payoff matrix for Player 1: [[3, 1], [2, 4]]. What is the Maximin value for Player 1?
A) 1
B) 2
C) 3
D) 4
20. When applying the Minimax principle, a player is acting in a manner that is:
A) Optimistic
B) Pessimistic
C) Aggressive
D) Defensive
21. When applying the Maximin principle, a player is acting in a manner that is:
A) Optimistic
B) Pessimistic
C) Aggressive
D) Cooperative
22. If a game is not strictly determined, what is the implication for the players' optimal strategies?
A) They will use pure strategies.
B) They must use mixed strategies.
C) There is no optimal strategy.
D) The game has no value.
23. What does it mean for a game to be 'strictly determined'?
A) It has a unique mixed strategy equilibrium.
B) It has a saddle point, and thus a pure strategy equilibrium.
C) It has multiple Nash equilibria.
D) The outcome is predictable with certainty.
24. In a zero-sum game, the payoff matrix for Player 1 is the negative of the payoff matrix for Player 2. This statement is:
A) Always true.
B) Sometimes true.
C) Never true.
D) True only for non-zero-sum games.
25. Which of the following best describes a 'non-zero-sum game'?
A) A game where the sum of payoffs is always negative.
B) A game where the sum of payoffs is always positive.
C) A game where the sum of payoffs is not necessarily zero; players' gains and losses are not directly opposed.
D) A game with only one player.
26. The Minimax principle aims to:
A) Maximize the opponent's minimum gain.
B) Minimize the opponent's maximum gain.
C) Maximize one's own minimum gain.
D) Minimize one's own maximum loss.
27. The Maximin principle provides a player with a strategy that guarantees:
A) The highest possible payoff.
B) The lowest possible loss.
C) A minimum payoff level (security level).
D) A win against any opponent.
28. What is the 'security level' of a player in game theory?
A) The maximum payoff achievable.
B) The minimum payoff guaranteed regardless of the opponent's strategy.
C) The payoff achieved at a Nash Equilibrium.
D) The average payoff over all possible outcomes.
29. If Player 1 uses a mixed strategy, what does this mean?
A) Player 1 chooses one action randomly.
B) Player 1 chooses a sequence of actions.
C) Player 1 assigns probabilities to each of their pure strategies and chooses accordingly.
D) Player 1 always plays the same action.
30. What is the primary challenge when a game does not have a saddle point?
A) It is impossible to determine the value of the game.
B) Players must resort to mixed strategies, making analysis more complex.
C) The game is no longer zero-sum.
D) Rational play is not assumed.
31. Consider Player 2 (column player) facing a payoff matrix (where payoffs are for Player 1). Player 2 calculates the maximum payoff Player 1 could receive for each of Player 2's possible columns. Then, Player 2 chooses the column that yields the minimum among these maximum payoffs. This is:
A) Maximin strategy
B) Minimax strategy
C) Pure strategy
D) Equilibrium strategy
32. Consider Player 1 (row player) facing a payoff matrix. Player 1 calculates the minimum payoff for each of their possible rows. Then, Player 1 chooses the row that yields the maximum among these minimum payoffs. This is:
A) Minimax strategy
B) Maximin strategy
C) Mixed strategy
D) Saddle point strategy
33. What is the significance of the 'minimax theorem' (for zero-sum games)?
A) It proves that mixed strategies are always necessary.
B) It states that every zero-sum game has a value, and the Maximin value equals the Minimax value.
C) It guarantees that a saddle point always exists.
D) It introduces the concept of Nash Equilibrium.
34. In a 3x3 payoff matrix for a zero-sum game, how many pure strategies does each player have?
A) 1
B) 2
C) 3
D) 9
35. What is the fundamental assumption about players in most classical game theory models?
A) Players are altruistic.
B) Players are irrational.
C) Players are rational and self-interested.
D) Players have incomplete information.
36. Which of the following is NOT a characteristic of a rational player in game theory?
A) They aim to maximize their own payoff.
B) They can perfectly predict the opponent's moves.
C) They act consistently based on their objectives.
D) They understand the rules and payoffs of the game.
37. For a game to have a saddle point, the following condition must be met:
A) Maximum of row minimums > Minimum of column maximums
B) Maximum of row minimums < Minimum of column maximums
C) Maximum of row minimums = Minimum of column maximums
D) Sum of all elements is zero.
38. If the Maximin value of a game is 5 and the Minimax value is 7, what can be concluded?
A) The game has a saddle point with value 5.
B) The game has a saddle point with value 7.
C) The game does not have a saddle point, and players must use mixed strategies.
D) The value of the game is 6.
39. What is the 'value of the game'?
A) The maximum payoff Player 1 can guarantee.
B) The minimum loss Player 2 can guarantee.
C) The expected payoff when both players play their optimal strategies.
D) The sum of all payoffs in the matrix.
40. Consider a payoff matrix where Player 1 is the row player and Player 2 is the column player. Player 1 wants to maximize their payoff, and Player 2 wants to minimize Player 1's payoff. Player 2 will choose the column that gives the minimum of the column maximums. This is applying the principle of:
A) Maximin
B) Minimax
C) Pure Strategy
D) Saddle Point
41. Consider a payoff matrix where Player 1 is the row player and Player 2 is the column player. Player 1 wants to maximize their payoff, and Player 2 wants to minimize Player 1's payoff (as it's a zero-sum game). Player 1 will choose the row that gives the maximum of the row minimums. This is applying the principle of:
A) Minimax
B) Maximin
C) Nash Equilibrium
D) Zero-sum
42. If a game has a saddle point, what does this imply about the optimal strategies?
A) Both players must use mixed strategies.
B) The optimal strategies are pure strategies.
C) The game has no unique solution.
D) The value of the game is zero.
43. What is a 'saddle point' in the context of a payoff matrix?
A) The element with the highest value in the matrix.
B) The element with the lowest value in the matrix.
C) An element that is the minimum in its row and the maximum in its column.
D) An element that is the maximum in its row and the minimum in its column.
44. In a two-person zero-sum game, if a saddle point exists, what can be said about the Maximin value and the Minimax value?
A) Maximin value is greater than Minimax value.
B) Minimax value is greater than Maximin value.
C) Maximin value equals Minimax value.
D) They are unrelated.
45. The Minimax principle is associated with which player's perspective in a zero-sum game?
A) The player who assumes the opponent will act to minimize their own payoff.
B) The player who seeks to maximize their minimum possible gain.
C) The player who wants to minimize their maximum possible payoff.
D) The player who assumes the opponent will always play randomly.
46. The Maximin principle is associated with which player's perspective in a zero-sum game?
A) The player who wants to maximize their payoff.
B) The player who wants to minimize their maximum possible loss.
C) The player who assumes the opponent will always play optimally.
D) The player who seeks to achieve a saddle point.
47. What is a 'mixed strategy' in game theory?
A) A strategy involving a combination of pure strategies with associated probabilities.
B) A strategy that is not optimal.
C) A strategy used only in non-zero-sum games.
D) A strategy where players take turns making moves.
48. What does the term 'pure strategy' refer to in game theory?
A) A strategy where a player randomly chooses between actions.
B) A strategy where a player always chooses a specific action with probability 1.
C) A strategy that guarantees a win for the player.
D) A strategy that considers the opponent's mixed strategies.
49. In a two-person zero-sum game, what is the sum of the gains and losses of the players?
A) Always positive
B) Always negative
C) Always zero
D) Can be any real number
50. What is the primary objective of Game Theory?
A) To find the optimal strategy for a single decision-maker in isolation.
B) To model and analyze strategic interactions between rational decision-makers.
C) To predict random events with certainty.
D) To optimize resource allocation in non-competitive scenarios.
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