Game theory - fundamental theory, maximin and minimax principles - One Line Questions

1. In a 3x3 payoff matrix for a zero-sum game, how many pure strategies does each player have? — 3
2. Consider a game with the following payoff matrix for Player 1: [[3, 1], [2, 4]]. What is the Maximin value for Player 1? — 2
3. 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)? — 3
4. Which of the following best describes a 'non-zero-sum game'? — A game where the sum of payoffs is not necessarily zero; players' gains and losses are not directly opposed.
5. What does it mean if the Maximin value is strictly less than the Minimax value in a zero-sum game? — Mixed strategies are required to find the optimal solution.
6. Which of the following scenarios is best modeled by game theory? — Two competing companies setting prices.
7. What is a 'mixed strategy' in game theory? — A strategy involving a combination of pure strategies with associated probabilities.
8. What does the term 'pure strategy' refer to in game theory? — A strategy where a player always chooses a specific action with probability 1.
9. In the context of game theory, 'fundamental theory' primarily deals with: — The basic concepts, assumptions, and principles of strategic interaction.
10. The concept of 'rationality' in game theory implies that players: — Act in accordance with their preferences and available information to achieve their goals.
11. In a two-person zero-sum game, what is the sum of the gains and losses of the players? — Always zero
12. In a zero-sum game, the payoff matrix for Player 1 is the negative of the payoff matrix for Player 2. This statement is: — Always true.
13. If a game has a saddle point, what does this imply about the optimal strategies? — The optimal strategies are pure strategies.
14. What does it mean for a game to be 'strictly determined'? — It has a saddle point, and thus a pure strategy equilibrium.
15. What is the primary challenge when a game does not have a saddle point? — Players must resort to mixed strategies, making analysis more complex.
16. What is the significance of the 'minimax theorem' (for zero-sum games)? — It states that every zero-sum game has a value, and the Maximin value equals the Minimax value.
17. 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: — Minimax
18. What is the main difference between the Maximin and Minimax principles when applied by the same player? — Maximin is about maximizing minimum gains, Minimax is about minimizing maximum losses.
19. 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: — Minimax strategy
20. In a two-person zero-sum game, if a saddle point exists, what can be said about the Maximin value and the Minimax value? — Maximin value equals Minimax value.
21. The Minimax principle aims to: — Minimize the opponent's maximum gain.
22. For a game to have a saddle point, the following condition must be met: —
23. 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: — Maximin
24. 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? — Maximin
25. 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: — Maximin strategy
26. What is the fundamental concept that connects the Maximin and Minimax principles in solving zero-sum games? — The Minimax Theorem
27. When applying the Maximin principle, a player is acting in a manner that is: — Pessimistic
28. When applying the Minimax principle, a player is acting in a manner that is: — Defensive
29. If Player 1 uses a mixed strategy, what does this mean? — Player 1 assigns probabilities to each of their pure strategies and chooses accordingly.
30. 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? — The game has a value of $5, and both players are playing optimally.
31. What is the fundamental assumption about players in most classical game theory models? — Players are rational and self-interested.
32. What does the term 'payoff' represent in a game theory context? — The utility or value a player receives from an outcome.
33. What is a 'saddle point' in the context of a payoff matrix? — An element that is the minimum in its row and the maximum in its column.
34. If the Maximin value of a game is 5 and the Minimax value is 7, what can be concluded? — The game does not have a saddle point, and players must use mixed strategies.
35. The Maximin principle provides a player with a strategy that guarantees: — A minimum payoff level (security level).
36. What is the 'security level' of a player in game theory? — The minimum payoff guaranteed regardless of the opponent's strategy.
37. What is the 'value of the game'? — The expected payoff when both players play their optimal strategies.
38. The Minimax principle is associated with which player's perspective in a zero-sum game? — The player who assumes the opponent will act to minimize their own payoff.
39. The Maximin principle is associated with which player's perspective in a zero-sum game? — The player who wants to minimize their maximum possible loss.
40. What is the relationship between a player's security level and the value of the game? — For optimal strategies, the security level equals the value of the game.
41. A payoff matrix is used to represent: — The payoffs for each player for every possible combination of strategies.
42. In a game theory context, 'optimal strategy' refers to: — A strategy that maximizes a player's expected payoff, assuming other players also play optimally.
43. Which of the following is NOT a characteristic of a rational player in game theory? — They can perfectly predict the opponent's moves.
44. If a game has multiple saddle points, what can be said about their values? — They must all have the same value.
45. If a game is not strictly determined, what is the implication for the players' optimal strategies? — They must use mixed strategies.
46. What is the role of probability in a mixed strategy? — To assign likelihoods to different pure strategies, making the choice unpredictable.
47. What is the primary goal when analyzing a game using the Maximin principle? — To find the strategy that offers the best worst-case scenario.
48. What is the primary objective of Game Theory? — To model and analyze strategic interactions between rational decision-makers.
49. What is the primary goal when analyzing a game using the Minimax principle? — To find the strategy that minimizes the maximum possible loss (or minimizes the opponent's maximum gain).
50. For the payoff matrix [[3, 1], [2, 4]], does the game have a saddle point? — No
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