Explanation: The foundational text 'Theory of Games and Economic Behavior' (1944) by John von Neumann and Oskar Morgenstern is credited with introducing the formal study of cooperative games and developing an axiomatic framework for expected utility theory.

Explanation: Contrary to the statement, numerous game theorists have received the Nobel Memorial Prize in Economic Sciences. As of 2020, fifteen game theorists had been recognized, underscoring the field's profound impact on economic thought.

Explanation: Ernst Zermelo's seminal 1913 paper demonstrated that games with perfect information, like chess, have optimal strategies that are strictly determined, utilizing foundational concepts from set theory.

Explanation: John von Neumann's significant 1928 paper established game theory as a distinct field, using the Brouwer fixed-point theorem to prove the minimax theorem. It did not focus solely on biological evolution.

Explanation: The 1950s were indeed a pivotal decade for game theory, witnessing the formalization and introduction of crucial concepts like the core, extensive form representations, and the Shapley value, alongside applications in philosophy and political science.

Explanation: John von Neumann and Oskar Morgenstern are widely recognized for establishing modern game theory with their collaborative work, most notably the book 'Theory of Games and Economic Behavior'.

Explanation: John von Neumann's pivotal contribution was the proof of the minimax theorem, which establishes the existence of optimal mixed strategies for two-person zero-sum games.

Explanation: The source indicates that fifteen game theorists had received the Nobel Memorial Prize in Economic Sciences by the year 2020, highlighting the field's significant influence on economic scholarship.

Explanation: Ernst Zermelo demonstrated that games with perfect information, like chess, possess strictly determined optimal strategies, meaning a player can always force a win or a draw if they play optimally.

Explanation: The 1950s saw the introduction of several important concepts; the Shapley value, a method for fairly distributing the gains generated by a coalition of players, is a prominent example from this era.

Explanation: The Nash equilibrium was developed by John Nash in 1950, providing a fundamental framework for analyzing strategic interactions in non-cooperative settings.

Explanation: The film 'A Beautiful Mind,' as referenced in the source material, depicts the life and groundbreaking work of the distinguished game theorist John Nash.

Explanation: The foundational definition of game theory posits it as the study of mathematical models analyzing strategic interactions among multiple agents, rather than isolated decision-making.

Explanation: Historical accounts indicate that early game theory predominantly concentrated on two-person zero-sum games. The expansion to non-zero-sum games occurred later, becoming a comprehensive science of rational decision-making.

Explanation: This assertion is incorrect. While the book addressed two-person zero-sum games, it significantly expanded the field by introducing concepts for cooperative games and developing an axiomatic theory of expected utility.

Explanation: This statement accurately defines a Nash equilibrium. It represents a state where each player's chosen strategy is the best response to the strategies chosen by the other players, precluding any unilateral incentive to deviate.

Explanation: This is contrary to the definition of cooperative games. Cooperative games are characterized by the possibility of players forming binding agreements and coalitions to achieve mutually beneficial outcomes.

Explanation: This description aligns more closely with cooperative game theory. Non-cooperative game theory typically analyzes individual player strategies and payoffs, often focusing on equilibria like the Nash equilibrium.

Explanation: In a symmetric game, players typically have identical strategy sets. The symmetry arises from the fact that the payoff structure is the same for all players; if player A plays strategy X and player B plays strategy Y, player A receives the same payoff as player B would if player B played X and player A played Y.

Explanation: A game can be asymmetric even if players have identical strategy sets. Asymmetry arises when the payoff functions differ for players choosing the same strategies, or when players have different strategy sets.

Explanation: This statement is incorrect. By definition, in a zero-sum game, the total gains and losses across all players must sum precisely to zero for every possible outcome.

Explanation: This is a mischaracterization. Non-zero-sum games allow for outcomes where all players can benefit, all can lose, or some can gain while others lose, but the sum of gains and losses is not necessarily zero.

Explanation: This acronym correctly identifies the fundamental components necessary to formally specify a game in game theory: the set of players, their available actions or strategies, the payoffs resulting from all possible outcomes, and the information available to players at each decision point.

Explanation: This is incorrect. The mutually beneficial outcome in the Prisoner's Dilemma occurs when both players cooperate (e.g., remain silent). The outcome where both betray each other is typically suboptimal for both compared to mutual cooperation.

Explanation: The source identifies game theory as the study of mathematical models that analyze strategic interactions, finding application across various fields to understand rational decision-making among multiple interacting agents.

Explanation: A Nash equilibrium is defined as a state where each player's chosen strategy is the best response to the strategies chosen by the other players. Consequently, no player can unilaterally improve their payoff by changing their strategy alone.

Explanation: The primary distinction lies in the enforceability of agreements. Cooperative games permit players to form binding commitments, often supported by external enforcement mechanisms. Non-cooperative games, conversely, mandate that agreements be self-enforcing, relying on credible threats and incentives, as external alliances are not permissible.

Explanation: A zero-sum game is characterized by the condition that the aggregate sum of all players' gains and losses equals zero for every possible combination of strategies. This implies that any advantage gained by one player is precisely matched by losses incurred by other players, resulting in no net change in total resources.

Explanation: The Prisoner's Dilemma illustrates a situation where the dominant strategy for each individual player (to betray) leads to a collectively suboptimal outcome (mutual betrayal) compared to the outcome achieved through mutual cooperation.

Explanation: The "Battle of the Sexes" game models a conflict where two players desire to coordinate their actions but hold differing preferences over the potential outcomes. It characteristically features two pure strategy Nash equilibria, occurring when both players select the same option, alongside potentially infinite mixed strategy equilibria, illustrating scenarios where coordination is sought despite divergent preferences.

Explanation: The Ultimatum Game demonstrates that human economic decisions are often influenced by factors beyond pure self-interest, such as perceptions of fairness and the desire for social acceptance, leading to behavior that deviates from predictions based solely on pure self-interest.

Explanation: This describes simultaneous games, not sequential ones. Sequential games involve players making decisions in a specific order, where later players are aware of the actions taken by earlier players.

Explanation: This statement is incorrect. A game of perfect information requires players to know the entire history of previous moves made by all players. Knowledge of payoffs and strategies pertains to complete information.

Explanation: This describes perfect information, not complete information. Complete information means players know each other's available strategies and payoff functions, but not necessarily the specific sequence of moves. Games can have complete information but imperfect information, like many card games.

Explanation: This is incorrect. Bayesian games are specifically designed to model situations with *incomplete* information, where players have uncertainty about certain characteristics (like payoffs or types) of other players.

Explanation: The primary source of complexity in combinatorial games stems from the sheer number of possible moves and sequences, rather than psychological factors or bluffing. Games like chess and Go exemplify this.

Explanation: This statement is incorrect. Continuous games involve strategy spaces or outcome spaces that are continuous, allowing for an infinite number of choices or results, unlike discrete games.

Explanation: This assumption is characteristic of classical game theory, not evolutionary game theory. Evolutionary game theory often relaxes the assumption of perfect rationality, focusing instead on strategy dynamics and fitness-based selection mechanisms.

Explanation: This is a standard technique for incorporating randomness into game models. The 'move by nature' player represents chance events, allowing for the analysis of games with uncertain outcomes.

Explanation: A game of perfect information is defined by the condition that at every decision point, the player whose turn it is knows the complete history of all previous moves made by all players in the game.

Explanation: Bayesian games are characterized by incomplete information, where players may have uncertainty regarding certain characteristics (like payoffs or types) of other players. This uncertainty is modeled using probability distributions over possible states of the world.

Explanation: The primary source of complexity in combinatorial games like chess and Go arises from the vast number of possible sequences of moves and resulting states, rather than from psychological factors or strategic bluffing.

Explanation: Stochastic outcomes are commonly modeled by introducing a hypothetical 'move by nature' player. This player's actions represent the random events, allowing for the integration of uncertainty into the game's structure.

Explanation: Evolutionary game theory often relaxes the strict assumption of perfect rationality found in classical game theory. Instead, it focuses on strategy dynamics driven by factors like imitation, adaptation, and fitness, which may not require players to be perfectly rational.

Explanation: Perfect information signifies complete knowledge of all past moves made within the game. Complete information pertains to players knowing each other's available strategies and payoff functions, without necessarily knowing the specific moves executed. It is possible for a game to possess complete information while lacking perfect information.

Explanation: This is incorrect. Payoff matrices (normal form) are typically used to represent simultaneous games. Sequential games are more commonly represented using game trees (extensive form).

Explanation: This is incorrect. While backward induction is standard for finite sequential games, differential games, which involve continuous time and state dynamics, are generally solved using optimal control theory methods.

Explanation: Simultaneous move games are most commonly represented using the normal form, which is typically depicted as a payoff matrix detailing players' strategies and their corresponding outcomes.

Explanation: The extensive form represents games using a sequential structure, typically a game tree, detailing players' moves and information sets. The normal form, conversely, uses payoff matrices to represent players' strategies and outcomes, primarily for simultaneous move games.

Explanation: Differential games are typically solved using principles derived from optimal control theory, such as the Pontryagin maximum principle or dynamic programming, rather than backward induction used for finite sequential games.

Explanation: Charles Waldegrave's 1713 work offered an early contribution to game theory by analyzing the game 'le her' and deriving a minimax mixed strategy solution, now referred to as the Waldegrave problem.

Explanation: This statement is factually inaccurate regarding the field of application. John Maynard Smith primarily applied game theory to biology in the 1970s, developing the concept of the evolutionarily stable strategy (ESS) to model biological evolution and behavior.

Explanation: John Maynard Smith applied game theory to the field of biology in the 1970s, developing evolutionary game theory and the concept of the evolutionarily stable strategy (ESS).

Explanation: The source explicitly mentions the application of game theory to the analysis of market structures, such as oligopolies, within the field of economics.

Explanation: In political science, game theory is utilized to model the strategic interactions among various political actors, including voters, interest groups, and politicians, providing insights into electoral processes, policy formation, and political stability.

Explanation: When evolutionary game theory is applied to biology, the concept of 'payoffs' is generally interpreted as biological fitness, representing the reproductive success of an organism possessing a particular strategy.

Explanation: Game theory provides foundational principles for multi-agent systems and artificial intelligence planning, particularly in scenarios involving uncertainty and strategic interaction among computational agents.