Game Theory Unit 3 Review: Simultaneous Move Games and Nash Equilibrium

Key Concepts

• Simultaneous move games involve players making decisions simultaneously without knowledge of the other players' choices
• Players in a simultaneous move game choose strategies based on their preferences and the expected payoffs
• Nash equilibrium is a key concept in simultaneous move games representing a stable state where no player has an incentive to unilaterally change their strategy
• Payoff matrices are used to represent the outcomes and payoffs for each combination of strategies chosen by the players
• Best response is a player's optimal strategy given the strategies of the other players
• Dominant strategies are those that provide the highest payoff for a player regardless of the strategies chosen by other players
• Mixed strategies involve players randomizing over their available strategies according to certain probabilities

Game Structure and Representation

• Simultaneous move games are typically represented using a normal form or strategic form representation
  • Normal form representation consists of a matrix listing the players, their available strategies, and the corresponding payoffs
  • Payoffs in the matrix represent the outcomes for each player based on the combination of strategies chosen
• The number of players and the number of strategies available to each player define the size and complexity of the game
• Payoffs can be represented as numerical values, ordinal rankings, or other measures of preference
• Zero-sum games are a special class of simultaneous move games where the gains of one player equal the losses of the other player(s)
• Symmetric games are those where the payoffs depend only on the strategies chosen, not on which player chooses each strategy
• Coordination games involve players trying to coordinate their actions to achieve mutually beneficial outcomes (e.g., choosing the same meeting location)
• Anti-coordination games involve players trying to differentiate their actions from those of other players (e.g., choosing different market niches to avoid competition)

Strategies and Payoffs

• Players in a simultaneous move game choose from a set of available strategies
  • Pure strategies are those where a player chooses a single action with certainty
  • Mixed strategies involve players randomizing over their available strategies according to certain probabilities
• Payoffs represent the outcomes or rewards players receive based on the combination of strategies chosen by all players
• Payoffs can be in various forms such as monetary rewards, utility values, or ordinal rankings
• Dominant strategies are those that provide the highest payoff for a player regardless of the strategies chosen by other players
  • Strictly dominant strategies always yield a higher payoff than any other strategy
  • Weakly dominant strategies yield payoffs at least as high as any other strategy
• Dominated strategies are those that provide lower payoffs compared to other available strategies regardless of the opponents' choices
• Best response is a player's optimal strategy given the strategies of the other players
• Iterated elimination of dominated strategies is a process of simplifying a game by removing dominated strategies

Nash Equilibrium Explained

• Nash equilibrium is a key concept in simultaneous move games representing a stable state where no player has an incentive to unilaterally change their strategy
• In a Nash equilibrium, each player's strategy is a best response to the strategies of the other players
• Nash equilibrium can be pure strategy Nash equilibrium if all players choose pure strategies
• Mixed strategy Nash equilibrium involves at least one player choosing a mixed strategy
• Nash equilibrium is self-enforcing because no player can benefit by deviating from their equilibrium strategy given the strategies of the other players
• A game can have one Nash equilibrium, multiple Nash equilibria, or no Nash equilibrium
• Pareto optimality is a related concept where no player can be made better off without making another player worse off
  • Nash equilibria are not always Pareto optimal, as there may be other outcomes that improve payoffs for all players

Finding Nash Equilibria

• To find Nash equilibria, we analyze the best responses of each player given the strategies of the other players
• In a 2x2 game (two players with two strategies each), Nash equilibria can be found by examining the payoff matrix and identifying the cells where both players' strategies are best responses to each other
• Underline method is a technique for finding Nash equilibria in 2x2 games by underlining the best responses for each player and identifying the cells where both players' best responses intersect
• In larger games, finding Nash equilibria can be more complex and may require solving systems of equations or using computational methods
• Existence of Nash equilibria is guaranteed in finite games with mixed strategies (Nash's Theorem)
• Pure strategy Nash equilibria may not always exist, but mixed strategy Nash equilibria always exist in finite games
• Dominant strategy equilibrium is a special case of Nash equilibrium where all players have dominant strategies
• Iterated elimination of dominated strategies can be used to simplify a game before finding Nash equilibria

Applications and Examples

• Prisoner's Dilemma is a classic example of a simultaneous move game illustrating the conflict between individual and collective interests
  • In the Prisoner's Dilemma, two suspects are interrogated separately and face the choice of confessing or remaining silent
  • The Nash equilibrium is for both players to confess, even though mutual silence would yield a better outcome for both
• Cournot duopoly model is an example of a simultaneous move game in economics where two firms compete by choosing their production quantities simultaneously
  • The Nash equilibrium in the Cournot model represents the production levels where each firm's output is a best response to the other firm's output
• Matching Pennies is a simple 2x2 game illustrating mixed strategy Nash equilibrium
  • In Matching Pennies, two players simultaneously choose heads or tails, with one player winning if the choices match and the other winning if they differ
  • The mixed strategy Nash equilibrium involves both players randomizing their choices with equal probabilities
• Battle of the Sexes is a coordination game illustrating the conflict between two players trying to agree on an outcome
  • In Battle of the Sexes, a couple wants to attend an event together but has different preferences for which event to attend
  • The game has two pure strategy Nash equilibria corresponding to each player's preferred outcome, and a mixed strategy Nash equilibrium

Common Pitfalls and Misconceptions

• Assuming that players always choose dominant strategies, even when they don't exist or when other factors influence decision-making
• Confusing Nash equilibrium with Pareto optimality or assuming that Nash equilibria are always efficient or desirable outcomes
• Failing to consider the possibility of multiple Nash equilibria or the existence of mixed strategy Nash equilibria
• Ignoring the role of communication, coordination, or external factors that may influence players' choices and the resulting equilibria
• Overreliance on simplistic models or assumptions that may not capture the complexity of real-world strategic interactions
• Misinterpreting the concept of best response as always leading to the best possible outcome for a player
• Neglecting the potential for players to make mistakes, act irrationally, or have incomplete information about the game structure or payoffs
• Assuming that all players have the same level of rationality, knowledge, or computational abilities when making their decisions

Advanced Topics and Extensions

• Subgame perfect equilibrium is a refinement of Nash equilibrium used in dynamic games or extensive form games
  • Subgame perfect equilibrium requires players' strategies to be optimal not only in the overall game but also in every subgame (a portion of the game tree)
• Bayesian Nash equilibrium is an extension of Nash equilibrium for games with incomplete information, where players have different types or private information
  • In a Bayesian Nash equilibrium, each player's strategy is a best response to the expected strategies of the other players based on their beliefs about the other players' types
• Correlated equilibrium is a generalization of Nash equilibrium that allows for players' strategies to be correlated through a mediator or external signal
  • In a correlated equilibrium, players' strategies are not necessarily independent, but they still have no incentive to deviate from the recommended strategies
• Evolutionary game theory studies the dynamics of strategy adaptation and selection in populations of players over time
  • Evolutionary stable strategies are those that, if adopted by a population, cannot be invaded by alternative strategies
• Repeated games involve players interacting over multiple rounds, allowing for the possibility of cooperation, punishment, and reputation effects
  • Nash equilibria in repeated games can differ from those in one-shot games due to the potential for future interactions and the role of discount factors
• Mechanism design is the study of designing game rules and incentive structures to achieve desired outcomes or equilibria
  • Revelation principle states that any equilibrium outcome achievable by a mechanism can also be achieved by a direct revelation mechanism where players truthfully report their types