What is Game Theory?
- Understanding how agents behave in scenarios where they have to make decisions
- Game Theory is a formal way to analyze
- interactions among
- a group of agents
- who are rational and behave strategically
- Game Theory does not guarantee winning, it is rather a framework for thinking about strategic interaction.
Which assumptions are made about the decision makers in game theory?
- Players are rational
- Maximizing own utility
- Sharing a common knowledge
Explain the difference between Game Theory vs Decision Theory
- Decision Theory: single decision maker, game is played against Nature. For example, private investing can be seen as a part of decision theory.
- Game Theory: multiple decision makers, strategic interaction between players. For example, prisoner dilemma.
What are the concepts that need to be defined for a game?
- Players: who is acting in the game
- Actions: set of actions each player can choose at each stage of the game
- Strategies: A strategy specifies what action player A will take next, for every feasible decision situation that player A might face
- Outcome and Payoff: Outcome of the game for every possible set of players strategies and players preferences regarding them.
- Rules: which set of rules define the game
Which types of rules can be defined?
- Nature of the conflict (cooperative vs. non cooperative)
- Nature of interaction (sequential vs. simultaneous)
- Time frame (repeated game vs. one-shot interaction)
- Available information (complete vs. incomplete, perfect vs. imperfect information)
Name the different degrees of information to be distinguished regarding completeness.
- Complete Information: Players preferences on outcomes are common knowledge, and players action spaces are common knowledge
- Incomplete Information: At least one player has limited information about the preferences of another player (payoff specification is incomplete)
Name the different degrees of information to be distinguished regarding knowledge of opponent actions.
- Perfect information: All previous moves are observed before the next move is chosen, no moves are simultaneous. A can chose his strategy based on actions of B during his turn.
- Imperfect information: Some players cannot observe the decisions of the other players.
Does complete information imply perfect information?
- No, complete does not imply perfect information: In contrast to games of perfect information, in games of complete information players may choose actions simultaneously, but know the preferences of each other. Example: Prisoners dilemma is a game with complete, but not perfect information.
Incomplete information vs imperfect information
- Incomplete information - preferences of opponents are unknown
- Imperfect information - actions of opponent required to chose best strategy in current turn is unknown
Which forms of interaction can be used in Game Theory?
- Normal form game: players choose their action simultaneously - like Prisoners Dilemma
- Extensive form game: players choose their action sequentially - like Tic-Tac-Toe
How are Normal Form Games described?
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Simultaneous move: Each player chooses a strategy without knowledge of others choices in this turn
What are the differences of Extensive Form Games to Normal Form Games?
- In Extensive Form Games turns are sequentially, in Normal Form Games - simultaneously
Explain the difference between pure and mixed strategies, and give an example for each type.
- Pure strategy: Any course of action open to a participant in a conflict situation (non randomized strategy)
- Mixed strategy: A probability distribution over the set of pure strategies (randomized choice of pure strategy)
Name and explain solutions to Normal Form Games
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Dominant strategy: everybody can choose his best strategy. “Which strategy should I use if opponent make move A? Which strategy should I use if opponent make move B?” → If the best strategy in all cases is the same, it is a dominant strategy. Weak dominant strategy for each player does always exist. If this strategy is a dominant strategy for all players, we have a dominant strategy equilibrium. It does not always exist. Dominant strategy does not consider what other player might do.
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Nash Equilibrium: In a Nash equilibrium we choose the best strategy, while taking the decisions of others into account. Is weaker, that the dominant strategy equilibrium. Dominant Strategy Equilibrium considers the utility maximization of each player, and Nash Equilibrium considers the best strategies of other players. It can be no ore more than one Nash Equilibrium
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Mixed Strategy Nash Equilibrium: in situations where there are several Nash Equilibriums, we can calculate the probability of each player to make the decision, based on his utility.
What is Nash Equilibrium?
- A Nash equilibrium is a set of strategies, one for each player, such that no player has incentive to change his or her strategy given what the other players are doing.
- A Nash equilibrium can be seen as a law that no one would break even in the absence of an police force or another punishment laws
Can Nash Equilibrium have no strategies, or more than one?
- Yes, left example has no Nash Equilibrium, right example has two Nash Equilibrium
What are the Solution Concepts to Extensive Form Games?
- Subgame Perfect Nash Equilibrium: Traverse decision tree bottom-up, chose the solutions that maximize the utility of player in subgames.
- Assume first Subgame (Monopolist)
- Player 2 (Monopolist) would play accommodate
- Assume second subgame (here full game)
- Player 1 (Challenger), knowing that left path is played, then plays enter
