Game Theory: Extensive Form and Subgame Perfection

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Questions and Answers

What concept refines Nash equilibrium by requiring that beliefs are formed optimally given decisions?

Perfect Bayesian Nash equilibrium (PBNE)

Which of the following statements about belief systems is correct?

A belief system assigns probabilities to every node in the game. (correct)

Belief systems are not necessary in dynamic games.

Probabilities in a belief system do not need to sum to 1.

Only informed players use belief systems.

A strategy profile is sequentially rational if the expected payoff of the player is minimal given the strategies played by others.

False

What is the purpose of a signal in signaling games?

To update beliefs about the informed player's unknown type. Signup and view all the answers

What type of equilibrium allows uninformed players to deduce the informed player's type based on the signal sent?

Separating equilibrium Signup and view all the answers

A ______ consists of player 1's strategy $x1^(θ)$ that assigns a decision $x1$ to each possible type of player 1 and player 2's strategy $x2^(x1)$ that assigns a decision $x2$ to each possible choice of player 2.

PBNE Signup and view all the answers

What do extensive-form games better represent?

Dynamic situations where players make decisions at different times. Signup and view all the answers

Which of the following is a characteristic of extensive-form games? (Select all that apply)

Tree representation of decisions Signup and view all the answers

In games of perfect information, players do not know where they are in the game.

False Signup and view all the answers

What is Zermelo's theorem related to?

Every finite game of perfect information has a pure-strategy equilibrium derived through backward induction. Signup and view all the answers

A subgame in an extensive-form game begins at a decision node that is the only decision node of the ________ it belongs to.

information set Signup and view all the answers

What does SPNE stand for?

Subgame Perfect Nash Equilibrium Signup and view all the answers

What kind of games involve incomplete information? (Select all that apply)

Bayesian games Signup and view all the answers

What is the role of 'Nature' in Bayesian games?

Nature chooses the state of the world which is partially revealed to players. Signup and view all the answers

Study Notes

Extensive-form Games

Dynamic situations are interpreted through extensive-form games, detailing the timing and decisions of players.
Key elements include: players involved, decision timing, possible decisions, prior knowledge among players, and payoff structure.
Initial decision node leads to branches representing choices until terminal nodes are reached, indicating game outcomes.

Games of Perfect Information

In games of perfect information, players are fully aware of the decision nodes, meaning information sets contain a single node.
Example: Simplified Stackelberg game with demand functions and marginal costs of zero.
Players can choose high output (qH) or low output (qL), leading to calculated profits based on respective choices.

Game Trees and Normal-form Representation

Extensive-form games can be visualized as game trees, where each terminal node shows player payoffs.
Strategies for players in a normal-form representation depend on decision nodes; example strategies for firms identify possible actions and decisions.
Pure-strategy Nash equilibria observed in the presented Stackelberg game, specifically three outcomes with one being the most credible through backward induction.

Subgame Perfect Nash Equilibrium (SPNE)

A Nash equilibrium is SPNE if it also qualifies as a Nash equilibrium in all subgames, excluding non-credible threats.
SPNE provides a reliable prediction in extensive-form games by iterating decision-making within established subgames.
Generalized backward induction can help determine SPNE in finite games, but is ineffective in infinitely repeated games.

Bayesian Games and Incomplete Information

Bayesian games incorporate private information, where players cannot observe certain payoff-relevant data of others, leading to strategies based on probability distributions.
Players have beliefs shaped by Nature's choice of state, affecting their decisions based on observed types and expected payoffs.

Perfect Bayesian Nash Equilibrium (PBNE)

In dynamic asymmetric information contexts, players can update their beliefs based on others' actions before making their own choices.
PBNE enhances understanding of equilibria by ensuring strategies remain optimal based on beliefs, and beliefs are informed by observed actions.
Key concepts for PBNE include belief systems, sequential rationality, and consistent belief formation through Bayes' rule.### Perfect Bayesian Equilibrium
Defined as a strategy profile and belief system that must be sequentially rational.
Belief system consistency is essential wherever possible according to the strategy profile.
Bayes’ rule applies to beliefs at information sets on the equilibrium path, allowing positive probability moves according to players' strategies.
Off the equilibrium path, beliefs are also determined by Bayes’ rule and players' strategies, leading to a wide range of potential equilibria.
Refinements to PBNE can restrict players' beliefs for off-equilibrium moves, focusing on types that could justify the observed move.
The intuitive criterion is a common refinement used in signalling games.

Signalling Games

A category of dynamic games characterized by incomplete information.
In these games, an informed player signals their type to an uninformed player through their action.
The uninformed player updates beliefs about the informed player’s type based on the signal received.
Stages of the game:
- Stage 1: The type θ of player 1 is realized with a known probability distribution ρ(θ).
- Stage 2: Player 1 chooses a strategy x1, which is publicly observed.
- Stage 3: Player 2 updates beliefs on player 1's type based on x1 and makes a decision x2.

Conditions for PBNE in Signalling Games

Player 1's strategy x1∗(θ) maps each type to a decision x1.
Player 2's strategy x2∗(x1) maps player 1's action to a decision x2.
Posterior beliefs of player 2 denoted as µ(θ|x1) assign probabilities to player 1's type given x1.
Requirements for PBNE:
- Posterior beliefs must be a probability distribution over types (sum to 1).
- Player 2 must maximize her payoff given her beliefs.
- Player 1 must maximize his payoff given player 2's equilibrium strategy.
- Posterior beliefs must utilize Bayes’ rule for all compatible types with the chosen action.

Types of Equilibria

Separating Equilibrium: Different types of the informed player send signals from distinct subsets of actions. This allows the uninformed player to deduce the informed player's type based on the signal.
Pooling Equilibrium: All informed player types take the same action, preventing the uninformed player from extracting useful information and updating prior beliefs based on the action.
A potential third type, hybrid equilibria, is not covered in detail.

Additional Resources

Gibbons' "A Primer in Game Theory" offers foundational concepts.
"Microeconomic Theory" by Mas-Colell, Whinston, and Green provides comprehensive material.
Fudenberg and Tirole's work offers an advanced perspective on game theory.

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Explore the intricacies of extensive-form games and subgame perfection in this quiz. Understand how players make decisions over time and the implications of these strategies in game theory. This quiz will deepen your knowledge of game dynamics and decision-making processes.