Game Theory Final

Perfect Bayesian Equilibrium

A strategy profile and a system of beliefs that yield a WPBE in every subgame

Nash's Theorem

Any finite game has at least one Nash Equilibrium, if mixed strategies are allowed

Ståhl Model

Bargaining continues at most a fixed finite number of periods T

Pure Strategy

Deterministic rule that tells a player what to choose at each information set belonging to that player

Separating Equilibria

Different types would choose different signals so signal reveals type

Common Value

Each bidder observes private signal and the value of the object is the same for each bidder

Independent Private Value

Each bidder's value is only known by that bidder

Static Games

Each player makes her choice without having observed the choice(s) of the other player(s)

Direct

I's message space is set of possible types t, so a strategy for player is to announce some strategy of all types

Intuitive Criterion

If there is a type(w) for whom some signal choice (B) is strictly dominated by the equilibrium and there is another type(s) for whom the signal B is not dominated by the equilibrium, then on observing B, 2 should put 0 probability on 1's type being W and probability 1 on 1's type being S

Mixed Strategies

Involve players making deliberate independent random choices from the set of pure strategies

Incentive-Compatible

It is a BNE for each type of each player to announce her true type

Information Sets

Partition set of player's decision nodes into subsets

Equivalent

Same allocation and payments

Imperfect Information

Structure of game (extensive form, payoffs) common knowledge, but some prior moves unknown

Signaling Games

2 players, the sender and the receiver. Timeline: Player 1 learns her type t and chooses a message m, Player 2 observes m but not t, and then chooses an action a

Finite Game

A game with a finite number of players each having a finite number of pure strategies

Subgame

A part of a game that can be analyzed as a game in its own right

Nash Equilibrium

A strategy list/profile such that each player's strategy is optimal given the other players' strategies.

Weak Perfect Bayesian Equilibrium

A strategy profile and a system of beliefs that satisfy 2 conditions: 1) At each information set the player's strategy choice is optimal given the player's beliefs and given the other player's strategies 2) At each information set along the equilibrium path the player's beliefs are consistent with other players' strategies via the condition probability formula

Subgame Perfect Equilibrium

A strategy profile that gives a Nash Equilibrium in that subgame

Revelation Principle

Any BNE of a mechanism can be replaced by an equivalent, incentive-compatible direct mechanism

Dynamic Games

At least one player gets to observe a move by some other player

Pooling Equilibria

Both types would choose the same signal so the signal is uninformative and posterior beliefs equals prior beliefs

Sequential Equilibrium

Consists of a strategy profile and a system of beliefs such that 1) the strategy profile is sequentially rational given the system of beliefs, and 2) there is a sequence of completely mixed strategy profiles

Grim Trigger Strategy (GTS)

Each player plays one strategy until another deviates away from that strategy, in which the other will play that same strategy forever

Revenue / Payoff Equivalence

In the IPV model with risk-neutral bidders and all assumptions hold, all auction equilibria in which the highest bidders wins and 1) all bidders use the same strictly increasing bid function, and 2) a bidder of lowest type a pays zero

Rubinstein Model

No limit on number of periods

Cooperative

Players can make and enforce binding agreements (including threats)

Noncooperative

Players cannot make or cannot enforce agreements

Incomplete Information

Some player not sure about some player's payoffs, or about some player's choice set

Bayesian Nash Equilibrium

Specifies a strategy choice for each type of each player that is optimal given the strategies of the various types of other players and given the beliefs about the types of other players

Winner's Curse

The fact that one has won the auction is bad news about the value of the price

Best Response

The strategy (or strategies) which produces the most favorable outcome for a player, taking other players' strategies as given

Game Theory

The study of multi-person decisions when the outcome is not entirely under one person's control