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