Micro 4: Game Theory Flashcards
What is a game? What do they all contain?
A game consists of:
- A set of players - at least two, but finitely many.
- Information that the players have about their payoffs, others’ payoffs, and others’ moves.
- Moves that each of these players can make
- Payoffs that players receive after a given set of moves
- Strategies that players can choose from
Precisely define the concept of a Nash Equilibrium.
In a Nash equilibrium, all players are playing a weak best response to the actions of all other players. Given what all other players are doing, no player can make themselves better off my deviating to a different strategy.
Do all finite games have Nash equilibria? What about infinite games?
Nash’s theorem (1950) states that every game with a finite number of players and a finite number of pure strategies has at least one Nash equilibrium. However, if there are an infinite number of strategies, this is possible; for example, consider the two-player game “name a higher number than your opponent”.
What is a strictly dominant strategy?
A strategy is strictly dominant if it yields a greater payoff to the player than any other strategy, no matter what the other players do. This is a stronger condition than Nash equilibrium; it must be superior to all other strategies regardless of what other players do, not just under optimal play from other players.
How does iterated elimination of strictly dominated strategies work?
A strategy is strictly dominated if there is some strategy which yields a higher payoff than it for any action of the other player. No rational player will ever play this strategy. Therefore, it can be eliminated. Eliminating this strategy will result in a changed set of possible payoffs for other players, so further strategies may now be strictly dominated and able to be eliminated. Under successful IESDS, this continues until one unique strategy set is identified.
What is a best response curve? Why are they used?
The best response curve plot the optimal probability of playing each strategy, as a function of the other player’s probability of playing each strategy. At their intersection points, we have Nash Equilibria.
What is a mixed-strategy Nash equilibrium? How can you find them?
A mixed-strategy Nash equilibrium is where one or more players chooses to play some strategy with probability strictly between 0 and 1. In order for this to be optimal, all pure strategies must output the same expected payoff; otherwise, one pure strategy would be preferred. Therefore, calculating the expected payoff of U and D (in terms of q) and setting them equal to each other gives the value of q that will lead to a mixed-strategy.
What is backward induction? How can it be used to solve dynamic games?
Start at the terminal nodes of a sequential game - ones which result in a payoff, not further play. Choose the payoff that is optimal for the player who gets to make this move, and replace the choice node with this payoff. Repeat until you reach the initial node. The remaining strategy set and payoffs will be the subgame-perfect Nash equilibrium.
How do you specify strategies in dynamic games?
You must specify both the strategy on the equilibrium path (eg Enter, Acquiesce), and the conditional strategy off the equilibrium path.
What is subgame-perfect equilibrium? Why is it a superior concept to Nash equilibrium?
For a Nash equilibrium to also be a subgame-perfect Nash equilibrium, each proper subgame must also be in Nash equilibrium. This removes the possibility of equilibria that include non-credible threats.
When the Prisoner’s Dilemma is repeated a finite number of times, why is ‘always defect’ the only SPE?
Consider the final repetition of the game. This is identical in form to a single repetition of the game, and so (D, D) will be the only NE. Then, consider the penultimate game. There is no reason to cooperate in this game; each knows that next period will involve mutual defection, so cooperation hurts both players. Therefore, (D, D) is the only NE in the penultimate subgame. Continuing this reasoning to the first game, always defect is the only SPE; any other strategy cannot be a NE in all subgames.
When the Prisoner’s Dilemma is repeated an infinite number of times, when is ‘grim trigger’ an SPE?
If players are ‘sufficiently patient’, ‘grim trigger’ is an SPE in the infinitely repeated Prisoner’s Dilemma.
What is the Folk Theorem? What does it state?
If players are sufficiently patient, there are an infinite number of sustainable Nash equilibria corresponding to any feasible and individually rational payoff.
