Game Theory

Standard Assumptions of Game Theory

1. All players are rational maximizers. 2. All players know everything about the game. 3. Assumptions 1 & 2 are common knowledge among the players, meaning that all players know the other players aren't going to make mistakes.

Sequential Move Game

A game in which players move after one another in sequence, meaning that a player gets to see the other player's move before making his/her own move.

Strategic Interaction

A strategic situation is one in which: • There are two or more "players" (agents) who are interacting in some way, and... • The payoff to each player from each of his/her strategies depends on the choices of the other agents.

Dominated Strategy

a. A strategy is dominated when the player has another strategy that has a greater payoff no matter what the other players do. b. For future reference, it's more important to know which strategy is dominated rather than which strategy is doing the dominating. c. Strategy s1 is strictly dominated by strategy s2 if and only if the payoff from s1 is always strictly less than the payoff from s2, no matter what strategies the other player(s) is/are playing. d. A strategy can only be dominated by one other specific strategy, meaning the dominated strategy definition can only apply to two strategies and no more, no less. e. Strategy s1 is weakly dominated by strategy s2 if and only if s2 yields a payoff at least as great as the payoff from s1 no matter what strategies the other player(s) is/are playing.

Best Response

a. One player's best response to a specific combination of the other players' strategies is/are the strategy/strategies that gets him or her the greatest payoff, given that the other players play that combination of strategies. b. "Player _________'s best response to strategy ___________ is ___________." i. When giving an answer, word it like the above example. ii. The goalkeeper's best response to Kick Left is Jump Left. iii. The goalkeeper's best response to Kick Straight is Stay Put. iv. The goalkeeper's best response to Kick Right is Jump Right. c. Figuring out what a given player's best response is to another player's strategy is classified as a best response analysis.

Rational Maximizer

a. Roughly definition: players always try to get the highest payoffs for themselves. b. Technical definition: agents' preferences are complete and transitive, and agents always choose the actions that give them the greatest (expected) payoff. Complete means that the agent prefers A to B, B to A, or indifferent between A and B (take chocolate ice cream vs. vanilla; it's complete to say that you don't care whether or not you have vanilla or ice cream, but it is incomplete to say you don't know which one you want). Transitive means if the agent prefers A to B and B to C, then the agent must also prefer A to C. Height is transitive, for example. If you're taller than me and I'm taller than my sister, then you must also be taller than my sister.

Nash Equilibrium

o A Nash equilibrium of a game is a strategy profile such that each player is playing a best response to the other players' strategies. • When a game is in Nash equilibrium, no player has an incentive to be the only player to change strategies. * To find Nash equilibrium on a game matrix, circle the players' best responses to each of his/her opponent's strategies. The cells in which both payoffs are circled would be Nash Equilibria. (also, if there happen to be three choices - let's say left, middle, right - and two of the choices have the same payoff, circle both as there will be no preference between those two strategies for that given player.

Backward Induction

o In backward induction, start with the last decision of every branch and consider what should occur given the payoffs (remember, each player in game theory is a rational maximizer). From this, we can roll back up the tree and figure out what the original player would do given the outcome of the payoffs (think of Sam and Frodo example). o The "rollback equilibrium" is (F,CS) for the Sam/Frodo example. The rollback equilibrium must contain something (letters) from each remaining branch. In short, you need the same amount of letters as you have branches or decision nodes on the tree.

First/Second Mover Advantage

o Roughly...a game exhibits first-mover advantage if and only if the player who moves first can be sure of achieving his or her most desired outcome. o The first-mover advantage offers the greater ability to control the flow of the game than the second player to move. o A game exhibits second-mover advantage if and only if the player who moves second can be sure of achieving his or her most desired outcome.

Rollback Equilibrium

o The "rollback equilibrium" is (F,CS) for the Sam/Frodo example. The rollback equilibrium must contain something (letters) from each remaining branch. In short, you need the same amount of letters as you have branches or decision nodes on the tree.

Complete Strategy

• A complete strategy specifies what a player does at each node he/she controls. Therefore, there must be one letter per node (leftover after backward induction) when giving a complete strategy. • Must know how to specify a complete strategy for exam and their must be as many letters as nodes for a player.

Zero-sum Game

• A game in which the sum of the payoffs must equal 0 for each strategy profile, meaning one player has to do worse than the other.

Simultaneous Move Game

• A simultaneous-move game is one where no player gets to move before knowing what the other player(s) has/have decided to do.

Strategy Profile

• A strategy profile is a combination of every player's strategies. It's a list of strategies that contains one and only one complete strategy for each player. • In a game of chicken, the strategy profiles are: • (straight, swerve) • (swerve, straight) • (straight, straight) • (swerve, swerve)