Static games of complete information

• a strategy is a complete (contingent) plan, which specifies how the player will act in every possible distinguishable circumstance in which she might be called upon to move
• a strategy may not be strictly dominated by any pure strategy, but be dominated by a mixed one
• if all players believe that others are rational, we can iteratively eliminate strictly dominated strategies
• Player's action ai is called rationalizable if it is a best response to some mix (usually player i's belief of what other players are going to play) of the opponent's rationalizable actions
• In finite games the set of correlatedly rationalizable actions coincides with the set of actions surviving the iterated elimination of strictly dominated strategies
• In two player finite games there is no need to think about correlations: all actions surviving IESDS are rationalisable and the other way round
• In a Nash equilibrium, nobody wants to deviate given all other players play the same strategies - each player is best-responding so there is no profitable unilateral deviation
• A set of 'intersections' of BR correspondences gives a full set of NE
• If deriving best responses is particularly troublesome, then guess and check may deliver a solution quicker
• Nash equilibrium implies that players' beliefs are correct in equilibrium
• common knowledge of rationality does not imply Nash equilibrium
• Mixed strategy NE is problematic, as players do not have any incentives to randomise in a specific way over the actions in the support of the mixed strategies, since all actions deliver exactly the same expected payoff
• any weighted average of any two NE will be a correlated equilibrium as well
• Finding the best correlated equilibrium: maximise players' total payoff subject to their incentive constraints
• Correlated equilibrium is a mechanism that picks a strategy profile according to a given joint distribution and recommends each player to play their part without revealing what others are told to do, and no player has a strict incentive to disobey
• Existence of correlated equilibrium is guaranteed in finite games by Nash's existence theorem and the fact that all Nash equilibria are also correlated equilibria
• IESDS expresses the notion of common rationality of players
• correlated rationalisability is equivalent to iterated strict dominance in finite games
• all strategies which survive IESDS are (correlated) rationalizable
• hybrid NE occur when a player is indifferent between strategies against a pure strategy of an opponent
• the best correlated equilibrium maximises total payoff s.t players' incentive constraints
• correlated equilibrium is a mechanism that picks a strategy profile according to a given joint distribution and recommends each player to play their part without revealing what others are told to do, and no player has a strict incentive to disobey
• correlated equilibrium relies on the existence of an external correlating device with the required informational structure