1. Introduction to Game Theory

How does game theory consider actions, outcomes, and payoffs?

• It recognises that these are sequential, leading to the other, but often omits outcomes in this consideration and examines how different actions directly lead to different payoffs

• The utility function is itself a function of actions which determine outcomes (ug(a)) — omitting the ‘outcome’ function gives simply, u(a)

• This function might be random, in which case we examine expected payoff

What assumptions do we make about the basic knowledge of players?

• They have knowledge of the set of actions and the set of outcomes available to them

• They know which actions map onto which outcomes

• An agent has an understanding of their own rankings of the outcomes

• An agent wishes to maximise their outcomes

What is strategic interdependence?

• The relationship which defines much of game theory; a person doesn’t operate in a vacuum, but must make decisions with respect to other peoples maximising efforts and their preferences

How do we define a game?

• A game has a set of players, i ∈ N

• It also has a set of strategies, s_i ∈ S_i

• There must be information regarding timing (the order players are asked to play in) and knowledge at every node at which the agent must play a game

• There must be payoffs, a function which assigns a numerical value to the outcomes of a players strategies, as a payoff function π

What is common knowledge?

• When everyone knows a fact, knows that others know the fact, know that others know that they know the fact and so on ad infinitum

What is complete information?

• In a game, information is complete where the facets of a game are common knowledge — are known to all people playing the games

  • Everyone knows the strategies, payoffs, etc

What is normal form?

• The simplest way of representing a game which shows all of the payoffs for agents dependent on the strategies that they use

• Contains a set of players, a set of strategies, and a set of payoffs

What role do solution concepts play?

• Solution concepts use assumptions to restrict strategies to be able to predict behaviour

• If the selection is too broad, we might have too many strategies to be able to make a sound prediction (a selection problem)

• If the strategies are too restrictive, we might prevent a solution from existing (existence problem)

• Assumptions we might use to restrict solution concepts (in increasingly demanding order) are rationality, intelligence, common knowledge (of rationality and intelligence) and self-enforceability

What does dominance refer to?

• If players are strictly rational, they will act only to self-maximise, regardless of their opponent

• A strategy s_i ∈ S_i is a strictly dominant strategy if, for each s_-i ∈ S_-i (other players strategies) and s’_i ∈ S_i (a strategy which is not dominant), π_i(s_i, s_-i) > π_i(s’_i, s_-i)

• If this strategy exists, then all rational players would play it

• A strategy s’_i is strictly dominated by another strategy s_i, if for each s_-i ∈ S_-i, π₁(s’_i, s_-i) < π₁(s_i, s_-i)

What is iterative elimination of strictly dominated strategies? (IESDS)

• The rationality assumption allows to conclude that players will never play dominated strategies

• Thus, in a game, we can predict the strategies chosen by discarding dominated strategies — this works for multi-stage games too

What is a best response?

• The strategy s_i ∈ S_i is player i’s best response to his opponents’ strategies s_−i ∈ S_−i if v_i(s_i, s_−i) ≥ vi(s′ _i, s_−i) ∀s′ _i ∈ S_i

• Rationality dictates that a player must always choose their best response to a strategy another player chooses

• If a strategy is dominated, then it can never be played as a best response

What is a belief?

• A possible profile of his opponents’ strategies, s−i ∈ S−i

What is best response correspondence?

• When a player has a belief about another, the set of best response strategies he has available are given by BR(s_-i); if there is a unique best response this set will only contain one strategy

• Selects for each s_−i ∈ S_−i a subset BR_i(s_−i) ⊂ Si where each strategy s_i ∈ BR_i(s_−i)is a best response to s_−i

Two main solution concepts

• One looks at what a rational person would not do — they would never play a strictly dominated strategy, leading to our iterative elimination of strictly dominated strategies

• The other concerns what a rational person would do, leading to our best response solution concept (this solution set is understood as ‘rationalizable’)