Intro to game theory

Intro To Game Theory:

Decision Theory:

Decision Theory - How individual agents (or firms) make isolated decisions given their objectives and constraints.

• Marginal-cost, marginal-benefit analysis is an example of decision theory.

For example:

• Agents making a choice from a given choice set

• Agents maximizing utility subject to a budget constraint

• Firms maximizing profit

• Firms minimizing costs.

Game Theory:

In contrast to decision theory, game theory is the study of how agents make choices in environments where the choices of others affect their outcomes and their choices.

• Game Theory is for analyzing situations in which

  • Your actions affect others

  • Others’ actions affect you.

  • All parties are aware of these effects and have to plan accordingly 

• Game Theory is the study of social interactions.

Strategic Interactions & Strategies:

• Strategic Interaction: The need for a player in a game to consider the outcomes, plans, and beliefs of the other players when deciding what to do.

  • And the other players have to consider the outcomes, plans, and beliefs of that player.

• The different possible plans of action for a player in a game are called Strategies.

Payoffs:

Think of players’ payoffs as utilities.

• They are not - in general - monetary payoffs

• Numbers generated after putting monetary payoffs into the players’ utility function.

Rationality:

In game theory, we assume rational behavior from all of the players.

• Meaning the players’ payoffs remain consistent over the outcome as the game progresses.

• The players are able to flawlessly calculate what strategy will help serve their best interests. 

Common Knowledge of Rationality:

• Game Theory assumes common knowledge of rationality.

  • That is, each player knows that the other player is rational, and so on.

• This is a highly problematic assumption but we’ll make the assumption anyway because if we didn’t, we wouldn’t be able to make the analysis we want to make. 

Equilibrium:

• An equilibrium will be a set of strategies from the players where each player would still use her strategy even if she knew what the other player were doing,

  • An equilibrium is described by the optimal strategies involved - not the payoffs.

• Another way to think about equilibrium is that at an equilibrium there is no tendency for change. 

  • There’s no reason for any of the players to change, given what the other player is doing. 

Nash Equilibrium:

• A Nash Equilibrium is a set of strategies such that each player would have no reason to change her strategy if she were to know with certainty what the other player is doing. 

• The players don’t necessarily achieve their highest possible payoffs at Nash Equilibrium.

  • Key is that each player can’t do any better given what the other player is doing.

Best-Response Analysis:

• To find Nash Equilibria is to go strategy-by-strategy, cell-by-cell to find each player’s best response given the other player’s actions.

• Any cell in which all players are best responding is a Nash Equilibrium. 

Dominant Strategy:

A dominant strategy is a single strategy of a player that always yields higher payoffs for the player no matter what the other player’s actions might be. 

• There can be games in which both (or all) players have a dominant strategy, only one player, or all of the players.

• Most games do not have any dominant strategies.