9: Principles of Game Theory Flashcards
What does Game Theory analyze?
Game Theory analyzes the strategic interaction between multiple individuals (“players”).
How is a game defined?
- The players i = 1, …, N who play the game
- The set of strategies S of the players:
Each player i has her or his individual set of strategies S of the various strategies s that he can choose to play. Thus, the set of strategies for player i is Si = (s1i, s2i,…) and hows us the different actions this player can choose from - An outcome function that determines for each combination of strategies that our players can play the next outcome (result) of the game
- A utility function ui (…) for each player. This utility function tells us for every potential outcome of the game how much this particular player likes or dislikes this outcome.
Economists use games in order to predict how people and firms behave. What do we want to find out?
Best-response function:
We want to know which optimal strategies a rational player chooses, given the strategies of the other player.
Nash-Equilibrium
We want to know if all players strategies that maximize their own utility (that is, all players play best-response functions), what is the result of the game?
What is the definition of “Best response”?
The best response is strategy s*i of Si, that maximizes the utility ui (si, s…i) for given strategies of other players.
ui(s*i,s…i)>/= ui(si,s…i)
Book page 77 for the formal way to write it down.
What does the best response of a player depend on?
It is important to understand that the best response of a player i usually depends on the combination of the strategies of all the other players s…i.
In other words, for each combination of strategies of other players s…i, our player i might have a different best response. The best-response-function tells us the best response for each strategy combination s…i.
What is the definition of the Best-Response function?
A function that gives the best response for all possible strategies of all other players s…i is a best response-function, sometimes also called reaction function.
Which problem does the Nash-Equilibrium solve?
We use our concept of the best-response-function to find out how rational players strategically react to what other players are likely going to do.
==> The concept of the Nash-Equilibrium solves this problem.
In a Nash-Equilibrium, all players play best responses - that is every player chooses a strategy that maximizes her or his own utility, given the strategies of all other players (who do exactly the same).
How is the Nash-Equilibrium defined and notated?
A strategy profile (sNE1,… SNEN) is a Nash equilibrium if each player’s strategy is a best response to the equilibrium strategies of all other players:
u1(sNEi, sNE-i) >/= ui (si, sNE-i)
for all si out of Si and i = 1, … N.
Each game with a well defined, finite number of players and strategies has at least one Nash equilibrium, maybe not in pure but in mixed strategies.
Why does no player have an individual incentive to unilaterally change his strategy?
Because in a Nash equilibrium everyone plays already a best response.
==> This also means that if you look at a combinatio of strategies and find at least one player who wants to change his strategy, then this combinatio of strategies can never be a Nash equilibrium!
What is one big problem with the Nash Equilibrium?
It can easily happen that a game has more than one Nash equilibrium.
==> In this case it is usually not possible to make a clear prediction which outcome would actually be the result if the game would be played in the real world.
