9: Principles of Game Theory

Question 1

What does Game Theory analyze?

Answer 1

Game Theory analyzes the strategic interaction between multiple individuals (“players”).

Question 2

How is a game defined?

Answer 2

• 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.

Question 3

Economists use games in order to predict how people and firms behave. What do we want to find out?

Answer 3

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?

Question 4

What is the definition of “Best response”?

Answer 4

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.

Question 5

What does the best response of a player depend on?

Answer 5

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.

Question 6

What is the definition of the Best-Response function?

Answer 6

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.

Question 7

Which problem does the Nash-Equilibrium solve?

Answer 7

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).

Question 8

How is the Nash-Equilibrium defined and notated?

Answer 8

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.

Question 9

Why does no player have an individual incentive to unilaterally change his strategy?

Answer 9

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!

Question 10

What is one big problem with the Nash Equilibrium?

Answer 10

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.

Question 11

When is a Nash Equilibrium pareto-efficient?

Answer 11

A Nash equilibrium is Pareto-efficient if there is no other combination of strategies (which do not have to be a Nash equilibrium) that increase the utility of at least one player while not decreasing the utility of all other players.

A nash equilibrium can be pareto-superior compared to an other Nash equilibrium.
==> This means that in the Pareto-superior Nash equilibrium, the utility of at least one player is higher and the utility of no player is lower compared to the other Nash equilibrium.

Question 12

What is a weakly dominant strategy?

Answer 12

A strategy that

• independently of the strategies of all other players, the strategy always gives the person at least the same utility as all other strategies that they could use
• for at least one strategy combination of other players one possible strategy of this player is strictly better than all other strategies this person could use

… is weakly dominant strategy for player.

Question 13

What is a strictly dominant strategy?

Answer 13

A strategy that independently of the other strategies possible of all other players, this strategy gives this player a strictly higher utility than all other strategies that this person could use is strictly dominant strategy for this player.

Question 14

What is the relation between the weakly and the strictly dominant strategies?

Answer 14

If a strategy is strictly dominant, it is always also weakly dominant.
==> However, a weakly dominant strategy is not automatically also a strictly dominant strategy!

Question 15

If an exam question only says “dominant” strategies without defining if its strict of weak, what should you assume?

Answer 15

Then you can usually assume that this means weak dominance. (But ask professor or assistant to be sure!)

Question 16

What is an Equilibrium in dominant strategies?

Answer 16

If all players have a (weakly or strictly) dominant strategy, we have an Equilibrium in dominant strategies.

Question 17

Games in normal form: What does this “normal form” mean?

Answer 17

The normal form is a special way of combining all information of a game.

We need:
1. the players,
2. their strategy sets,
3. and the utility of the players depending on the combination of all strategies.

Question 18

Why is the “best” option in the prisoner’s dilemma not a Nash equilibrium?

Answer 18

We define an equilibrium with the strategies the players choose, never with the outcome (utility of the two players)!
==> Thus, (2,2) is NOT the Nash equilibrium.

Question 19

How do you find the Nash equilbria?

Answer 19

Mark the best responses of the players.
Each cell which you have highlighted both payoffs shows you a combination of strategies that is a Nash equilibrium.

Question 20

When do you use a game tree instead of cells?

Answer 20

If players choose their strategies sequentially while the game evolves instead of simultaneously.

But you can also display it in cells.

Question 21

What is “subgame perfection”?

Answer 21

Players already anticipated how opponents who maximize their own utility will decide at later points in the game and choose their own strategy accordingly.
==> We can use the concept of backwards induction to find such a subgame perfect Nash equilibrium.

Question 22

What is Backwards induction?

Answer 22

To solve the game “backwards”:
Look at the node with a decision that is the closest to the end of the game.
Now assume for a moment that the game has reached this node. What is the optimal decision for the player at this node? Now move one node towards the beginning of the game and look at this decision.
What is the optimal decision for the player at this node if she anticipates the decision of the other player at the very last node? Continue with this procedure until you’ve reached the first node of the game.
In the subgame perfect Nash equilibrium all players play a strategy in which they behave optimally once they reach a particular node and anticipate that all other players behave optimally in the same way.