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Fabienne Keller
Karten 67 Karten
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Stufe Universität
Erstellt / Aktualisiert 21.02.2013 / 22.07.2020
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Fenster schliessen

A Static Bayesian Game consists of:

...?

- The set of players I

- For each player i:

   a set of possible types T.i & a set of actions A.i

- payoffs which depend on the actions taken by all players and types

- Probability measure on types p: T --> (0, 1)

- All these elements are common knowledge

Fenster schliessen

How to solve a game with incomplete information!?

--> What was John Harsanyi's great idea?

To transform a game of incomplete information into a game with imperfect information.

To do so he introduced a new player, nature, who moves at the beginning

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Definition of a Bayesian Nash Equilibrium!?

In a static Bayesian Game the strategies s* form a (pure-) strategy bayesian NE if for each player i and for each of i's types t.i no player wants to change her strategy, even if the change involves only one action by one type.

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Give some examples for games with incomplete information. !?

- auction: Bidders do not know how much the other bidders value the object

- an insurance company does not know for sure the characteristics of the client

- we do not know if a second hand car is good or not

- a country does not know the exact military strenght of the enemy

- entrant / incumbent game

- etc....

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If mü is the belief of player 2: probability that he is at the left note, given that he is at his information set. 

How do we find mü?

--> picture

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a) with what probability is P2's info set reached?

b) with what probability is p2's left node reached? and the right one?

c) What belief mü should player 2 have?

d) given those beliefs, which one is p2's optimal strategy?

a) 2/3

b) left: 2/3, right: 0

c) mü = 1

d) always play L

Fenster schliessen

Assume now that P1 plays a mixed strategy which attaches equal probability to each of her pure strategies. Given this stragey for p1, 

a) With what prob is P2's info set reached?

b) prob for left & right node?

c) mü?

a) 5/6

b) left node: 2/3, right node: 1/6

c) mü = 4/5

Fenster schliessen

....give those beliefs, wich one is p2's optimal strategy?

b) is that strategy, together with p1's strategy, a NE?

c) expected value of L? and of R?

a) ????

b) ????

c) L: 4 x (4/5) + 0 x (1/5) = 16/ 5

R: 3 x (4/5) + (5/5) = 17 / 5

--> R > L

Fenster schliessen

What is the unique solution to all bargaining problems?

The solution, that satisfies Nash's four axioms!

If...

- "you do not leave anything on the table"

- the outcome doesn't depend on the utility function

- one doesn't gets more than the other one

- Assume that you think (x.y) should be the solution and that I tell you that a set of partitions cannot be chosen but (x,y) is still possible. --> you still want (x,y) to be the solution!

Fenster schliessen

What are Nash's four axioms, that lead to the Nash bargaining solution?

1.) Invariance to equivalent utility representations (If we change the utilities according to positive affine transformation, the solution should not change!)

2.) pareto efficiency 

3.) symmetry (if the bargaining problem is symmetric, then both players should be treated the same)

4.) Independance of irrelevant alternatives (options which are not selected should not change the solution of they are no longer available)

Fenster schliessen

To what do this four nash axioms lead?

(Formel für Problemlösung im bargaining)

The solution assigns to the bargaining problem the pair of payoffs that solves the problem:

 

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R / F ?

a) If we make on of the player more risk averse (make his utility more concave) the bargaining problem changes and the less risk averse bargainer gets a higher share.

b) Improving a bargainer's outside option will make him better off in the bargaining

a) true

b) true

Fenster schliessen

Describe the Rubinstein's Bargaining Game

 

P1 & P2 play:

- a cake of size one has to be divided. In a starting round, P1 makes a proposal tha P2 can accept or turn down

- if he accepts the cake is split according to this

- if he declines the games moves one round forward and it is now P2 who proposes a share

---> This game can last forever, but time is valuable!!

payoff z, after n round is worth:

delta ^n x Z

(set of Nash equilibria is infinitely large in this game!)

Fenster schliessen

What's rubinsteins bargaining solution?

we solve first for a finite number of periods --> find the SPNE

--> it can be shown that for infinite T, there is a unique SPNE in which:

1 gets 1/ (1+ delta) and 2 gets: delta / (1+delta)

Fenster schliessen

Compare the Nash bargaining solution with the NE in general.

Does the NE fulfill the 4 axioms of the bargaining solution?

1.) Invariance to equ. utility representations: YES

(NE are invariant to affine (=equivalent) utility representations)

2.) pareto efficiency: NO

3.) Symmetry: NO

4.) Independance of irrelevant alternatives: ??

Fenster schliessen

a) Describe a game with perfect information.

 

b) And give some examples.

(belongs to "extensive form games")

a) a player moves
his action is observed by the other player who then moves
and his action is observed by the other player and so on, until the game finishes. 

 

b) - removing sticks

   - chess

  - game played with myself (alarm clock)

  - incrumbent and challenger (price fight or not?)

 

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How do we usually solve a game with perfect information?

 

⇒ Games of perfect information are typically solved backwards starting at the end, and are very important to understand credibility of threats and commitment. 

Fenster schliessen
Describe a normal form game.
normal form is a description of a game. Unlike extensive form, normal-form representations are not graphical per se, but rather represent the game by way of a matrix. While this approach can be of greater use in identifying strictly dominated strategies and Nash equilibria, some information is lost as compared to extensive-form representations
Fenster schliessen

Describe an equilibrum in a normal form game.

 

You have played an equilibrium if your choice was a best reply to the other player’s choice and this is also true for the other player, namely her/his choice is a best reply to yours.
In an equilibrium you cannot gain anything by choosing another strategy under the assumption that the other player does not change hers/his. 

Fenster schliessen

a) describe a simultaneous move game

 

b) and give some examples

a) simultaneous games are typically represented by the normal form; a game where both players have to decide/move simultaneously

 

b) prisoner's dilemma 

Fenster schliessen

decision or game:

a) a pair of teenage girls choosing dresses for their prom

b) a group of grocery shoppers in the diary section with each shopper choosing flavor of yogurt to purchase

 

a) game - girls choice affects the others choice

b) decision - no interaction among players

Fenster schliessen

decision or game:

a) a college student considering what type of postgraduate education to pursue

b) the new york times and the wall street journal choosing prices for their online subscriptions this year

a) decision - doesn't affect other people

b) game - (Betrand; price competition; strategic)

Fenster schliessen

What kind of decision theory is game theory?

 

Game theory is interactive decision theory

(It studies the behavior of players whoce decisions affect each other)

Fenster schliessen

What's the difference between a) complete and b) incomplete information?

a) In a game with complete information all players are common knowledge. Example: prisoner's dilemma

b) In a game of incomplete information at least one player is uncertain about another player0s payoffs. Example: an auction ("true men do not eat quiche")

Fenster schliessen

What's the difference between a

a) zero-sum game &

b) non zero-sum game?

a) In a zero-sum game what one players win is what the other looses. --> The players interests are completely opposed. The sum of the payoffs is always O. 

Example: Rock-Scissors-Paper

b) In non zero-sum games both players can gain (think of trade), both can loose (think of nuclear war), there are some element of conflict but also scope for cooperation (think of bargaining)

Fenster schliessen

difference between:

a) one shot game

b) repeated game

a) a one shot game is played only once by the players

b) In a repeated game, the "same" game is played repeatedly by the "same" players

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What kind of game is "removing sticks"?

a dynamic game of perfect information

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An extensive form of a game consists of:

(6)

- Players

- Orders of moves

- Actions. What players can do when they have to move

- Information sets. What each player knows when he has to move.

- Payoffs received by each player for each combination of moves that could be chosen by the players

- The probability distribution over exogenous events. New player: Nature

Fenster schliessen

If a player does not know the moves a previous player had made, he cannot distinguish the corresponding nodes.

--> we speak of....

 

imperfect information

(a player does not know in which node he is)

Fenster schliessen

Whats an information set?

An information set is a collection of decision nodes which cannot be distinguished by the player with the move.

(Entscheidungsknoten, die nicht unterschieden werden können)

--> in a game of perfect information each information set contains one and only one decision node.