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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
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
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.
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
If mü is the belief of player 2: probability that he is at the left note, given that he is at his information set.
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: ??
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
- game played with myself (alarm clock)
- incrumbent and challenger (price fight or not?)
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.
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
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.
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
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
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)
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)
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")
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.
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)
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
What kind of game is "removing sticks"?
a dynamic game of perfect information
An extensive form of a game consists of:
- 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
If a player does not know the moves a previous player had made, he cannot distinguish the corresponding nodes.
--> we speak of....
(a player does not know in which node he is)
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.