Game Theory VL

- 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

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

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.

- 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....

--> picture

a) 2/3

b) left: 2/3, right: 0

c) mü = 1

d) always play L

a) 5/6

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

c) mü = 4/5

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

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!

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)

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

a) true

b) true

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!)

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)

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: ??

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

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

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

b) prisoner's dilemma 

a) game - girls choice affects the others choice

b) decision - no interaction among players

a) decision - doesn't affect other people

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

Game theory is interactive decision theory

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

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")

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)

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

a dynamic game of perfect information

- 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

imperfect information

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

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.