Histories and subhistories A terminal history is a listing of every play in a possible course of the game all the way to the end A proper subhistory is a listing of every play in the course of the game up to some point before the end ID: 531158
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Slide1
More on Extensive Form Games
Slide2
Histories and subhistories
A terminal history is a listing of every play in a possible course of the game, all the way to the end.
A
proper
subhistory
is a listing of every play in the course of the game up to some point before the end.
Every proper
subhistory
induces a game, called a
subgame
which is defined by the remaining possibilities for play and resulting payoffs.Slide3
Proper Subgames
For any proper
subhistory
, there is a well-defined extensive form game that follows this
subhistory
.
A
subgame
following any non-empty proper
subhistory
is called a proper
subgame
.Slide4
Subgame perfect Nash Equilibrium
A strategy specifies what each person will do at any possible point in the game where it is his turn.
A strategy profile
(i.e. list of strategies chosen by each player)
then determines the course of play in every possible
subgame
.
A
subgame
perfect Nash equilibrium (SPNE) is a strategy profile such that each person’s play in each
subgame
is a best response to the other players’ actions in that
subgame
.Slide5
Berlin or Havana? Prob
156.2 cSlide6
Histories and subgames
Terminal histories:
Proper
subhistories
:
Player functions:
Proper
subgames
:Slide7
How many proper
subgames
does the game on the
the
blackboard have?
6
10
4
3
5+Slide8
In the game on the blackboard, what is the payoff to Player 2 in a
subgame
perfect Nash equilibrium?
0
1
2
3
There are two
subgame
perfect
equilibria
. In one of them he gets 2 and in one of them he gets 1.Slide9
Choosing Sides Game Ex 174.1
Two choosers, 3 players, a, b and c.
Chooser 1 gets to choose first, then 2 chooses, then 1 gets a second choice.
Player
Value to Chooser 1
Value to Chooser 2
a
3
1
b
2
3
c
1
2Slide10
Game Tree: Choosing SidesSlide11
Analysis
Player 2 never gets his last choice.
Therefore it never makes sense for Chooser 1 to choose Chooser 2’s last choice first. Chooser 1 is always going to get that player anyway.
Chooser 1’s first choice should be the one that he likes better of the two who are not Chooser 2’s last choice.Slide12
Variant of All-Pay Auction: 175.2
Two bidders compete for an object that is worth $2.50 to each of them.
They bid sequentially. They must bid an integer number of dollars. When it is your turn you must either raise the bid by $1 or pass. Nobody can afford to bid more than $3.
If you pass, other bidder gets object. Both must pay the amount they bid.Slide13
Game treeSlide14
What if nobody can bid more than $4?Slide15
Repeated Prisoners’ Dilemma
Backwards induction solution?
Does this solution seem reasonable if game
is repeated
100 times?