of Reduction a Game in Extensive Form to Strategic
Form.
Pure strategy. A pure strategy is a players complete
plan for playing the game. It should cover every
contingency.
A pure strategy for a Player is a rule that tells him exactly
what move to make in each of his information sets. It
should specify a particular edge leading out from each
information set.
Example:
Player I has 3 information sets. The set of edges coming
out from the information sets are{A,B}, {E,F}, {C,D}.
Player II has two information sets. The set of edges coming
out from the information sets are {a,b}, {c,d}.
Set of Pure Strategies for I: {A,B}x{C,D}x{E.F}
={ACE, ACF, ADE, ADF, BCE, BCF, BDE, BDF}
Set of Pure Strategies for II: {a,b}x{c,d}={ac,ad,bc,bd}
Remark: Even for a simple game, there may be a large
number of pure strategies.
Example: (Two move Tic-Tac-Toe) Player I and II take
turn to make a move on the Tic-Tac-Toe board. The game
stops after each player has made one move. How many
pure strategies are there for Player I and for Player II?
Example: (Two move chess) There are 20 possible moves
for Player I and 20 possible moves for Player II. How many
pure strategies are there for Player II?
Reduced pure strategy: Specification of choices at all
information sets except those that are eliminated by the
previous moves.
Remark: It is not too easy to find the set of reduced pure
strategies. Also we need to use full pure strategies for
another important concept.
Example:
Set of Reduced Pure Strategies for I:{AE, AF, BC, BD}
Set of Reduced Pure Strategies for II: {ac,ad,bc,bd}
Now that we have set of pure strategies for each player, we
need to find the payoffs to put the game in strategic form.
Random payoffs. The actual outcome of the game for
given pure strategies of the players depends on the chance
moves selected, and is therefore a random quantity. We
represent random payoffs by their average values.
Example:
Suppose Player I uses BCF and Player II uses ac.
Then, the payoff will be (-5, 1) with probability 0.7 and (4,5)
with probability 0.3. Then, the expected payoff is
0.7(-5,1) + 0.3(4,5) = (-2.3, 2.2)
Remark: In representing the random
payoffs by their averages, we are making
a rather subtle assumption. We are saying
that receiving $5 outright is equivalent to
receiving $10 with probability 0.5. The
proper setting for this concept is Utility
Theory developed by von Neumann and
Morgenstern.
Game in strategic form:
Set of players: {1,,n}
A set of pure strategies for player i, Xi i=1,,n.
Payoff function for the ith player.
ui : X1 xx Xn R
Payoff function tabulated in bimatrix form:
ac ad
ACE
ACF
ADE
ADF
BCE
BCF
BDE
BDF
bc bd
Set of reduced strategies for I:{AE, AF, BC, BD}
Set of reduced strategies for II:{ac, ad, bc, bd}
Straegic Form:
ac
AE
AF
ad
bc
bd
(10, 0)
(0, 20)
(10, 0)
(0, 20)
(0,30)
(40, 0)
(0,30)
(40, 0)
(2.6,3.7)
(2.6,3.7)
(2.3, 2.2)
(0.5, 2.3)
(2.6,3.7)
(2.6,3.7)
BC (2.3, 2.2)
BD (0.5, 2.3)
Example:
Strategic Form:
r1r2
R1R2
(3,3)
r1d2
(1,4)
d1r2
(0,3)
d1d2
(0,3)
R1D2
(2,2)
(2,2)
(0,3)
(0,3)
D1R2
(1,1)
(1,1)
(1,1)
(1,1)
D1D2
(1,1)
(1,1)
(1,1)
(1,1)
Definition. A vector of pure strategy choices (x1, x2, . . . , xn)
with xi Xi for i = 1, . . . , n is said to be a pure strategic
equilibrium, or PSE for short, if for all i = 1, 2, . . . , n, and
for all x Xi,
u i (x1, . . . , xi-1, xi, xi+1, . . . , xn) u i (x1, . . . , xi-1, x,
xi+1, . . . , xn). (1)
Equation (1) says that if the players other than player i use
their indicated strategies, then the best player i can do is to
use xi. Such a pure strategy choice of player i is called a
best response to the strategy choices of the other players.
The notion of strategic equilibrium may be stated: a
particular selection of strategy choices of the players forms a
PSE if each player is using a response best to the strategy
choices of the other players.
face.jpg
Finding All PSEs. (2 Person)
Put an asterisk after each of Player Is payoffs that is a
maximum of its column.
Put an asterisk after each of Player IIs payoffs that is a
maximum of its row.
Then any entry of the matrix at which both Is and IIs
payoffs have asterisks is a PSE, and conversely.
Example:Prisoner's Dilemma
Confess
Confess (9* , 9* )
Silent
*
(10, 0 )
Silent
(0* , 10)
( 1, 1)
Remark:Many games have no PSE's.
Example:
*
*
(2, 2 ) (3 , 3)
*
*
(3 , 3) (4, 4 )
Games of perfect information always have at least one PSE
that may be found by the method of backward induction.
Method of Backward Induction: Starting from any
terminal vertex and trace back to the vertex leading to it.
The player at this vertex will discard those edges with lower
payoff. Then, treat this vertex as a terminal vertex and
repeat the process. Then, we get a path from the root to a
terminal vertex
Theorem: The path obtained by the method of backward
induction defines a PSE.
SREN KIERKEGAARD (1813-1855)
Life can only be understood backwards;
Life
but it must be lived forwards
but
Example:
Zermelos Theorem
Zermelo
Theorem (Zermelo (1912)): In chess
either white can force a win, or balck
can force a win, or both can force at
least a draw.
In fact, the PSE obtained by the method of backward
induction satisfies stronger properties so that it is called a
perfect pure strategy equilibrium.
Definition: A subgame of a game presented in extensive
form is obtained by taking a vertex in the Kuhn tree and all
the edges and paths originated from this vertex.
Definition: A PSE of a game in extensive form is
called a Perfect Pure Strategy Equilibrium (PPSE)
if it is a PSE for all subgames.
Theorem: The path obtained by the method of
backward induction defines a PPSE.
Remark: Subgame perfect implies that when
making choices, a player look forward and
assumes that the choice that will subsequently be
made by himself and by others will be rational.
Threats which would be irrational to carry through
are ruled out. It is precisely this kind of forwardlooking rationality that is most suited to economic
applications.
Example: Incredible Threats
Example:
Strategic Form
Monopolist
no entry
fight
(0,2)
no fight
(0,2)
entry
(-1,-1)
(1,1)
Entrant
Example:
Strategic Form
U
u
(1,1)
d
(1,1)
D
(-1,-1)
(2,0)
Incredible Threats
Incredible
?
PSE allow players to make noncredible
threats provided they never have to carry
them out. Decisions on the equilibrium
path are driven in part by what the
players expect will happen off the
equilibrium path.
Example:
Strategic Form
a
AC
AD
B
b
(20,7)
(20,7)
(5,-20)
(3,6)
(4,8)
(5,-20)
PPSE: (BD, b)
Remark: PPSE is not so perfect.
Example: Centipede Game
I
II
(1,1) (0,3)
I
II
I I
(2,2)
(1,4)
(3,3)
II
(99,99) (98,101)
Player I will go down in the 1st move.
(100,100)
Application of Backward Reasoning
Application
Sophisticated Voting
Sincere Voting: Vote for most preferred
outcomes.
Sophisticated Voting: Individual actors vote
against their preferences in the earlier votes if
they can anticipate the outcome of later votes.
Example: Assume there are three alternatives- x,
y, and z- and three voters- Player 1, 2, and 3with the following preferences:
Player 1: x y z
Player 2: y z x
Player 3: z x y
The three alternatives are voted in pairwise
comparisons: first x versus y, and then the
winner of that vote versus z.
At the last round of voting, sincere voting is the
dominated strategy.
Then, we can figure out the Voting Tree.
Therefore, Player 1 has a sophisticated voting
strategy in round 1 by voting for y. The outcome
is then y which Player 1 prefers over z.
Player 2 will vote for y in round 1 and 2.
Player 3 does not has the option of sophisticated
voting.