Transcript Game - Mathematics

Game Theory
Jacob Foley
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http://www.youtube.com/watch?v=HCinK2
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http://www.youtube.com/watch?v=l0ywiYb
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Overview
1.
2.
3.
4.
Introduction and history
Total-conflict games
Partial-conflict games
Three-person voting game
What is Game Theory
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Game- two or more individuals compete to
try to control the course of events
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Uses mathematical tools to study
situations involving both conflict and
cooperation
History
The first known discussion of game theory
occurred in a letter written by James
Waldegrave in 1713
 Theory of Games and Economic Behavior
by John von Neumann in 1944
 Eight game theorists have won Nobel
prizes in economics
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Definitions
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Player-maybe be people, organizations or
countries
Strategies- course of action they may take
based on the options available to them
Outcomes- the consequences of the strategies
chosen by the players
Preferences- each player has a perfered
outcome
Game theory analyzes the rational
choice of strategies

Areas Applied
 Bargaining
tactics in labor-management
disputes
 Resource allocation decisions
 Military Choices in international crises
What makes it different
Analyzes situations in which there are at
least two players
 The outcome depends on the choices of
all the players
 Players can cooperate but it is not
necessary
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Why is it important?
Provided theoretical foundations in
economics
 Applied in political science (study of
voting, elections, and international
relations)
 Given insight into understanding the
evolution of species and conditions under
which animals fight each other for territory
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Two-Person Total-Conflict
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Location Game
Two Young Entrepreneurs with a new restaurant
in the mountains
Lisa likes low elevations
Henry likes higher elevations
Routes A, B, and C run east-west
Highways 1, 2 and 3 run north-south
Henry selects one of the routes
Lisa selects one of the Highways
Selection is made simultaneously
Heights of the intersections
Highways
Routes
1
2
3
A
10
4
6
B
6
5
9
C
2
3
7
How do they choose
Maximin- the maximum value of the
minimum numbers in the row of a table
 Minimax- the minimum value of the
maximum numbers in the columns of a
table
 Saddlepoint- the outcome when the row
minimum and the column maximum are
the same
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Highways
Routes
1
2
3
A
10
4
6
Row
Minima
4
B
6
5
9
5
C
2
3
7
2
Column
Maxima
10
5
9
Solution
In total-conflict games, the value is the
best outcome that both players can
guarantee
 In our example the value is 5
 The value is given by each player
choosing their maximin and minimax
strategies
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Example 2: Restricted-Location
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Use the same information from previous
problem
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However, the county officials outlaw
restaurants on Route B and Highway 2
Highways
Routes
1
3
A
10
6
Row
Minima
6
C
2
7
2
Column
Maxima
10
7
Results
There are no saddlepoints
 If both choose their minimax and maximin
strategy, we will result in 7
 However, they could try to out think the
other which could result in 10 or 2
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Duel Game
Pitcher
Batter
F
C
F
.300
.200
Row
Minima
.200
C
.100
.500
.100
Column
Maxima
.300
.500
Flawed Approach
Pitcher- If I choose F I hold the batter
down to .300 or less but the batter is likely
to guess F which gives him at least .200
and actually .300
 Batter- Because the pitcher will try to
surprise me with C, I should guess C. I
would then average .500.
 Pitcher- But if batter guess C, I should
really throw F. Thus leading to an average
of .100 for the batter
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As we see…we can keep going
over and over…
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Pure Strategy- Each of the definite courses of
action that a player can choose
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Mixed Strategy- Course of action is randomly
chosen from one of the pure strategies by:
 Each
pure strategy is assigned some probability,
indicating the relative frequency with which that pure
strategy will be played
 The specific strategy used in any given play of the
game can be selected at using some appropriate
random device
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Expected Value of E
 In
each of the n payoffs, s1, s2, ……, sn, will
occur with the probability p1, p2, ………pn,
respectively.
 The expected Value E
E=p1s1 +p2s2+………..+ pn*sn
 And we assume p1+p2+……+pn=1
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Matching Pennies
Two players
 Each has a penny
 They both show either heads or tails at the
same time
 If the match, player 1 gets the pennies
 If they are not a match, player 2 gets the
pennies
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Payoff Matrix
Player 2
Player 1
H
T
H
1
-1
T
-1
1
Results
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H & T are pure strategies for both players
There is no way one player can outguess the
other
Each player should use a mixed strategy
choosing H half the time and T half the time
For player 1:
 E(h)=
½(1) + ½(-1) = 0
 E(t)= ½(-1) + ½(1) =0
Cont.
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The expected value for player 2 is the same
This means the game is fair, which means the
expected value = 0 and therefore favors neither
player when at least one player uses an optimal
mixed strategy
If one player does not use the 50-50 strategy the
player that does gains an advantage
Another example
Player 2
Player 1
H
T
H
5
-3
T
-3
1
Results
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Player 1
E(H) = 5*(p) + (-3)(1-p) = 8p-3
E(T) = (-3)(p) +(1-p)=-4p +1
 8p-3=-4p+1
 12p = 4
 P=1/3
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Therefore,
 E(H)
= 8(1/3) -3 = E(T) = -4(1/3) + 1 =-1/3 => p=1/3
 So their optimal mixed straigy is (1/3, 2/3) with
expected value of 1/3
Cont.
Using same calculations for player 2 we
get the same optimal mixed stratigy of
(1/3, 2/3)
 However, the expected value for player 2
is 1/3
 Therefore, we have a zero-sum game.
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Lets go back to the baseball game
Pitcher
Batter
F
C
F
.300
.200
Row
Minima
q
C
.100
.500
1-q
Column
Maxima
p
1-p
What should the pitcher do?
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E(f)= (0.3)p + (0.2)(1-p) = 0.1p + 0.2
E(c)= (0.1)p + 0.5(1-p) = -0.4p + 0.5
Solution is at the intersection of these two lines
-0.4p + 0.5 = 0.1p + 0.2
p = 0.6
Giving E(f)=E(c)=E=0.26
Thus, the Pitcher should pitch F with p = 3/5 and
C with p=2/5 so the batter will not be better than
.260
What should the batter do?
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E(f)= (0.3)q + (0.1)(1-q) = 0.2q + 0.1
E(c)= (0.2)q + (0.5)(1-q) = -0.3q + 0.5
0.2q + 0.1 = -0.3q + 0.5
q=0.8
E(f) = E(c) = E = 0.260
Therefore, he should guess F with p=4/5 and C
with p=1/5 which gives him a batting average of
0.260
So this gives us an outcome of 0.260
Partial-Conflict Games
These are games in which the sum of
payoffs to the players at different
outcomes varies
 There can be gains by both players if the
cooperate but this could be difficult
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Prisoners’ Dilemma
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Two-person variable-sum game
Shows the workings behind arms races, price
wars, and some population problems
In these games, each player benefits from
cooperating
There is no reason for them to cooperate without
a credible threat of retaliation for not cooperating
Albert Tucker, Princeton mathematician, named
the game the Prisoners’ Dilemma in 1950
So the actual game
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Two people are accused of a crime
Each person has a choice:
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Claim their innocence
Sign a confession accusing the partner of committing the crime
It is in their interest to confess and implicate their partner
to receive reduce sentence
However, if both confess, both will be found guilty
As a team, their best interest is to deny having
committed the crime
Apply it to the real world army race
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Two nations, Red and
Blue
A: Arm in preparation
for war
D: Disarm or
negotiate an armscontrol agreement
Rank from best to
worse (41)
Blue
A
Red
D
A
(2,2) (4,1)
D
(1,4) (3,3)
What should they do?
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Red
 If
Blue selects A- Red receives a payoff of 2
for A and 1 for D, so choose A
 If Blue select D- Red receives a payoff of 4 for
A and 3 for D, so choose A
Red has a dominate strategy of A
 So a rational Red nation will choose A
 Similarly, Blue will choose A
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Results
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If the nations work independently, we get an
outcome of (A,A) with payoff of (2,2)
This is a Nash Equilibrium- where no player can
benefit by departing by itself from its strategy
associated with an outcome
So, each player can corporate, play
independent, or defect
Defect dominates cooperate and playing
independent for both players
However, defect by both players results in a
worse outcome than the mutual-cooperation
outcome
Another Example “Chicken”
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Two Drivers coming at each other at high speeds
Driver 2
Driver 1
Swerve
Not Swerve
(3,3)
(2,4)
Not Swerve (4,2)
(1,1)
Swerve
Results
Neither player has a dominate strategy
 The Nash Equilibrium are (4,2) and (2,4)
 This means that getting the result of (3,3)
will be unlikely because each players has
an incentive to deviate to get a high payoff
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Larger Games
Lets look for a 3x3x3 game
 We find the optimal solution by looking at
individuals dominant strategy
 Reducing it to a 3x3 game and we solve
like a 2 person games we have been
doing
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Example: Truel
A duel with 3 people
 Each player has a gun and can either fire
or not fire at either of the other players
 Goal is to survive 1st and survive with as
few other players as possible
 http://www.youtube.com/watch?v=rExm2F
bY-BE&feature=related
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Game Tree