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Game Theory
Jacob Foley
http://www.youtube.com/watch?v=HCinK2PUfyk
http://www.youtube.com/watch?v=l0ywiYboCLk
Overview
1. Introduction and history
2. Total-conflict games
3. Partial-conflict games
4. Three-person voting game
What is Game Theory
Game- two or more individuals compete to try to control the course of events
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
Definitions
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 AppliedBargaining tactics in labor-management
disputesResource allocation decisionsMilitary 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
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
Two-Person Total-Conflict 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
Highways
Routes 1 2 3 Row Minima
A 10 4 6 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
Example 2: Restricted-Location
Use the same information from previous problem
However, the county officials outlaw restaurants on Route B and Highway 2
Highways
Routes 1 3 Row Minima
A 10 6 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
Duel GamePitcher
Batter F C Row Minima
F .300 .200 .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
As we see…we can keep going over and over… Pure Strategy- Each of the definite courses of
action that a player can choose
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
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
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
Payoff Matrix
Player 2
H T
Player 1 H 1 -1
T -1 1
Results
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.
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
H T
Player 1 H 5 -3
T -3 1
Results
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
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.
Lets go back to the baseball gamePitcher
Batter F C Row Minima
F .300 .200 q
C .100 .500 1-q
Column Maxima
p 1-p
What should the pitcher do?
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?
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
Prisoners’ Dilemma
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
Two people are accused of a crime Each person has a choice:
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
Two nations, Red and Blue
A: Arm in preparation for war
D: Disarm or negotiate an arms-control agreement
Rank from best to worse (41)
Blue
A D
Red A (2,2) (4,1)
D (1,4) (3,3)
What should they do?
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
Results 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” Two Drivers coming at each other at high speeds
Driver 2
Swerve Not Swerve
Driver 1 Swerve (3,3) (2,4)
Not Swerve (4,2) (1,1)
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
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
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=rExm2FbY-BE&feature=related
Game Tree