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