By: Donté Howell Game Theory in Sports

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By: Donté Howell Game Theory in Sports




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Presentations text content in By: Donté Howell Game Theory in Sports

Slide1

By:Donté Howell

Game Theory in Sports

Slide2

What is Game Theory?

It is

a tool used to analyze strategic behavior

and trying

to maximize

his/her payoff

of the game by anticipating the actions of the other players and responding to them

correctly

Slide3

History Of Game Theory

Was invented by

was invented by John von Neumann and Oskar Morgenstern 

in 1944

Back then Game Theory was only limited to certain circumstances, but as the years go by the are more and situations where Game Theory can be applied

Slide4

Where Game Theory can be applied

Game theory can be applied in any situation that calls for strategic thinking, such as:

Getting a person involved in a crime to confess

Businesses competing in a market

War

Sports

Etc.

Slide5

Elements of Game Theory

1)

The agent:

Also known as the player, which

refers to a

person, company, or nation who

have their own goals and

preferences

2

)

The

 

utility:

The

amount of

satisfaction, or payoff that

an agent

receives

from

the situation or

an

event

3

)

The

 

game:

The situation or event that all the agents/players involved will be participating in

4

)

The

 

information:

What

a player knows about what has already happened in the game, and

what

can be used to come up with a good

strategy

5

)

The

 

representation:

Describes the

order of play employed in the

game

6

)

The

 

equilibrium:

An outcome

of or a solution to the game.

Slide6

Type Of Games Used In Game Theory For Sports

 

S

imultaneous game:

This is the type of game where all

players

come up with a strategy without knowing the strategy

that

that the other player/players are choosing. The

game is simultaneous because each player has no information about the decisions of

the other player/players; therefore the decisions were

made simultaneously. Simultaneous games are

solved

using the concept of a 

Nash Equilibrium.

Zero Sum Game:

When one player’s loss is equal to anther player’s gain. There the sum of the winnings and losses equal to zero

Slide7

What is Nash Equilibrium?

When a player can

receive no

positive

benefit from changing actions, assuming other players remain constant in their strategies. A game may have multiple Nash equilibria or none at all

.

There are two different types strategies used to try to achieve a Nash Equilibrium:

Pure Strategy

Mixed Strategy

Slide8

The Strategies

Pure

strategy

 

Having the complete knowledge of

how a player will play a

game. It

determines the move a player will make for any situation he or she could face.

Mixed

strategy

 

The probability of which a pure strategy will be used. This allows a player to keep an opponent guessing by randomly choosing a pure strategy.

Since probabilities are continuous, there are infinitely many mixed strategies available to a player, even

if the amount of pure strategies is

finite

.

Slide9

The Formula For Calculating the Mixed Strategies

Player 2

Player

1

A

B

A

a,b

c,d

B

e,f

g,h

Equation used

=>

Let q, be the probability

q × a

+ (1-q

) × c

=

probability of player 2 doing A

q × e + (1-q) x g

= probability of player 2 doing B

q × a + (1-q) ×

c =

q × e + (1-q) x g

->To find probability of maximizing player 2’s payoff

p

× a

+ (

1-p)

×

e

=

probability of player

1

doing

A

p

×

c

+ (

1-p)

x

g

= probability of player

1

doing B

p

× a + (

1-p)

×

e

=

p

×

c

+ (

1-p)

x

g

-> To find probability of maximizing of player 1’s payoff

Slide10

Game Theory In Tennis

Server

Receiver

Forehand

Backhand

Forehand

90,10

20,80

Backhand

30,70

60,40

In this example the payoff for the Receiver is the probability of saving,

and the payoff for the Server is the probability of scoring,

Let’s consider the potential strategies for the

Server:

If

the Server always aims Forehands then the

Receiver (anticipating

the

Forehand serve

) will always

move Forehands, and

the payoffs will be (90,10) to Receiver

and Server respectively.

If

the Server always aims Backhands then the

Receiver (anticipating

the Backhand serve) will always

move Backhands

and the payoffs will be (60,40).

Slide11

Game Theory In Tennis

Obviously the server wants the probability to be more in his favor. So the next step would be to find the best mixed strategy for the server to have his best possible performance.

Suppose the Server aims Forehands with q probability

and Backhands

with 1-q probability. Then the Receiver’s payoff

is:

q x 90 + (

1-q

) x 20

= 20 + 70q if she moves Forehands

q x 30 + (

1-q

) x 60

= 60 - 30q if she

moves Backhands

.

Slide12

Game Theory In Tennis

From these solutions the server sees that the receiver is going to want to maximize their chance of a payoff. Therefore the receiver would move:

Forehands if 20 +

70q > 60

-

30q

Backhands

if 20 +

70q < 60

-

30q

Either

one if 20 + 70q = 60 - 30q.

The

Receiver’s payoff is the larger of 20+70q

and 60-30q

Slide13

Game Theory In Tennis

In order for the server to maximize his payoff, he has to minimize the payoff of the receiver. He can do that by setting the two probabilities equal to each other.

20 +

70q = 60 − 30q => 100q = 40 => q = 0.4 = 40%

This solution tells the

server that

in

order to maximize his payoff the Server should

aim Forehands

40% of the time and Backhands 60% of the time.

In this

case the Receiver’s payoff will

be:

20 +

70 × 0.4

= 60

– 30 × 0.4 = 48%

Slide14

Game Theory In TennisThe result of Server mixing his serves 40-60

then the

Receiver’s payoff

will be

48%chance of saving it

whether

he/she

moves Forehands or

Backhands, or

mixes between

them.

Therefore

the Server’s payoff will

be:

100-48

=

52% of successfully scoring

Slide15

Game Theory In TennisNOW, lets do the same thing but for the receiver

Lets say the receiver doesn’t mix up their strategy, then the server will move its strategy to the side

more

favorable to

them.

Suppose

the Receiver moves

for Forehands

with p

probability. Then their

payoff

is:

p × 90

+ (1-p

) × 30

= 30 + 60p if the Server aims

Forehands

p × 20 + (1-p) × 60 = 60 - 40p if the Server aims Backhands.

Slide16

Game Theory In TennisThe server will look to minimize the receiver's payoff so, they will aim for the smaller side:

Forehands if

30 +

60p

< 60

-

40p

Backhands

if 30 +

60p >

60

-

40p

Either

one if 30 + 60p = 60 -

40p

To maximize the receiver’s payoff they have to set them equal to each other:

30 + 60p = 60 − 40p => 100p = 30 => p = 0.3 = 30%

Slide17

Game Theory In Tennis

For the receiver to maximize their payoff they should move for a forehand 30% of the time and backhand 70% of the time.

In this

case the Receiver’s her payoff will be 30 +

(60 × 0.3)

= 60

– (40 × 0.3)

= 48. Therefore the Server’s payoff will be 100-48 =

52%

Mixed Strategy Result

Receiver: 0.3F +

0.7B

Server: 0.4F +

0.6B

At this point both players have found their best possible strategy to maximize their chance of getting their payoff, and don’t have anything to gain by changing their strategies now, therefore this is mixed strategy Nash Equilibrium

Slide18

Work Cited

http://

dictionary.reference.com/browse/zero-sum+game

http://

www.siliconfareast.com/game-theory.htm

http://

www.znu.ac.ir/members/afsharchim/lectures/MixedStrategy.pdf


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