- By
**liam** - Follow User

- 234 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about 'Game Theory' - liam

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

Game Theory

- Game theory models strategic behavior by agents who understand that their actions affect the actions of other agents.
- Useful to study
- Company behavior in imperfectly competitive markets (such as Coke vs. Pepsi).
- Military strategies.
- Bargaining/Negotiations.
- Biology
- A game consists of players, strategies, and payoffs.

- We want to model the Battle of the Bismarck Sea.
- Two Admirals: Imamura (Japan) and Kenny (US).
- Japan is in retreat.
- Imamura wants to transport troops in a convoy
- from Rabaul to Lae
- Kenny wants to bomb Japanese troops. .
- North route is two days, Southern route is three days.
- It takes one day for Kenny to switch searching routes.

Imamura

North

South

-2

-2

North

2

2

Kenny

-3

-1

South

1

3

- This representation is called a Normal form Game.
- Imamura wants to transport troops.
- Kenny wants to bomb Japanese troops. .
- North route is two days, Southern route is three days.
- It takes one day for Kenny to switch routes.

Imamura

Players

North

South

-2

-2

North

2

2

Kenny

-3

-1

South

1

3

- Imamura wants to transport troops.
- Kenny wants to bomb Japanese troops. .
- North route is two days, Southern route is three days.
- It takes one day for Kenny to switch routes.

Imamura

Players

North

South

Imamura’s

Strategies

-2

-2

North

2

2

Kenny

-3

-1

South

1

3

- Imamura wants to transport troops.
- Kenny wants to bomb Japanese troops. .
- North route is two days, Southern route is three days.
- It takes one day for Kenny to switch routes.

Imamura

Players

Kenny’s

Strategies

North

South

Imamura’s

Strategies

-2

-2

North

2

2

Kenny

-3

-1

South

1

3

- Imamura wants to transport troops.
- Kenny wants to bomb Japanese troops. .
- North route is two days, Southern route is three days.
- It takes one day for Kenny to switch routes.

Imamura

Imamura’s

Payoffs:

Each day of

Bombing =

-1 in payoff

North

South

-2

-2

North

2

2

Kenny

-3

-1

South

1

3

- Imamura wants to transport troops.
- Kenny wants to bomb Japanese troops. .
- North route is two days, Southern route is three days.
- It takes one day for Kenny to switch routes.

Kenny’s

Payoffs:

Each day of

Bombing =

1 in payoff

Imamura

Imamura’s

Payoffs:

Each day of

Bombing =

-1 in payoff

North

South

-2

-2

North

2

2

Kenny

-3

-1

South

1

3

- Imamura wants to transport troops.
- Kenny wants to bomb Japanese troops. .
- North route is two days, Southern route is three days.
- It takes one day for Kenny to switch routes.

Imamura

North

South

-2

-2

North

2

2

Kenny

-3

-1

South

1

3

Notice that Kenny guessing wrong costs him a day.

- Imamura wants to transport troops.
- Kenny wants to bomb Japanese troops. .
- North route is two days, Southern route is three days.
- It takes one day for Kenny to switch routes.

Clyde

S

C

-5

-1

S

-5

-15

Bonnie

-10

-15

C

-1

-10

Bonnie and Clyde are caught.

They can confess or be silent.

Nash equilibrium

- A Nash equilibrium is a set of strategies (s1,s2)
- Where each player has no incentive to deviate given the other players’ strategies.
- (or) Given other equilibrium strategies a player would choose his equilibrium strategy.
- (or) A best response to a best response.
- u1(s1,s2)>=u1(z,s2) for all possible z and u2(s1,s2)>=u2(s1,z) for all possible z.
- A pure strategy equilibrium is where each player only chooses a particular strategy with certainty.
- What are the pure-strategy equilibria in Prisoners’ dilemma?

Trench Warfare: a prisoners’ dilemma

Mars

Not Shoot

Shoot

-5

Not Shoot

-1

-5

-15

Venus

-10

-15

Shoot

-1

-10

Are we doomed to the bad outcome?

Not in trench warfare of WWI.

This happens since the game is repeated –that is, played several times.

Repeated games: forever punish

- Forever punish strategy: if someone cheats, punish forever.
- Cheating gives short term gain but long term loss.
- Gain of 4 for the time of cheating.
- Loss of 5 from the next period on.
- Whether this stops cheating depends upon how you value today compared to tomorrow (discount rate).
- Forever punish is not so great if there is uncertainty. Why?
- Many tribal religions do not have explicit ethical code as part of the religion: no stealing, no murder, etc.

Jim

VHS

Beta

1

0.5

VHS

1

0

Sean

2

0

Beta

0.5

2

- Jim and Sean want to have the same VCR.
- Beta is a better technology than VHS.

Information Technology

- Phones, Faxes, e-mail, etc. all have the following property:
- Network externalities: The more people using it the more benefit it is to each user.
- Computers, VCRs, PS2s, also have this property in that both software can be traded among users and the larger the user market, the larger number of software titles are made.
- How do markets operate with such externalities?

Discussion points

- Competitors: VHS vs. Beta, Qwerty vs. Dvorak, Windows vs. Mac, Playstation vs. Xbox.
- Does the best always win?
- Standardization helps with network externalities.
- Drive on left side vs. right side. Out of 206 countries 144 (70%) are rhs.
- Left is more nature for an army: swords in right hand, mounting horses. (Napolean liked the other way.)
- Sweden switched from left to right in 1967.
- Lots of networks: Religions and Languages.

Goalie

Dive L

Dive R

1

-1

Kick L

-1

1

Kicker

1

-1

Kick R

1

-1

- A Kicker can kick a ball left or right.
- A Goalie can dive left or right.

Mixed Strategy equilibrium

- Happens in the Penalty kick game.
- Notice that if the Kicker kicks 50-50 (.5L+.5R), the Goalie is indifferent to diving left or right.
- If the Goalie dives 50-50 (.5L+.5R), the Kicker is indifferent to kicking left or right.
- Thus, (.5L+.5R,.5L+.5R) is a mixed-strategy N.E.
- Nash showed that there always exists a Nash equilibrium (which includes mixed strategies).

Do you believe it?

- Can we empirically test this theory?
- Yes!
- Study was done with the Italian football league.
- Step 1: See if the strategies are really left or right.
- Step 2: Calculate payoffs. How? If when the goalie guesses correctly, there is no goal 100% of the time the payoffs are 0 for the kicker and 100 for the goalie. If there no goal 80% of the time, then the payoffs are 20 for the kicker and 80 for the goalie, etc…
- Step 3: Calculate the Nash equilibrium.
- Step 4: Compare.

Do you believe it?

- Do they really choose only L or R? Yes. Kickers 93.8% and Goalie 98.9%.
- Kickers are either left or right footed. Assume R means kick in “easier” direction. Below is percentage of scoring.
- Nash prediction for

(Kicker, Goalie)=(41.99L+58.01R, 38.54L+61.46R)

- Actual Data =(42.31L+57.69R, 39.98L+60.02R)

Dive L

Dive R

58.3

94.97

Kick L

Kick R

92.91

69.92

Student Driver

Park OK

Park in Staff

-5

-95

Check

-5

5

University

5

-5

Don’t

5

0

- Student can decide to park in staff parking.
- University can check cars in staff parking lot.

What happens?

- If the University checks, what do the students do?
- If the students park ok, what does the Uni do?
- If the uni doesn’t check, what do the students do?
- If the students park in the staff parking, what does the uni do?
- What is the equilibrium of the game?
- What happens if the university makes it less harsh a punishment to only –10. Who benefits from this? Who is hurt by this?

Answer

- Student parks legally 1/3 of the time and the uni checks 1/10 of the time.
- With lower penalty, student parks legally 1/3 of the time and the uni checks 2/3 of the time.
- Who’s expected payoff changes? No one.

Download Presentation

Connecting to Server..