Game Theory

1
Game Theory

• Game theory models strategic behavior by agents
  who understand that their actions affect the
  actions of other agents.
• Been used to study
• Market competition Cournot Oligopoly.
• Military strategies.
• Bargaining/Negotiations
• Biology
• A game consists of players, strategies, and
  payoffs.

2
Battle of Bismarck Sea
Imamura
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.

3
Imamura wants to run convoy from Rabaul to Lae
4
The Prisoners Dilemma
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.
5
War
Mars
Not Shoot
Shoot
-5
-1
Not Shoot
-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. Tit-for-tat can work. Not so easy if
uncertainty of action.
6
Coordination Problem
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.

7
Nash equilibrium

• A Nash equilibrium is a set of strategies
• Where each player has no incentive to deviate.
• Given other equilibrium strategies a player would
  choose his equilibrium strategy.
• A best response to a best response.
• A pure strategy equilibrium is where each player
  only chooses a particular strategy with
  certainty.
• What are the pure-strategy equilibria in
  Prisoners dilemma and the coordination game.
• Is there always a pure strategy equilibrium?

8
Penalty Kick
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.

9
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.

10
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)(41.99L58.01R, 38.54L61.46R)
• Actual Data (42.31L57.69R,
  39.98L60.02R)

Dive L
Dive R
58.3
94.97
Kick L
Kick R
92.91
69.92