Pamela Schmitt
United States Naval Academy
Game Theory
 REVIEW payoff matrix
 REVIEW definition and determination of
dominant strategies
 NASH EQUILIBRIA with dominant strategies
 NASH EQUILIBRIA without dominant strategies
(cover and underline best response method)
 Applications: Oligopolies, Prisoner dilemma
(suboptimal outcomes), Battle of the Sexes,
Chicken, Hotelling's Beach
The Payoff Matrix:
Dominant Strategy Equilibrium
Danny
Lily
Left
Right
Top
4,7
5, 8
Bottom
2, 1
3, 6
The “row” player  Lily
Lily has two strategies “Top” and “Bottom”
Danny
“row”
Lily
Left
Right
Top
4, 7
5, 8
Bottom
2, 1
3, 6
The “column” player  Danny
Danny has two strategies “Left” and “Right”
“column” Danny
Lily
Left
Right
Top
4, 7
5, 8
Bottom
2, 1
3, 6
The Payoff Matrix
 The first number in each cell is the payoff the
row player (Lily) receives if both players
choose the action that leads to that cell.
 Similarly, the second number in each cell is the
payoff the column player (Danny) receives.
If Lily chooses “Top”: Lily earns
4 if Danny chooses “Left” and
5 if Danny chooses “Right” Danny
Lily
Left
Right
Top
4, 7
5, 8
Bottom
2, 1
3, 6
If Lily chooses “Bottom”: Lily earns
2 if Danny chooses “Left” and
3 if Danny chooses “Right” Danny
Left
Right
Lily
Top
4, 7
5, 8
Bottom
2, 1
3, 6
Dominant strategies
 A dominant strategy is the best strategy
regardless of what the other player chooses.
 If both players have a dominant strategy, the
outcome is a dominant strategy equilibrium.
 All dominant strategy equilibrium are Nash
Equilibrium.
Lily has a dominant strategy: choosing Top
always leads to a higher payoff regardless of
what Danny chooses:
Danny
4>2 and 5>3
Lily
Left
Right
Top
4, 7
5, 8
Bottom
2, 1
3, 6
Dominant strategies
 But not all Nash Equilibrium are
dominant strategy equilibrium.
 A Nash Equilibrium is the outcome in
which neither player has a desire to
choose a different strategy given the
choice of the other player. (mutual best
responses)
Let’s try it with AP questions
Let’s try it with AP questions
Let’s try it with AP questions
Let’s try it with AP questions
Let’s try it with AP questions
Note: Neither has a dominant strategy.
But, we can now answer (a): if Red Shop chooses
“South” Blue Mart chooses “North” (1 pt).  b/c
4000>1000 (1 pt.)
And for (b): “South” is not a dominant strategy for Red Shop
chooses (1 pt.) If Blue Mart chooses south, Red Shop is better
off choosing north. (Red Shop’s best response depends on
Blue Mart’s move.) (1 pt.)
Part (c): the highest combined payoff are at (S,N): (5,000
+4,000) > 6,5000 > 2,7000,> 2,500. (1 pt.) Stating that Red Shop
chooses south and Blue Mart chooses north
Part (d) redraw such that +$2,000 are added
to “South” payoffs
Blue Mart
Red
Shop
North
South
North
900, 1800
3000,5500
South
7000,4000
3500, 3000
http://gametheory.tau.ac.il/
 When teaching game theory, I prefer to have students
start with their own intuition.
 Ariel Rubinstein has an online resource that allows
teachers to use simple games (and more complex
ones!) to build this intuition.
 This is following Rubinstein, A. (1999). “Experience
from a Course in Game Theory: Pre- and Post-class
Problem Sets as a Didactic Device” Games and
Economic Behavior 28, 155 – 170.
http://gametheory.tau.ac.il/
Vecon Lab Games
Basic Attacker/Defender Game
Two Person/Binary Decision Game of Strategy
Multi-Site Attacker/Defender Game
New! Defaults Implement a Simple Profiling
Game
Centipede
Alternating Two-person "Pass or Take" Game
Coordination
Minimum-Effort Game, with Incentive Pay
Options
Guessing Game
With Incentive to Guess Others' Decisions
2x2 Matrix Game
Prisoner's Dilemma, Battle of Sexes, etc.
Asymmetric Matrix Game
"Large" Setup, e.g. Coordination with 7 Effort
Choices
Symmetric Matrix Game
NxN Matrix Game with Symmetric Payoffs
Security Coordination Game
Coordination of Security Investment Decisions
Traveler's Dilemma
Social Dilemma with No Dominant Strategy
2-Stage Game
Generic Two-Stage Extensive-Form Game
View Results
View Results of Any Prior Setup