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Exploring Nash Equilibria Without Dominant Strategies
Pamela SchmittUnited States Naval Academy
Game TheoryREVIEW payoff matrixREVIEW definition and determination of
dominant strategiesNASH EQUILIBRIA with dominant strategiesNASH 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
Left Right
Top 4,7 5, 8
Bottom 2, 1 3, 6
Danny
Lily
The “row” player Lily Lily has two strategies “Top” and “Bottom”
Left Right
Top 4, 7 5, 8
Bottom 2, 1 3, 6
Danny
“row”Lily
The “column” player Danny Danny has two strategies “Left” and “Right”
Left Right
Top 4, 7 5, 8
Bottom 2, 1 3, 6
“column” Danny
Lily
The Payoff MatrixThe 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”
Left Right
Top 4, 7 5, 8
Bottom 2, 1 3, 6
Danny
Lily
If Lily chooses “Bottom”: Lily earns 2 if Danny chooses “Left” and 3 if Danny chooses “Right”
Left Right
Top 4, 7 5, 8
Bottom 2, 1 3, 6
Danny
Lily
Dominant strategiesA 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: 4>2 and 5>3
Left Right
Top 4, 7 5, 8
Bottom 2, 1 3, 6
Danny
Lily
Dominant strategiesBut 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
North South
North 900, 1800 3000,5500
South 7000,4000
3500, 3000
Blue Mart
Red Shop
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/
Basic Attacker/Defender GameTwo Person/Binary Decision Game of Strategy
Multi-Site Attacker/Defender GameNew! Defaults Implement a Simple Profiling Game
CentipedeAlternating Two-person "Pass or Take" Game
CoordinationMinimum-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 GameNxN Matrix Game with Symmetric Payoffs
Security Coordination GameCoordination of Security Investment Decisions
Traveler's DilemmaSocial Dilemma with No Dominant Strategy
2-Stage Game Generic Two-Stage Extensive-Form Game
View Results View Results of Any Prior Setup
Vecon Lab Games