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