🔎 Interesting Examples
Prisoner’s Dilemma
Section titled “Prisoner’s Dilemma”Wikipedia page on Prisoner’s Dilemma
| Prisoner’s Dilemma | Cooperate | Defect/Rat |
|---|---|---|
| Cooperate | 3,3 | 1,4 |
| Defect/Rat | 4,1 | 2,2 |
✏️ Find the Nash Equilibria
✔ Click here to view answer
| Prisoner’s Dilemma | Cooperate | Defect/Rat |
|---|---|---|
| Cooperate | 3,3 | 1,4 |
| Defect/Rat | 4,1 | 2,2 |
NE is Defect, Defect. ✅
Matching Pennies
Section titled “Matching Pennies”Wikipedia page on Matching Pennies
With matching pennies, both players choose heads or tails. If they choose the same thing, the row player wins. If they choose the opposite, the column player wins.
| Matching Pennies | H | T |
|---|---|---|
| H | 1,-1 | -1,1 |
| T | -1,1 | 1,-1 |
✏️ Find the Nash Equilibria
✔ Click here to view answer
| Matching Pennies | H | T |
|---|---|---|
| H | 1,-1 | -1,1 |
| T | -1,1 | 1,-1 |
There is no Nash Equilibrium. ✅
Coordination Game
Section titled “Coordination Game”Wikipedia page on the Coordination Game
This type of game is used to model situations in which people have multiple options and it’s vital for them to coordinate. Suppose that the row and column players both prefer Ballet to Boxing, but that neither will have any fun if they are alone:
| Simple Coordination Game | Boxing | Ballet |
|---|---|---|
| Boxing | 1,1 | 0,0 |
| Ballet | 0,0 | 2,2 |
✏️ Find the Nash Equilibria
✔ Click here to view answer
| Simple Coordination Game | Boxing | Ballet |
|---|---|---|
| Boxing | 1,1 | 0,0 |
| Ballet | 0,0 | 2,2 |
There are two Nash Equilibria. In one, both play boxing and in the other, both play ballet.
Both players would prefer the NE in which both play Ballet. However, the NE in which they both play Boxing is still a NE. This is true because if the other player is playing boxing, it is still rational for both of them to play boxing, because neither of them want to be alone. We say that neither of them can “profitably deviate” from the NE in which they both play boxing. ✅
Battle of the Sexes
Section titled “Battle of the Sexes”Wikipedia page on Battle of the Sexes
A battle of the Sexes is a Coordination game in which the two players have different preferences regarding the two Nash Equilibria. For example, both partners want to see each other in the evening. However, one partner would rather that they meet at the ballet and the other partner would rather meet at boxing.
| Battle of the Sexes | Boxing | Ballet |
|---|---|---|
| Boxing | 1,2 | 0,0 |
| Ballet | 0,0 | 2,1 |
✏️ Find the Nash Equilibria
✔ Click here to view answer
| Battle of the Sexes | Boxing | Ballet |
|---|---|---|
| Boxing | 1,2 | 0,0 |
| Ballet | 0,0 | 2,1 |
2 NE: Boxing, Boxing and Ballet, Ballet.
The Row player prefers the (Ballet,Ballet) NE and the Column player prefers (Boxing,Boxing) NE. However, once they are both playing one of the Nash Equilibria, neither can unilaterally change their strategy without ensuring a payoff of zero for both of them. ✅
Chicken
Section titled “Chicken”Wikipedia page for Chicken Game
The 80s movie Footloose had a version involving tractors.
| Chicken | Swerve | Straight |
|---|---|---|
| Swerve | -1,-1 | -1,1 |
| Straight | 1,-1 | -10,-10 |
✏️ Find the Nash Equilibria
✔ Click here to view answer
| Chicken | Swerve | Straight |
|---|---|---|
| Swerve | -1,-1 | -1,1 |
| Straight | 1,-1 | -10,-10 |
Two Nash equilibria.
In each Nash equilibrium, one player is scaring/bullying the other and “winning.” The other player loses but can’t change the situation. ✅
Complete Indifference
Section titled “Complete Indifference”No one made a Wikipedia page about “Complete Indifference” because they were completely indifferent. (Actually, this one isn’t famous like the others - I do think it’s an interesting one to think about, though.)
| Indifference Game | Left | Right |
|---|---|---|
| Up | 1,1 | 1,1 |
| Down | 1,1 | 1,1 |
✏️ Find the Nash Equilibria
✔ Click here to view answer
| Indifference Game | Left | Right |
|---|---|---|
| Up | 1,1 | 1,1 |
| Down | 1,1 | 1,1 |
All 4 cells are NE. ✅
How would you describe this game?
| 3x3 | Left | Center | Right |
|---|---|---|---|
| Top | 2,3 | 1,4 | 0,0 |
| Middle | 4,1 | 3,2 | 0,0 |
| Bottom | 0,0 | 0,0 | 5,5 |
✏️ Find the Nash Equilibria
✔ Click here to view answer
| 3x3 | Left | Center | Right |
|---|---|---|---|
| Top | 2,3 | 1,4 | 0,0 |
| Middle | 4,1 | 3,2 | 0,0 |
| Bottom | 0,0 | 0,0 | 5,5 |
The players can be caught in (center, center) even though they would rather be in (bottom, right). ✅
Rock Paper Scissors
Section titled “Rock Paper Scissors”Wikipedia page for Rock Paper Scissors
| Rock | Paper | Scissors | |
|---|---|---|---|
| Rock | 0,0 | -1,1 | 1,-1 |
| Paper | 1,-1 | 0,0 | -1,1 |
| Scissors | -1,1 | 1,-1 | 0,0 |
✏️ Find the Nash Equilibria
✔ Click here to view answer
| Rock | Paper | Scissors | |
|---|---|---|---|
| Rock | 0,0 | -1,1 | 1,-1 |
| Paper | 1,-1 | 0,0 | -1,1 |
| Scissors | -1,1 | 1,-1 | 0,0 |
Like with Matching Pennies, there is no Nash Equilibrium. That’s what makes it a fun game to play. If there were a Nash Equilibrium, people would always play the same thing once they got into a habit, but with this it is very unpredictable. ✅
Feedback? Email munger.e1010@gmail.com 📧. Be sure to mention the page you are responding to.