🔎 Finding Nash Equilibria
A Nash equilibrium is a situation in which both players are playing best responses to the other player.
The first step is to find all of the best responses for each player. We do this by role playing. We take the role of one player at a time, and we imagine that the other player is taking a specific action. What is our best response? We then underline the payoff we get when we play our best response to their action.
Each box in the table represents a pair of strategies, so a box with two underlines is a Nash Equilibrium (because both players are playing best responses to the other player).
We’ll use the following game as an example:
| Rob | |||
|---|---|---|---|
| Don’t Confess | Confess | ||
| Bruce | Don’t Confess | uR = -1 uB = -1 | uR = 0 uB = -10 |
| Confess | uR = -10 uB = 0 | uR = -5 uB = -5 |
The hard part is underlining (or circling) the best responses. We break it down into four steps.
Bruce’s best responses:
- What is Bruce’s best response if Rob doesn’t confess?
- What is Bruce’s best response if Rob confesses?
Rob’s best responses:
- What is Rob’s best response if Bruce doesn’t confess?
- What is Rob’s best response if Bruce confesses?
First we roleplay as the person who chooses which row we are in (Bruce). While doing so, we imagine the specific actions that might be chosen by their opponent (Rob). This allows us to underline two best responses.
Next, we roleplay as the person who chooses what column we are in (Rob). While doing so, we imagine the specific actions that might be chosen by their opponent (Bruce). This allows us to underline two more best responses.
Rob’s Best Responses
Section titled “Rob’s Best Responses”Suppose Bruce doesn’t Confess. What is Rob’s best response?
Section titled “Suppose Bruce doesn’t Confess. What is Rob’s best response?”| Rob | |||
|---|---|---|---|
Don’t Confess | Confess | ||
| Bruce | Don’t Confess | uR = -1 uB = -1 | uR = 0 uB = -10 |
| Confess | uR = -10 uB = 0 | uR = -5 uB = -5 |
If Bruce doesn’t Confess, my only best response is to Confess. Therefore, we underline uR = 0 going forward.
Suppose Bruce Confesses. Which of Rob’s responses is a best response?
Section titled “Suppose Bruce Confesses. Which of Rob’s responses is a best response?”| Rob | |||
|---|---|---|---|
Don’t Confess | Confess | ||
| Bruce | Don’t Confess | uR = -1 uB = -1 | uR = 0 uB = -10 |
| Confess | uR = -10 uB = 0 | uR = -5 uB = -5 |
If Bruce Confesses, my only best response is to Confess. Therefore, we underline uR = -5 going forward.
Bruce’s Best Responses
Section titled “Bruce’s Best Responses”Suppose Rob Doesn’t Confess. Which of Bruce’s responses is a best response?
Section titled “Suppose Rob Doesn’t Confess. Which of Bruce’s responses is a best response?”| Rob | |||
|---|---|---|---|
Don’t Confess | Confess | ||
| Bruce | Don’t Confess | uR = -1 uB = -1 | uR = 0 uB = -10 |
| Confess | uR = -10 uB = 0 | uR = -5 uB = -5 |
If I Don’t Confesses, Bruce’s only best response is to Confess. Therefore, we underline uB = 0 going forward.
Suppose Rob does confess. Which of Bruce’s responses is a best response?
Section titled “Suppose Rob does confess. Which of Bruce’s responses is a best response?”| Rob | |||
|---|---|---|---|
Don’t Confess | Confess | ||
| Bruce | Don’t Confess | uR = -1 uB = -1 | uR = 0 uB = -10 |
| Confess | uR = -10 uB = 0 | uR = -5 uB = -5 |
If I Confesses, Bruce’s only best response is to Confess. Therefore, we underline uB = -5 going forward.
Finding the Nash Equilibria:
Section titled “Finding the Nash Equilibria:”Finally, we copy the best responses down into one table.
| Rob | |||
|---|---|---|---|
Don’t Confess | Confess | ||
| Bruce | Don’t Confess | uR = -1 uB = -1 | uR = 0 uB = -10 |
| Confess | uR = -10 uB = 0 | uR = -5 uB = -5 |
The only cell where both players are playing a best response is when we both confess. Therefore, (Confess, Confess) is the only Nash Equilibrium.
A Nash Equilibrium is when you have two underlines in one box.
A game can have 0, 1, 2, 3, or 4 Nash Equilibria. In other words, it’s possible to write down games with pairs of underlines in 0, 1, 2, 3, or 4 boxes.
Why is “look for two underlines in the same box” a good solution concept?
Section titled “Why is “look for two underlines in the same box” a good solution concept?”| Rob | |||
|---|---|---|---|
Don’t Confess | Confess | ||
| Bruce | Don’t Confess | uR = -1 uB = -1 | uR = 0 uB = -10 |
| Confess | uR = -10 uB = 0 | uR = -5 uB = -5 |
If you have two underlines, it means that both players are simultaneously choosing optimal strategies. This makes a good solution concept because we assume that in business, both parties will do their absolute best to choose optimal strategies. Therefore, as analysts, we need to look for solution concepts in which both players are simultaneously optimizing. This is exactly what Nash Equilibrium does, with “two underlines in one box.”
Let’s try it!
Section titled “Let’s try it!”| Tommy | |||
|---|---|---|---|
| Call | Not Call | ||
| Jenny | Give Number
| uT = 8 uJ = 6 | uT = 7 uJ = 5 |
| Not Give Number | uT = 3 uJ = 0 | uT = 9 uJ = 9 |
There are two Nash Equilibria.
First one is: (Jenny Gives Number, Tommy Calls)
Second one is: (Jenny Doesn’t Give Number, Tommy Doesn’t Call)
With two NE, our solution concept doesn’t tell us which of the Nash Equilibria will happen. We don’t know.
Nash Equilibrium vs Dominant Strategy Equilibrium
Section titled “Nash Equilibrium vs Dominant Strategy Equilibrium”If a game has a dominant strategy equilibrium, then that equilibrium will also be a Nash Equilibrium. Further, it will be the only Nash Equilibrium.
However, most Nash Equilibria won’t also be Dominant Strategy Equilibria.
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