6.5 Nash Equilibria

Game theory is the study of strategic interactions between rational decision-makers. In this assignment, I’ll introduce you to some of the main ideas:

Games in Matrix Form

A payoff matrix shows the outcomes (payoffs) for each player based on different combinations of their choices. For two players, it’s typically shown as a grid where:

  • Rows represent Player A’s choices
  • Columns represent Player B’s choices
  • Each cell contains (Player A’s payoff, Player B’s payoff)

Game 1

So the interpretation of the matrix above is that there are two players (player A and player B), each of them have two strategies they can take (player A can choose top or bottom; player B can choose left or right). If player A chooses top and player B chooses left, player A gets a payoff of 1 while player B gets a payoff of 2.

Question 1) Interpreting a game in matrix form

Fill in the blanks to complete the interpretation of the matrix above:

  • If player A chooses top and player B chooses right, player A gets a payoff of ___ while player B gets a payoff of ___.

  • If player A chooses bottom and player B chooses left, player A gets a payoff of ___ while player B gets a payoff of ___.

  • If player A chooses bottom and player B chooses right, player A gets a payoff of ___ while player B gets a payoff of ___.

Best Response

If player A thinks that player B will pick left, what is player A’s best response? Draw an arrow over player B picking left and consider the potential payoffs for player A: they can pick top to get a payoff of 1, or they can pick bottom to get a payoff of 2. They would rather get 2, so player A’s best response is to play bottom. Put a star next to the payoff for player A’s best response to player B picking left.

Game 1

I’ll repeat the process and find best responses for each player’s potential actions, adding four stars in total:

Game 1

Question 2) Best Responses

Consider a new game: I’ll call it Game 2:

Game 2

Find all four best responses and label the corresponding payoffs with stars.

Dominant Strategy

A dominant strategy is a strategy that is optimal regardless of what other players do. If you have a dominant strategy, you should always play it.

For example, going back to Game 1, notice that player A wants to pick bottom not only when player B picks left, but also when player B picks right! Player A has a dominant strategy, which is to pick bottom.

Game 1

Question 3) Dominant Strategies

  1. In Game 1, does player B have a dominant strategy? If so, what is it?

  2. In Game 2, does player A have a dominant strategy? If so, what is it?

  3. In Game 2, does player B have a dominant strategy? If so, what is it?

Nash Equilibria

A Nash Equilibrium (NE) is a situation where no player can benefit by unilaterally changing their strategy. Both players are choosing their best responses (there are stars next to both payoffs), so a Nash Equilibrium represents a stable outcome in a game.

For example, in game 1, there is one Nash Equilibrium, where player A goes to the bottom and player B goes left:

Game 1

It’s the cell where there are stars next to both payoffs. At a NE, neither players can benefit by unilaterally changing their strategy. Let’s verify this is true in Game 1:

  • If player A instead switches to playing top, they get a payoff of 1 instead of 2, which is a loss.
  • If player B instead switches to playing right, they get a payoff of 0 instead of 1, which is a loss.

(Bottom, Left) is a NE, and it’s the only NE in Game 1.

Question 4) Nash Equilibria

Is there a NE in Game 2? If so, explain how any unilateral deviation only creates losses.

Game 2

Example: The Ice Cream Vendor Game (Continuous Strategy Set)

Imagine two ice cream vendors deciding where to locate on a beach that’s 1 mile long. Customers will go to the closest vendor.

  • If they locate at different spots, they split the market unevenly
  • If they locate at the same spot, they split the market equally

What’s the Nash equilibrium? Both vendors end up in the middle of the beach. Why? If one vendor isn’t in the middle, the other can always move closer to capture more customers. This is one reason you often see similar businesses clustered together, like two gas stations in the same intersection.

Question 5) The Technology Standards Battle

Two companies are deciding whether to make their products compatible with each other, like adopting USB-C as a universal charging standard versus developing a proprietary port.

  • If both choose compatibility, they each earn $10M
  • If both choose incompatibility, they each earn $5M
  • If one chooses compatibility and the other doesn’t, the incompatible company earns $12M and the compatible one earns $2M

  1. Draw the game matrix, label best responses, find dominant strategies if they exist, and find any NE.

  2. Fill in the blanks to interpret: even though firms could cooperate and both choose ___, the incentive to defect on that cooperative agreement is too strong, which makes the only stable outcome for both firms to choose ___. This should remind you of the instability of the Cartel agreement under perfect collusion. When this happens in a 2x2 game, it’s called a Prisoner’s Dilemma: there is tension between individual rationality and collective benefit. Here’s the classic version: Two suspects are arrested for a crime and held in separate cells. The prosecutor doesn’t have enough evidence to convict them of the main crime unless one confesses. So each prisoner is offered a deal:

  • If one confesses (defects) and the other stays silent (cooperates), the confessor goes free, and the silent one gets 10 years in prison.
  • If both confess, they each get 5 years in prison.
  • If both stay silent, they both get only 1 year in prison for a minor charge.

It’s a dilemma because each prisoner reasons:

  • “If the other stays silent, I should confess and go free.”
  • “If the other confesses, I should also confess to avoid 10 years.”

So both confess—even though they’d both be better off staying silent. The dominant strategy is to confess, but the socially optimal outcome is mutual silence.

This story reflects real-world situations where rational decisions by individuals can lead to worse outcomes for the group.

Question 6) The Restaurant Quality Game

Two restaurants in a small town decide whether to invest in high-quality ingredients. If both choose high quality ingredients, the town gets a reputation for nice restaurants and both restaurants can earn more. But if one restaurant chooses high quality and the other restaurant chooses low quality, the town doesn’t gain a reputation for nice restaurants, and the high quality restaurant just has higher costs.

  • If both maintain high quality: Each gets $8K/month
  • If both choose low quality: Each gets $5K/month
  • If one chooses high and other low: High gets $4K, Low gets $10K
  1. Draw the game matrix, label best responses, find dominant strategies if they exist, and find any NE. Is this a prisoner’s dilemma?

Now suppose this game is played repeatedly, with no known end date. Firm 1 decides to use a “tit-for-tat” strategy:

  • Round 1: Firm 1 starts by choosing high quality
  • All later rounds: Firm 1 copies whatever Firm 2 did in the previous round

Let’s analyze what happens if Firm 2 considers two strategies:

  • Always choose low quality
  • Always choose high quality

  1. If Firm 2 always chooses low quality, what would the payoffs look like in:

    • Round 1?
    • Round 2 and all future rounds?
    • Find the value of the infinite geometric series assuming the discount factor (common ratio) is 0.9.

  2. If Firm 2 always chooses high quality, what would the payoffs look like in:

    • Round 1?
    • Round 2 and all future rounds?
    • Find the value of the infinite geometric series assuming the discount factor (common ratio) is 0.9.

  3. After 3 rounds, which strategy is better for Firm 2? Could tit-for-tat succeed in maintaining high quality from both restaurants in the long run?

Key Insights to Remember:

  1. Not all Nash equilibria maximize joint payoffs - sometimes the incentive to defect makes it so that players can’t cooperate, so they settle on lower joint payoffs. This is called a prisoner’s dilemma.

  2. Repeated games can support cooperation that would be impossible in one-shot games through the possibility of retaliation.

Game theory helps explain many real-world phenomena:

  • Why gas stations cluster at intersections
  • Price wars
  • How international treaties can be enforced without a global police force
  • Why some industries maintain high prices while others race to the bottom

Question 7) Review

Define these terms in your own words:

  • Best Response:

  • Dominant Strategy:

  • Nash Equilibrium:

  • Prisoner’s Dilemma:

  • Tit-for-tat: