25 Game Theory
For more information on these topics, see Varian Chapter 28: Game Theory.
Game theory is the study of strategic interactions between rational decision-makers. In this chapter, I’ll introduce you to some of the main ideas:
- Payoff Matrix
- Best Responses
- Dominant Strategies
- Nash Equilibria
25.1 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)
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.
Classwork 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 ___.
25.2 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.
I’ll repeat the process and find best responses for each player’s potential actions, adding four stars in total:
Classwork Question 2) Best Responses
Consider a new game: I’ll call it Game 2:
Find all four best responses and label the corresponding payoffs with stars.
25.3 Dominant Strategy
A dominant strategy is a strategy that’s 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.
Classwork 3) Dominant Strategies
In Game 1, does player B have a dominant strategy? If so, what is it?
In Game 2, does player A have a dominant strategy? If so, what is it?
In Game 2, does player B have a dominant strategy? If so, what is it?
25.4 Nash Equilibria
A Nash Equilibrium (NE) is a situation where each player’s strategy is optimal given what others are doing. No player can benefit by unilaterally changing their strategy. In other words, both players are playing their best responses (there are stars next to both payoffs).
For example, in game 1, there is one Nash Equilibrium, where player A goes to the bottom and player B goes left:
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.
Classwork 4) Nash Equilibria
Is there a NE in Game 2? If so, explain how any unilateral deviation only creates losses.
25.5 More Examples
Example 1: 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
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 explains why you often see similar businesses clustered together, like two gas stations in the same intersection.
Classwork 5) The Technology Standards Battle
Two companies are deciding whether to make their products compatible with each other:
- 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
Draw the game matrix (use a table in quarto or include a picture in your upload), label best responses, find dominant strategies if they exist, and find any NE.
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.
Classwork 6) The Restaurant Quality Game
Two restaurants in a small town decide whether to invest in high-quality ingredients:
- 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
- Draw the game matrix (use a table in quarto or include a picture in your upload), label best responses, find dominant strategies if they exist, and find any NE.
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
If Firm 2 always chooses low quality, what would the payoffs look like in:
- Round 1?
- Round 2 and all future rounds?
If Firm 2 always chooses high quality, what would the payoffs look like in:
- Round 1?
- Round 2 and all future rounds?
Based on these calculations, which strategy is better for Firm 2? Explain why tit-for-tat might succeed in maintaining high quality from both restaurants in the long run.
Key Insights to Remember:
Not all Nash equilibria are good outcomes - sometimes the incentive to defect makes it so that players can’t cooperate, so they settle on lower joint payoffs. This should remind you of cartel agreements breaking down because of the firms’ incentives to cheat.
Repeated games can support cooperation that would be impossible in one-shot games through the possibility of retaliation.
The beauty of game theory is that it helps explain many real-world phenomena:
- Why gas stations cluster at intersections
- How price wars start and end
- How international treaties can be enforced without a global police force
- Why some industries maintain high prices while others race to the bottom
25.6 Practice Questions