Class 2: Game Theory I: Simultaneous Moves
Textbook reading: Kreps, Chapter 2, “Noncooperative Game Theory,” through Section 2.2 (dominance) and Section 2.4 (Nash equilibrium). Section 2.3 (backward induction) is for Class 3. You may skip the subsection on mixed strategies.
Key Ideas
Last Class
- Relationships with your suppliers, customers, and rivals determine your profitability
- We’ll think about situations from every party’s points of view at once: Game Theory is the study of strategic interaction
Games
- Define players
- Define strategies those players can take
- Define payoffs for each player at every combination of strategies
- Today: players choose simultaneously and independently
- Next time: players choose in sequence
Problems of the Day
Q3: Sam and Jan
Sam and Jan are friends. Each is deciding, independently and without consulting the other, where to go on Tuesday evening: a bar called Old Pros, the art museum, or a coffee house called Cafeen. Each would rather be together than apart, but they disagree about where. The table gives their payoffs; Sam picks the row, Jan picks the column, and Sam’s payoff is listed first in each cell. A higher number is a better outcome for that person.
| Jan: Old Pros | Jan: Art Museum | Jan: Cafeen | |
|---|---|---|---|
| Sam: Old Pros | 6, 4 | 4, 3 | 4, 2 |
| Sam: Art Museum | 2, 1 | 5, 5 | 2, 2 |
| Sam: Cafeen | 1, 1 | 1, 3 | 3, 6 |
Here are the steps to solve this game:
Step 1: Sam’s best responses get stars. If Sam suspects Jan will go to Old Pros, where does Sam want to go? Put a star next to his highest payoff in that case. If Sam suspects Jan will go to the art museum, find out where he will want to go, and add another star. Finally, if Sam suspects Jan will go to Cafeen, find out where he will want to go, and add another star.
Step 2: Dominance. Does Sam have a dominant strategy? That is, does Sam want to go to a certain place, no matter where he suspects Jan might go? Are any of Sam’s strategies dominated? That is, no matter where he suspects Jan might go, is there any strategy Sam absolutely will not take?
Step 3: Jan’s best responses get stars. Repeat steps 1 and 2, but from Jan’s perspective. If Jan suspects Sam will go to Old Pros, where does Jan want to go? Add a star. Repeat for the art museum and then for Cafeen. You should have six stars: three on Sam’s payoffs and three on Jan’s payoffs. Does Jan have a dominant strategy, or are any of her strategies dominated?
Step 4: Nash Equilibria. Are there any cells where both payoffs have stars next to them? These are called Nash Equilibria: a combination of strategies, one per player, such that no player can do better by changing their own strategy alone, given what everyone else is doing. Each player is playing a best response to the others. If there is more than one Nash Equilibrium, what settles it in practice is communication before the game, a shared history, social conventions, or a focal point: a way of playing that stands out to everyone for reasons outside the payoff table. When none of those is present, and especially when the safe choice and the best coordinated choice differ, players often fail to reach any equilibrium at all.
So the recipe is: model the situation as a game, eliminate what dominance can eliminate, find the Nash equilibria, and then ask the question the math cannot answer, which is whether the people involved have a reason to see one of them as the obvious way to play.
Q4: How Many Steps Can You Count On?
Alice picks a row and Bob picks a column, simultaneously. Alice’s payoff is listed first.
| Bob: A | Bob: B | Bob: C | |
|---|---|---|---|
| Alice: 1 | 7, 3 | 3, 1 | 0, 5 |
| Alice: 2 | 5, 1 | 5, 3 | 2, 2 |
Step 1: Alice’s Best Responses. Put stars next to Alice’s best responses, given what she suspects Bob might do. Does she have a dominant strategy?
Step 2: Bob’s Best Responses. Put stars next to Bob’s best responses, given what he suspects Alice might do. Does he have a dominant strategy?
Step 3: Iterated Dominance. Bob will never choose one of his strategies. Alice should realize this. How does this change how Alice should play? How should Bob respond? Are there any Nash Equilibria, and does iterated dominance bring players toward them or away from them?
Extension. Kreps asked 330 first-year MBA students, before they had studied any game theory, how they would play this game for money. Here is what happened. Rows: 92% chose Row 2 and 8% chose Row 1. Columns: 1% chose Column A, 36% chose Column B, and 63% chose Column C. Explain the pattern: how many steps ahead were the MBA students thinking?
Q5: The Classic Prisoner’s Dilemma
Two suspects, Carla and Dev, are arrested for a crime and held in separate cells. The prosecutor lacks evidence for the main charge, so each is offered the same deal: confess and testify against the other, or stay silent. Neither can communicate with the other before deciding.
The outcomes (years in prison, Carla’s sentence listed first):
| Dev stays silent | Dev confesses | |
|---|---|---|
| Carla stays silent | −1, −1 | −10, 0 |
| Carla confesses | 0, −10 | −5, −5 |
Step 1. Put stars next to Carla’s best responses. Does she have a dominant strategy?
Step 2. Put stars next to Dev’s best responses. Does he have a dominant strategy?
Step 3. Are there Nash Equilibria? Explain how the players get there even though it’s a tragic one.
Extension. The players get to the Nash Equilibrium with no steps of iterated dominance, so it should be the most reliable prediction in game theory. Yet when the same 330 students played this game for money against a classmate in Kreps’s class, with the strategies relabeled “cooperate” and “fink,” 74% said they would cooperate. What went wrong: the theory, or the model?
Q6: Obvious to Whom?
Each of the four games below is played once, simultaneously, by two strangers who cannot talk beforehand. Row’s payoff is listed first.
(a) Easy Coordination
| Farid: left | Farid: right | |
|---|---|---|
| Elena: top | 0, 0 | 5, 5 |
| Elena: bottom | 15, 15 | 1, 1 |
(b) Risky Coordination
| Hugo: left | Hugo: right | |
|---|---|---|
| Greta: top | 5, −10 | 10, 10 |
| Greta: bottom | 15, 15 | −10, 5 |
(c) Harder Coordination
| Jamal: left | Jamal: middle | Jamal: right | |
|---|---|---|---|
| Isla: top | −5, −5 | 10, 10 | −5, −5 |
| Isla: middle | 15, 5 | −5, −5 | −5, −5 |
| Isla: bottom | −5, −5 | −5, −5 | 0, 30 |
(d) Chicken
| Lucia: left | Lucia: right | |
|---|---|---|
| Kenji: top | 0, 0 | 0, 20 |
| Kenji: bottom | 20, 0 | −10, −10 |
For each game: find all the pairs of choices from which neither player wants to move (these are the Nash equilibria).
Extension: Answer the question that matters more: if you were Row, playing against a random classmate for real money, what would you do, and how confident are you about what your opponent would do? Rank the four games from “obvious how to play” to “no idea.”
Kreps’s MBA students chose as follows.
- Easy coordination: 96% chose bottom or left.
- Risky coordination: 72% chose bottom or left.
- Harder coordination: For rows,
- 44% chose row 1
- 40% chose row 2
- 16% chose row 3
- Chicken: 60% chose to dare.
Does the data match your ranking?
Answers
- Q3:
- Step 1, Sam’s best responses: if Jan goes to Old Pros, Sam goes to Old Pros (6). If Jan goes to the art museum, Sam goes to the art museum (5). If Jan goes to Cafeen, Sam goes to Old Pros (4 beats 3). Stars on 6, 5, and the 4 in the top right cell.
- Step 2: Sam has no dominant strategy; his best choice depends on Jan. Cafeen is dominated for Sam: Old Pros beats it against every column (6 > 1, 4 > 1, 4 > 3). Art Museum is not dominated (it is best against Jan’s Art Museum).
- Step 3, Jan’s best responses: against Sam’s Old Pros, Old Pros (4). Against Sam’s Art Museum, Art Museum (5). Against Sam’s Cafeen, Cafeen (6). Stars on 4, 5, 6 down the diagonal. Jan has no dominant strategy and, on the full table, no dominated strategy either. Once Sam’s Cafeen row is deleted, though, Jan’s Cafeen column is dominated by Art Museum (3 > 2, 5 > 2), so iterated dominance shrinks the game to a 2 by 2 in the top left.
- Step 4: two cells carry two stars, (Old Pros, Old Pros) with payoffs (6, 4) and (Art Museum, Art Museum) with payoffs (5, 5). (Cafeen, Cafeen) is not an equilibrium because Sam would move to Old Pros. Neither equilibrium is better for both: Sam prefers Old Pros, Jan prefers the museum. Dominance removes Cafeen but cannot settle the rest. What settles it is history (where they usually go), a convention, or a conversation beforehand. Without one, a miscoordination such as Sam at Old Pros and Jan at the museum, (4, 3), is a real possibility, since Old Pros is Sam’s safe choice (worst case 4) and the museum is Jan’s (worst case 3).
- Q4:
- Step 1: against A, Alice plays 1 (7 > 5); against B, 2 (5 > 3); against C, 2 (2 > 0). No dominant strategy.
- Step 2: against Row 1, Bob plays C (5); against Row 2, B (3). No dominant strategy, but A is dominated by C (3 < 5 and 1 < 2).
- Step 3: Bob never plays A. Delete it and Alice’s Row 2 dominates Row 1 (5 > 3, 2 > 0), so Alice plays 2. Bob’s best response to Row 2 is B. The only cell with two stars is (2, B), payoffs (5, 3), and iterated dominance leads straight to it: three steps, Bob, then Alice, then Bob.
- Extension: rows needed one step (notice A is dominated, or just notice that Row 2 is safer) and 92% got there. Columns needed two steps (Alice will realize A is gone, so she plays 2, so B) and only 36% got there; 63% stopped after one step and chose C, the best response to Row 1 and also the column with the better worst case. Almost nobody played the dominated strategy A. Each extra round of “he knows that I know” loses a large share of players, so the equilibrium prediction weakens with every step of iteration. Note the cost: the most common pair, (2, C), pays (2, 2), worse for both than the equilibrium (5, 3).
- Q5:
- Step 1: if Dev is silent, Carla confesses (0 > −1); if Dev confesses, Carla confesses (−5 > −10). Confess is dominant.
- Step 2: by symmetry, confess is dominant for Dev.
- Step 3: one Nash equilibrium, (confess, confess) at (−5, −5). Each player gets there without thinking about the other at all: confessing is better whatever the other does. Both would prefer (−1, −1), but neither can reach it alone, and each has a private incentive to defect from it.
- Extension: the model. The theory says players choose dominant strategies given their payoffs; that part is not in doubt. What the table misses is that the students’ payoffs were not just the money: they were playing a classmate they would see again, with reputations, guilt, and a label (“cooperate”) that invoked a norm. Put those into the payoffs and the game they were actually playing may not have been a prisoner’s dilemma. The lesson is about modeling: a dominance prediction is only as reliable as the payoff table behind it. In anonymous, one shot, high stakes settings, cooperation rates fall sharply, which is what the theory predicts once the payoffs are right.
- Q6:
- Two equilibria, (top, right) at (5, 5) and (bottom, left) at (15, 15). Bottom left is better for both and carries no real risk, so it is the obvious way to play.
- Two equilibria, (top, right) at (10, 10) and (bottom, left) at (15, 15). Bottom left is better for both, but bottom risks −10 if Hugo plays right, while top guarantees at least 5 (the same holds for Hugo). Bottom is the better bet only if Greta thinks Hugo plays left with probability above one half. With 72% going for it, bottom’s expected payoff is 0.72(15) + 0.28(−10) = 8.0 versus 6.4 for top, so the payoff dominant equilibrium is still the right target, but it is no longer safe.
- Three equilibria on three different cells: (top, middle) at (10, 10), (middle, left) at (15, 5), and (bottom, right) at (0, 30). None is best for both; each player prefers a different one, and the fair split competes with self interest. No focal point, so the row split of 44/40/16 is exactly what miscoordination looks like: most pairs land off the diagonal at (−5, −5).
- Two equilibria, (bottom, left) at (20, 0) and (top, right) at (0, 20). Each equilibrium hands everything to one player, so each wants the other to swerve and neither outcome stands out between strangers. Daring beats swerving only if fewer than two thirds of opponents dare; at 60%, daring earns 0.4(20) + 0.6(−10) = 2 versus 0 for swerving, with a 36% chance of a crash.
- Ranking, most to least obvious: (a), (b), (d), (c). The data match: the share choosing the modal action falls from 96% to 72% to 60% to 44%. Having a Nash equilibrium is not what makes play predictable; having one equilibrium that everyone can see is the obvious choice is.