Week 12 — Graph & Model Workshop · "The Pricing Game: Finding the Nash Equilibrium"

Course: Principles of Microeconomics (ECON 1) · Silver Oak University (fictional sample) · Prof. Kessler
Objective 7 — game theory, oligopoly & the prisoner's dilemma · SLO A
Worth 50 points · Model Workshops group = 15% of the grade · Workshop 12
Format: work through a two-firm payoff matrix in a spreadsheet (Google Sheets or Excel, free), complete a short scaffold, interpret the result in words, then catch the AI's mistakes.

This is the course's signature weekly component — the economics analog of a lab. Every instructional week has one workshop: you set up a model, solve it, and explain what it means. All tools are links to free external sites — nothing to buy or download.

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Part 1 — The Big Picture

This week you learned the prisoner's dilemma — one of the most famous results in all of social science. Two players are both better off cooperating, but each has a private incentive to defect, so they both defect and end up worse off. It explains cartels breaking down, arms races escalating, and firms stuck in price wars. Today you'll work through the logic step by step on a real payoff matrix — finding dominant strategies, verifying the Nash equilibrium, and putting into words exactly why the cooperative outcome falls apart.

The tool: 🔗 Google Sheets — https://sheets.google.com (free, sign in with any Google account) — or Microsoft Excel / Numbers. You'll build a 2×2 payoff table and annotate it to find the equilibrium.

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Part 2 — The Guiding Question

Two streaming services, StreamPrime and Vaultflix, each set their monthly subscription price. Both know their profits depend on what the other does. Can they cooperate to keep prices high — or are they trapped?

The scenario. StreamPrime and Vaultflix each choose between a High subscription price ($15/month) or a Low price ($10/month). The table below shows each firm's monthly profit (in millions of dollars) for every combination.

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Part 3 — Set Up the Model (in your spreadsheet)

1. Open a new spreadsheet. Build the payoff matrix exactly as shown:

	Vaultflix: High	Vaultflix: Low
StreamPrime: High	(10, 10)	(3, 12)
StreamPrime: Low	(12, 3)	(5, 5)

Each cell = (StreamPrime profit, Vaultflix profit) in millions of dollars.

2. Label your rows (StreamPrime's choices) and columns (Vaultflix's choices) clearly. Use borders or shading to make the four cells distinct.

3. Add a column to the right: for each of StreamPrime's rows, note which payoff is higher (you'll use this in Part 4). Add a row at the bottom for Vaultflix's comparison.

Verified numbers (pre-computed — use exactly as given):
- (High, High) = (10, 10)
- (Low, High) = (12, 3) — StreamPrime defects to Low while Vaultflix holds High
- (High, Low) = (3, 12) — Vaultflix defects to Low while StreamPrime holds High
- (Low, Low) = (5, 5)

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Part 4 — Solve (complete this scaffold)

Fill in the blanks. Show the numerical comparisons that justify each answer.

StreamPrime's dominant strategy:

Fix Vaultflix on…	StreamPrime earns __ if High	StreamPrime earns __ if Low	StreamPrime prefers…
High (left column)	______	______	______
Low (right column)	______	______	______

StreamPrime's dominant strategy: _ (because it is higher in both rows above)

Vaultflix's dominant strategy (do the same for columns — fix StreamPrime's row, compare Vaultflix's two payoffs):

Fix StreamPrime on…	Vaultflix earns __ if High	Vaultflix earns __ if Low	Vaultflix prefers…
High (top row)	______	______	______
Low (bottom row)	______	______	______

Vaultflix's dominant strategy: _

The Nash equilibrium:

Both firms play their dominant strategy → outcome = (_, _) = (__, ____)

Verify it's a Nash equilibrium: if Vaultflix plays _ and StreamPrime switches to _, StreamPrime earns _ instead of . Is switching beneficial? (yes / no). Conclusion: ____ is a Nash equilibrium / is NOT a Nash equilibrium. (circle one)

The cooperative outcome:

The jointly best outcome for both firms is (High, High) = (_, _). At this outcome, does StreamPrime want to deviate to Low? If Vaultflix plays High and StreamPrime switches to Low, StreamPrime earns _ instead of ___. Is that a gain? (yes / no). Conclusion: the cooperative outcome (is / is not) a Nash equilibrium. (circle one)

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Part 5 — Interpret in Words (this is the SLO-A skill)

In 3–4 sentences:
- State the Nash equilibrium and explain why it is stable (use the phrase "no incentive to deviate unilaterally").
- Explain why the cooperative outcome (High, High) is better for both firms but is not stable.
- Name the pattern — this is a _ — and explain in plain English why individually rational behavior produces a collectively bad outcome.

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Part 6 — Analysis Questions

1. The defection temptation. If Vaultflix holds to High ($15), StreamPrime earns $12M by switching to Low vs. $10M by staying at High. In one sentence, name the gain from defecting and explain why this private temptation is the structural reason cartels (formal collusion agreements) break down even when cooperation is jointly profitable.

2. Repeated interaction. Some economists argue that if the two firms play this pricing game every month, indefinitely, cooperation can be sustained through "tit-for-tat": cooperate today; if your rival defects, punish them next month by also defecting. Does the possibility of a long-run relationship change your analysis of whether StreamPrime and Vaultflix can sustain (High, High)? Explain in 2–3 sentences why repeated play can help — and why it is not guaranteed to work.

3. Connect it: name one real-world situation — in business, international relations, or everyday life — where two parties are stuck in the equivalent of (Low, Low) = (5, 5) when they could both be better off at (High, High) = (10, 10). Why can't they just agree to cooperate?

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Part 7 — AI-Critique Moment (required — the BYOAI step)

Bring in your approved chatbot (Gemini, Claude, or ChatGPT) and be the economist who checks its work.

1. Paste this to the chatbot: "Two streaming services, StreamPrime and Vaultflix, play a pricing game. The payoff matrix (StreamPrime profit, Vaultflix profit) is: (High, High) = (10, 10); (Low, High) = (12, 3); (High, Low) = (3, 12); (Low, Low) = (5, 5). What is each firm's dominant strategy, and what is the Nash equilibrium? Is (High, High) a Nash equilibrium?"

2. Audit every claim against your own work:
   - Did it correctly identify Low as the dominant strategy for both firms? (12 > 10 and 5 > 3 — check both comparisons.)
   - Did it correctly identify the Nash equilibrium as (Low, Low) = (5, 5)?
   - Did it correctly explain that (High, High) = (10, 10) is NOT a Nash equilibrium — specifically because StreamPrime would earn 12 > 10 by switching to Low when Vaultflix plays High?
   - Common chatbot errors: calling (High, High) the Nash because it "looks better for both"; confusing "jointly preferred" with "stable equilibrium"; forgetting to check the deviation incentive; stating the Nash as (High, High) because both payoffs are higher numbers without verifying no-deviation.

3. Write 2–3 sentences naming what the AI got right and at least one specific thing you had to correct or watch carefully. (If it got everything right, explain exactly how you verified each claim — that verification IS the skill.)

The habit all term: the tool drafts, you judge. Chatbots are particularly prone to calling the jointly preferred outcome the Nash equilibrium — catching that confusion is the whole point.

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Part 8 — What to Submit

One document (or text entry) with: your spreadsheet screenshot or table (Part 3), your Part 4 scaffold (with all comparisons filled in), your Part 5 interpretation, your Part 6 answers, and your Part 7 AI-critique paragraph. Due Sun, Nov 22, 11:59 p.m. (50 points).

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Instructor answer key — REMOVE BEFORE PUBLISHING TO STUDENTS

Every number pre-computed and independently verified against the build brief's verified payoff matrix.

Part 4 — StreamPrime's dominant strategy:
- Fix Vaultflix on High: StreamPrime earns 10 (High) or 12 (Low) → prefers Low (12 > 10). ✓
- Fix Vaultflix on Low: StreamPrime earns 3 (High) or 5 (Low) → prefers Low (5 > 3). ✓
- StreamPrime's dominant strategy: Low. ✓

Part 4 — Vaultflix's dominant strategy (by symmetry, confirmed independently):
- Fix StreamPrime on High: Vaultflix earns 10 (High) or 12 (Low) → prefers Low (12 > 10). ✓
- Fix StreamPrime on Low: Vaultflix earns 3 (High) or 5 (Low) → prefers Low (5 > 3). ✓
- Vaultflix's dominant strategy: Low. ✓

Part 4 — Nash equilibrium:
- Both firms play Low → (Low, Low) = (5, 5). ✓
- Verify: if Vaultflix plays Low and StreamPrime switches to High, StreamPrime earns 3 < 5. No incentive to deviate. (Low, Low) IS a Nash equilibrium. ✓

Part 4 — Cooperative outcome:
- (High, High) = (10, 10). ✓
- If Vaultflix plays High and StreamPrime switches to Low, StreamPrime earns 12 > 10. YES, a gain. So (High, High) is NOT a Nash equilibrium — StreamPrime has an incentive to deviate. ✓

Part 5 — Interpretation (model answer):
The Nash equilibrium is (Low, Low) = (5, 5). It is stable because, given that Vaultflix plays Low, StreamPrime earns only 3 by switching to High — no incentive to deviate unilaterally. (High, High) = (10, 10) is better for both firms, but it is not stable: if Vaultflix holds to High, StreamPrime can earn 12 by defecting to Low — a gain of 2 million. Both firms face this temptation simultaneously, so both defect. This is the prisoner's dilemma: individually rational choices produce a collectively bad outcome.

Part 6 — Analysis answers:
1. Defecting from High to Low earns StreamPrime $12M vs. $10M — an extra $2M when Vaultflix holds to High. That $2M gain is the structural reason cartels break down: each member can always earn more by quietly cheating on the agreement than by holding to it.
2. Repeated play can help because future punishment (the rival also plays Low forever) reduces the value of defecting today — if StreamPrime expects Vaultflix to match any defection immediately, the short-term gain of $2M may be outweighed by the long-run loss from mutual Low pricing. But it is not guaranteed: if one firm is impatient (discounts the future heavily) or uncertain about whether the rival will actually punish, the temptation to defect can still dominate.
3. Any well-reasoned real-world example earns credit: arms races (both countries prefer mutual disarmament but each fears the other's first strike), international climate agreements (each country gains from others cutting emissions while it free-rides), studying together vs. not (each group member tempted to let others do the work). Credit requires naming why the binding commitment problem prevents cooperation.

Part 7 — AI-critique: The most common chatbot errors on this task are (a) calling (High, High) the Nash equilibrium because both payoffs are larger, without checking the deviation incentive, and (b) confusing "both players prefer it" with "neither wants to deviate" — which are not the same thing. A correctly verified answer must state that from (High, High), StreamPrime earns 12 by switching to Low while Vaultflix plays High — and 12 > 10, so the deviation IS profitable, confirming that (High, High) is not a Nash equilibrium.

Grading rubric — 50 points

Criterion	Full	Partial	None
Scaffold (Part 4) — all comparisons shown; dominant strategies correct for both firms; Nash identified and verified; cooperative outcome correctly ruled not-Nash, with the deviation payoff (20)	20	10–16	0–8
Interpretation (Part 5) — Nash stable, cooperative not stable, prisoner's dilemma named and explained in plain English (12)	12	6–10	0–5
Analysis (Part 6) — defection temptation quantified; repeated-play logic engaged; real-world example with binding-commitment explanation (10)	10	5–8	0–4
AI-critique (Part 7) — names at least one specific thing the AI got right AND at least one specific thing to correct/verify (8)	8	4–6	0–3

Quality gate (self-checked):
- Quantitative gate — all four payoff comparisons re-verified: 12>10 (Low beats High when rival is High); 5>3 (Low beats High when rival is Low); Nash=(Low,Low)=(5,5) confirmed (deviation earns 3<5); cooperative=(High,High)=(10,10) NOT Nash (deviation earns 12>10). All PASS. ✓
- Graph-logic check — dominant strategy: Low wins BOTH columns for both firms ✓; Nash equilibrium: (Low,Low) stable (no deviation profitable) ✓; cooperative outcome: (High,High) not Nash (deviation earns 12>10) ✓; prisoner's dilemma structure: Nash payoff (5,5) < cooperative payoff (10,10) ✓. All PASS. ✓