Speaking French during defense cycles will cost your group points! Slide 4 of 5

Class Debrief & Reflection

PLENARY REFLECTION

Q1

Risk vs. Expected Returns

Why does standard deviation act as a good mathematical measurement for "risk" in investment models? How does it differ from expected return?

Q2

The Logic of Monty Hall

Why does your gut tell you that switching doors shouldn't matter (50/50), when the formal mathematics proves that switching wins 2/3 of the time?

Excellent work in Session 1! Get ready for Game Theory in Session 2. Slide 5 of 5

SESSION 1

STATION 1: THE WARRANTY DILEMMA Topic: Expected Value

Part A (Distributions): Without warranty (\(X_1\)): \(P(X_1 = 4500) = 0.08\), \(P(X_1 = 0) = 0.92\).
With warranty (\(X_2\)): \(P(X_2 = 500) = 1.00\) (constant cost).

Part B (Expected value): \(E(X_1) = 4500 \times 0.08 + 0 \times 0.92 = \mathbf{360}\) euros.
\(E(X_2) = \mathbf{500}\) euros. Based purely on expected value, **self-insuring is optimal** since \(360 < 500\) (expected savings of €140).

STATION 2: THE STARTUP SYNDICATE Topic: Volatility & Risk Analysis

Part A (Distributions): Alpha (\(A\)): \(P(A=60000) = 0.20\), \(P(A=0) = 0.80\).
Beta (\(B\)): \(P(B=15000) = 0.60\), \(P(B=8000) = 0.40\).

Part B (Expectations & Volatility):
\(E(A) = 60000 \times 0.20 + 0 = \mathbf{12000}\) euros. Net expected profit: \(12000 - 10000 = \mathbf{2000}\) euros.
\(E(B) = 15000 \times 0.60 + 8000 \times 0.40 = 9000 + 3200 = \mathbf{12200}\) euros. Net expected profit: \(12200 - 10000 = \mathbf{2200}\) euros.
Startup Beta has both a **higher expected value** and **lower volatility (safer)** since its minimum outcome is €8,000 (Alpha's is €0).

STATION 3: CLINICAL DIAGNOSTICS Topic: Conditional Probability & Bayes

Part A (Tree probabilities): Top path: \(P(D) = 0.02\). Branch to \(T^+ = 0.95\), Branch to \(T^- = 0.05\).
Bottom path: \(P(\overline{D}) = 0.98\). Branch to \(T^+ = 0.10\) (1 - 0.90 specificity), Branch to \(T^- = 0.90\).

Part B (Posterior Probability calculation):
Total Positive test probability: \(P(T^+) = P(D \cap T^+) + P(\overline{D} \cap T^+) = (0.02 \times 0.95) + (0.98 \times 0.10) = 0.019 + 0.098 = 0.117\).
Apply Bayes' rule: \(P_{T^+}(D) = \frac{P(D \cap T^+)}{P(T^+)} = \frac{0.019}{0.117} \approx \mathbf{0.1624}\) (or **\(16.24\%\)**).

STATION 4: THE MONTY HALL SHOWDOWN Topic: Conditional Logic

Part A (Strategies): Initially, your chosen door (Door 1) has a \(1/3\) chance of having the car. The other two doors combined have a \(2/3\) chance.

Part B (Rigorous Proof): If you **Stay**: You only win if your initial choice was correct. Probability = \(\mathbf{1/3}\).
If you **Switch**: Since Monty always opens a door with a goat, switching wins if your initial choice was a goat (which forces Monty to open the *other* goat door, leaving the car behind the remaining closed door). Since the probability of choosing a goat initially is \(\mathbf{2/3}\), the probability of winning by switching is \(\mathbf{2/3}\).

The Decision Lab • Chances & Choices Guide Page 2 of 2

A Analyze Strategic Dominance

Do the players have dominant strategies in this game? Explain why or why not, citing the mathematical definitions.

B Find Multiple Nash Equilibria

How many pure Nash Equilibria exist in this game? Write down the strategy profile pairs and explain why they qualify.

Oral Challenge

English Speech Prompts

Practice defending coordination solutions aloud. Speak clearly and use these frames:

Phrases for Coordination

• "The outcome depends entirely on..."
• "Neither player has a dominant..."
• "Both (Alpha, Alpha) and (Beta, Beta) are..."

Oral Question to Defend

"If both outcomes are equilibria, which one is 'better' for the market (Pareto dominant) and how do companies communicate to select it?"

The Strategist's Playbook • Station Packet

STATION 3: THE CLIMATE ACCORD

Focus: Public Goods, Free-Riding & Global Game Theory

Session 2

Unit: Game Theory Topic: Public Goods Target: Collective Action

The Free-Rider Dilemma

Two neighboring countries, Country A and Country B, are deciding on climate policies. They can either invest heavily to Reduce Emissions or choose to Do Nothing. Reducing emissions costs 6 units, but provides a clean atmosphere benefit of 5 units to BOTH countries.

Country B (Column Player)

Country A

A The Math Behind the Numbers

Briefly verify why the payoffs in the matrix are correct. For example, explain why the profile (Reduce, Do Nothing) yields a payoff of -1 for Country A and 5 for Country B.

B The Global Equilibrium

Identify the Nash Equilibrium of this game. Explain the term "Free-Rider" using the strategies and payouts.

Oral Challenge

English Speech Prompts

Verbally defend ecological incentives in English. Work with these templates:

Environmental English

• "The cost of reducing emissions is..."
• "A country has an incentive to..."
• "To free-ride on the other's effort..."

Oral Question to Defend

"How does game theory prove that voluntary agreements are highly likely to fail on carbon emissions? Suggest a mathematical mechanism to fix this."

The Strategist's Playbook • Station Packet

STATION 4: HAWK-DOVE CONFLICT

Focus: Anti-Coordination Games & Evolutionary Dynamics

Session 2

Unit: Game Theory Topic: Hawk-Dove Target: Anti-Coordination

To Fight or To Yield

Two firms are competing for a lucrative government contract worth 10 units. They can choose to be aggressive (Hawk strategy) or peaceful (Dove strategy). If both are Hawks, they fight, causing a loss of 8 units each (net payoff: \(5 - 8 = -3\)). If one is Hawk and one is Dove, the Hawk wins all.

Competitor Y (Column Player)

Competitor X

A Find Pure Strategy Nash Equilibria

Identify the pure Nash Equilibria of this game. Explain why this is referred to as an "anti-coordination" game.

B Biological & Economic Analogy

Compare this game to the "game of chicken" or animal territory disputes. Write a brief reflection on how playing Hawk (or Dove) depends on what you believe the other will do.

Oral Challenge

English Speech Prompts

Verbally defend anti-coordination models aloud. Use the templates below:

Phrases for Anti-Coordination

• "If I expect them to act aggressively, I..."
• "The worst possible outcome is..."
• "This anti-coordination equilibrium..."

Oral Question to Defend

"In animal fights, how does evolutionary biology prevent the lethal (Hawk, Hawk) outcome? Why do animals perform ritualized threats instead?"

Public Goods Game

STATION 4

Hawk-Dove

To fight aggressively or play passive. How does territorial conflict translate to corporate tenders and pricing models?

Anti-Coordination

Use the oral prompt cards at each station to defend your solutions aloud in English. Slide 4 of 5

Strategic Plenary: Oral Challenge

DEBATE & DISCUSS

D1

Solving the Free Rider Crisis

How can we rewrite the payoff matrices of the Climate Accord game so that the Nash Equilibrium changes from "Do Nothing" to "Reduce Emissions"? Explain the impact of carbon taxes or international fines.

D2

Biology vs. Business

Compare the Hawk-Dove game in biology to pricing wars in business. Is acting like a "Hawk" (aggressive low pricing to push others out) always a successful long-term strategy?

Congratulations! You have completed the Decision Lab sequence. Keep thinking strategically! Slide 5 of 5

STATION 2: TECH COORDINATION Type: Coordination Game

Part A (Dominant Strategies): Neither Company X nor Company Y has a dominant strategy. If X plays Alpha, Y wants to play Alpha (12 vs 0). If X plays Beta, Y wants to play Beta (8 vs 0). Optimal moves depend entirely on beliefs.

Part B (Nash Equilibria): Two pure Nash Equilibria exist: **(Alpha, Alpha)** with payoffs **(12, 12)** and **(Beta, Beta)** with payoffs **(8, 8)**. Both are stable, but (Alpha, Alpha) is Pareto-dominant since it yields higher utility for both players.

STATION 3: THE CLIMATE ACCORD (FREE RIDER) Type: Public Goods

Part A (Payoff Verification): Under (Reduce, Do Nothing), Country A pays 6 but both get 5. Thus, A gets \(5 - 6 = -1\) and B gets \(5 - 0 = 5\).

Part B (Free-Rider Equilibrium): Dominant strategy for both is **Do Nothing** (if other reduces: 5 vs 4; if other does nothing: 0 vs -1). Nash Equilibrium is **(Do Nothing, Do Nothing)** with payoffs **(0, 0)**. Country B "free-rides" when A reduces and B does nothing.

STATION 4: HAWK-DOVE CONFLICT Type: Anti-Coordination

Part A (Nash Equilibria): Two pure Nash Equilibria exist: **(Hawk, Dove)** with payoffs **(10, 0)** and **(Dove, Hawk)** with payoffs **(0, 10)**. It is "anti-coordination" because players want to choose different actions (one aggressive, one passive).

Part B (Analogy): If you believe the other will behave as a Hawk (fight), your best response is to play Dove (0 vs -3). If you believe they will play Dove, your best response is to play Hawk (10 vs 5). This explains territorial posturing.

The Decision Lab • Game Theory Guide Page 2 of 2