Bayesian Games

🎯 Learning Objectives

• Model games of incomplete information using types and beliefs
• Apply Harsanyi transformation to convert incomplete to imperfect information
• Define and compute Bayesian Nash equilibrium (BNE)
• Solve auction and signalling problems
• Understand common prior assumption

8.1 Intuition: What You Don't Know Matters

🔑 Key Insight: When you don't know someone's payoffs, you form beliefs about their type, and these beliefs shape your strategy. Real-world examples:

• Auctions: You don't know how much others value the item
• Job market: Employer doesn't know worker's ability
• Price competition: Firms don't know each other's costs

8.2 Formal Definition

• Players: $N = \{1, \dots, n\}$
• Types: $\Theta_i$ — private information for player $i$
• Beliefs: $p_i(\theta_{-i} \mid \theta_i)$ — probability distribution over others' types
• Actions: $A_i$ — available actions
• Payoffs: $u_i(a_1, \dots, a_n; \theta_1, \dots, \theta_n)$

Harsanyi Transformation

1. Nature draws types $\theta = (\theta_1, \dots, \theta_n)$
2. Each player $i$ observes their own $\theta_i$ but not others'
3. Players choose actions simultaneously (Diagram)

8.3 Bayesian Nash Equilibrium

Key Insight

8.4 Example: Two-Type Entry Game

• Incumbent (Player 2) has either High cost (type $H$, probability 0.4) or Low cost (type $L$, probability 0.6)
• Entrant (Player 1) doesn't know which; Incumbent knows its own type Payoffs:
• Entrant: Enter → if Incumbent accommodates (2), if fights (-1). Stay Out → 0.
• Incumbent H: Accommodate Entrant → 1, Fight → 0, No entry → 3
• Incumbent L: Accommodate Entrant → 1, Fight → 3, No entry → 4 Incumbent's strategy (type-contingent):
• Type H: Accommodate (1 > 0)
• Type L: Fight (3 > 1) Entrant's expected payoff from Enter:
• With prob 0.4: Incumbent H accommodates → 2
• With prob 0.6: Incumbent L fights → -1
• Expected: $0.4(2) + 0.6(-1) = 0.8 - 0.6 = 0.2 > 0$ BNE: Entrant enters. Incumbent H accommodates, Incumbent L fights.

8.5 Common Prior Assumption

8.6 Signalling Games

1. Sender (informed player) observes type and sends a signal
2. Receiver (uninformed player) observes signal and takes action
3. Payoffs depend on type, signal, and action

Types of Equilibria

Spence's Education Model

• Worker has ability $\theta \in \{L, H\}$ (low/high)
• Worker chooses education level $e$
• Employer observes $e$ and pays wage $w(e)$ Intuition: If education is less costly for high-ability workers, a separating equilibrium exists where $H$ gets education and $L$ doesn't, and employers pay higher wages to educated workers.

📊 Formula Summary

✅ Practice Questions

Solution

Incumbent's behaviour unchanged: High cost accommodates, Low cost fights.

Entrant's expected payoff from Enter: $0.8(2) + 0.2(-1) = 1.6 - 0.2 = 1.4 > 0$

Still enters. The expected payoff is even higher because it's more likely the incumbent will accommodate. Q2: If entering costs 1 (so payoff is 1 if accommodated, -2 if fights), would the entrant enter with p(H)=0.4? Solution

Expected payoff: $0.4(1) + 0.6(-2) = 0.4 - 1.2 = -0.8 < 0$ Entrant stays out. BNE: Entrant stays out. Incumbent H accommodates (if entered), L fights (if entered). Q3: In Spence's model, what two conditions must hold for a separating equilibrium? Solution

1. Incentive compatibility for type L: $w(e_L^*) - c_L(e_L^*) \geq w(e_H^*) - c_L(e_H^*)$ — L type prefers its own education
2. Incentive compatibility for type H: $w(e_H^*) - c_H(e_H^*) \geq w(e_L^*) - c_H(e_L^*)$ — H type prefers its own education

Where $c_\theta(e)$ is the cost of education for type $\theta$, and $w(e)$ is the wage paid by employers who observe $e$. Join Discord PreviousFolk TheoremNextSignalling Games