30 Session 21: Mean-Field Games (Concepts Only)

Unit	4 — Math Intuition Bridges
Book Chapter	13 (concepts only — full derivation in Session 26 Track 2)
Track	Common core (both tracks)

30.1 Learning Objectives

By the end of this session, students will be able to:

State the key challenge of mean-field games: each agent optimizes, but the optimum depends on what other agents do.
Define a McKean-Vlasov SDE in plain language and recognize the McKean-Vlasov coupling term.
Articulate the fixed-point structure of MFG equilibrium: distribution → individual optimum → aggregate distribution.
Explain the collective externality in private markets — why individual LP exit decisions create costs for all LPs.
Connect MFG to game-theoretic frameworks students may know (Nash equilibrium, congestion games, mechanism design).

30.2 Pre-Class Assignment

Read: Book Chapter 13 (concepts only — skim mathematical detail)

30.3 In-Class Outline (75 minutes)

Time	Segment	Format
0:00–0:05	Recap Session 20 · Today: when everyone solves HJB	Lecture
0:05–0:20	The mean-field intuition — many agents, common environment	Lecture
0:20–0:40	McKean-Vlasov SDEs — what they are and why they matter	Lecture
0:40–0:55	The fixed-point structure of MFG equilibrium	Lecture
0:55–1:10	The collective externality in GE-LAV	Lecture
1:10–1:15	Connection to other game-theoretic frameworks	Discussion

30.4 Discussion Questions

The McKean-Vlasov externality is calibrated at ~2.3%/yr aggregate welfare loss across the private capital market. As a fraction of total private market AUM ($13T), that’s ~$300B/year. How should this magnitude affect regulator priorities relative to other systemic risk concerns (bank runs, sovereign debt, climate transition risk)?
Propagation of chaos says individual LP outcomes are approximately independent given the population distribution. Does this break down in concentrated markets (few large LPs)? How would you test this empirically?
MFG equilibrium has a stability condition $\kappa > \gamma$. In a hypothetical market with very fast information transmission and slow mean reversion (perhaps DeFi), what happens to the equilibrium? Multiple? Unstable?

30.5 Worked Example: The McKean-Vlasov Externality, Quantified

Setup: Consider 1,000 LPs each with $\$100M$ in PE positions during the 2008–2010 GFC.

Step 1: Individual exit decision

LP A exits $\$100M$ at a 30% discount: saves $\$70M$ in stress capital but takes $\$30M$ loss vs. NAV.

Step 2: Externality

LP A’s exit increases secondary supply marginally. Result: secondary discounts widen by ~0.1 basis point for all other LPs.

For 999 other LPs each with $\$100M$: - Additional loss: $999 \times \$100M \times 0.001\% = \$1M$ in aggregate

LP A captured a $\$70M$ private benefit. They imposed a $\$1M$ aggregate externality.

Step 3: When everyone exits simultaneously

Now 1,000 LPs all exit: - Each one creates a 0.1 bp externality - Aggregate widening: 100 bp = 1% additional discount on top of the original - $\$100M$ positions all worth $\$1M$ less each because of collective behavior - Aggregate externality: $\$1B$

Step 4: Welfare gap

Of the $\$30B$ in losses from the collective exit, approximately $\$1-2B$ is externality cost — losses that wouldn’t exist if exits had been coordinated.

Generalization to the calibrated 2.3%/yr figure:

Over a typical decade with one or two stress episodes, aggregate externality losses accumulate to approximately 2.3% of average private market AUM per year. At $$13T AUM = $$300B/year welfare loss.

30.6 What to Expect Next Session

Session 22 covers Fokker-Planck and the master equation — the computational tools for tracking how distributions evolve over time. These are the PDEs that the MFG framework requires.

Topics: - The Fokker-Planck equation: how the distribution of $L_t$ evolves - The master equation: how the value function evolves on the space of distributions - Why these are intuitive (heat-equation analogues)

Reading: Book Chapters 14, 15 (concepts only).

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