Module I·Article III·~4 min read
Game Theory: Nash Equilibrium, Strategic Interactions, and Mechanism Design
Microeconomic Analysis
Turn this article into a podcast
Pick voices, format, length — AI generates the audio
Game theory is the mathematical language of strategic interactions. When each agent’s outcome depends on the actions of others, classical optimization is insufficient. Game theory, developed by Nash, Aumann, and Shapley (all Nobel laureates), has become a universal tool in economics, political science, biology, and computer science.
Normal Form Games
A game in normal form: (I, {Sᵢ}ᵢ∈I, {uᵢ}ᵢ∈I):
- I = {1,...,n} — set of players
- Sᵢ — strategy set of player i
- uᵢ: S₁×...×Sₙ → ℝ — payoff function
Nash Equilibrium (1950): a strategy profile (s₁*,...,sₙ*) such that no player can improve their payoff by unilaterally changing their strategy:
uᵢ(sᵢ*, s_{-i}) ≥ uᵢ(sᵢ, s_{-i}) for all sᵢ ∈ Sᵢ, i ∈ I
Explanation: s_{-i}* = strategies of all players except i. Nash equilibrium is a "rest point": no one wants to deviate.
Nash’s Theorem (1950): Every finite normal form game has an equilibrium in mixed strategies.
Classic Examples
Prisoner’s Dilemma:
| Keep Silent | Betray | |
|---|---|---|
| Keep Silent | (3,3) | (0,5) |
| Betray | (5,0) | (1,1) |
Dominant strategy for both is "betray" (payoff 5>3 or 1>0). NE: (Betray, Betray) = (1,1). Pareto inefficient — (Keep Silent, Keep Silent) = (3,3) is better for both. A fundamental illustration: individual rationality ≠ collective efficiency.
Real-life examples: arms race, price wars, fisheries (everyone catches as much as possible → resource collapse).
Coordination Games:
| Left Side | Right Side | |
|---|---|---|
| Left Side | (1,1) | (0,0) |
| Right Side | (0,0) | (1,1) |
Two NEs: (Left, Left) and (Right, Right). Without coordination — a disaster is possible. Solution: social norms (in Russia, right-hand traffic), laws.
Cournot Equilibrium (Oligopoly): n firms with costs c, market demand P = a − bQ. NE: each firm produces q* = (a−c)/((n+1)b). Price: P* = (a+nc)/(n+1). As n→∞: P* → c (competition). If n=1 (monopoly): P* = (a+c)/2.
Extensive Form and Subgame Perfection
Game in extensive form: decision tree with information sets. Strategy = complete plan of action at every information set.
Backward induction: in games with complete information, we solve "from the end": first the optimum in the last node, then in the penultimate, etc.
Subgame Perfect Equilibrium (SPE): NE in every subgame (not just the whole game). Eliminates incredible threats.
Example: monopoly entry. Entrant: enter or not. Incumbent: fight or concede. NE: many, including "Enter, Concede." The threat "I will fight" is eliminated in SPE — incumbent will prefer to concede in this subgame (if fighting is costlier than conceding).
Mechanism Design: Reverse Game Theory
Problem: the designer wishes to implement a certain social choice (allocation, rule), but agents have private information (types). How to create "rules of the game" so that agents voluntarily reveal their true types?
Revelation Principle: any implementable outcome can be achieved by a direct revelation mechanism (each agent reports their type), in which truth-telling is a dominant strategy (strategy-proof mechanism).
VCG (Vickrey-Clarke-Groves) mechanism: in problems of efficient allocation of public goods: each agent receives a transfer tᵢ = h(θ_{-i}) + Σⱼ≠ᵢ vⱼ(k*, θⱼ). With truthful reporting: maximizes total welfare. Truth-telling is a dominant strategy. Example: second-price auction (Vickrey) — highest bidder wins, pays the second-highest bid.
Vickrey Auction: buyers have valuation θᵢ. Dominant strategy — bid exactly θᵢ. Winner: highest bid, pays the second. Proof of truthfulness: if θᵢ > max θⱼ (j≠i), overstating the bid doesn’t help, understating risks losing.
Numerical Example: Nash Equilibrium in Mixed Strategies
Matching Pennies matrix: player 1 chooses H/T, player 2 — H/T. If matched — 2 pays 1 (payoff (1,−1)), mismatched — 1 pays 2 (payoff (−1,1)). No pure NE. Mixed NE: each chooses H with probability p*. Indifference condition: 1·p* + (−1)(1−p*) = (−1)·p* + 1·(1−p*) → p* = 0.5. Expected payoff for each = 0.
Assignment: Game: two firms choose prices p₁,p₂ ∈ [0,10]. Demand for firm i’s product: Dᵢ = 12−2pᵢ + pⱼ. Costs: cᵢ = 2. (1) Find best response functions. (2) Compute Bertrand NE. (3) Compare with cooperative equilibrium (joint profit maximization). (4) If the game is infinitely repeated with discount δ=0.9 — under what conditions can cooperation be sustained by grim trigger strategy?
§ Act · what next