Mathematics for Economics — Game Theory and Optimization

Game Theory Basics

Why game theory matters:

the mathematical analysis of strategic interdependence
what’s optimal for me depends on the other player’s choice
applied in economics, international relations, biology, and computer science

Normal-form games:

components: players, strategies, payoffs
represented with a payoff matrix

Dominant strategy:

a strategy that’s always best regardless of the opponent’s strategy
weakly dominant strategy: always at least as good, sometimes better

Nash equilibrium:

a strategy combination where no player can gain by unilaterally deviating
definition: (s₁, s₂) is a Nash equilibrium
- if each player’s strategy is a best response given the other’s strategy
Nash’s theorem: an equilibrium always exists (including mixed strategies)

Best response:

BR₁(s₂): player 1’s best response to s₂
Nash equilibrium: s₁* ∈ BR₁(s₂*) AND s₂* ∈ BR₂(s₁*)
the intersection of best-response functions is the Nash equilibrium

Mixed-strategy Nash equilibrium:

mixing strategies probabilistically
when no pure-strategy equilibrium exists, a mixed-strategy equilibrium does
choosing probabilities that make the opponent indifferent

Common Game Types

The prisoner’s dilemma:

Payoff matrix (cooperate/defect):

both cooperate: (3, 3)
player 1 defects, player 2 cooperates: (5, 0)
player 1 cooperates, player 2 defects: (0, 5)
both defect: (1, 1)

Analysis:

each player’s dominant strategy: defect
Nash equilibrium: (defect, defect) → payoff (1, 1)
Pareto optimum: (cooperate, cooperate) → (3, 3)
the clash between social optimum and individual rationality explains market failure

Applications:

arms races, environmental agreements, price competition, cartel collapse

Coordination games:

what matters is reaching the same equilibrium (a standardization problem)
example: which side of the road to drive on, technology standards

Sequential games:

represented in extensive form (a game tree)
backward induction: solving from the end of the game backward
subgame perfect Nash equilibrium

Stackelberg equilibrium:

the leader (first mover) decides first
first-mover advantage: capturing the opportunity of moving first

Chicken (a game of brinkmanship):

both swerve: a small payoff
one goes straight: the one going straight gets a large payoff
both go straight: the worst outcome
two pure-strategy Nash equilibria, one mixed-strategy equilibrium

Repeated Games and Cooperation

Repeated games:

the same game played multiple times
reputation and retaliation become possible → creates an incentive to cooperate

Finitely repeated games:

in the last round, it’s identical to a one-shot game → defect
backward induction: defection in every round → cooperation becomes impossible
if the end point is uncertain, cooperation can survive

Infinitely repeated games:

discount factor δ (0 < δ < 1): the present value of future payoffs
tit-for-tat: cooperate first, then mirror the opponent’s last move
the Folk Theorem: if δ is large enough (the future matters enough), many cooperative equilibria are possible condition: cooperative payoff > one-time gain from defecting × (1-δ)/δ

Applications:

implicit collusion among oligopolistic firms
compliance with international trade agreements
trust built through repeated transactions

Auction theory:

English auction: price rises → the highest bidder wins
Dutch auction: price falls → the first bidder to accept wins
sealed-bid, first-price: the highest bidder pays their own bid
sealed-bid, second-price (Vickrey): the highest bidder wins, but pays the second-highest bid truthful bidding is the dominant strategy (incentive compatible)

Multivariable Optimization

Optimizing multivariable functions:

Extrema of a two-variable function f(x, y):

First-order conditions:

∂f/∂x = 0
∂f/∂y = 0
satisfying both simultaneously gives a critical point

Second-order conditions:

the Hessian matrix: H = [fxx, fxy; fyx, fyy]
det(H) > 0 AND fxx < 0 → a local maximum
det(H) > 0 AND fxx > 0 → a local minimum
det(H) < 0 → a saddle point

Constrained optimization — the Lagrange multiplier method:

objective: maximize f(x, y)
constraint: g(x, y) = c
the Lagrangian: L = f(x, y) - λ[g(x, y) - c]

Conditions:

∂L/∂x = ∂f/∂x - λ∂g/∂x = 0
∂L/∂y = ∂f/∂y - λ∂g/∂y = 0
∂L/∂λ = -(g(x,y) - c) = 0

Interpreting λ:

the change in the objective function when the constraint is relaxed by one unit (a shadow price)
if λ = 1, relaxing the constraint by one unit increases the objective function by λ

Economic applications:

utility maximization: maximize U(x,y) subject to pₓx + pᵧy = M
cost minimization: minimize wL + rK subject to Q = f(L, K)

Linear Programming

The basic form of a linear program

objective function: maximize (or minimize) Z = c’x
constraints: Ax ≤ b (or = b)
non-negativity: x ≥ 0

Graphical solution (two variables)

graph the constraint lines
identify the feasible region
shift the objective-function line to find the optimal vertex → the optimal solution always occurs at a vertex

The simplex method

an algorithm that systematically searches vertices
starts with non-basic variables at 0 and basic variables > 0
repeatedly moves to the vertex that increases the objective function

The dual problem

every LP (the primal problem) has a corresponding dual LP
the primal’s maximum equals the dual’s minimum (the strong duality theorem)
the dual variables equal the Lagrange multipliers λ — the shadow prices of the constrained resources

Economic applications

production planning: deciding the optimal output for multiple products
transportation problems: minimizing shipping costs from factories to warehouses
portfolio optimization: maximizing return subject to a risk constraint

Integer programming

used when variables must be integers (number of factories, number of employees)
the branch-and-bound algorithm
more computationally complex than standard LP

Frequently Asked Questions

Q. Is a Nash equilibrium always socially optimal? A. No. As the prisoner’s dilemma shows, a Nash equilibrium is the outcome of individually rational strategies, but it isn’t necessarily optimal from society’s point of view. When both players defect at the Nash equilibrium, they get a payoff of (1, 1), even though both cooperating would produce the better outcome of (3, 3). This mismatch is a source of market failure, and it explains a wide range of economic phenomena — environmental pollution, the underprovision of public goods, and the collapse of cartels.

Q. What’s the economic meaning of the Lagrange multiplier? A. The Lagrange multiplier λ is a shadow price — it tells you how much the objective function changes when the constraint is relaxed by one unit. In a utility-maximization problem under an income constraint, for instance, λ tells you how much maximum utility rises when income increases by one dollar. In a production problem, it tells you how much more you could produce if the budget rose by one dollar. This concept is useful for valuing the marginal value of a resource, and it carries the same meaning as the dual variables in linear programming.