Stanford ECON50 - Game Theory, Competitive Strategy & Market Equilibrium Flashcards

Question 1

First Fundamental Theorem of Welfare Economics

Accepted Answer

States that under complete markets, perfect competition, zero transaction costs, and no externalities or public goods, any competitive equilibrium allocation is **Pareto efficient**.

Question 2

Lagrangian Optimization for Utility Maximization

Accepted Answer

The constrained optimization problem of maximizing utility $u(x,y)$ subject to budget constraint $p_x x + p_y y = I$ is formulated as:
$$\mathcal{L}(x, y, \lambda) = u(x,y) - \lambda(p_x x + p_y y - I)$$
where $\lambda$ represents the marginal utility of income.

Question 3

First-Order Conditions (FOCs) for Consumer Choice

Accepted Answer

Differentiating the utility Lagrangian $\mathcal{L} = u(x,y) - \lambda(p_x x + p_y y - I)$ yields:
1. $\frac{\partial \mathcal{L}}{\partial x} = MU_x - \lambda p_x = 0 \implies MU_x = \lambda p_x$
2. $\frac{\partial \mathcal{L}}{\partial y} = MU_y - \lambda p_y = 0 \implies MU_y = \lambda p_y$
3. $\frac{\partial \mathcal{L}}{\partial \lambda} = -(p_x x + p_y y - I) = 0 \implies p_x x + p_y y = I$

Question 4

Marginal Rate of Substitution (MRS) Condition

Accepted Answer

Dividing the first-order conditions for $x$ and $y$ gives the tangency condition:
$$MRS_{x,y} = \frac{MU_x}{MU_y} = \frac{p_x}{p_y}$$
At equilibrium, the consumer's willingness to substitute good $y$ for good $x$ equals the market exchange rate of prices.

Question 5

Pure Strategy Nash Equilibrium

Accepted Answer

A strategy profile $(s_1^*, s_2^*, \dots, s_n^*)$ where no player $i$ has a unilateral incentive to deviate:
$$u_i(s_i^*, s_{-i}^*) \ge u_i(s_i, s_{-i}^*) \quad \forall s_i \in S_i$$
Every player's chosen strategy is a best response to the opponents' strategies.

Question 6

Strictly Dominant Strategy

Accepted Answer

A strategy $s_i^*$ for player $i$ that strictly yields a higher payoff than any other strategy $s_i$, regardless of choices made by other players:
$$u_i(s_i^*, s_{-i}) > u_i(s_i, s_{-i}) \quad \forall s_i 
eq s_i^*, \forall s_{-i}$$

Question 7

Weakly Dominant Strategy

Accepted Answer

A strategy $s_i^*$ that yields at least as high a payoff as any alternative strategy against all opponent profiles, and a strictly higher payoff against at least one profile:
$$u_i(s_i^*, s_{-i}) \ge u_i(s_i, s_{-i}) \quad \forall s_i, s_{-i}$$ with strict inequality for at least one $s_{-i}$.

Question 8

Mixed Strategy Nash Equilibrium (MSNE)

Accepted Answer

An equilibrium where at least one player randomizes over pure strategies using a probability distribution. In an MSNE, a player must be indifferent between all pure strategies assigned positive probability.

Question 9

Cournot Duopoly Model

Accepted Answer

An oligopoly game where two firms simultaneously choose **quantities** ($q_1, q_2$). The market-clearing price is determined by total output $Q = q_1 + q_2$ via the inverse demand function $P(Q)$.

Question 10

Cournot Best-Response Function

Accepted Answer

Firm 1's best response function $q_1^*(q_2)$ maximizes profit given firm 2's output $q_2$. For linear demand $P = a - b(q_1 + q_2)$ and constant marginal cost $c$:
$$q_1^*(q_2) = \frac{a - c - b q_2}{2b}$$

Question 11

Cournot Nash Equilibrium

Accepted Answer

The intersection of firms' best-response curves. For symmetric firms with demand $P = a - b Q$ and cost $c$:
- Individual quantity: $q_1^* = q_2^* = \frac{a - c}{3b}$
- Total quantity: $Q^* = \frac{2(a - c)}{3b}$
- Market price: $P^* = \frac{a + 2c}{3}$

Question 12

Bertrand Duopoly Model (Homogeneous Goods)

Accepted Answer

An oligopoly game where firms simultaneously set **prices** ($p_1, p_2$). The firm offering the lower price captures the entire market demand $D(p)$; if prices are equal, firms split demand evenly.

Question 13

Bertrand Paradox

Accepted Answer

In a homogeneous-good Bertrand duopoly with identical constant marginal cost $c$, price competition drives the unique Nash equilibrium to:
$$p_1^* = p_2^* = c$$
Firms earn zero economic profit despite there being only two competitors.

Question 14

Cournot vs. Bertrand Equilibrium Comparison

Accepted Answer

- **Cournot (Quantity)**: $P > MC$, positive economic profits, lower total output, creates deadweight loss.
- **Bertrand (Price with identical goods)**: $P = MC$, zero economic profit, maximum total output, zero deadweight loss (allocatively efficient).

Question 15

Stackelberg Duopoly Model

Accepted Answer

A sequential-move game where a **leader** firm chooses quantity $q_1$ first, and a **follower** firm observes $q_1$ before choosing $q_2$. Solved via backward induction using the follower's best-response function.

Question 16

Stackelberg First-Mover Advantage

Accepted Answer

The leader commits to a larger output $q_1^* = \frac{a - c}{2b}$, forcing the follower to contract output $q_2^* = \frac{a - c}{4b}$. The leader earns strictly higher profits than in Cournot, while the follower earns lower profits.

Question 17

Pareto Efficiency (Pareto Optimality)

Accepted Answer

An allocation of resources such that no individual can be made strictly better off without making at least one other individual strictly worse off.

Question 18

Edgeworth Box & Contract Curve

Accepted Answer

A graphical representation of allocations of two goods between two consumers. The **Contract Curve** is the set of all Pareto-efficient allocations where consumers' indifference curves are tangent:
$$MRS_A^{x,y} = MRS_B^{x,y}$$

Question 19

Second Fundamental Theorem of Welfare Economics

Accepted Answer

States that any Pareto-efficient allocation can be supported as a competitive equilibrium through a suitable initial lump-sum redistribution of endowments, provided consumer preferences are convex.

Question 20

Prisoners' Dilemma

Accepted Answer

A classic normal-form game illustrating how individual rational incentives lead to a Pareto-inefficient outcome. The unique dominant strategy equilibrium is (Defect, Defect), yielding lower payoffs for both players than mutual cooperation.

Question 21

Grim Trigger Strategy

Accepted Answer

A repeated game strategy: cooperate until the opponent defects once, then defect forever after. Supports mutual cooperation in infinitely repeated games if discount factor $\delta$ is sufficiently close to 1.

Stanford ECON50 - Game Theory, Competitive Strategy & Market Equilibrium

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