Normal-Form Games & Equilibria

Objective

A normal-form game is the workhorse model for strategic interaction, and quant interviews lean on it for pricing-game intuition, auction reasoning, and "what would a rational counterparty do" questions. By the end you should be able to read a payoff matrix, eliminate dominated strategies, find every pure-strategy Nash equilibrium by best response, solve a $2\times2$ game for its mixed equilibrium using the indifference principle, and compute the minimax value of a zero-sum game. We close with correlated equilibrium and the five canonical $2\times2$ games.

Intuition

A game is a situation where your best move depends on what others do. There is no single "optimal" action in isolation; there is only a best response to a belief about opponents. An equilibrium is a consistency condition: every player is best-responding to the others, so no one regrets their choice given what actually happened. That mutual no-regret property is what makes a prediction stable.

Definitions

A normal-form game has:

• A finite set of players $i = 1, \dots, n$.
• For each player, a finite set of pure strategies $S_i$. A strategy profile is $s = (s_1, \dots, s_n)$.
• A payoff function $u_i(s)$ giving player $i$'s utility for each profile.

Write $s_{-i}$ for the strategies of everyone except $i$. For two players we display payoffs in a matrix with entries $(u_1, u_2)$: rows are player 1's strategies, columns are player 2's.

Strategy $a$ strictly dominates $b$ for player $i$ if $u_i(a, s_{-i}) > u_i(b, s_{-i})$ for every $s_{-i}$. A rational player never plays a strictly dominated strategy, which licenses iterated deletion: remove a dominated strategy, then re-examine the smaller game, repeat. Strict-dominance deletion never destroys an equilibrium and the surviving set is order-independent.

A best response to $s_{-i}$ is any $s_i$ maximizing $u_i(s_i, s_{-i})$. A pure-strategy Nash equilibrium (PSNE) is a profile $s^*$ where each $s_i^*$ is a best response to $s_{-i}^*$: no unilateral deviation helps.

A mixed strategy $\sigma_i$ is a probability distribution over $S_i$. A mixed Nash equilibrium is a profile of mixed strategies where each $\sigma_i$ maximizes expected payoff against the others. Nash's theorem guarantees at least one exists in any finite game.

Key formulas and the indifference principle

The central tool for $2\times2$ games: if a player mixes (puts positive probability on two pure strategies), those strategies must yield equal expected payoff against the opponent's mix. Otherwise the player would shift all weight to the better one. So the opponent's mixing probabilities are pinned down by your indifference, not your own.

For a zero-sum two-player game ($u_2 = -u_1$), define the value

$v = \max_{\sigma_1} \min_{\sigma_2} \; \mathbb{E}[u_1] = \min_{\sigma_2} \max_{\sigma_1} \; \mathbb{E}[u_1].$

The minimax theorem says these two are equal; $v$ is the guaranteed payoff player 1 can secure and player 2 can hold them to. Assumption: payoffs are known, players are expected-utility maximizers, and the game is played once simultaneously.

A correlated equilibrium generalizes Nash: a trusted device draws a profile from a joint distribution $p(s)$ and privately recommends $s_i$ to each player. It is an equilibrium if no player gains by deviating from their recommendation given the conditional distribution over others. Every Nash equilibrium is a correlated equilibrium (take $p$ as the product of the mixed strategies), but correlation can achieve payoffs outside the Nash set.

Worked example: Battle of the Sexes (mixed equilibrium)

Two players coordinate on Opera (O) or Football (F). Player 1 prefers O, player 2 prefers F, but both prefer agreeing to disagreeing. Payoffs $(u_1, u_2)$:

| | O | F | |---|---|---| | O | (2, 1) | (0, 0) | | F | (0, 0) | (1, 2) |

Pure equilibria. At (O, O): player 1 gets 2 (deviating to F gives 0), player 2 gets 1 (deviating gives 0). Neither deviates, so (O, O) is a PSNE. By symmetry (F, F) is also a PSNE.

Mixed equilibrium. Let player 1 play O with probability $p$, player 2 play O with probability $q$. Use the indifference principle on player 2: their O and F payoffs must match. Player 2's payoff from O is $p\cdot 1 + (1-p)\cdot 0 = p$. From F it is $p\cdot 0 + (1-p)\cdot 2 = 2(1-p)$. Set equal:

$p = 2(1-p) \implies p = 2 - 2p \implies 3p = 2 \implies p = \tfrac{2}{3}.$

By symmetry, player 1's indifference gives player 2's $q$: O payoff $= 2q$, F payoff $= 1\cdot(1-q)$, so $2q = 1 - q \implies q = \tfrac{1}{3}$. So player 1 plays O with prob $\tfrac{2}{3}$, player 2 plays O with prob $\tfrac{1}{3}$.

Each player's own expected payoff: player 1 gets $2q = 2\cdot\tfrac{1}{3} = \tfrac{2}{3}$. Notice this is worse than either pure equilibrium, the classic miscoordination cost. A correlated equilibrium fixes this: a coin flip recommending (O, O) or (F, F) each with probability $\tfrac{1}{2}$ gives each player $\tfrac{1}{2}(2) + \tfrac{1}{2}(1) = 1.5$, beating the mixed Nash, with no incentive to deviate from the recommendation.

The five canonical 2x2 games

• Prisoner's Dilemma: Defect strictly dominates Cooperate for both; the unique Nash (Defect, Defect) is Pareto-dominated by mutual cooperation. Dominance solves it outright.
• Matching Pennies: zero-sum, no PSNE; unique equilibrium is each player mixing $(\tfrac{1}{2}, \tfrac{1}{2})$, with value $0$.
• Stag Hunt: two PSNE, a payoff-dominant (Stag, Stag) and a risk-dominant (Hare, Hare); coordination under risk.
• Battle of the Sexes: worked above; two pure plus one mixed equilibrium.
• Hawk-Dove (Chicken): two asymmetric PSNE (Hawk, Dove) and (Dove, Hawk) plus a mixed one; models entry and contest games.

Edge cases and interview insight

Watch the distinction between strict and weak dominance: iterated deletion of weakly dominated strategies can be order-dependent and can delete equilibria, so quote results carefully. A pure equilibrium is just a degenerate mixed one, so "find all Nash equilibria" means check pure profiles by best response and solve the indifference conditions. In zero-sum games, equilibrium, minimax, and maximin all coincide, which is why "what value can I guarantee" is well-posed. Common trap: assuming a unique equilibrium; coordination games have several, and selecting among them needs extra criteria (payoff vs risk dominance). A follow-up lesson would add extensive-form games, subgame-perfection, and backward induction.