Normal-Form Games and Nash Equilibrium

Representing a Simultaneous-Move Game

In a strategic setting, an outcome depends not only on what one decision-maker chooses but also on what others choose. A firm’s payoff from setting a high price, for example, may depend on whether a rival also sets a high price. A normal-form game gives a compact description of these interdependent decisions. It is useful in econometrics because a model of strategic behavior specifies how observed choices relate to players’ incentives.

The focus here is a finite, simultaneous-move game with known payoffs: each player chooses an action without observing the other players’ choices, and the payoff associated with each resulting action profile is specified. The normal form does not, by itself, describe how an equilibrium is selected if more than one equilibrium exists.

Definition: A finite normal-form game consists of a finite set of players $i=1,\ldots,I$, a finite, nonempty action set $A_i$ for every player $i$, and a payoff function $u_i:A\to\mathbb{R}$ for every player, where $A=A_1\times\cdots\times A_I$ is the set of action profiles. An action profile $a=(a_1,\ldots,a_I)$ specifies one action for each player. The number $u_i(a)$ is player $i$’s payoff at that profile.

The requirement that every action set is finite is part of the definition here. It ensures that there are finitely many action profiles and that, when a player compares payoffs across their actions, a maximum is attained. The notation $a_{-i}$ denotes the actions of all players other than $i$. Thus $u_i(a_i,a_{-i})$ is the payoff to player $i$ when they choose $a_i$ and the other players choose $a_{-i}$.

A pure strategy in this simultaneous-move setting is simply an action $a_i\in A_i$. A player’s best response to the other players’ actions is any action that maximizes their payoff given those actions:

BR_i(a_{-i})=\arg\max_{a_i\in A_i} u_i(a_i,a_{-i}).

Because $A_i$ is finite and nonempty, at least one action attains this maximum. There may be more than one best response if several actions yield the same highest payoff.

Pure-Strategy Nash Equilibrium

A profile is a Nash equilibrium when each player’s action is a best response to the actions of the others. This definition checks each player’s incentives separately: holding everyone else’s action fixed, no player can gain by changing their own action alone.

Definition: A pure-strategy profile $a^*=(a_1^*,\ldots,a_I^*)$ is a Nash equilibrium if, for every player $i$ and every alternative action $a_i\in A_i$, $$ u_i(a_i^*,a_{-i}^*)\geq u_i(a_i,a_{-i}^*). $$ Equivalently, $a_i^*\in BR_i(a_{-i}^*)$ for every player $i$.

The comparison is unilateral: when player $i$ considers a deviation, the other players’ actions are held fixed. An equilibrium need not maximize total payoff, and it need not be the outcome that all players would jointly prefer. It describes mutual consistency of individual incentives, not a guarantee of efficiency.

Worked Example: Finding Pure Equilibria in a Coordination Game

Two players each choose $A$ or $B$. Their hypothetical payoffs are shown as (Player 1, Player 2).

Player 1 \ Player 2	A	B
A	(4, 4)	(0, 0)
B	(0, 0)	(2, 2)

If Player 2 chooses $A$, Player 1 receives 4 from $A$ and 0 from $B$, so $A$ is Player 1’s best response. If Player 2 chooses $B$, Player 1 receives 0 from $A$ and 2 from $B$, so $B$ is the best response. The same comparisons hold for Player 2 by symmetry.

At $(A,A)$, neither player can improve by switching alone: a switch changes that player’s payoff from 4 to 0. At $(B,B)$, a unilateral switch changes the deviator’s payoff from 2 to 0. Both profiles are therefore pure-strategy Nash equilibria. At either mismatched profile, a player can switch to match the other player and raise their payoff from 0, so neither mismatched profile is an equilibrium.

Mixed Strategies and Expected Payoffs

A player may randomize over actions. A mixed strategy assigns a probability to each action, with nonnegative probabilities that sum to one. In a mixed-strategy profile, players’ randomizations are taken to be independent. The payoff to each player is evaluated as an expected payoff over the resulting action profiles.

Definition: A mixed strategy for player $i$ is a probability distribution $\sigma_i$ over the finite set $A_i$. For a profile of mixed strategies $\sigma=(\sigma_1,\ldots,\sigma_I)$, the expected payoff to player $i$ is $$ U_i(\sigma)=\sum_{a\in A}u_i(a)\prod_{j=1}^{I}\sigma_j(a_j). $$ A mixed-strategy profile is a Nash equilibrium if no player can raise their expected payoff by unilaterally replacing their own probability distribution with another distribution over their actions.

A pure strategy is included as a special case: it puts probability one on a single action. When a player randomizes between two actions with positive probability in equilibrium, those actions must give the same expected payoff against the other players’ strategies. Otherwise, the player could increase expected payoff by shifting probability toward the action with the higher payoff. Actions given zero probability need not yield the same payoff.

Worked Example: A Mixed Equilibrium in the Coordination Game

Use the payoff table from the previous example. Suppose each player chooses $A$ with probability $p$ and $B$ with probability $1-p$, with both players using the same probability. To find a symmetric mixed equilibrium, make each player indifferent between $A$ and $B$, given that the opponent uses this distribution.

If the opponent chooses $A$ with probability $p$, a player choosing $A$ receives payoff 4 when the opponent chooses $A$, and 0 otherwise. The expected payoff is $4p+0(1-p)=4p$. Choosing $B$ yields 0 against $A$ and 2 against $B$, so its expected payoff is $0p+2(1-p)=2-2p$. Indifference requires

4p=2-2p \quad\Longrightarrow\quad 6p=2 \quad\Longrightarrow\quad p=\frac{1}{3}.

At $p=1/3$, either action gives expected payoff $4/3$, since $4(1/3)=4/3$ and $2(2/3)=4/3$. Any other mixture between these two actions also yields $4/3$, so neither player can improve by changing their randomization. Thus both players choosing $A$ with probability $1/3$ and $B$ with probability $2/3$ is a mixed-strategy Nash equilibrium. The pure equilibria $(A,A)$ and $(B,B)$ remain equilibria as well.

When No Pure Action Profile Is Stable

Randomization can be essential rather than merely another equilibrium possibility. In some games, every pure action profile gives at least one player an incentive to switch. A mixed equilibrium can balance those incentives: the probabilities chosen by one player make the other player indifferent between the actions they use.

Worked Example: Matching Pennies

Two players simultaneously choose $H$ or $T$. Player 1 receives 1 if the choices match and $-1$ if they differ; Player 2 receives the opposite payoff. The hypothetical payoff table is:

Player 1 \ Player 2	H	T
H	(1, -1)	(-1, 1)
T	(-1, 1)	(1, -1)

There is no pure equilibrium. If the choices match, Player 2 can switch and improve their payoff from $-1$ to 1. If the choices differ, Player 1 can switch and improve their payoff from $-1$ to 1.

Let $q$ be the probability that Player 2 chooses $H$. Player 1’s expected payoff from choosing $H$ is $q(1)+(1-q)(-1)=2q-1$. From choosing $T$, it is $q(-1)+(1-q)(1)=1-2q$. Indifference requires

2q-1=1-2q \quad\Longrightarrow\quad q=\frac{1}{2}.

Now let $p$ be the probability that Player 1 chooses $H$. Player 2’s expected payoff from choosing $H$ is $p(-1)+(1-p)(1)=1-2p$. From choosing $T$, it is $p(1)+(1-p)(-1)=2p-1$. Indifference gives $1-2p=2p-1$, so $p=1/2$. Each player therefore chooses $H$ and $T$ with probability $1/2$. Each action gives each player expected payoff zero, so neither player can improve by changing their randomization. This is a mixed-strategy Nash equilibrium.

Asymmetric Mixing

The probabilities that make players indifferent need not be the same. In a game with asymmetric payoffs, each player’s mixing probability is calculated from the other player’s payoff comparisons. The calculation is not an instruction to use a particular probability in every situation; it solves the indifference conditions for the specified payoff table.

Worked Example: Different Mixing Probabilities in a Coordination Game

Two players choose Opera or Football. Both prefer coordinating, but Player 1 prefers Opera and Player 2 prefers Football. Payoffs are shown as (Player 1, Player 2).

Player 1 \ Player 2	Opera	Football
Opera	(2, 1)	(0, 0)
Football	(0, 0)	(1, 2)

Let $q$ be the probability that Player 2 chooses Opera. Player 1’s expected payoff from Opera is $2q$; from Football it is $1-q$. For Player 1 to mix between both actions, set these equal:

2q=1-q \quad\Longrightarrow\quad q=\frac{1}{3}.

Let $p$ be the probability that Player 1 chooses Opera. Player 2’s expected payoff from Opera is $p$; from Football it is $2(1-p)$. Player 2 is indifferent when

p=2(1-p) \quad\Longrightarrow\quad 3p=2 \quad\Longrightarrow\quad p=\frac{2}{3}.

The mixed profile with Player 1 choosing Opera with probability $2/3$ and Player 2 choosing Opera with probability $1/3$ is an equilibrium: each player is indifferent between their two actions, so neither can gain by changing their own mixture. The probabilities differ because the players’ payoffs differ.

A Reliable Equilibrium Check

For a small finite game, a systematic calculation prevents a common error: identifying an action that looks attractive without checking every player’s incentives. A useful sequence is:

List the actions and payoffs.
State each player’s finite action set and record each player’s payoff at every action profile.

Find best responses.
For each possible choice by the other player or players, compare the payoffs from all of the player’s own actions.

Check pure profiles.
A pure profile is an equilibrium only if every player’s action is a best response to the others’ actions at that profile.

Check candidate mixtures.
Calculate expected payoffs against the other players’ probability distributions. For any actions used with positive probability, verify that they are best responses.

The expected-payoff calculations rely on the stated independent randomization assumption. For example, in a two-player game, the probability of action profile $(a_1,a_2)$ under independent mixed strategies is $\sigma_1(a_1)\sigma_2(a_2)$. Using a different joint distribution would describe correlated choices rather than the mixed-strategy profiles defined here.

Why the Distinction Matters

In structural models of strategic behavior, the normal form separates three ingredients: the available actions, the payoff consequences of action profiles, and the equilibrium restriction connecting incentives to choices. A payoff function is a model ingredient; Nash equilibrium is a restriction on which profiles are consistent with the players’ incentives. Observing a profile alone does not establish which payoffs generated it.

A common mistake is to call a profile an equilibrium because it gives the players high combined payoffs, or because it seems like a reasonable outcome. That is not the equilibrium test. For example, in the first coordination game, the mismatched profiles give both players zero, but each player can improve by switching to match the other. Conversely, an equilibrium can be less attractive to the players than another profile they could reach through coordinated changes. The correction is to hold the other players’ choices fixed and check every unilateral deviation.

Another frequent error in mixed-strategy calculations is to set the probabilities of the two players equal automatically. The matching-pennies example happens to have equal probabilities because of its symmetry. In the Opera–Football example, the indifference conditions instead give $p=2/3$ and $q=1/3$. Always derive each player’s mixing probability from the other player’s payoff comparisons.

Key takeaway: A Nash equilibrium is a profile of mutual best responses. In a mixed equilibrium, each player’s randomization makes the other players’ actions used with positive probability optimal in expected-payoff terms. Check incentives player by player; do not infer equilibrium from coordination, fairness, or high total payoff.

Check Your Understanding

Use the definitions and calculations above to answer these questions.

Why must the action set be finite in the definition of a finite normal-form game used here?
In a two-player payoff table, what must be true at a pure-strategy Nash equilibrium for each player?
In the first coordination game, calculate a player’s expected payoff from choosing $A$ when the opponent chooses $A$ with probability $1/3$.
In matching pennies, why does neither player’s equilibrium probability depend on their own payoff from choosing $H$ alone?
In the Opera–Football game, which probability makes Player 1 indifferent, and whose strategy is that probability describing?
Explain why an action profile that maximizes the sum of players’ payoffs need not be a Nash equilibrium.