Nash Equilibrium as Mutual Best Response

Why Mutual Best Responses Define Stability

In a strategic setting, one player’s payoff can depend on what other players choose. A decision that looks attractive in isolation may cease to be attractive once those choices are taken into account. The central question is therefore not whether a profile gives every player the highest payoff that could ever be obtained. It is whether any player, taking the others’ actions as fixed, can improve their own payoff by changing their action alone.

A pure-strategy Nash equilibrium captures this form of stability. Each player’s action must be a best response to the actions of the others. This definition provides a precise test for a candidate outcome in a simultaneous-move strategic-form game. It does not require players to coordinate a change, and it does not say that the outcome maximizes total payoff.

Definition: A strategic-form game consists of players $i=1,\ldots,n$, a strategy set $S_i$ for each player $i$, and a payoff function $u_i(s_i,s_{-i})$ for each player. Here $s_i\in S_i$ is player $i$’s action, and $s_{-i}$ denotes the actions of all other players. A strategy profile $s^*=(s_1^*,\ldots,s_n^*)$ is a pure-strategy Nash equilibrium if, for every player $i$ and every available action $s_i\in S_i$, $$ u_i(s_i^*,s_{-i}^*)\geq u_i(s_i,s_{-i}^*). $$ In words, no player can obtain a strictly higher payoff by changing only their own action.

A best response is an action that maximizes a player’s payoff, given the other players’ actions. The best-response set of player $i$ to $s_{-i}$ is $BR_i(s_{-i})=\operatorname{argmax}_{s_i\in S_i}u_i(s_i,s_{-i})$, when a maximizing action exists. A profile is a pure-strategy Nash equilibrium exactly when $s_i^*\in BR_i(s_{-i}^*)$ for every player $i$. If more than one action achieves the maximum, each is a best response; a tie does not prevent equilibrium.

The definition applies to finite or infinite strategy sets, provided the stated best responses exist. In a finite game, each player has finitely many available actions, so a maximum payoff exists for every fixed profile of the other players. In a continuous-choice game, existence of a maximizing action must be checked from the payoff function and feasible set rather than assumed automatically.

From Best Responses to the Equilibrium Condition

The equivalence between mutual best responses and the “no profitable unilateral deviation” test follows directly from maximization. It is useful to keep both forms in view: the best-response form describes what each player is choosing, while the inequality form makes the test explicit.

Fix the candidate actions of the other players.
For a proposed profile $s^*$, hold $s_{-i}^*$ fixed when evaluating player $i$. Do not change other players’ actions during this player’s test.

Compare the player’s payoff across their own available actions.
The chosen action $s_i^*$ is a best response if $u_i(s_i^*,s_{-i}^*)\geq u_i(s_i,s_{-i}^*)$ for every $s_i\in S_i$.

Repeat for every player.
If every chosen action is a best response to the others’ chosen actions, the profile is a pure-strategy Nash equilibrium. If at least one player has a strictly profitable unilateral deviation, it is not.

For the justification, suppose first that $s_i^*\in BR_i(s_{-i}^*)$ for every player. By the definition of a maximum, no alternative $s_i$ gives player $i$ more than $u_i(s_i^*,s_{-i}^*)$, so the equilibrium inequalities hold for all players. Conversely, if those inequalities hold for every player and every action, then each chosen action achieves at least as much payoff as any alternative against the fixed actions $s_{-i}^*$. Thus each chosen action is a best response. The two statements are equivalent.

s^*\text{ is a pure-strategy Nash equilibrium} \quad\Longleftrightarrow\quad s_i^*\in BR_i(s_{-i}^*)\text{ for every }i \quad\Longleftrightarrow\quad u_i(s_i^*,s_{-i}^*)\geq u_i(s_i,s_{-i}^*)\text{ for every }i,s_i.

Testing a Candidate in a Discrete Game

A candidate test is local in a specific sense: for each player, compare the chosen action with all of that player’s alternatives while holding everyone else’s actions fixed. The profile must pass every player’s test. One profitable deviation is enough to reject it; a player’s inability to improve does not compensate for another player’s profitable deviation.

Worked Example: Checking a Candidate Pricing Profile

Two firms simultaneously choose either a regular price $R$ or a discount price $D$. Payoffs are in thousands of dollars, with the first number belonging to Firm 1. The payoff table is

Firm 1 \ Firm 2	R	D
R	(10, 6)	(12, 4)
D	(7, 8)	(5, 9)

Test the candidate profile $(R,R)$. Method: check each firm’s unilateral alternative at the candidate. Common mistake avoided: treating a payoff that belongs to the other firm as the deviator’s payoff.

Step 1: Firm 1’s response. At $(R,R)$, Firm 1 earns $10$. If Firm 1 alone changes to $D$, the outcome is $(D,R)$, where Firm 1 earns $7$. Since $7<10$, this deviation is not profitable.

Step 2: Firm 2’s response. At $(R,R)$, Firm 2 earns $6$. If Firm 2 alone changes to $D$, the outcome is $(R,D)$, where Firm 2 earns $4$. Since $4<6$, this deviation is not profitable.

Step 3: Conclusion. Both firms’ chosen actions are best responses to the other firm’s action. Therefore $(R,R)$ is a pure-strategy Nash equilibrium. The two payoff comparisons are separate: Firm 1’s payoff is the first number in each ordered pair, and Firm 2’s is the second.

Continuous Choices: Best Responses from Payoff Functions

When actions are quantities, prices, or other continuous variables, it is usually impractical to list every possible deviation. Instead, derive each player’s best response from their payoff function, then check whether the candidate actions satisfy the best-response conditions simultaneously. A first-order condition identifies an interior maximum only when the payoff’s shape and feasibility conditions support that conclusion.

Worked Example: Mutual Best Responses in a Quantity Game

Two firms simultaneously choose quantities $q_1\geq0$ and $q_2\geq0$. Market price is $P=\max\{100-q_1-q_2,0\}$. Firm 1 has constant marginal cost $20$, and Firm 2 has constant marginal cost $30$. There are no fixed costs. Find and verify a pure-strategy equilibrium. Method: derive each firm’s best response, solve them together, and check the resulting payoffs. Common mistake avoided: treating an unconstrained first-order condition as a best response without checking nonnegative quantities or the demand range.

Step 1: Write the profit functions. When total quantity is below $100$, the price is $100-q_1-q_2$. In that range, profits are

\pi_1=(100-q_1-q_2-20)q_1=(80-q_1-q_2)q_1, \qquad \pi_2=(100-q_1-q_2-30)q_2=(70-q_1-q_2)q_2.

If the rival’s quantity is fixed and the firm chooses a quantity beyond the point where price reaches zero, its profit is negative, while producing zero gives zero profit. Thus such a quantity cannot be a best response. Within the positive-price range, each profit is a strictly concave quadratic in the firm’s own quantity.

Step 2: Derive Firm 1’s best response. For an interior choice, differentiate Firm 1’s profit with respect to $q_1$:

\frac{\partial \pi_1}{\partial q_1}=80-2q_1-q_2=0 \quad\Longrightarrow\quad q_1=\frac{80-q_2}{2}.

Accounting for $q_1\geq0$, the best response is $q_1=\max\{0,(80-q_2)/2\}$. Firm 2’s corresponding first-order condition gives $q_2=\max\{0,(70-q_1)/2\}$.

Step 3: Solve the two best-response equations. The positive interior solution satisfies $2q_1+q_2=80$ and $q_1+2q_2=70$. Subtracting the second equation from the first gives $q_1-q_2=10$, so $q_1=q_2+10$. Substituting into $q_1+2q_2=70$ gives $3q_2+10=70$, hence $q_2=20$ and $q_1=30$.

Step 4: Check feasibility and payoffs. Total quantity is $30+20=50$, so price is $100-50=50$, which is positive. Firm 1’s profit is $(50-20)30=30\cdot30=900$. Firm 2’s profit is $(50-30)20=20\cdot20=400$. At the other firm’s quantity, each firm’s strictly concave profit is maximized by its calculated quantity. Therefore neither can gain by changing quantity alone, and $(30,20)$ is a pure-strategy Nash equilibrium. The quantities are nonnegative and satisfy the demand range used in the calculation.

Ties, Joint Changes, and the Limits of the Test

A best response need not be unique. If a player’s chosen action gives the same maximum payoff as another available action, both actions are best responses. A Nash equilibrium requires that no unilateral change strictly improve a player’s payoff; it does not require that every other action give a strictly lower payoff.

The unilateral qualification also matters. A change in which two or more players alter their actions at once is not the deviation tested in the definition. A joint change may produce a higher payoff for one or more players without invalidating a profile’s Nash equilibrium status. The definition does not assert that players could not benefit from coordinating a change, nor does it rank profiles by total payoff.

Worked Example: Tied Best Responses and a Joint Change

Two organizations choose operating plans $L$ or $H$ and $C$, $D$, or $E$, respectively. Payoffs are listed in the order Organization 1, Organization 2.

Organization 1 \ Organization 2	C	D	E
L	(6, 4)	(2, 4)	(3, 1)
H	(6, 2)	(9, 9)	(1, 5)

Test $(L,C)$, assuming the listed actions are exhaustive. Method: compare each player’s payoff with all of that player’s alternatives, keeping the other action fixed. Common mistake avoided: rejecting a candidate because both players could gain after changing actions together.

Step 1: Organization 1’s best responses to $C$. If Organization 1 chooses $L$, its payoff is $6$. If it chooses $H$ while Organization 2 stays at $C$, its payoff is also $6$. These are all Organization 1’s available actions, so $L$ is a best response, tied with $H$.

Step 2: Organization 2’s best responses to $L$. Given $L$, Organization 2 receives $4$ from $C$, $4$ from $D$, and $1$ from $E$. Its maximum is $4$, so $C$ is a best response, tied with $D$.

Step 3: Conclusion and joint-change check. Both chosen actions are best responses, so $(L,C)$ is a pure-strategy Nash equilibrium. The joint change to $(H,D)$ gives payoffs $(9,9)$, higher for both than $(6,4)$. That comparison changes both actions at once, so it does not contradict the unilateral-deviation inequalities. The ties at the candidate also do not disqualify it.

Common Exam Traps

Changing more than one action during a player’s test. A player’s deviation must hold the other players’ actions fixed. A payoff from a joint change cannot be substituted for a unilateral-deviation payoff.
Reading the wrong player’s payoff. In an ordered payoff pair, only the deviating player’s own payoff determines whether that deviation is profitable. For example, if a candidate payoff is $(4,3)$ and a unilateral change by Player 1 leads to $(2,8)$, Player 1’s payoff changes from $4$ to $2$, not from $3$ to $8$.
Checking only one player. A candidate is an equilibrium only if every player’s action is a best response. One player’s profitable unilateral deviation is enough to reject it.
Assuming a tie rules out equilibrium. A player can be indifferent between the chosen action and another best response. The condition is that no alternative yields a strictly higher payoff.
Checking only the alternatives that happen to be mentioned. For an equilibrium claim, the comparison must cover every available action. In a finite game, check the full action set; in a continuous game, verify that the chosen action maximizes the payoff over the feasible set.

Key takeaway: A pure-strategy Nash equilibrium is a profile of mutual best responses. Fix the other players’ actions, compare each player’s payoff from every feasible own action, and accept the profile only if no player can improve by acting alone.

Check Your Understanding

For each problem, apply the best-response test to the specified candidate. Payoff pairs are ordered by player unless stated otherwise.

Two firms each have only actions $A$ and $B$. At $(A,A)$ the payoffs are $(5,4)$; if only Firm 1 switches, the payoffs are $(3,7)$; if only Firm 2 switches, they are $(6,2)$. Is $(A,A)$ an equilibrium? (Answer: Yes; Firm 1’s unilateral payoff falls from 5 to 3, and Firm 2’s falls from 4 to 2.)
At $(R,X)$, payoffs are $(4,6)$. If Player 1 alone switches to $S$, the outcome is $(S,X)$ with payoffs $(3,9)$. If Player 2 alone switches to $Y$, the outcome is $(R,Y)$ with payoffs $(8,5)$. Each player has only the stated two actions. Is $(R,X)$ an equilibrium? (Answer: Yes; Player 1’s payoff falls from 4 to 3 and Player 2’s falls from 6 to 5.)
At a candidate profile, Player 1 has exactly two actions, and receives 7 from the chosen action and 7 from the alternative, holding the other player fixed. Player 2’s chosen action gives 5, while its only alternative gives 4. Is the candidate an equilibrium? (Answer: Yes; both chosen actions are best responses, with Player 1 indifferent between its two actions.)
Two players have nonnegative quantity choices, and their best responses are $q_1=\max\{0,(60-q_2)/2\}$ and $q_2=\max\{0,(40-q_1)/2\}$. Find the mutual best-response quantities. (Answer: $q_1=80/3$ and $q_2=20/3$.)
At $(A,X)$, payoffs are $(5,4)$. Player 1’s only alternative, holding $X$ fixed, gives Player 1 payoff 5. Player 2’s only alternative, holding $A$ fixed, gives Player 2 payoff 3. A joint change gives payoffs $(9,8)$. Is $(A,X)$ an equilibrium? (Answer: Yes; the joint change is not a unilateral deviation, and neither player has a strictly profitable unilateral deviation.)

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