Finding Equilibria Cell by Cell
A payoff matrix lists the outcomes available when two players choose actions. Knowing how to define a Nash equilibrium is not enough when a matrix has several rows and columns: a reliable solution must test every relevant action combination, account for ties, and confirm that no equilibrium has been missed. These skills are useful for analyzing strategic choices by firms, including hypothetical decisions about product launches, pricing, or capacity.
This tutorial focuses on pure strategies: each player selects one action with certainty. It develops a systematic way to locate every such equilibrium in a finite payoff matrix. The method is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
In a payoff matrix, rows represent the row player’s actions and columns represent the column player’s actions. Each cell contains an ordered pair: the first payoff belongs to the row player, and the second belongs to the column player. A best-response comparison always holds the other player’s action fixed. For example, to find the row player’s best response to a particular column, compare the first payoffs down that column.
A Systematic Best-Response Method
For each column, identify the largest row-player payoff in that column. For each row, identify the largest column-player payoff in that row. A cell is an equilibrium exactly when it is among the row player’s best responses to that column and among the column player’s best responses to that row.
Here \(r^*\) and \(c^*\) are the selected row and column actions, while \(u_1\) and \(u_2\) are the row and column players’ payoffs. The notation \(\operatorname*{arg\,max}\) means the action or actions that attain the highest payoff. The method applies to finite payoff matrices with specified player preferences represented by the payoffs. It does not assert that every such game has a pure-strategy equilibrium.
In each column, find the highest first payoff and mark every cell attaining it.
In each row, find the highest second payoff and mark every cell attaining it.
Each such cell is a pure-strategy equilibrium; cells with only one mark are not.
Confirm that every row and every column was compared, and report all cells with both marks.
The justification follows directly from the definition. At a cell with both marks, the row player cannot increase their payoff by switching rows while the column remains fixed, and the column player cannot increase their payoff by switching columns while the row remains fixed. Thus neither has a profitable unilateral deviation, which is precisely the equilibrium condition. Conversely, if a cell lacks one player’s best-response mark, that player can switch to an action with a strictly higher payoff; the cell therefore cannot be an equilibrium.
Worked Examples
Worked Example: One Pure-Strategy Equilibrium
Two hypothetical firms choose whether to launch a product or delay. A second decision-maker chooses a discount or a premium price. The row player’s action is the firm’s choice; the column player’s action is the pricing choice. Payoffs are in thousands of dollars, with the row player’s payoff listed first. Find every pure-strategy equilibrium.
| Discount | Premium | |
|---|---|---|
| Launch | (4, 3) | (1, 2) |
| Delay | (2, 1) | (3, 0) |
Method. Compare the row player’s payoffs within each column, then compare the column player’s payoffs within each row.
Row player’s best responses. If the column player chooses Discount, the row payoffs are 4 for Launch and 2 for Delay, so Launch is the best response. If the column player chooses Premium, the row payoffs are 1 for Launch and 3 for Delay, so Delay is the best response.
Column player’s best responses. If the row player chooses Launch, the column payoffs are 3 for Discount and 2 for Premium, so Discount is the best response. If the row player chooses Delay, the column payoffs are 1 for Discount and 0 for Premium, so Discount is the best response.
At (Launch, Discount), both players are choosing best responses: 4 exceeds 2 for the row player, and 3 exceeds 2 for the column player. The other three cells fail at least one comparison. In particular, (Delay, Premium) is not an equilibrium: although Delay is the row player’s best response to Premium, the column player would switch to Discount because 1 exceeds 0.
Answer and check. The unique pure-strategy equilibrium is (Launch, Discount), with payoffs (4, 3). Neither player can improve their payoff by changing their own action alone. This check avoids the common mistake of treating a best response by just one player as sufficient.
Worked Example: Two Equilibria in a Three-Column Matrix
A hypothetical retailer (row player) chooses one of two operating policies, A or B. A supplier (column player) chooses one of three contract terms, L, M, or R. Payoffs are in thousands of dollars and are ordered retailer first, supplier second. Find all pure-strategy equilibria.
| L | M | R | |
|---|---|---|---|
| A | (4, 1) | (2, 2) | (1, 3) |
| B | (2, 1) | (5, 4) | (0, 2) |
Method. Mark the retailer’s best response in each column and the supplier’s best response in each row. Keeping the two payoff coordinates separate prevents a frequent source of errors.
Retailer’s comparisons by column. In column L, 4 for A exceeds 2 for B, so A is the best response. In column M, 5 for B exceeds 2 for A, so B is the best response. In column R, 1 for A exceeds 0 for B, so A is the best response.
Supplier’s comparisons by row. In row A, the supplier’s payoffs are 1 at L, 2 at M, and 3 at R, so R is the best response. In row B, they are 1 at L, 4 at M, and 2 at R, so M is the best response.
The cell (A, R) has both best-response marks: against R, the retailer prefers A because 1 exceeds 0; against A, the supplier prefers R because 3 exceeds 2 and 1. Its payoffs are (1, 3). The cell (B, M) also has both marks: against M, the retailer prefers B because 5 exceeds 2; against B, the supplier prefers M because 4 exceeds 2 and 1. Its payoffs are (5, 4).
Answer and check. The two pure-strategy equilibria are (A, R), with payoffs (1, 3), and (B, M), with payoffs (5, 4). Checking both player comparisons in each candidate cell is essential: overlooking either comparison at (B, M) would incorrectly omit an equilibrium.
Worked Example: No Pure-Strategy Equilibrium
Two hypothetical producers choose High or Low output simultaneously. The first payoff in each cell belongs to the row producer, and the second belongs to the column producer. Find every pure-strategy equilibrium, if one exists.
| Column: High | Column: Low | |
|---|---|---|
| Row: High | (2, 1) | (0, 2) |
| Row: Low | (1, 2) | (2, 0) |
Method. Compare best responses at every fixed action of the other producer. Checking all four cells is a useful safeguard against concluding that an equilibrium exists merely because one player is responding optimally.
If the column producer chooses High, the row producer prefers High because 2 exceeds 1. If the column producer chooses Low, the row producer prefers Low because 2 exceeds 0. For the column producer, when the row producer chooses High, Low is preferred because 2 exceeds 1. When the row producer chooses Low, High is preferred because 2 exceeds 0.
The four cells can now be checked. At (High, High), the row producer is responding optimally but the column producer prefers Low. At (High, Low), the column producer is responding optimally but the row producer prefers Low. At (Low, High), the column producer is responding optimally but the row producer prefers High. At (Low, Low), the row producer is responding optimally but the column producer prefers High.
Answer and check. There is no pure-strategy equilibrium: every cell gives at least one producer a strictly profitable unilateral change. The conclusion is limited to pure strategies; it does not mean that the game has no equilibrium of any kind.
Ties, Completeness, and Common Traps
A best response need not be unique. If two actions yield the same highest payoff against the other player’s fixed action, both are best responses. Therefore, a tied maximum must be marked in every cell where it occurs. For instance, if the row player receives 3 from either row when the column player chooses M, both rows are best responses to M. A cell involving either tied row may qualify as an equilibrium, but only if the column player is also best responding there.
A practical way to ensure completeness is to count the comparisons: with \(m\) rows and \(n\) columns, inspect the row player’s payoffs in each of the \(n\) columns, and the column player’s payoffs in each of the \(m\) rows. Then test every cell with both marks. Do not stop after finding one equilibrium, because another cell may also contain mutual best responses.
- Using the wrong payoff coordinate. For the row player, compare first payoffs down a column. For the column player, compare second payoffs across a row.
- Holding the wrong action fixed. A player’s best response is evaluated while the opponent’s action stays unchanged; a comparison across unrelated cells does not establish a deviation.
- Requiring a strict advantage. A tied maximum is still a best response. An equilibrium does not require each action to be the player’s unique favorite.
- Marking only one player’s best responses. A cell with one mark may be stable for one player but is not a Nash equilibrium unless both players are best responding.
- Reporting only the first equilibrium found. A complete answer reports every cell with both marks, or explicitly states that none exists.
The equilibrium test concerns incentives to change one player’s action while the other player’s action is fixed. It does not by itself rank equilibria by social desirability, explain how players coordinate on an outcome, or establish what happens when choices occur in sequence. Those are distinct questions. For the present task, the decisive test is simply whether either player can gain from a unilateral deviation.
Check Your Understanding
For each matrix, identify every pure-strategy equilibrium and report the payoff pair in row-player, column-player order.
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(Answer: (U, X) with payoffs (3, 2), and (D, Y) with payoffs (4, 3).)X Y U (3, 2) (1, 0) D (2, 1) (4, 3) -
(Answer: (A, X) with payoffs (2, 1).)X Y A (2, 1) (2, 0) B (1, 1) (0, 2) -
(Answer: (Up, Left) with payoffs (2, 2).)Left Right Up (2, 2) (1, 1) Down (2, 0) (0, 2) -
(Answer: No pure-strategy equilibrium.)High Low High (2, 1) (0, 2) Low (1, 2) (2, 0) -
(Answer: (B, M) with payoffs (4, 3).)L M R A (3, 2) (1, 0) (0, 1) B (2, 1) (4, 3) (1, 2)