Why Strategic-Form Representation Matters
In an oligopoly, a firm’s outcome can depend not only on its own choice but also on the choices of other firms. A strategic-form game provides a compact way to record that dependence. It identifies who chooses, what each player can choose, and the payoff each receives for every combination of choices. This representation is useful before analyzing which choices are attractive or what outcomes may result.
This tutorial focuses on representing a simultaneous-move situation, not on finding an equilibrium. The concepts and methods here are aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
Players, Actions, Profiles, and Payoffs
A player is a decision-maker whose choices can affect outcomes. A player may be an individual, a firm, a government, or another decision-making unit. The players need not be people; for example, two competing firms can be the players in a model of price or output decisions.
An action is a choice available to a player in the situation being modeled. If a firm can select either a low or a high output, those are its actions. The list of actions is a simplification of the decision problem: it should include the choices the model is intended to compare, and it should be stated clearly.
A strategy profile combines one action from each player’s action set. With two players, if Player 1 has actions \(L\) and \(H\), and Player 2 has actions \(L\) and \(H\), there are four profiles: \((L,L)\), \((L,H)\), \((H,L)\), and \((H,H)\). In this one-shot simultaneous setting, the chosen action is also the player’s strategy. In more complex settings, a strategy can specify what a player would do in different circumstances; that distinction is not needed to construct the simple matrices considered here.
A payoff is a numerical measure of an outcome for a particular player. Depending on the model, it may be profit, revenue, cost (where a lower value is better), or a more general measure of benefit. A payoff must be interpreted in context: the number alone does not say what has been measured or in what units.
A payoff matrix displays the same information in a table. Rows usually list Player 1’s actions and columns list Player 2’s actions. Each cell contains the payoff pair for the profile formed by that row and column. The order of the pair must stay consistent across the entire matrix; it does not switch to put the row player’s payoff first and the column player’s payoff second.
Worked Example: Calculating Payoffs from an Output Model
Worked Example: Calculating Payoffs from an Output Model
Two firms simultaneously choose either 10 or 20 units of output. Market price is \(p=100-Q\), where \(Q\) is total output, and each firm has constant marginal cost of $20 per unit and no fixed cost. Construct the payoff matrix, with payoffs in dollars and Firm 1’s payoff first.
Method. For each profile, add the firms’ quantities to obtain \(Q\), calculate the market price, and then calculate each firm’s profit as \((p-20)q_i\). This avoids assuming that each firm faces a different price or confusing total-market output with an individual firm’s output.
Both firms choose 10. Total output is \(Q=10+10=20\), so price is \(p=100-20=80\). The margin above marginal cost is \(80-20=60\) dollars per unit. Each firm’s profit is \(60(10)=600\) dollars, giving \((600,600)\).
Firm 1 chooses 10 and Firm 2 chooses 20. Total output is \(Q=10+20=30\), so price is \(p=100-30=70\). The margin is \(70-20=50\) dollars per unit. Firm 1 earns \(50(10)=500\) dollars and Firm 2 earns \(50(20)=1{,}000\) dollars. The payoff pair is \((500,1{,}000)\).
Firm 1 chooses 20 and Firm 2 chooses 10. Total output remains \(Q=20+10=30\), so price is again \(70\) and the margin is \(50\). Firm 1 earns \(50(20)=1{,}000\) dollars and Firm 2 earns \(50(10)=500\) dollars. The payoff pair is \((1{,}000,500)\).
Both firms choose 20. Total output is \(Q=20+20=40\), so price is \(p=100-40=60\). The margin is \(60-20=40\) dollars per unit. Each firm earns \(40(20)=800\) dollars, giving \((800,800)\).
The completed matrix uses rows for Firm 1 and columns for Firm 2. Every entry is in dollars, and Firm 1’s profit is always listed first.
| Firm 1 \ Firm 2 | 10 units | 20 units |
|---|---|---|
| 10 units | (600, 600) | (500, 1,000) |
| 20 units | (1,000, 500) | (800, 800) |
Check. In each cell, the two firm outputs add to the stated total output, and both firms face the same market price. For example, at \((10,20)\), the profits sum to $1,500, equal to the market margin of $50 times total sales of 30 units.
How to Construct and Check a Payoff Matrix
A matrix is not just a table of numbers: its labels and conventions are part of the model. Before calculating entries, identify the players and specify each one’s actions. Then define the payoff measure. For a profit model, for example, state whether the payoff is profit per period, total profit, or profit in thousands of dollars. Finally, evaluate the payoff for each profile and place the ordered payoff pair in the corresponding cell.
The reason a cell contains a pair is that one action profile produces an outcome for each player. For a two-player game, define the payoff functions \(\pi_1\) and \(\pi_2\). Once the row action \(a_1\) and column action \(a_2\) are selected, the cell records both functions evaluated at that same profile:
This also explains why the payoff order cannot be changed from cell to cell: \(\pi_1\) always refers to Player 1, and \(\pi_2\) always refers to Player 2. The matrix records consequences of profiles; it does not, by itself, tell us which profile will occur.
State who makes decisions and list the available choices for each player. If the game is simultaneous, make clear that the players choose without observing the other player’s current choice.
For example, specify that a payoff is net profit in thousands of dollars for one year. Do not leave a reader to infer whether a number is a cost, a profit, or a probability.
For each row-column combination, calculate the outcome for both players using the same assumptions and the same profile.
Put one player’s actions on rows and the other’s on columns, maintain the stated payoff order, and check that the units and arithmetic agree in every cell.
Worked Example: Translating Advertising Choices
Worked Example: Translating Advertising Choices
Two local firms simultaneously choose whether to advertise. Their payoffs are net profits in thousands of dollars for the period. Without advertising by either firm, each earns $40,000. If only one firm advertises, its additional sales contribute $20,000 before an $8,000 advertising cost, while its rival loses $10,000 in profit. If both advertise, each gains $6,000 in sales contribution before paying its own $8,000 advertising cost. Construct the matrix in thousands of dollars, with Firm A’s payoff first.
Method. Start with each firm’s $40,000 baseline, then add the effects that apply at the action profile. Convert to thousands of dollars as stated. A common mistake is to report values such as 52 as dollars rather than as thousands of dollars.
Neither advertises. Each earns \(40\) thousand dollars, so the payoff is \((40,40)\).
Only Firm A advertises. Firm A’s net addition is \(20-8=12\) thousand dollars, giving \(40+12=52\). Firm B loses \(10\) thousand dollars, giving \(40-10=30\). The payoff is \((52,30)\).
Only Firm B advertises. By the same stated effects, Firm A earns \(40-10=30\) and Firm B earns \(40+20-8=52\). The payoff is \((30,52)\).
Both advertise. Each earns \(40+6-8=38\) thousand dollars, so the payoff is \((38,38)\).
| Firm A \ Firm B | Do not advertise | Advertise |
|---|---|---|
| Do not advertise | (40, 40) | (30, 52) |
| Advertise | (52, 30) | (38, 38) |
Check. At the one-advertiser profiles, the advertising firm’s $12,000 net gain and the other firm’s $10,000 loss are both applied to the $40,000 baseline. All four entries are in thousands of dollars, not dollars.
Worked Example: Representing an Investment Decision
Worked Example: Representing an Investment Decision
Two competing service providers simultaneously choose whether to upgrade a shared customer-facing system. Payoffs are net operating profits in thousands of dollars. If neither upgrades, each earns 50. If both upgrade, each receives an additional 20 from expanded demand and pays an upgrade cost of 12. If only one upgrades, that provider receives an additional 24 and pays 12, while the provider that does not upgrade loses 10 relative to its baseline. Construct the matrix, with Provider 1’s payoff first.
Method. Treat each profile separately and apply only the effects specified for that profile. The payoff is a net amount, so the upgrade cost must be subtracted from the upgrading provider’s baseline-plus-benefit total.
Neither upgrades. The payoff is \((50,50)\).
Both upgrade. Each payoff is \(50+20-12=58\), so the pair is \((58,58)\).
Only Provider 1 upgrades. Provider 1 earns \(50+24-12=62\). Provider 2 earns \(50-10=40\). The pair is \((62,40)\).
Only Provider 2 upgrades. Provider 1 earns \(50-10=40\), and Provider 2 earns \(50+24-12=62\). The pair is \((40,62)\).
| Provider 1 \ Provider 2 | Do not upgrade | Upgrade |
|---|---|---|
| Do not upgrade | (50, 50) | (40, 62) |
| Upgrade | (62, 40) | (58, 58) |
Check. The two cells in which just one provider upgrades are mirror images because the providers have the same baseline and the same specified effects. The order of the pair still follows the fixed convention: Provider 1 first, Provider 2 second.
Common Matrix Errors and Why the Representation Matters
A payoff matrix makes strategic interdependence visible: each player’s payoff can vary with the full action profile. This is why a firm’s profit cannot always be calculated from its own action alone. In the output example, the price used to calculate each firm’s profit depended on total output, which included both firms’ choices.
- Reversing payoff order. If rows are Firm 1 and columns are Firm 2, entries should still list Firm 1 first. Do not reverse the pair in cells where Firm 2’s action is on the row or where its profit is larger.
- Mixing units. A payoff of 52 in a matrix defined in thousands of dollars means $52,000. It does not mean $52. Write the unit in the problem statement or directly above the table.
- Confusing an action with a profile. “Advertise” is one firm’s action. “Firm A advertises and Firm B does not” is a two-player strategy profile.
- Leaving out an action combination. A complete matrix has a cell for every row-column combination. Each cell must give payoffs for both players.
- Calling a payoff a prediction. A matrix lists the payoffs associated with each possible profile. It does not establish which profile players will choose.
Check Your Understanding
For each problem, identify the relevant profiles and check that the payoff order and units are consistent.
- Two firms choose quantities of 5 or 15 units. Inverse demand is \(p=90-Q\), where \(Q\) is total output, and each firm has marginal cost of $30 per unit with no fixed cost. Find the payoff pair in dollars when Firm 1 chooses 5 and Firm 2 chooses 15. (Answer: (200, 600) dollars)
- Two businesses choose whether to advertise. Each earns $25,000 if neither advertises. If only one advertises, it gains $14,000 before its $6,000 advertising cost, and the nonadvertiser loses $7,000. If both advertise, each gains $5,000 before its own $6,000 cost. Give the four payoff pairs in thousands of dollars, listing Business A first. (Answer: Neither: (25, 25); A only: (33, 18); B only: (18, 33); both: (24, 24), all in thousands of dollars)
- Two providers each earn 8 thousand dollars at baseline. If both select a shared standard, each receives a 5-thousand-dollar benefit. If exactly one selects it, that provider receives a 3-thousand-dollar benefit and the other loses 2 thousand dollars. Neither selecting it has no effect. Give the matrix entries, listing Provider 1 first. (Answer: Both select: (13, 13); only Provider 1: (11, 6); only Provider 2: (6, 11); neither: (8, 8), all in thousands of dollars)
- Two firms have actions \(L\) and \(H\). Rows represent Firm A, columns represent Firm B, and each cell lists Firm A’s profit first in dollars. The cell in row \(H\), column \(L\) contains \((720, 410)\). State the action profile and identify each firm’s payoff. (Answer: The profile is (Firm A chooses H, Firm B chooses L); Firm A earns $720 and Firm B earns $410)
- A two-player simultaneous game gives Player 1 three actions and Player 2 two actions. How many payoff pairs are required for a complete matrix, and what does each pair represent? (Answer: Six payoff pairs; each gives both players’ payoffs for one of the six action profiles)