Why Careful Game Analysis Matters
Strategic-game errors often arise from applying a valid idea in the wrong way: comparing the wrong player’s payoff, calling an action “dominant” after checking only one opponent choice, or treating a socially better outcome as if no player could profitably depart from it. In sequential games, a further mistake is to accept a threat without asking whether the player making it would actually carry it out. These checks are aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
The reliable method is to identify the players and their available strategies, read payoffs in the stated order, find best responses, and then check equilibrium conditions. If decisions occur in sequence, also examine incentives at each later decision point before inferring earlier choices.
Best Responses Are Not the Same as Dominant Strategies
A best response is conditional: it answers, “What should this player choose given the other player’s current action?” A dominant strategy is stronger: it must remain a best response no matter which action the other player chooses. Finding one best response does not establish dominance. To test for a dominant strategy, compare the player’s payoffs across their own actions for every possible action of the opponent.
Worked Example: Checking Dominance and Reading Payoff Order
Two firms choose whether to upgrade a service platform. Firm A chooses Upgrade or Maintain, and Firm B chooses Promote or Stay Quiet. Payoffs are listed in the order (Firm A, Firm B). Find each firm’s best responses, identify any dominant strategy, and determine the pure-strategy Nash equilibrium.
| Firm A / Firm B | Promote | Stay Quiet |
|---|---|---|
| Upgrade | (4, 3) | (1, 5) |
| Maintain | (3, 2) | (2, 4) |
Step 1: Find Firm A’s best responses. If B Promotes, A compares 4 from Upgrade with 3 from Maintain, so Upgrade is the best response. If B Stays Quiet, A compares 1 with 2, so Maintain is the best response. A’s best response changes with B’s action; neither action is dominant.
Step 2: Find Firm B’s best responses. If A Upgrades, B compares its own payoffs, 3 from Promote and 5 from Stay Quiet; Stay Quiet is better. If A Maintains, B compares 2 with 4; again, Stay Quiet is better. Therefore Stay Quiet is strictly dominant for B.
Step 3: Find the equilibrium. The only cell where both players are choosing best responses to one another is (Maintain, Stay Quiet), with payoffs (2, 4). Neither player can improve by changing their own action alone.
Common mistake and correction. If B’s payoff is mistakenly read as the first number, then at (Upgrade, Promote) one might compare 4 with 1 and wrongly conclude that B prefers Promote. The stated payoff order is (A, B), so B must compare the second numbers: 3 with 5. The correct best response is Stay Quiet. Checking both of B’s rows confirms it is dominant, and the equilibrium is (Maintain, Stay Quiet).
Why Pareto Superiority Does Not Establish Equilibrium
An outcome can give every player a higher payoff than another outcome and still fail to be a Nash equilibrium. The reason is that Pareto comparisons ask whether players are collectively better off across two outcomes, whereas equilibrium asks whether any one player can gain by changing their action while others hold theirs fixed. These are different tests, so one cannot substitute for the other.
Worked Example: A Better Joint Outcome That Is Not Stable
Two service providers independently choose High Effort or Low Effort. Their payoffs, in order (Provider 1, Provider 2), are shown below. Determine whether High Effort by both is a Nash equilibrium, compare it with Low Effort by both, and find the equilibrium.
| Provider 1 / Provider 2 | High Effort | Low Effort |
|---|---|---|
| High Effort | (4, 4) | (1, 6) |
| Low Effort | (6, 1) | (2, 2) |
Step 1: Compare the two joint outcomes. At (High Effort, High Effort), both receive 4. At (Low Effort, Low Effort), both receive 2. Since 4 is greater than 2 for each provider, High Effort by both is a Pareto improvement over Low Effort by both.
Step 2: Check whether High Effort by both is an equilibrium. If Provider 2 chooses High Effort, Provider 1 receives 4 from High Effort but 6 from Low Effort. Provider 1 can profit by switching. The same comparison applies to Provider 2 by symmetry: switching alone raises its payoff from 4 to 6. Thus (High Effort, High Effort) is not a Nash equilibrium.
Step 3: Check Low Effort by both. If the other provider chooses Low Effort, a provider receives 1 from High Effort and 2 from Low Effort. Neither wants to switch. Therefore (Low Effort, Low Effort) is a Nash equilibrium. In fact, Low Effort is strictly dominant for each provider: against High Effort, 6 exceeds 4; against Low Effort, 2 exceeds 1.
Common mistake and correction. “Both players prefer the High Effort outcome” does not imply “neither player wants to deviate from it.” The relevant equilibrium comparison holds the other player’s action fixed. Here, a unilateral switch from High to Low raises the deviator’s payoff from 4 to 6, so the Pareto-superior outcome is not stable.
This condition follows directly from the definition of a profitable unilateral deviation. If some alternative \(s_i'\) gave a higher payoff while \(s_{-i}\) stayed fixed, player \(i\) would have an incentive to change strategies, and the profile would not be an equilibrium. Conversely, if the inequality holds for every player and every alternative, no single player can gain by changing strategy alone. This is why equilibrium checks require comparisons of each player’s own payoff, not just an assessment of whether the outcome seems desirable.
Payoff Order: Always Identify Whose Number You Are Comparing
A payoff pair is meaningful only with its order specified. If a cell is written as \((a,b)\), the analyst must know which player receives \(a\) and which receives \(b\) before finding best responses. For a given player, compare only that player’s payoffs across the available actions. Do not compare one player’s payoff in one cell with the other player’s payoff in another.
A useful check is to label the payoff order in words before doing any calculations: for example, “(entrant, incumbent)” or “(Firm A, Firm B).” Then, when considering a deviation by one player, hold the other player’s action fixed and compare the correct coordinate. This simple discipline prevents errors in both simultaneous and sequential games.
Noncredible Threats in Sequential Games
A sequential game requires more than checking whether the players’ choices fit together along the realized path. A strategy in such a game specifies what a player would do at each decision point they may face, including points that are not reached. A threat is credible only if carrying it out would be optimal for the player when the relevant decision point is reached.
Worked Example: Testing an Incumbent’s Threat
A potential entrant chooses Enter or Stay Out. If it enters, an incumbent chooses Accommodate or Fight. If the entrant stays out, payoffs are (0, 5). If it enters and the incumbent accommodates, payoffs are (2, 2); if the incumbent fights, payoffs are (-3, -1). Payoffs are ordered (entrant, incumbent). Use backward induction to find the subgame-perfect outcome, and check whether a Nash equilibrium can rely on a threat to Fight.
Step 1: Solve the incumbent’s decision. After entry, the incumbent compares its payoff of 2 from Accommodate with -1 from Fight. Since \(2>-1\), it chooses Accommodate. A plan to Fight after entry is not credible: when the decision actually arises, the incumbent would prefer to accommodate.
Step 2: Predict the entrant’s choice. The entrant anticipates accommodation. Enter yields 2, while Stay Out yields 0. Since \(2>0\), the entrant chooses Enter. The subgame-perfect outcome is therefore Enter followed by Accommodate, with payoffs (2, 2).
Step 3: Check the apparent threat in strategic form. Consider the strategy profile Stay Out and Fight, where “Fight” describes what the incumbent would do if entry occurred. Given that plan, the entrant would receive 0 from staying out and -3 from entering, so Stay Out is its best response. If the entrant stays out, the incumbent receives 5 regardless of whether its unused response is Accommodate or Fight, so Fight is also a best response to Stay Out. Thus the profile can be a Nash equilibrium.
Step 4: Check credibility at the later decision. In the subgame after entry, Fight gives the incumbent -1 and Accommodate gives it 2. The incumbent would change its action to Accommodate, so the plan to Fight is not optimal in that subgame. The Nash equilibrium based on the threat is not subgame-perfect.
Common mistake and correction. It is incorrect to accept Stay Out and Fight as the sequential prediction simply because neither player gains by deviating from the full strategy profile as written. The off-path response must also be optimal when reached. Backward induction rules out Fight, after which the entrant compares 2 from Enter with 0 from Stay Out and enters.
A Reliable Error-Checking Routine
Before finalizing an analysis, work through the following sequence. It is especially useful when a question includes several plausible outcomes or a proposed threat.
Write down who receives each coordinate in every payoff pair before comparing outcomes.
For each opponent action, compare one player’s payoffs across that player’s own actions. Do this separately for every player.
Call an action dominant only if it is a best response for every possible combination of the other players’ actions.
At a proposed Nash equilibrium, verify that no player can improve by changing their own strategy alone.
Work backward from later decision points. Reject a threat if the player would choose differently when that decision point is reached.
Check Your Understanding
For each problem, keep player labels and unilateral-deviation comparisons explicit.
- Player A chooses X or Y and Player B chooses L or R. Payoffs, in order (A, B), are X-L: (3, 2), X-R: (1, 4), Y-L: (2, 1), and Y-R: (4, 3). Identify any dominant strategy and the pure-strategy Nash equilibrium. (Answer: B has the strictly dominant strategy R; the unique pure-strategy Nash equilibrium is (Y, R), with payoffs (4, 3).)
- Two operators choose Reliable or Cut Corners. Payoffs are (Reliable, Reliable): (7, 7), (Cut, Cut): (4, 4), (Cut, Reliable): (9, 1), and (Reliable, Cut): (1, 9), with payoffs ordered by operator. Is (Reliable, Reliable) a Nash equilibrium, and what is its Pareto comparison with (Cut, Cut)? (Answer: It is Pareto-superior to (Cut, Cut), but not a Nash equilibrium; each operator can switch alone from Reliable to Cut and increase its payoff from 7 to 9.)
- In a two-action game, the payoff pairs are Red-Left: (6, 1), Red-Right: (2, 5), Blue-Left: (4, 3), and Blue-Right: (3, 2), ordered (Row, Column). Find Column’s best response to each Row action and determine whether there is a pure-strategy Nash equilibrium. (Answer: Column’s best response to Red is Right and to Blue is Left; there is no pure-strategy Nash equilibrium.)
- An entrant chooses Enter or Stay Out. If it enters, an incumbent chooses Accommodate or Fight. Payoffs (entrant, incumbent) are (3, 1) after Enter and Accommodate, (-2, -1) after Enter and Fight, and (0, 4) after Stay Out. Find the subgame-perfect outcome. (Answer: The incumbent accommodates because \(1>-1\); anticipating this, the entrant enters because \(3>0\).)
- In a sequential entry game, Enter followed by Accommodate gives payoffs (1, 2), Enter followed by Fight gives (-4, -2), and Stay Out gives (0, 6), ordered (entrant, incumbent). Explain why Stay Out with a plan to Fight may be a Nash equilibrium but is not subgame-perfect, and identify the subgame-perfect outcome. (Answer: Stay Out/Fight can be Nash because entry against Fight gives -4 and the incumbent’s unused response does not affect its payoff of 6; it is not subgame-perfect because the incumbent would choose Accommodate after entry, where 2 exceeds -2. The subgame-perfect outcome is Enter/Accommodate.)