Introduction: Match the Tool to the Strategic Problem
Strategic-choice questions often present a payoff matrix or a sequence of decisions and ask for an outcome. The calculation is only one part of the task: first identify whether the situation calls for dominance, best responses, mixed strategies, repeated-game incentives, or backward induction. Applying a correct method to the wrong kind of game can produce a plausible but incorrect answer.
In a payoff matrix, each cell lists the payoffs to the row player and column player, in that order. Unless the problem says otherwise, assume each player knows the payoffs, chooses rationally to maximize their own payoff, and understands that the other player does too. In a simultaneous game, neither player observes the other’s current action before choosing. In a sequential game, later players observe earlier actions as specified by the game tree. These assumptions matter: a hidden simultaneous choice is not analyzed in the same way as an observed sequential choice.
Worked Example 1: Eliminate a Dominated Action
Worked Example: Choosing Whether to Launch
Two firms choose simultaneously. Firm A chooses Launch or Wait; Firm B chooses Cut Price or Hold Price. Payoffs are in thousands of dollars, with Firm A’s payoff first. Find the Nash equilibrium using dominance and best responses.
| Firm A | Cut Price | Hold Price |
|---|---|---|
| Launch | (3, 2) | (6, 5) |
| Wait | (4, 3) | (5, 4) |
Method. Compare each player’s payoffs across their own actions, holding the other player’s action fixed. Firm B’s payoff from Hold Price is 5 rather than 2 if Firm A Launches, and 4 rather than 3 if Firm A Waits. Thus Hold Price strictly dominates Cut Price.
Once Firm B’s dominated action is eliminated, compare Firm A’s payoffs when Firm B holds price. Launch gives Firm A 6, while Wait gives 5. Firm A therefore chooses Launch. The resulting strategy profile is (Launch, Hold Price), with payoffs (6, 5).
Check. Given Hold Price, Firm A cannot gain by switching to Wait, because 5 is less than 6. Given Launch, Firm B cannot gain by switching to Cut Price, because 2 is less than 5. Both actions are best responses, so this is a Nash equilibrium.
Common mistake avoided: Do not call an action dominated merely because it earns less in one cell. Strict dominance requires a lower payoff in every relevant comparison. Here, Hold Price beats Cut Price against both Launch and Wait.
Mixed Strategies: Make the Opponent Indifferent
If no pure-action profile is stable, a player may randomize. A mixed strategy assigns probabilities to available actions. In a two-action game, an opponent’s mixing probability makes a player willing to choose either action only when the player’s expected payoffs from those actions are equal. This indifference condition is the key calculation; it does not mean that the player is indifferent between the opponent’s actions.
Why does this condition work? Suppose one action gave a player a strictly greater expected payoff than another action that the player used with positive probability. The player could shift all probability to the better action and increase expected payoff. That contradicts equilibrium. Therefore, any actions used with positive probability must give equal expected payoffs.
Worked Example: Solving a Mixed-Strategy Equilibrium
A supplier chooses Up or Down, while a buyer chooses Left or Right. Payoffs are (supplier, buyer). Find the mixed-strategy equilibrium.
| Supplier | Left | Right |
|---|---|---|
| Up | (4, 1) | (0, 3) |
| Down | (1, 3) | (3, 0) |
Method. Let \(q\) be the probability that the buyer chooses Left. The supplier’s expected payoff from Up is \(4q+0(1-q)=4q\). From Down it is \(1q+3(1-q)=3-2q\). Indifference requires
Now let \(p\) be the probability that the supplier chooses Up. The buyer’s expected payoff from Left is \(1p+3(1-p)=3-2p\). From Right it is \(3p+0(1-p)=3p\). Thus
The equilibrium is for the supplier to choose Up with probability \(3/5\) and Down with probability \(2/5\), and for the buyer to choose Left with probability \(1/2\) and Right with probability \(1/2\). The supplier’s expected payoff is \(4(1/2)=2\). The buyer’s expected payoff is \(3-2(3/5)=9/5=1.8\); choosing Right gives the same \(3(3/5)=1.8\).
Check and common mistake avoided. Both calculated probabilities are strictly between 0 and 1, as required for an interior mixed equilibrium. Do not use the supplier’s probability \(p\) to make the supplier indifferent: the opponent’s probability \(q\) determines the supplier’s expected payoffs.
Coordination: Several Equilibria Can Be Stable
In a coordination game, players’ payoffs depend on whether their choices align. There may be more than one pure Nash equilibrium, because each player’s best response can depend on which coordinated outcome the other player expects. A mixed equilibrium may also exist. Finding the equilibria identifies stable strategy profiles; it does not, by itself, determine which one players will select.
Worked Example: Selecting a Shared Operating Protocol
Two firms independently choose protocol X or protocol Y. Payoffs are in thousands of dollars. Find all pure equilibria and the interior mixed equilibrium.
| Firm A | Firm B: X | Firm B: Y |
|---|---|---|
| Firm A: X | (8, 7) | (1, 2) |
| Firm A: Y | (2, 1) | (6, 8) |
Pure equilibria. If Firm B chooses X, Firm A prefers X because 8 is greater than 2. If Firm A chooses X, Firm B prefers X because 7 is greater than 2. Therefore (X, X) is a Nash equilibrium. If Firm B chooses Y, Firm A prefers Y because 6 is greater than 1; if Firm A chooses Y, Firm B prefers Y because 8 is greater than 1. Thus (Y, Y) is also a Nash equilibrium. At either mismatched outcome, at least one firm can improve by switching to match the other.
Mixed equilibrium. Let \(q\) be the probability that Firm B chooses X. Firm A’s expected payoff from X is \(8q+1(1-q)=1+7q\); from Y it is \(2q+6(1-q)=6-4q\). Setting these equal gives \(1+7q=6-4q\), so \(q=5/11\).
Let \(p\) be the probability Firm A chooses X. Firm B’s expected payoff from X is \(7p+1(1-p)=1+6p\); from Y it is \(2p+8(1-p)=8-6p\). Equating them gives \(1+6p=8-6p\), so \(p=7/12\). The interior mixed equilibrium therefore has Firm A choosing X with probability \(7/12\) and Firm B choosing X with probability \(5/11\).
Check and common mistake avoided. Each probability lies between 0 and 1, and each was calculated from the other firm’s mixing probability. Do not report only one coordinated outcome: both pure equilibria satisfy the best-response test. Which outcome is selected depends on expectations or other features not specified in this game.
Repeated Cooperation: Compare Cooperation With Deviation
A repeated game can change incentives because a current action affects future payoffs. Consider an infinitely repeated game with the same stage-game payoffs each period and a common discount factor \(\delta\), where \(0\leq\delta<1\). A payoff one period in the future is worth \(\delta\) times as much as a current payoff. Under a grim-trigger strategy, a player cooperates while both players have always cooperated; after any defection, both choose the punishment action forever.
Worked Example: Testing a Grim-Trigger Agreement
Two firms have the following per-period payoffs: mutual cooperation gives each 4; a firm that defects while the other cooperates receives 7, while the cooperator receives 0; mutual defection gives each 2. The firms use grim trigger. Find the minimum discount factor that makes cooperation sustainable.
Method. If a firm cooperates forever, its present value is \(4+4\delta+4\delta^2+\cdots=4/(1-\delta)\). If it defects once while the other cooperates, it earns 7 immediately and then receives 2 each period under the permanent punishment. Its deviation value is \(7+2\delta+2\delta^2+\cdots=7+2\delta/(1-\delta)\).
Cooperation is at least as good as a one-time deviation when
Since \(1-\delta>0\), multiply both sides by \(1-\delta\): \(4\geq7(1-\delta)+2\delta=7-5\delta\). Hence \(5\delta\geq3\), or \(\delta\geq3/5\). At the threshold, both choices have the same value; above it, cooperation has the higher value.
Check and common mistake avoided. Under the punishment path, mutual defection is a stage-game Nash equilibrium: against defection, choosing defection gives 2 rather than 0. This makes the threatened continuation credible. Do not compare only the immediate cooperative payoff 4 with the deviation payoff 7; that omits the future reward for cooperation and the future punishment after deviation.
Sequential Games: Use Backward Induction to Test Threats
In a sequential game, a later player chooses after observing earlier actions. Backward induction solves the game from its final decisions: identify the best action at each final decision point, then use those choices to evaluate earlier decisions. A threat is not credible if carrying it out would be worse for the player when the relevant decision actually arrives.
Worked Example: Is a Price War a Credible Threat?
A new firm chooses whether to Enter or Stay Out. If it stays out, its payoff is 0 and the incumbent’s payoff is 5. If it enters, the incumbent then chooses Accommodate or Fight. Entry followed by accommodation gives payoffs (2, 3), and entry followed by fighting gives payoffs (-2, 1), with the entrant’s payoff first. Find the subgame-perfect outcome.
Method: backward induction. First consider the incumbent’s decision after entry. Accommodation gives the incumbent 3, while fighting gives 1. The incumbent therefore accommodates. Knowing this, the entrant compares Enter, which leads to a payoff of 2, with Stay Out, which gives 0. The entrant enters. The predicted outcome is Entry followed by Accommodation, with payoffs (2, 3).
Check and common mistake avoided. The incumbent’s announced threat to fight is not credible: after entry, the incumbent would choose accommodation because \(3>1\). The strategy profile must specify what the incumbent would do after entry, even if a different action is threatened in advance. Backward induction requires optimal choices at each decision point, not just an attractive prediction for the first mover.
A Practical Sequence for Exam Problems
The order of analysis helps avoid mixing tools or overlooking a condition. Use the following sequence, adapting it to the information and timing in the problem.
Record the players, available actions, payoff order, whether choices are simultaneous or sequential, and whether the interaction repeats.
Compare a player’s own payoffs across actions for every relevant opponent action. Mark best responses before naming a Nash equilibrium.
For an interior mixed equilibrium, equate the expected payoffs of the actions used with positive probability, then verify all probabilities are valid.
For sequential games, solve later decisions first. For repeated games, compare discounted continuation values as well as current payoffs.
A final equilibrium check should be stated in context: ask whether any player can improve by changing their own strategy while the others’ strategies remain fixed. In a repeated or sequential problem, also check whether the prescribed continuation actions are rational when those decision points occur.
Check Your Understanding
For each problem, identify the relevant strategic tool and verify the result using the players’ incentives.
- Firm A chooses Expand or Hold; Firm B chooses Cut or Maintain. Payoffs are (Expand, Cut) (2, 1), (Expand, Maintain) (6, 4), (Hold, Cut) (3, 2), and (Hold, Maintain) (4, 3). Find the equilibrium using dominance. (Answer: Maintain strictly dominates Cut; the equilibrium is (Expand, Maintain) with payoffs (6, 4).)
- In a two-action game, Row’s payoffs are (5, 0) for Up against Left or Right, and (1, 3) for Down against Left or Right, respectively. More explicitly, Row’s payoffs are Up-Left 5, Up-Right 0, Down-Left 1, Down-Right 3. Find the probability that Column chooses Left that makes Row indifferent. (Answer: \(3/7\).)
- Two firms choose A or B. Their payoffs are (A,A) (7, 6), (A,B) (1, 2), (B,A) (2, 1), and (B,B) (5, 8). Identify the pure Nash equilibria. (Answer: (A,A) and (B,B).)
- A repeated game gives each firm 5 per period for cooperation, 9 to a firm that defects against a cooperator, 0 to the cooperator facing a defection, and 1 each under mutual defection. Under grim trigger, find the minimum discount factor for cooperation to be sustainable. (Answer: \(\delta\geq1/2\).)
- An entrant chooses Enter or Stay Out. Staying out gives (0, 4). After entry, the incumbent chooses Accommodate, giving (3, 3), or Fight, giving (-1, 5), with entrant payoff first. Use backward induction to find the subgame-perfect outcome. (Answer: The incumbent fights after entry because \(5>3\); anticipating a payoff of -1 rather than 0, the entrant stays out.)