Game Theory: Payoff Matrix, Dominant Strategy and Nash Equilibrium

Two gas stations, Red and Blue, share a corner. If both keep their prices high, both earn more. So why do they end up cutting? This is Module 4.5A, Oligopoly and Game Theory: the payoff matrix. One small table will answer that.

An oligopoly is a market with a few firms behind high barriers to entry. With so few, each firm's profit depends on what the others do: they're interdependent. The smallest oligopoly has two firms, a duopoly, like Red and Blue. When Blue changes its price, Red's profit moves. Oligopoly is also inefficient: like a monopolist, an oligopolist prices above marginal cost.

Quick refresher from Modules 4.1 and 4.2: a firm with market power must cut its price to sell more, so MR is below price, and producing where MR = MC leaves price above MC. At a $4 price and $3 MC, some gallons worth more than they cost go unsold.

Game theory studies interdependence, in markets and beyond. A game is any situation where each player's payoff depends on their own choice and everyone else's. Every game has three parts. Players: Red and Blue. Strategies, each player's complete plan of action: here, High or Cut. Payoffs: each station's daily profit.

A payoff matrix, the game's normal form, shows the payoffs for every combination of strategies: Blue's choices in the rows, Red's in the columns. On the AP exam, it's always two players with two choices each. Each cell holds two numbers: Blue's daily profit first, then Red's. Exam matrices have no colors; released questions say whose number comes first, so read that line first. Both keep prices high: Blue earns $800 a day, Red $700. Both cut: Blue $500, Red $400. If Blue cuts while Red stays high, Blue steals customers: Blue $1,000, Red $200. Flip it, and Red cashes in: Blue $300, Red $900.

So what just happened? One: Blue's choices are the rows, Red's the columns. Two: in each cell, Blue's number comes first, Red's second. Three: four cells, one per pair of choices.

Now the key move: a best response, a player's most profitable choice given what the other player picks.

Quick refresher from Unit 1: judge a choice against the next best alternative, its opportunity cost. Offered a $60 shift or a $45 shift, take the $60; you give up the $45.

A best response is that same comparison. Say Red stays high: cover its Cut column. Blue's numbers in the High column: $800 for High, $1,000 for Cut. Blue's best response: cut. If Red cuts, slide the cover over: $300 for High, $500 for Cut. Cutting wins again. Red reads across a row, using the second numbers. Blue high: $700 for High, $900 for Cut. Blue cuts: $200 for High, $400 for Cut. Red cuts both times.

So what just happened? One: fix the rival's choice by covering a column or row. Two: compare only your own two payoffs. Three: circle the higher one. Both circled Cut twice.

Blue picked Cut both times. A strategy that pays more no matter what the other player does is a dominant strategy. Proving it takes two comparisons, one for each of Red's choices, using Blue's own payoffs. Red high: $1,000 beats $800. Red cuts: $500 beats $300. So yes, Cut is dominant for Blue. Same test for Red: Blue high, $900 beats $700; Blue cuts, $400 beats $200. Cut is dominant for Red too.

So what just happened? One: with Red high, Blue's Cut beat High, $1,000 to $800. Two: with Red cutting, $500 to $300. Three: two wins, so Cut is dominant, for Red too.

Exam trap: on a recent AP exam, fewer than half of students earned the dominant strategy point. One comparison isn't enough, and the wrong numbers fail too. "$500 beats $400"? That compares Blue with Red, inside one cell. "$800 beats $500"? That jumps columns, changing Red's choice too, and points the wrong way. Fix the rival's choice; compare your own two numbers.

So where do they end up? A Nash equilibrium is a cell where no player can raise their payoff by switching alone, given the other player's choice. Test Both Cut. If Blue alone switches to High, its profit falls from $500 to $300. If Red alone switches, it falls from $400 to $200. Neither gains, so Both Cut is a Nash equilibrium. Shortcut: a cell with two circles, one from each player, is a Nash equilibrium. Only Both Cut has two. Both High fails: Blue could switch to Cut and jump from $800 to $1,000. One profitable switch is enough for a no.

That answers the corner. Both High would pay $800 and $700, not $500 and $400. But whatever the rival does, each station earns more by cutting. So both cut.

So what just happened? One: from Both Cut, Blue's switch drops $500 to $300, Red's $400 to $200. Two: so it's the Nash equilibrium. Three: Both High pays more, but each gains by cutting.

Exam trap: "yes, both have dominant strategies" doesn't earn the point, and neither does one check. It needs both switches, each with its two numbers.

Not every game has a dominant strategy. Same corner, new choice: each station adds a car wash or a snack shop. Matching means splitting customers. Both car washes: Blue $200, Red $300. Blue wash, Red snacks: $600 and $500. Blue snacks, Red wash: $400 and $700. Both snack shops: $300 and $400. Exam trap: if Blue builds a car wash, what's Red's best response? Red's $700, under Wash, is the biggest number here, but that cell needs Blue to pick snacks. Cover the Snacks row: Red's own payoffs are $300 for a wash and $500 for snacks. Its best response: snacks.

Circle every best response. Blue: against a wash, snacks, $400 over $200; against snacks, a wash, $600 over $300. Blue's choice flips, so no dominant strategy. Red's flips too: against snacks, a wash, $700 over $400. Two cells hold two circles: Blue wash with Red snacks, and Blue snacks with Red wash. Two Nash equilibria, with no dominant strategy.

So what just happened? One: each station's best choice flips with its rival's, so no dominant strategy. Two: each does best being different. Three: two cells with two circles, two Nash equilibria.

Exam trap: "no dominant strategy, so no Nash equilibrium" is wrong, and so is naming one of two. On a recent AP exam, only about half named both.

Exam moment: two AP-style multiple-choice questions, then one free-response part.

Orion and Vega, the only two phone makers, each pick a selling point at once, without cooperating. The first entry in each cell is Orion's profit; the second is Vega's. Which firm has a dominant strategy, and what is it? A) Orion, Thin B) Orion, Tough C) Vega, Camera D) Vega, Battery E) Neither firm has one.

The answer is D. Whatever Orion picks, Vega's Battery pays more: $4 million against $2 million, then $8 million against $6 million. Orion has none: Thin wins against Camera, Tough against Battery. A grabs Orion's $9 million, the biggest number, after one comparison. B and C read the other firm's numbers. E changes Orion's choice inside Vega's comparisons.

Volt and Amp, the only two charger makers, each pick a Round or Flat plug at once, without cooperating. The first entry in each cell is Volt's profit; the second is Amp's. Identify all Nash equilibria. A) Only both Round B) Only both Flat C) Both Round, and both Flat D) Volt Round with Amp Flat, and the reverse E) None, as neither has a dominant strategy.

The answer is C. From Both Round, a lone switch drops Volt from $12 million to $3 million, Amp from $7 million to $5 million. From Both Flat: Volt $9 million to $4 million, Amp $11 million to $2 million. A and B each stop at one of the two. D names the mismatched cells, where a switch pays. E assumes an equilibrium needs a dominant strategy; it doesn't.

Free response. Peak and Core, the only two gyms in town, each choose Expand or Stay at once, without cooperating. The first entry in each cell is Peak's yearly profit; the second is Core's. Is Peak choosing Expand and Core choosing Stay a Nash equilibrium? Explain using numbers from the payoff matrix.

To earn the point: yes, plus both switches with numbers. If Peak switched to Stay, its profit would fall from $80,000 to $60,000. If Core switched to Expand, its profit would fall from $30,000 to $20,000. The costly mistake: checking only one firm, or "yes, because of a dominant strategy" with no switches. On a recent AP exam, fewer than one student in three earned this point.

So what earned the point? One: yes. Two: Peak switching to Stay drops $80,000 to $60,000. Three: Core switching to Expand drops $30,000 to $20,000.

Recap: a best response compares your own two payoffs, with the rival's choice fixed. A dominant strategy wins both comparisons. A Nash equilibrium survives both switch tests, and there can be two. So why can't Red and Blue just agree to keep prices high? Next: Module 4.5B, Oligopoly and Game Theory: collusion, changing payoffs and the prisoner's dilemma.