Game Theory and Strategic Choices – Chapter 18

Introduction to Game Theory and Strategic Decisions

• Game theory = science of making good decisions in situations with strategic interaction.
• Strategic interaction: your best choice depends on what others choose and vice-versa (interdependence principle).
• Applies to both competition and collaboration.
• Chapter roadmap:
– How to Think Strategically.
– Prisoner’s Dilemma & Cooperation Challenges.
– Multiple Equilibria & Coordination Problems.
– Advanced Strategy: First- & Second-Mover Advantages.
– Advanced Strategy: Repeated Games & Punishments.

Four-Step Recipe for Strategic Thinking

• 1 Consider all possible outcomes (list every combination of players’ choices in a payoff table).
• 2 Think about the “what-ifs” separately (avoid the infinite “I think they think…” loop; evaluate scenarios one at a time).
• 3 Identify your best response in each scenario (highest payoff given the rival’s move).
• 4 Put yourself in the other player’s shoes (redo Steps 1–3 from their perspective to predict their move).

Prisoner’s Dilemma & the Challenge of Cooperation

• Thought experiment: 2 suspects face choices Deny or Confess with payoffs (years in prison):
– Both deny \Rightarrow (-1,-1).
– One confesses, other denies \Rightarrow (0,-10) / (-10,0).
– Both confess \Rightarrow (-2,-2).
• Analysis with the 4 steps:
– Both players’ best response in every “what-if” is Confess.
– Two check-marks in (Confess, Confess) cell ⇒ Nash equilibrium.
• Key insights:
– Defecting (confessing) is a dominant strategy.
– Nash equilibrium ≠ socially optimal (joint optimum is both deny ⇒ only 1 year each).

Key Concepts

• Nash equilibrium: each player’s choice is a best response to others.
• Dominant strategy: best response regardless of rival’s action.
• Failure of cooperation arises from temptation to defect.

Cola Advertising Example (Real-world Prisoner’s Dilemma)

• Facts: Coke & Pepsi each spend \$1\text{ b} on ads, earn \$1\text{ b} profit.
• “Deal” to abolish ads would save \$1\text{ b} ⇒ profits \$2\text{ b} each.
• If one defects while other cooperates ⇒ defector earns \$3\text{ b}, cooperator \$0.
• Payoff table yields dominant strategy “Defect” for both ⇒ Nash equilibrium \$1\text{ b} each (status quo).
• Social optimum \$2\text{ b} each unattained because agreement is not credible.

Markets Can Deliver Bad Outcomes

• Competitive markets are efficient only in perfectly competitive settings without strategic interdependence.
• Prisoner’s-Dilemma settings show free markets may fail to maximize surplus.

Multiple Equilibria & Coordination Games

• Coordination game: players share common interest in matching choices (complementarity).
• Difficulty: multiple Nash equilibria; failure to coordinate yields off-equilibrium losses.

Battle of the Sexes Example

• Payoffs (Man, Woman):
– Match/Match =(10,7).
– Ballet/Ballet =(7,10).
– Off-diagonal =(0,0).
• Two equilibria: (Match,Match) & (Ballet,Ballet); no dominant strategies.

Business Hours Example

• Store wants to open when customers shop; customers shop when store open. Early & Late are both equilibria.

Anti-coordination Games

• Best response is to do the opposite of rival.
• Example: Phone-tag after dropped call – equilibria (You call, They wait) and (You wait, They call).
• Example: Traffic routes – equilibria (You highway, Others backroads) & vice versa.

Good vs. Bad Equilibria (Booms & Busts)

• Boom: Firms hire ↑ ⇒ workers spend ↑ ⇒ firms revenue ↑.
• Bust: Firms hire ↓ ⇒ workers spend ↓ ⇒ revenue ↓.
• Need policies (e.g., government stimulus) to coordinate on the good equilibrium.

Solving Coordination Problems

• Solution 1 Communication – viable when interests align.
• Solution 2 Focal points, culture, norms – external cues (e.g., meet at noon).
• Solution 3 Laws & regulations – enforce a single equilibrium (drive on right side).

Advanced Strategy: Sequential Games & Move Advantages

• Sequential game: players move in order; visibility of prior moves matters.
• Represented by a game tree (trunk → branches → leaves).
• Use backward induction: “look forward & reason backward.”

Airline Scheduling Example

• Simultaneous game equilibrium: both run 2 flights ⇒ \$30m each.
• Sequential version: American moves first; choosing 3 flights forces United to 1 ⇒ American \$45m (first-mover advantage).

First-Mover Advantage

• Gain from committing early to an aggressive action that constrains rival.
• Requires credible commitment (e.g., buy aircraft, lease gates).

Second-Mover Advantage

• Value of flexibility; adapt after observing rival.
• Examples: under-cut pricing, niche product positioning, cake-cutting (“they cut, you pick”).

Repeated Games & Punishments

• One-shot game: interacts once ⇒ classic Prisoner’s outcome.
• Repeated game: same rivals, same payoffs across periods ⇒ today’s decision affects future.
– Finitely repeated (known N periods).
– Indefinitely repeated (unknown horizon).

Finitely Repeated Prisoner’s Dilemma

• Backward induction: final round is one-shot ⇒ both defect; unravels backward ⇒ defect every round.

Indefinitely Repeated Prisoner’s Dilemma

• No known last round ⇒ cooperation can be sustained with threat of punishment.
• Strategic plan = list of contingent actions for every history.
• Grim Trigger Strategy:
– Cooperate as long as everyone has cooperated in past.
– Upon any defection, switch to defection forever.
• Cooperation viable if present value of future profits outweighs one-time gain from cheating:
– Let G = one-time gain from defection, M = per-period cooperative profit, \delta = discount factor.
– Cooperate if M + \delta M + \delta^2 M + \dots \ge G ⇒ \frac{M}{1-\delta} \ge G.

CVS–Walgreens Concept Check

• Under Grim Trigger, two equilibria: (Cooperate,Cooperate) yielding \$16b vs \$12b per period; (Defect,Defect) yields 0 each.
• (Cooperate,Cooperate) is self-enforcing when future is valued highly (large \delta).

Summary of Key Take-Aways

• Strategic thinking: enumerate outcomes, analyze “what-ifs,” pick best response, empathize with rival.
• Nash equilibrium: mutual best responses; may be inefficient.
• Prisoner’s Dilemma: dominant strategies produce non-cooperative equilibria.
• Coordination/anti-coordination: multiple equilibria ⇒ need communication, focal points, or rules.
• Sequential play: timing creates first- or second-mover advantages; analyze with game trees & backward induction.
• Repeated interactions: future punishments (Grim Trigger) can sustain cooperation in indefinite horizon games.
• Markets with strategic interaction may fail to maximize surplus ⇒ policy or institutional design can improve outcomes.