18.18 : Sequential Game: Backward Induction

Backward induction is a technique for solving sequential games. It involves analyzing the game starting from the end and working backwards to the beginning. This method helps players determine their best strategies by anticipating how others will react at each stage of the game, ultimately leading to the Nash equilibrium.

Imagine two beverage companies, FreshFizz and CoolBrew, deciding whether to enter a new market. FreshFizz moves first and must choose to enter or stay out. If FreshFizz enters, CoolBrew must decide whether to enter the market or remain out.

Using backward induction, the analysis begins with CoolBrew's options. If CoolBrew also enters, it earns a payoff of $900, while FreshFizz gets $100 due to increased competition. However, if CoolBrew stays out, it earns $0, and FreshFizz secures a payoff of $700 from the new market alone. Since $900 is better than $0, CoolBrew will enter the market if FreshFizz does.

Knowing this, FreshFizz looks at its initial choice. If it enters the market, it will ultimately earn only $100, as CoolBrew will enter, too. Alternatively, by staying out, FreshFizz avoids the competition and maintains a stable profit of $300 from its current operations. Understanding that entering leads to a much lower payoff, FreshFizz chooses to stay out.

Backward induction allows FreshFizz to anticipate CoolBrew's move and choose the strategy that maximizes its own payoff, leading both companies to the equilibrium point. This method is valuable for predicting outcomes in sequential games, ensuring that each player's actions are informed by what they know their competitor will rationally do.