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Key points
- Backward induction is a decision strategy that considers the end of a problem first.
- By reasoning backward in time, the decision-maker can arrive at an optimal choice for the beginning.
- This strategy can be used in situations where the outcome relies on a series of interdependent choices.
Have you ever played the 21 Game? It’s a deceivingly easy numbers game, which goes like this: Starting at 0, two players take turns adding 1, 2, or 3. For instance, the first player could start by adding 2, then the second player could continue by adding 3, then the first player could add 2, and so on. The game ends when the sum of the added numbers reaches (or surpasses) 21. The player, who is forced to make the final move and pass the number 21 loses the game.
Sound simple enough? What if I told you that it’s possible to win the 21 Game in a single move?
Imagine the first player has made their initial move and added 3 to the starting value of 0. Now it’s your turn and the number you add could determine the game’s overall winner. If you choose strategically, you could set yourself up for a sequence of optimal choices, ultimately forcing the other player to cross the game’s threshold of 21. The surprising solution lies in a decision-making approach called backward induction.
Backward Induction
The term backward induction was coined by scientists Von Neumann and Morgenstern from the field of game theory, although the idea dates back to much earlier. It describes a reasoning strategy that starts by considering the end of a problem and reasons backward in time to arrive at an optimal approach for the beginning.
Let’s illustrate this abstract process using the 21 Game. Imagine again that you are trying to decide on a move following the other player's initial choice of 3. Rather than only considering the immediate situation, backward induction reasoning would involve considering the game’s end first.
Both players want to avoid crossing the threshold of 21. This can be achieved by being the player, who reaches 20, because no matter the next person’s choice, it will invariably take them to 21 or higher.
But how can you ensure to be the player who calls 20? You need to take another step backward in your reasoning. The only certain way of getting to 20 is to reach the number 16 in your preceding move. This is because no matter the other person’s subsequent choice (adding 1, 2, or 3), the highest they could get to is 19, thus allowing you to bring it up to 20 on your following move.
Again, you’ve reached a new conundrum. How can you ensure to be the player who calls 16? It’s simple. You need to be the one to call 12, because whatever your opponent does, you can bring the total up to 16 on your following move. In this vein, you can work your way backward until the very start of the game.
Backward induction leaves you with a clear solution. The winning strategy is to reach a multiple of 4 with each move, guaranteeing that you can ultimately get to 20. Following the other player's initial choice of 3, your optimal response would therefore be to add 1 point, thus reaching a total of 4 and taking the upper hand in the forthcoming decision sequence.
If you don’t believe me, why don’t you try this strategy with a friend?
Backward Induction in Real Life
Backward induction is an approach that can be applied to repeated or sequential decision situations, where the outcome relies on a series of interdependent choices. It requires the skill to think ahead and take different perspectives. It has been studied extensively in laboratory experiments involving abstract decision scenarios and games.
In addition to helping you win the 21 Game, backward induction may come in handy during many real-life decision challenges. One example is presented in the sitcom "Friends," where the character Rachel Green uses a similar approach to achieve her life goal of becoming a mother.
Her 30th birthday party prompts Rachel to reflect on future plans and ambitions. One day, Rachel would like to be a mother of three children. Reasoning backward, she realises that the restraints of nature require her to have her first child before the age of 35. This means she’ll have to get pregnant by the age of 34.
Rachel considers marriage a prerequisite to having children and wants to be married for at least one year prior to getting pregnant, meaning she’d have to get married at 33. Assuming she’ll need 1.5 years to plan the wedding and another 1.5 years to get to know the guy before saying “yes,” she arrives at a difficult conclusion. This very moment, at the age of 30, is the time to get serious and meet her future husband. Only a few hours later, she heavy-heartedly breaks up with scooter-loving boyfriend Tag, who is six years her junior and not ready for commitment.
Rachel’s example illustrates how using a future goal as a reference point allows you to work your way backward in time and arrive at the most rational strategy for the present moment. Backward induction can thus be a helpful approach for tackling long-term aims that might appear overwhelming or unattainable.
Having said that, the whimsical nature of life means that even the best plans may become obsolete. If you’re a die-hard fan of "Friends" like me, you will know that Rachel’s wish for a child turned into reality much sooner than expected. Yet it certainly wouldn’t have happened if she had still been dating Tag.
