A Simple But Proven Strategy from Game Theory: Backward Reasoning
Recently, I’ve been enjoying reading content and books related to Game Theory, and one concept that stood out is backward reasoning.
It’s useful when you “play” a sequential game. And, how does it work?
You start at the end of all the sequences with your desired outcome and reason backward. It is that simple. But it can be powerful.
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| Photo by jeshoots.com on Unsplash |
Let me show you an example: the flag game.
Flag game
Instruction
The game is played with two teams.
There are 20 flags and the teams make alternating turns. In each turn, a team can pick 1, 2, or 3 flags - no more and no less. The winner is the team that picks the last flag.
Solution
At first glance, it might seem like strategy only matters somewhat—and mostly depends on what the other team does.
You’re partially right. But with the right strategy, your team can always win—as long as you let the other team start.
Let’s apply backward reasoning. The winning team is the one that picks the last flag. That means if the opposing team is ever faced with 4 flags, they’re guaranteed to lose. Whether they take 1, 2, or 3 flags, your team will always be able to take the remaining ones and win.
Working backward from that, if they face 8 flags, you can ensure they end up with 4 on their next turn:
- If they take 1, you take 3.
- If they take 2, you take 2.
- If they take 3, you take 1.
This same logic applies all the way back. The key losing positions are: 4, 8, 12, 16, 20. If you force the other team into any of these counts on their turn, you’re guaranteed to win.
Since the game starts with 20 flags and the other team goes first, you’ve already won—as long as you follow the strategy.
There is a variant to the game: if you pick 21 flags your team has to start and begin by taking exactly 1 flag. This puts the other team at 20, and from there, the same winning sequence applies.
The takeaway
As you can see you’ll always win if you play the right strategy in this game. In most games in real life winning isn’t that certain because games are different and there is a probability in between making it uncertain.
However, backward reasoning remains an extremely useful method in making strategic decisions.
Now it’s your turn. Try out backward reasoning yourself and play the flag game with a friend (who isn’t a game theory expert).