Consider this childhood game with two players, A and B. There are 11 lit candles. The...

Consider this childhood game with two players, A and B. There are 11 lit candles. The players take turns blowing out 1, 2, or 3 candles, with A going first. The player that blows out the last candle wins. It is possible to solve this game using backwards induction. The numbers below may help you to organize your thoughts.

1 2 3 4 5 6 7 8 9 10 11

a. In equilibrium, which player wins the game?

b. On this player’s first move, how many candles should they blow out?

c. Pick a number of candles greater than 11 for which the other player would definitely be able to win the game. If no such number of candles exists (i.e. the same player would win for any number of candles 11 or greater), say so.

Solutions

Expert Solution

a. In equilibrium, A wins the game.

Any player who blows out the last (11th) candle wins. So starting from 11th candle and applying backward induction. Any player will be able to blow out 11th candle if he's can blow out 7th candle, because then other player can only blow out either 8th or 9th or 10th candle. With same reasoning, any player will be able to blow out 7th candle if he's able to blow out 3rd candle, because then other player can only blow out either 4th or 5th or 6th candle. Knowing that if he can blow out 3rd candle, any player can win, a rational player will try to blow out 3rd candle. Given that player A is going first, he can easily blow out 3rd candle and win the game.

b. As explained above, A should blow out 3 candles on his first move.

c. Other player (i.e. player B) can definitely win any game in which number of candles are a multiple of 4. So any game consisting 12, 16, 20, 24 and so on, can be won by player B.

Reason is, if its a multiple of 4, then any player who can blow out 4th candle will win. Since 4th candle can be blown only by player B (if he plays rationally), he'll win. Take the example of 16 candles: any player who blows out 16th will win, which again turns out that any player who blows out 12th candle will win (by reasoning in part a), which again turns out that any player who blows out 8th candle will win (by reasoning in part a) and and 8th candle is blown by player who blows 4th, which in our case will be player B.

Rahul Sunny answered 1 year ago

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