First mover advantage? Not always…

The order in which players make their decisions in a game can make a huge difference to the outcome.

Take the simple game of Nim. In this game there are two players who take it in turns to play. There are two piles of matches and at each player’s turn they take as many matches as they want but they can only take from one pile at each turn. The winner is the player that takes the last match.

Take the situation where there are ten matches in each pile, would you want to go first or second?

Try to come up with your answer before you scroll down







To work this out think about the simpler case where there is one match in each pile. Player 1 must take a match from one of the piles, but as they can only take from one pile, they must leave the one match that is in the other pile. Player 2 then takes the last match and wins.

If we start with two matches in each pile then Player 1 can take either one or two matches from one of the piles. If they take two matches then Player 2 takes the two matches that are in the other pile and wins. If Player 1 only took one match then Player 2 takes one from the other pile and we have a situation where there is one match in each pile, Player 1 must take one and Player 2 takes the other one to win. Again Player 2 wins.

We can now follow the same kind of logic to get the answer for two piles with ten matches each. Whatever Player 1 does, Player 2 can always take the same number of matches from the other pile to leave two piles with the same number of matches in. This way Player 2 is guaranteed to win.

If you now think about the situation where we start with two piles with different numbers of matches in then Player 1 will always win. This is because they can always make the two piles even with their first go and then from there they will win.

So we have a game where the order of play determines the winner.

This is true for lots of situations, particularly negotiations. Always think about whether it is better for you to move first or not.


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