Game Theory - First move vs second move advantage?

Question

This question came up in a lunchtime discussion with coworkers. None of us are professional mathematicians or teachers of math, and we weren't sure how to get the answer. I apologize in advance if my question is not rigorous or uses the wrong terminology.

Is there any game (like NIM, etc) where the player making the second move has an advantage?

Additional question: Can anyone give me an example of such a game?

Answer 1

I'm sure there's plenty of practical solutions out there. Here's a trivial solution that proves the existence of such games:

Let's say we play a game where starting from 0, each person gets to add a number from 1 to 9 to the running total. The winner is the person who makes the total be ten.

In this case, the second player ALWAYS wins, because they just pick 10 minus the number the first person picked.

Answer 2

Consider the chess position with white king on a5, black king on c4, white pawn on b4 and black pawn on b5. The player on turn loses.

Answer 3

Consider any game along the lines of tic-tac-toe, where two players compete to form certain configurations, except that, opposite to tic-tac-toe, the player who forms one of the specified configurations is the loser. It should be clear that it is generally advantageous to move last in such a game.

For a specific example, consider the game of Sim. The board is six dots. Two players, Red and Blue, take turns. A turn consists of a player connecting two not-connected dots, with a line of that player's color. A player who completes a triangle of their own color loses immediately.

It's easy to see that this game cannot be a first-player win, by an argument very similar to usual strategy-stealing argument: in summary, if the first-player had a winning strategy, the second player could claim the win by ignoring the actual first player's move, pretending to be the first player, and using the hypothetical first-player strategy. The extra first-player move on the board can't hurt them, and might help. Once one shows that Sim never ends in a tie, this completes the proof that the game is a forced win for the second player to move, although the second player's actual winning strategy is quite intricate.

Answer 4

A simple example is Nim with two equal piles. In this game, the second player has the advantage. The advantage to moving second is that you can mimic whatever your opponent does to one pile on the other pile. Thus, he'll have to clear one of the piles first, and then you'll clear the second pile and win.