First-Mover Advantage/Disadvantage: Should You Make the First Move?
There's a common misconception that moving first in a strategic situation provides an advantage over the opponent under all circumstances. While this advantageous is quite obvious in simple settings like the game of Tic Tac Toe, other cases, such as in the game of chess, it is difficult to prove that first mover has an advantage or disadvantage or if it does not matter at all. The goal of this article is to analyze a simple game to illustrate the concept of first-mover advantage/disadvantage, i.e., whether making first move is advantageous or disadvantageus. We will also show how a small change can convert first mover advantage to disadvantage or vice-versa.
In a simple board game, first-mover advantage means the player who moves first might gain a strategic edge by securing key positions on the board. This can put the second player in a reactive position, limiting their options. However, the first player may also face a disadvantage if their move reveals their strategy, allowing the second player to adapt and counter effectively. The impact of this advantage or disadvantage depends on the game's rules and how much flexibility the second player has to respond to the first player's moves.
Game: Coin on Board
Two players, R and B, play on a square board. Player R has an unlimited supply of red coins, and player B has an unlimited supply of blue coins. Both coins are circular with same radius. The game begins with player R placing a red coin on the board. Player B then places a blue coin, and they alternate turns. A coin must be placed flat on the board without overlapping or disturbing other coins. The game continues until one player cannot place a coin. Does player R or player B have a guaranteed winning strategy? Justify your answer with a strategy for thw winning player.
Solution
Without loss of generality, assume player R plays first. We claim that there exists a guaranteed winning strategy for the player R. The winning strategy is as follows. Player R starts with placing a red coin at the center of the table. Now, wherever the player B places a coin, player R should place a red coin on the table such that the mid-point of the line connecting the last blue coin and current red coin lies on the center of the table. Using this strategy, if player B can find a place to put a blue coin on the table, player R will definitely find a place to put another red coin.
Modified Game: Board with a Hole
Make a circular hole, greater than the radius of coins, in the center of the board. In the modified board, does player R or player B have a guaranteed winning strategy? Justify your answer with a strategy for thw winning player.
Solution
In the modified game, there exists a guaranteed winning strategy for player B. Note that player R can not place a coin at the center of the board. Therefore, in the modified game, player B can replicate the winning strategy of the player R in the original game.
Author
Anurag Gupta is an M.S. graduate in Electrical and Computer Engineering from Cornell University. He also holds an M.Tech degree in Systems and Control Engineering and a B.Tech degree in Electrical Engineering from the Indian Institute of Technology, Bombay.
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