Chapter 3. The Dangers of Price Competition

5.

Assume that three people secretly write down a dollar amount on a piece of paper. They must pick a whole dollar amount between $0 and $100. The person who writes down the lowest number wins the amount she wrote down. If there is a tie, the winners split the total. Thus if:

Person one writes down $53 and

Person two writes down $22 and

Person three writes down $30,

then person two wins $22. If person three had also written down $22 rather than $30, then persons two and three would have each received $11 because they would have split the $22. Find the reasonable outcome in this game when all players are rational.

Answers

5.

Answer: All three people choose $1.

First, note that no one should ever write down $100. If you choose $100, the best possible result for you is if your two opponents also write down $100. (If either one of them wrote down any other number, you would get nothing.) In this case you would win $33.33. If everyone else wrote down $100, however, you would be better off writing down $99 because if your two opponents wrote down $100, you would win $99. Thus, in the only circumstance where writing down $100 wins you money, you would have been better off writing down $99. Consequently, writing down $99 always gives you a better or equal payoff than writing down $100. Thus, you should never choose $100, and you should assume that no one else will ever pick $100.

Since $100 will never be chosen, no player should write down $99. If you were to choose $99, your only hope to win would be if both of your opponents also wrote down $99 (since neither of your foes will pick $100). In this case, however, you would only win $33. If both of your opponents wrote down $99, you would have been better off writing down $98 because then you wouldn’t have to split the money. Thus, in the only possible circumstance when you could win with $99, you’re better off writing down $98. Consequently, you should never write down $99. Can you see the pattern emerging?

Should you ever write down $98? Well, since no one is going to pick $99 or $100, the only chance to win with $98 is if both of the others also choose $98. In this case, however, you would have been better off writing down $97. Thus, $98 is out. This process continues all the way down to $1. You don’t want to pick $0 because then you would always get nothing. You would rather split $1 than get zero. Therefore, the only reasonable outcome in this game is where everyone chooses $1. It seems very wasteful that all of you split $1 when you could all have split $100. The logic of game theory, however, compels the players of this game to bid against each other and throw away almost all of the available money.