This is another game theory problem called “war of attrition”:
You and a competitor will battle in rounds for a prize worth $5. Each round you may choose to either fight or fold. So may your competitor. The first one to fold wins $0. If the other player doesn’t also fold he wins the $5. In the case you both fold in the same round you each win $0. If you both choose to fight you both go on to the next round to face a fight or fold choice again. Moving on to each round after round 1 costs $0.75 per round per player (that is, both players pay $0.75 per round in which they both choose to fight onward).
Example 1: In round 1 you and your competitor both fold. The game is over. Both you and your competitor incur no cost and win no money.
Example 2: In round 1 you fold but your competitor fights. The game is over. You lose nothing and gain nothing. Your competitor incurs no cost and wins the $5.
Example 3: In round 1 both you and your competitor fight. Therefore you move on to round 2 and each pay $0.75. In round 2 you both fight again. Therefore you move on to round 3 and each pay another $0.75. In round 3 you fight and your competitor folds. The game is over. You have spent $1.50 and win $5 for a net gain of $3.50. Your competitor has lost $1.50.
How many rounds of fighting would you be willing to go? How would your answer change with the size of the prize? With the size of the per-round fee? I’ll post analysis next week followed by interpretations and applications.