A team of 10 people are going to play a cooperative game. While their eyes are closed, a pit boss will place either a red or green hat on each of their heads, chosen by fair coin flip. They then all open their eyes, and will be able to see the hats on everyone's heads except their own.
The players will then simultaneously bet a number of dollars that the hat they are wearing is green. Specifically, on the pit boss's signal, the players will each call out an integer, which may be positive, negative or zero. Calling a negative number is effectively betting that your hat will be red.
The total winnings of the players is calculated by adding up the bets of people with green hats, and subtracting the bets of people with red hats. The team of players wins if and only if their total winnings are more than zero.
Before the game begins, the players may agree on a strategy, but once the hats are placed, no communication between the players is possible.
What strategy maximizes the players' chances of winning? Why can't they do better?