A small company is losing money and the boss is looking for ways to cut costs. The boss is a foolish gambler, so he offers two employees the choice of either taking flat 10% pay cut, or playing the following coin flip game for each paycheck:

- The employees each flip a fair coin – 50% heads and 50% tails; the boss will ensure they are not pulling any tricks.
- They can see their own outcome but not the outcome of the other employee’s coin flip. They must then guess the outcome of the other employee’s coin flip.
- If at least one of them guesses correctly, they get their full paycheck.
- If both of them guess incorrectly, they get nothing.

The two employees take some time to work out a strategy, and then confidently accept the offer to play the coin flip game.

What strategy did they come up with that made them confident the coin flip game will give them more money?

#### Solution

One employee looks at their coin flip outcome and guesses the same outcome – for example, if they got tails, they would guess heads. The other employee looks at their coin flip outcome and guesses the opposite outcome – for example, if they got tails, they would guess tails.

Since the outcomes of two coin flips can only be the same or opposite, this ensures that one of them always guesses correctly. This means the employees guarantee they will always get their full paychecks, despite the boss’s efforts. Better than a 10% pay cut!

## 2 replies on “Game the Coin Flip Game”

Although we can use this strategy, we can also say that regardless of strategy, they would always enter the game. If we put the payoff of full paycheck as +100, payoff of 10% cut as +90, and the payoff of nothing as -100 (could also be 0, but put -100 as we can talk about utility; getting no pay equates to negative utility). If you list all the outcomes and calculate the expected values, you can see the expected payoff from the game is 100; which is greater than 90. Hence, given that they are either risk averse or risk neutral, they will always enter the game.

or similarily, put the cut of 10% as being 0, the full wage as being 1, and getting nothing as being -1. You can see that expected payoff of game is 1, which is greater than 0, hence we would rather enter game than take flat out pay cut.