Expected value I Nick right? Dan's payoff probability Eric right? 1/8 112 1/2 Dan right? $1 118 112 18 112 1/2 1/2N 1/2 1/8 112 N 1/2 $2 1/8 112 112 In the"payoff"column, were accounting for the fact that Dan has to put in $2 just to play So, for example, if he guesses correctly and Eric and Nick are wrong, then he takes all $6 n the table, but his net profit is only $4. Working from the tree diagram, Dans expected payoff is: Ex( payoff=0·3+1·+1·+4·+(-2)·。+(-2)·+(-2)·3+0 So the game perfectly fair! Over time, he should neither win nor lose money. The trick is that Nick and Eric are collaborating, in particular, they always make oppo site guesses. So our assumption everyone is correct independently is wrong; actually the events that Nick is correct and Eric is correct are mutually exclusive! As a result, Dan can never win all the money on the table. When he guesses correctly, he always has to split his winnings with someone else. This lowers his overall expectation, as the correct ted tree diagram below shows2 Expected Value I 1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8 probability Dan right? Eric right? Nick right? Dan’s payoff Y N Y Y Y Y Y Y N N N N N N $0 −$2 −$2 −$2 $4 $1 $1 $0 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 In the “payoff” column, we’re accounting for the fact that Dan has to put in $2 just to play. So, for example, if he guesses correctly and Eric and Nick are wrong, then he takes all $6 on the table, but his net profit is only $4. Working from the tree diagram, Dan’s expected payoff is: 1 1 1 1 1 1 1 1 Ex (payoff) = 0 · + 1 · + 1 · + 4 · + (−2) · + (−2) · + (−2) · + 0 · 8 8 8 8 8 8 8 8 = 0 So the game perfectly fair! Over time, he should neither win nor lose money. The trick is that Nick and Eric are collaborating; in particular, they always make oppo￾site guesses. So our assumption everyone is correct independently is wrong; actually the events that Nick is correct and Eric is correct are mutually exclusive! As a result, Dan can never win all the money on the table. When he guesses correctly, he always has to split his winnings with someone else. This lowers his overall expectation, as the corrected tree diagram below shows: