# Dice roll game fair price

You and I each roll 1 die each at the same time. I win if I roll a six on one roll, and then a five on the next. You win if you roll two sixes in a row. Who would you bet your money on? Note that You: 1, 1, 5 and I: 1, 6, 5 is a winning sequence for me, i.e. the six and the five (or two sixes) can appear as a subsequence in any position.

Isnt the fair price zero if payoffs are 1-1? Theres equal probability in getting "6-5" sequence and "6-6" sequence?

• Is the game played in distinct "games" of two rolls, or is the "win" condition just detected whenever it occurs in a unstructured sequence of rolls? If the latter, would the second player win twice if they rolled 6,6,6 in three rolls? Jan 18, 2014 at 23:04
• I believe its the first. If the second case is true, then my EV for the game is negative since it always takes at least 4 rolls for me to win twice and for the "6-6" sequence, it takes only 3 rolls to win twice. Jan 18, 2014 at 23:23
• what is your definition of fairness here? What are your assumptions on the risk aversion of the agents, or more generally the form of their utility function? Jan 19, 2014 at 22:47

$6-5$ is more likely to win. The expected occurrence of the first $6$ is equal for both sides, but then if you miss your second $6$, that throw was definitely wasted; whereas if I miss my $5$, I might have rolled a $6$ instead, in which case I stand better.