The Price is Right: Big Wheel

I was off in Ottawa for Canada Day and ended up catching an episode of the Price is Right in the hotel room. (I finished reading all the books I'd brought so I was left being bored and turned to daytime tv.) The first person to spin the wheel landed on 75 and paused to think if they should spin again or not. They decided to stop and Drew told them they'd made a good choice and hurried them over to stand under the sign. But it got me thinking... Did they really make a good choice? Or was Drew just trying to make the person feel good? 75 felt like it was probably just a really bad opening spin. Spinning again brings with it a guaranteed 75% chance of failure, but stopping gives your opponents 4 spins with a 25% chance each to beat you. Rough off my head math told me stopping was right, but your chances of advancing were not good.

There does have to be a point where it's right to spin again though... And now I need to know what that point is! It also feels like going 3rd is strong since you have more information, but just how strong is it going to be?

Start with the simple case... Player one busted out and player two is up. If you come up with a 50 or lower you need to spin again. If you come up above 55 you need to stop. (This is all ignoring any potential gain from the 1 in 20 chance of hitting exactly 1 dollar. I'm asserting the value of getting the showcase showdown outweighs that.) Starting with a spin of 70 or lower means you expect to lose even with proper spin again play. 75 is actually the first number that has a more than 50% chance to win! Overall it works out that player 2 will only win 45.76% of the time after player 1 busts out, which does mean a spin of 70 is better than expected since it will win 46.48% of the time.

I can use that information to determine player two's optimal play when player 1 spins low. Player two doesn't care who beats him, so even if player one opens with 5 into 5 and player two opens with 50 he should still spin the second time. I did some crunching and starting on a 75 is actually only slightly below expected for the first player, with a 28.42% chance of winning off the 75. Overall the first player is only 30.83% to win, so the 75 isn't all that bad. You should certainly stay on 75 (you only win 15.36% of the time if you spin again) and you should even stay on 70. 65 or lower and the first player should spin again, but the choices at 65 aren't very appealing. 16.05% to win if you stay, 17.86% to win if you spin. The best number to start with after the 70 is actually the 5. Because once you accept that you're spinning a second time you want to reduce the chance of busting. Even if you start on 5-5 that still has a theoretical chance to win where going bust is a guaranteed loss!

As far as the division of wins between the different positions assuming optimal play as defined on my sheet (which may be a little wonky in terms of if player three should spin when his first spin ties both other players) we have player one at 30.83% chance to win, player two at 32.73% chance to win, and player three at 36.45% chance to win. So there's a pretty significant edge in going last! Player one should spin at 65 or higher. Player two should spin when he's behind player one or when he spun at 50 or below. Player three should spin if behind, or if tied under 50.