WILY: Wild Card Risk Management

Last night we had the in-laws over for dinner.  Nothing fancy, just burgers on the grill.

After dinner, we played a few games of six-handed Uno.  I won more games that anyone else, but that's not the point of this story.

Toward the end of one of the games, my wife had two cards left.  On her turn, she played a Wild card and named a color.  Assuming she'd picked the color of her last card, the rest of us immediately schemed to change colors on her.

We failed.  And she went out on her next turn.  But not by playing a card of the same color.  She just happened to have a matching number to the card at the top of the discard pile.

I told her I'd never seen anyone make that play before - to use their Wild card to make it a color other than the one they had left.  

She shrugged and said, "you would have just changed the color anyway.  This way I had a 1 in 3 chance you'd change it to the one I needed."  She figured that if she played it straight and someone did change the color, it absolutely would be the wrong color for her.

Risk management at the Uno table.  It was an aggressive move, and your appetite may vary, but there's no denying that she performed a quick decision tree analysis before playing the Wild card.  She decided that the probability of someone changing the color to the color she needed was greater than the odds of it going around the table unchanged.  

At the risk of a bunch of emails from statistics majors, the decision tree could have looked something like this.  Five of the six players would be trying to change the color during the next round.  Each color-change scenario has its own impact on whether or not she could still win on that turn.  Multiply the chance of the color changing by the chance of winning with the color change, and calculate the best way to mitigate the risk.  

Ironically, she won despite the color not changing.