Posted April 13, 2011 By Presh Talwalkar. Read about me , or email me .

Marilyn vos Savant is most known for being listed as the “Highest IQ” in the Guinness Book of World Records.

She is also famous for her column “Ask Marylin” in Parade magazine (that she has written since 1986) that tackles interesting questions and puzzles, the most controversial being the Monty Hall problem.

The following question appeared in her column on March 31, 2002, and it is an interesting game theory problem:

.

.

"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon. .

.



Say you’re in a public library, and a beautiful stranger strikes up a conversation with you. She says: ‘Let’s show pennies to each other, either heads or tails. If we both show heads, I pay you $3. If we both show tails, I pay you $1. If they don’t match, you pay me $2.’ At this point, she is shushed. You think: ‘With both heads 1/4 of the time, I get $3. And with both tails 1/4 of the time, I get $1. So 1/2 of the time, I get $4. And

with no matches 1/2 of the time, she gets $4. So it’s a fair game.’ As the game is quiet, you can play in the library. But should you? Should she? – Edward Spellman, Cheshire, Connecticut. Source: “Ask Marilyn,” Parade, March 31 2002, quoted in Siam News June 2003.

Here is Marylin’s answer which appeared in the following issue:

The woman in the library said: ‘Let’s show pennies to each other, either heads or tails. If we both show heads, I pay you $3. If we both show tails, I pay you $1. And if they don’t match, you pay me $2.’ Should you play? No. She can win easily. One way: If she shows you twice as many tails as heads, she wins an average of $1 for every six plays.

While the answer is on the right track, and the logic is proper, I would still have to say the answer is wrong because it is incomplete. It only specifies one possible best response and the reasoning stops.

At best, that answer would get partial credit in a game theory class.

The complete answer would specify the equilibrium strategies for both players along with their payoffs. Then it would be fair to answer whether the game is worth playing.

The challenge to you: Can you figure out the complete solution? Does the game favor you or the beautiful stranger, or is it fair?

Write your thoughts / ideas / guesses in the comments.

I will post my writeup of the solution in a few days in the comments section.