Our challenge this week was suggested by Laura Feiveson, an M.I.T.-trained economist at the Federal Reserve Board and a frequent contributor to Numberplay. Here’s Laura:
The upcoming presidential election has put me in the mood for some game theory. (I’m an economist, after all.) As we were inundated with news of the workings of our daft primary system a few months ago (ahem, caucuses?), I was reminded of some of the voting systems I learned about in economics graduate school. Most voting mechanisms will occasionally lead to some odd outcomes, and those that I find the most fun to study are those for small groups — in which voters are aware of each other’s preferences and act strategically. As the voting mechanism I write about below highlights, small rules can sometimes make big differences, and forward-looking behavior can lead to seemingly unintuitive results. To keep the mood upbeat and drama-free, I set the election to be in Westeros rather than in the United States.
After years of fighting and tragedy without any resolution, the leaders of Westeros decide to convene to vote among themselves for the next Ruler of the Iron Throne. Tyrion, Dany, Sansa, Cersei and Jon are the five contenders. The vote will proceed as follows: Tyrion will first nominate one of the five of them. They will all vote yea or nay for his nominee, and if that person gets at least three “yea’s”, he/she will be named the ruler. Otherwise, Dany will have an opportunity to nominate someone, and then, if that person doesn’t receive at least three votes, Sansa will choose a nominee to put up for a vote, and then Cersei will. Finally, if Cersei’s nominee doesn’t pass with three votes, Jon will get to appoint the next ruler (with no voting necessary).
Their preferences are as follows:
Tyrion: T > D > S > J > C (Tyrion hates Cersei.)
Dany: D > T > S > C > J
Sansa: S > C > J > T > D
Cersei: C > J > S > D > T (Cersei hates Tyrion.)
John: J > D > S > T > C
Suppose that all five of them know each other’s preferences and are strategic and forward-looking. Furthermore, suppose that they cannot commit to vote against their own preferences in the future. Finally, if indifferent between possible nominees, everyone would prefer to nominate the ultimate winner. Who will Tyrion nominate and who will win?
Now suppose that Tyrion has a chance to institute a rule that no one is allowed to nominate (or appoint, in the case of Jon) himself or herself. Would he want to institute that rule? Who would he nominate and who would win?
Thank you, Laura!
That’s it for the week. As always, send your favorite puzzles to gary.antonick@NYTimes.com.
Check reader comments on Friday for solutions and commentary by Laura Feiveson.
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