## Tuesday, October 19, 2010

### My Trip to Denmark

Last week I had a pleasant short trip to Aarhus, Denmark for the inauguration of the new Center for Research in the Foundations of Electronic Markets. Kevin Leyton-Brown had a more interesting trip to the center.

The Center focuses on real-world pricing mechanisms helped by secure computation building on research of Aarhus cryptographer Ivan Damgård. We heard several talks from government agencies and energy companies, seeing how the electricity network flows through Scandinavia and Northern Europe. Denmark with its many windmills is often an electricity exporter and messy pricing mechanisms come to play.

Dale Mortensen, the Northwestern Economics professor who won the Nobel prize last Monday, spends the falls in Aarhus so the Nobel prize was already big news there. When the local dean officially inaugurated the center, he noted that the center had international partners at Northwestern (Nicole Immorlica and myself) and used that fact to brag about Dale being at Aaarhus not once but twice. He never mentioned the other international partners. Take that Harvard. Dale himself was nowhere to be found.

At the workshop someone ran one of these weird rule auctions that on large scales is essentially a lottery. It costs 5 DKK (about US\$1) for each bid. Each bid must be a multiple of 5 DKK. The player with the lowest unique bid pays that bid and gets an iPod Touch. I tried with a random bid. Game theorist Hervé Moulin won the auction. His strategy: Bid on every number from 5 to 75 DKK. A seemingly crazy strategy but with 15 DKK as his winning bid, he got the iPod for all of \$18.

1. I don't understand the rules of the auction. If his winning bid was 15 DKK (~ 3 USD), why did he have to pay \$18?

2. Each bid costs 5 DKK (~1\$) and he placed 15 bids.

3. The *lowest* bid? Why is 5 DKK not the optimal bid?

4. lowest *unique* bid

5. Everybody bids 5 and nobody gets it, seems to be the equilibrium. Why risk bidding more?

6. Duh, I get it, a higher bid wins if it's unique. So bidding a spread makes sense.

7. This ends up being a psychological game. Play with a group of n people, choosing random positive integers, and you'll get a distribution for the winning bid. Play with only primes, and you get a completely different spread with regard to the indexes (so even if 10 is the optimal bid with 50 people, the 10th prime isn't necessarily the optimal bid for only primes).