Suppose Alice, Bob, and Charlie want to decide who has to take out the garbage by playing the following game. Each of the players independently and simultaneously raises his hand or not. Alice loses if exactly one player raises his hand, whereas Bob loses if exactly two players raise their hands, and Charlie loses if either all or no players raise their hand. A normal-form representation of the situation looks as follows (Alice chooses rows, Bob columns, and Charlie matrices).


The game exhibits some peculiar phenomena despite the existence of a unique Nash equilibrium: Alice raises her hand, Bob does not raise his hand, and Charlie randomizes with equal probability. Charlie couldn’t be happier with the equilibrium as he will never have to take out the garbage (and could even decide who has to do the job by playing a pure strategy instead).

The security level of all players is 0.5 and the expected payoff in the Nash equilibrium is (0.5, 0.5, 1). However, the minimax strategies of Alice and Bob are different from their equilibrium strategies, i.e., they can guarantee their equilibrium payoff by not playing their respective equilibrium strategies (a phenomenon that was also observed by Aumann)! The solution in which all players play their minimax strategies obviously suffers from the fact this strategy profile fails to be an equilibrium: Both Alice and Bob would want to deviate. On top of that, the unique equilibrium is particularly weak in the sense that it fails to be quasi-strict, i.e., all players could as well play any other strategy without jeopardizing their payoff.

Quasi-strict equilibrium is an equilibrium refinement proposed in 1973 by Harsanyi and requires that all pure best responses are played with positive probability. Harsanyi showed that in almost all games all equilibria are quasi-strict. Indeed the three-player game above (taken from this paper) is one of the very few exceptions. Quasi-strict equilibrium is rather attractive from an axiomatic perspective. For example, it has been shown that the existence of quasi-strict equilibrium is sufficient to justify the assumption of common knowledge of rationality when players are ‘cautious’ (for more details see here and here).

In 1999, Henk Norde proved that every two-player game contains a quasi-strict equilibrium (via a rather elaborate proof using Brouwer’s fixed-point theorem), strengthening earlier results which showed existence in zero-sum games, bimatrix games with a finite number of equilibria, 2×n games, etc. Norde’s existence result implies that computing a quasi-strict equilibrium is PPAD-hard (while this problem was shown NP-hard for games with at least three players). Curiously, however, membership in PPAD for two-player games remains open due to the intricate existence proof by Norde (see also this review of Norde’s paper by Bernhard von Stengel).

Coming back to the example, it seems as if Charlie has to live with the deficiencies of Nash equilibrium and prepare to take out the garbage with positive probability.

Sperner’s Lemma hits the popular press

Here’s an article that has been trending on the New York Times site. It’s about Sperner’s Lemma–and, amazingly, they get the technical details right! The article describes a pretty practical scheme for pairing n indivisible goods with n agents; it’s motivated by the example of matching roommates with rooms, each of which has different pros and cons that the roommates each value differently. There’s a nice discussion of the idea of fair division, a pretty thorough description of a paper by Francis Su, a shout-out to Turing’s Invisible Hand Blogger Emeritus Ariel Procaccia and his web site spliddit, a quote from Stephen Brams, an online rent division calculator, and a very nice interactive graphic of how Sperner’s Lemma works.

So, what do you all think? Do you buy it? And, have you ever used a formal fair-division algorithm to make a real-life decision?

Blogger emeritus and COMSOC co-chair Ariel asked me to announce that the registration for COMSOC 2014 is open. The list of accepted papers can be found here. The invited speakers are Edward Felten, Jean-Francois Laslier, and Mark Satterthwaite.

Registration for the temporally co-located Meeting of the Society for Social Choice and Welfare (SCW) is open as well.


The organizers of the AdAuctions workshop have extended the submission deadline to April 27. Updated information on the workshop is as follows.

The Tenth Ad Auction Workshop  (here is the call for papers)

  • Date: June 8, 2014, 8:30-15:30
  • Submission deadline: April 27, 2014, (23:59 Hawaii time)
  • Organizing Committee: adauctionsworkshop2014@gmail.com
    Itai Ashlagi (MIT Sloan)
    Patrick Jordan (Microsoft)
    De Liu (University of Kentucky)
    Renato Paes Leme (Google)
    Chris Wilkens (Yahoo)
  • Keynote speakers: Hal Varian (Google) and Paul Milgrom (Stanford).

SAGT 2014

The 7th International Symposium on Algorithmic Game Theory (SAGT) will take place in Haifa, Israel, from September 30 to October 2, 2014.  The submission deadline is May 1st, and the Call for Papers is here.  This would be the place to send all your papers that were rejected from EC…

This year, EC will feature three workshops. All of them are inviting contributed papers. I hope you will consider submitting some of your work to one of these!

  1. The Tenth Ad Auction Workshop  (here is the call for papers)
    • Date: June 8, 2014, 8:30-15:30
    • Submission deadline: April 22, 2014, (23:59 Hawaii time)
    • Organizing Committee: adauctions2014@gmail.com
      Itai Ashlagi (MIT Sloan)
      Patrick Jordan (Microsoft)
      De Liu (University of Kentucky)
      Renato Paes Leme (Google)
      Chris Wilkens (Yahoo)
    • Keynote speakers: Hal Varian (Google) and Paul Milgrom (Stanford).
  2. The Fourth Workshop on Social Computing and User-Generated Content  (here is the call for papers)
    • Dates: June 8-9, 2014, 14:00-15:30 on June 8 and 8:30-12:30 on June 9.
    • Submission deadline: April 25, 2014
    • Organizing Committee: scugc2014@easychair.org
      Yan Chen (University of Michigan)
      Yiling Chen (Harvard University)
      Arpita Ghosh (Cornell University)
      Ashish Goel (Stanford University)
      Alex Slivkins (Microsoft Research)
  3. The Second Workshop on Crowdsourcing and Online Behavioral Experiments (here is the call for papers)
    • Date: June 8, 2014, 16:00-18:00
    • Submission deadline: April 30, 2014
    • Organizing Committee:
      Siddharth Suri (Microsoft Research NYC)
      Winter A. Mason (Facebook)
      Daniel G. Goldstein (Microsoft Research NYC & London Business School)


The list of accepted papers for ACM EC 2014 has been posted.

PS: Since I accepted the intimidating invitation below, I’ve been thinking of what my first post should be about. Now I have to admit I took the easy way out. Thanks for letting me participate. I’ll try to think of something more original next time.