Athey et al. (2005)

Original article (link) posted: 07/11/2005

Athey, Atkeson and Kehoe (2005) "The Optimal Degree of Discretion in Monetary Policy" Econometrica

The paper examines a monetary policy game in which the monetary authority has private information about the state of the economy. In the literature, two seminal papers, Taylor (1983) and Canzoneri (1985), established no discretion should be left to the monetary authority if there is no such private information; the best outcomes can be achieved by rules that specify the action of the monetary authority as a function of observables. The introduction of private information creates a tension between discretion and time inconsistency. Tight constraints on discretion mitigate the time inconsistency problem in which the monetary authority is tempted to claim repeatedly that the current state of the economy justifies a monetary stimulus to output. However, tight constraints leave little room for the monetary authority to time-tune its policy to its private information. In short, loose constraints allow the monetary authority to do that fine tuning, but they also allow more room for the monetary authority to stimulate the economy with surprise inflation.

Making use of dynamic mechanism design techniques, they find that the optimal mechanism is quite simple; for a broad class of economics, the optimal mechanism is static (policies depend only on the current report by the monetary authority) and can be implemented by setting an inflation cap, an upper limit on the permitted inflation rate. It is also shown that the optimal degree of monetary policy discretion turns out to shrink as the severity of the time inconsistency problem increases relative to the importance of private information.

The results they derive seem to be very interesting. In particular, the “inflation cap” result is directly related to the optimal inflation targets and has strong importance with actual monetary policies. As they mention, their work provides one theoretical rationale for the type of constrained discretion advocated by Bernanke and Mishkin (1997). This paper might become a must-read paper for those who work in monetary policies.

Bernanke and Mishkin (1997) "Inflation Targeting: A New Framework for Monetary Policy?" J. of Econ, Perspectives, 11
Canzoneri (1985) "Monetary Policy Games and the Role of Private Information" AER, 75
Taylor (1983) "Rules, Discretion, and Reputation in a Model of Monetary Policy: Comments" JME, 12


Echenique (2003)

Echenique, F. (2003), The Equilibrium Set of a Two Player Game with Complementarities is a Sublattice, Economic Theory, 22: 903-905.  Link 
Summary  I prove that the equilibrium set in a two-player game with complementarities, and totally ordered strategy spaces, is a sublattice of the joint strategy space.

It is widely known that in games with strategic complementarities (GSC), i.e., best reply correspondings are monotone increasing for all players, the set of pure-strategy Nash equilibria forms a non-empty complete lattice. This result implies the existence of smallest and largest Nash equilibria.

The current paper looks further into the structure of the equilibrium set of GSC, and shows that under certain restrictions the equilibrium set becomes not just a complete lattice but also a sublattice, as is written down in the Summary above.

The practical importance of this result, according to the author, is:
If the equilibrium set of a game is a sublattice, then we can find new equilibria from knowing that two profiles are equilibria, and by taking the componentwise join and meet of players’ strategies.

Note that we cannot obtain such strong property by a complete lattice structure alone. In this sense, a sublattice is indeed critical. The proof is extremely simple, which makes use of the observation that (a) when there are only 2 players and (b) their strategy spaces are completely ordered, (c) the other player's strategy must be completely ordered. The property (c) does not hold either (a) or (b) is not satisfied. Once (c) is verified, the rest of the proof is just to follow the definition of GSC.

My random thought: A sublattice property also arises in one-to-one two-sided matching markets (but neither in one-to-many nor many-to-many markets). I'm wondering if the idea of this paper can be somehow connected to the sublattice property of the set of stable matchings.

A final remark: A nicely written but a bit uninformative note.


Sotomayor (1996)

Sotomayor, M. (1996), A non constructive elementary proof of the existence of stable marriages, Games and Economic Behavior, 13: 135–7.  Link 
Abstract  Gale and Shapley showed in their well known paper of 1962 (Amer. Math. Monthly 69, 9–14) that stable matchings always exist for the marriage market. Their proof was constructed by means of an algorithm. Except for the existence of stable matchings, all the results for the marriage market which were proved by making use of the Gale and Shapley algorithm could also be proved without the algorithm. The purpose of this note is to fill out this case. We present here a nonconstructive proof of the existence of a stable matching for the marriage market, which is quite short and simple and applies directly to both cases of preferences: strict and nonstrict. 

In this short note, the author establishes the existence of stable matchings for the marriage market, i.e., one-to-one matching market, without employing any algorithms. Her key idea is a new concept that she calls a simple matching, which is defined as follows:
A matching is simple if, in the case a blocking pair (m, w) exists, w is single.

Since the set of simple matchings is nonempty and finite, there must exist a specific matching that I call here (for notational convenience) an M-simple matching:
A simple matching is M-simple if it cannot be (weakly) Pareto dominated for men by any other simple matching.

Then, she shows that such M-simple matching is stable. The basic logic is that the existence of blocking pair necessarily induces Pareto improvement for men (since w is single), which contradicts to the Pareto efficiency of M-simple matching for men.

As the author claims, the proof is very shout and simple, yet quite novel I think. To explain her proof strategy in a different way, she

  1. focuses on the subset of stable matchings
  2. characterize it by "M-simple matching," and
  3. shows that the set is non-empty. 

It is immediate that M-simple matching is unique and coincides with M-optimal stable matching when all the preferences are strict. In this way, we can see that her idea is somewhat related to the existence of one-sided optimal stable matchings, a widely known result in the literature.

A minor remark: The definition of matching in the paper incorporates individual rationality, and consequently that of of stable matchings pays attention only on blocking pairs. This looks a bit non-standard while nothing is lost in her analysis.

A final remark: I like this paper very much :)


How did GS algorithm come out?

Roth, A. (2008), Deferred acceptance algorithms: history, theory, practice, and open questions, International Journal of Game Theory, 36: 537-569.

In this survey article on the Gale-Shapley's deferred acceptance algorithm, Al Roth mentions (in footnote 3) an amazing story about how GS discovered the algorithm (which of course established the theory of two-sided matchings):

At his birthday celebration in Stony Brook on 12 July 2007, David Gale related the story of his collaboration with Shapley to produce GS by saying that he (Gale) had proposed the model and definition of stability, and had sent to a number of colleagues the conjecture that a stable matching always existed. By return mail, Shapley proposed the deferred acceptance algorithm and the corresponding proof