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The B.E. Journal of Theoretical Economics

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Macro Meets Micro: Stochastic (Calvo) Revisions in Games

Jan Libich
  • Corresponding author
  • School of Economics, La Trobe University, Melbourne, VIC, 3086, Australia
  • Department of Economics, VŠB-TUO, Sokolská trída 33, Ostrava, Czech Republic; and CAMA
  • Email:
/ Dat Thanh Nguyen
  • School of Economics, La Trobe University, Melbourne, VIC, 3086, Australia
  • Email:
Published Online: 2013-12-20 | DOI: https://doi.org/10.1515/bejte-2013-0042


The timing of moves in conventional games is deterministic. To better capture the uncertainty of many real world situations, we postulate a stochastic timing framework. The players get a revision opportunity at a pre-specified time (common to them) with some known probability (different across them). The probabilistic revisions resemble the Calvo (1983) timing widely used in macroeconomics, and by nesting the standard simultaneous move game and Stackelberg leadership they can serve as a “dynamic commitment” device. The analysis shows how the revision time and probabilities affect the outcomes in games with multiple and/or inefficient equilibria. Unsurprisingly, we show in the Battle of the sexes that commitment – low revision probability relative to the opponent – improves the player’s chances to uniquely achieve his preferred outcome (i.e. to dominate). What may, however, seem surprising is that the less committed (higher revision probability) player may dominate the game under some circumstances (for which we derive the necessary and sufficient conditions). This is in contrast to the intuition of Stackelberg leadership where the more committed player (leader) always does so. The paper then applies the framework to the strategic interaction between monetary and fiscal policies in the aftermath of the Global financial crisis. It is modelled as the Game of chicken in which a double-dip recession and deflation can occur when both policies postpone stimulatory measures – attempting to induce the other policy to carry them out. In order to link our theoretic results to the real world, we develop new indices of monetary and fiscal policy leadership (pre-commitment) and quantify them using institutional characteristics of high-income countries. This exercise shows that the danger of the undesirable deflationary scenario caused by a monetary–fiscal policy deadlock may be high in some major economies.

Keywords: timing of moves; revisions; Stackelberg leadership; Battle of the sexes; monetary–fiscal interactions

JEL Classification: C71; C73; E63


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About the article

Published Online: 2013-12-20

Published in Print: 2014-01-01

In Calvo’s framework, each agent faces an exogenously given probability, independent across periods, that they will be able to revise their existing action. While the context of our timing differs from Calvo’s, the time of the probabilistic revision is pre-determined, which implies a stochastic duration of actions in both frameworks.

Specifically, the random revision element may express technological factors, e.g. the probability that firms will be able to convert their R&D investment into a new invention allowing them more frequent production rounds. Alternatively, the revision probability may express physical or environmental constraints, e.g. different weather conditions in various parts of the country affecting the farmers’ relative ability to supply to the market. Macroeconomic factors can also be captured by the revision probabilities, for example, a recession in one country reduces the probability (relative to a competitor in a well-performing country) that local firms will be able to launch a new marketing campaign. Stochastic revisions may also represent political and legal developments that can never be predicted with certainty, but affect the ability of economic agents to make decisions at will. Section 5 will compare the analysis with alternative timing frameworks we have examined. One is a fully-stochastic setup from Basov, Libich, and Stehlík (2013) in which the revision of one player can arrive at any time (but the other player cannot revise or can only revise prior to that). Another is a fully-deterministic setup of Libich and Stehlík (2010) in which each player i moves with a constant frequency – every periods.

For example, the central bank may prefer to avoid additional rounds of quantitative easing for fear of a difficult “exit strategy”. Similarly, the government may be reluctant to engage in additional fiscal packages to avoid debt problems.

Let us note five related issues. First, it is apparent that our setup nests the standard simultaneous move game and the Stackelberg leadership game as special cases. Specifically, the former is represented, for all by , whereas the latter by . Second, we consider the case of a common to keep it closer to the standard setting and to only focus on one type of heterogeneity (regarding s). It is, however, easy to show that the intuition of the case is analogous. Third, the timing can be endogenized. Libich and Stehlík (2011) do so in a different timing framework, for an alternative avenue, see Leshem and Tabbach (2012). Fourth, we do not examine a repeated version of this game, since the effects of repetition in improving coordination are widely known. This is both under standard and asynchronous timing, see Mailath and Samuelson (2006) or Wen (2002). Fifth and similarly, the players’ discounting has the standard effects and is, therefore, disregarded for parsimony.

The respective condition is

It is apparent that the players’ discounting would have the conventional effects. A greater amount of impatience would make it harder for the more committed player to achieve his dominance region, as it would decrease the present value of his ratio.

Fully examining the multiplicity region is beyond the scope of the paper. Let us just note that the “probability” of coordinating on one of the efficient outcomes may differ within this region. Consider for example the case of . If , then M knows that if he initially plays S he will surely achieve his preferred coordinated regime after time – provided F gets a revision opportunity. If instead , this is no longer the case, and hence may occur throughout the whole game even if F does get a revision opportunity.

In the general game, is further increasing in z and c, whereas it is decreasing in , and b. This is because (i) higher z and lower y lead to a more costly conflict for F, and (ii) higher c reduces M’s benefit from coordination with the opponent (payoffs a and b increase this benefit and, therefore, have the opposite effect on ).

The companion scenario of the incomplete information game in Libich, Nguyen, and Stehlík (2012) features the long-term sustainability perspective and the threat of an unpleasant monetarist arithmetic. We do not cover it here, as it has been studied extensively since Sargent and Wallace (1981).

This incorporates the experience of the post-Nasdaq bubble, whereby it is commonly accepted that “The Fed’s decision to hold interest rates too low for too long from 2002 to 2004 exacerbated the formation of the housing bubble”, see Taylor and Ryan (2010). Rajan (2011) and many others voice similar concerns about the policy actions during the Great Recession.

The and inequalities in eq. [4] may be reversed without affecting our conclusions.

Obviously, there are some differences between the two games (in addition to one falling in the coordination and the other anti-coordination class). While these differences may be relevant for some purposes, e.g. equilibrium stability, they do not play any role in our analysis.

While we have included the countries using/pegged to the Euro (indicated in blue), their monetary leadership values, and their position in the figure, should be interpreted with extreme caution. This is because they do not have an independent monetary policy, and thus the policy interaction is more complex. This is even more the case due to a free-riding problem in a monetary union, for details, see Libich, Nguyen, and Stehlík (2012).

It is apparent that our framework with stochastic timing is different from – but compatible with – the stochastic games by Shapley (1953).

The same is true of Libich and Stehlík (2012), which uses their special case of . The paper’s application is the long-term view of monetary–fiscal interactions, namely the sustainability of public finances and the likelihood of an unpleasant monetarist arithmetic of Sargent and Wallace (1981). The paper also formally studies policy interactions in a monetary union composed of independent fiscal authorities with differing sizes and revision probabilities and provides an empirical application to the Eurozone.

Let us mention that both the fully-stochastic and fully-deterministic revision frameworks have some similarities to timing games developed by Simon and Stinchcombe (1989). In their framework, each player chooses the time(s) of his move from a finite set of options, implying that asynchronous timing and commitment can arise in such games (and can thus be used in various contexts, e.g. Monte 2010). For exploration of alternative timing structures and probability distributions using time scales calculus, see recent research in mathematics, e.g. Stehlík and Volek (2013).

Note that this is simply Condition A for the opponent, player

This is naturally the case for the Pure coordination game as well.

Citation Information: The B.E. Journal of Theoretical Economics, ISSN (Online) 1935-1704, ISSN (Print) 2194-6124, DOI: https://doi.org/10.1515/bejte-2013-0042.

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