Abstract
We relax the Kajii and Morris (Econometrica 65:1283–1309, 1997a) notion of equilibrium robustness by allowing approximate equilibria in close incomplete information games. The new notion is termed “approximate robustness”. The approximately robust equilibrium correspondence turns out to be upper hemicontinuous, unlike the (exactly) robust equilibrium correspondence. As a corollary of the upper hemicontinuity, it is shown that approximately robust equilibria exist in all two-player zero-sum games and all two-player two-strategy games, whereas (exactly) robust equilibria may fail to exist for some games in these categories.
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Notes
Note that this assumption does not rule out that the players may actually play an exact Bayesian Nash equilibrium, as it is a particular instance of \(\varepsilon \)-equilibrium, for every \(\varepsilon >0\).
Following up Weinstein and Yildiz (2007), Chen et al. (2014) utilize approximate best responses to discuss selection of rationalizable actions in incomplete information games via perturbing higher order beliefs continuously. The approximate robustness also utilizes approximate best responses, but it accommodates discontinuous changes in higher order beliefs, like KM-robustness.
Such an extension is a much needed step, as there are games without a KM-RE, and there are only limited KM-RE existence results (see, e.g., KM, Ui 2001; Morris and Ui 2005). The general scope of this extension remains an open question, however, and some games of interest are obviously left out. For instance, Oyama and Takahashi (2011) describe an open set of two-player supermodular games without a KM-RE, but non-existence of ARE in these games can also be established in a similar way.
The countability assumption on \(\Omega \) is made just as in KM. When the information partitions \(\left\{ Q_{i}\right\} _{i\in \mathcal {I}}\) are at most countable, this entails no loss of generality. Indeed, the positive measure elements of \(\wedge _{i\in \mathcal {I}}Q_{i}\) (\(\equiv \)the coarsest partition that refines each \(Q_{i}\)) can be equivalently viewed as states of nature instead of the elements of \(\Omega \), and the payoff function can be redefined by taking the conditional expectation of the original payoffs at each of the new states. See the discussion in Sect. 6.3 concerning more general assumptions on \(\Omega .\)
This simplifying assumption is also made in KM.
One reason for insisting on there being a common prior P in the game \(\mathcal {U}\) is the need for a well-defined action distribution to be associated with any \(\hat{\sigma }\in BE_{\varepsilon }\left( \mathcal {U}\right) .\) Action distributions of \(\varepsilon \)-BE in incomplete information games play a central role in our notion of approximate robustness that is introduced in the next section.
Our \(\delta \)-elaboration is a slight extention of the original notion of KM, who require that \(P\left( \Omega \left( \mathcal {U},g\right) \right) =1-\delta \). It is easy to see that both definitions of \(\delta \)-elaboration would give rise to identical concepts of robustness (introduced in Definition 1 below). However, our notion streamlines several mathematical arguments in the sequel, as any \(\delta \)-elaboration is also a \(\delta ^{\prime }\)-elaboration in our sense, for any \(\delta ^{\prime }>\delta \).
One could define a weaker concept by restricting elaborations to canonical elaborations, as in Kajii and Morris (1997b) and Ui (2001). It will become clear that all the results and comments we report in this paper remain valid for the weaker notion. It is however an open question whether this is a strictly weaker notion.
For \(\varepsilon \ge 0,\) a product distribution \(\mu \in \Delta \left( A\right) \) is an \(\varepsilon \) -Nash equilibrium of g if for each \(i\in \mathcal {I}, \mu _{i}\) is an \(\varepsilon \)-best response of i to \(\mu _{-i}.\)
With one minor modification, explained in Footnote 7.
I.e., the claim holds for all payoff matrices in some dense and open (w.r.t. the Euclidean topology) subset of the space of all real-valued \(\left| A^{1}\right| \times \left| A^{2}\right| \) matrices.
Note that with our specification of payoffs, the incomplete information game \(\mathcal {U}_{\delta }\) is zero-sum in every state of nature. Thus, our method will show that the game g has no KM-robust equilibrium even under the restriction that all nearby incomplete information elaborations must be of zero-sum nature. However, the same proof would work if we assumed that, for both \(i=1,2\), player i’s payoffs are given by his payoffs in \(\tilde{g}\) when \(t_{i}=0\), and by his payoffs in g when \(t_{i}>0.\) Thus, the elaborations we use can be given another special feature (in addition to being canonical in the sense of Kajii and Morris (1997b) and Ui (2001), as every player i is “committed” to his strictly dominant action given \(t_{i}=0\)) – it can be assumed that each player i is “commited” to some strategy precisely when he knows that his payoffs are not given by g, i.e., when \(\Omega _{i}\left( \mathcal {U}_{\delta },g\right) \) does not occur.
See Footnote 3.
See Kajii and Morris (1997b) for a precise definition.
See also proposition 5 of Kajii and Morris (1998, p. 271).
This idea was communicated to us by one of the referees.
For instance, a generic two-player zero-sum game has a unique correlated equilibrium, by the proof of Corollary 5 and the discussion following Proposition 3.2 of KM. Then, the unique correlated equilibrium must be the only ARE (since all limit points of \(\varepsilon \)-BE action distributions in incomplete information elaborations converging to g must be \(\varepsilon \)-correlated equilibria, and any limit point of the latter, when \(\varepsilon \rightarrow 0,\) is an (exact) correlated equilibrium).
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Acknowledgments
Kajii acknowledges financial support from JSPS Grant-in-Aid for Scientific Research No.(S)20223001. We wish to thank two anonymous referees for comments that helped us to improve the introduction and formal presentation of our notion of approximate robustness.
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Haimanko, O., Kajii, A. Approximate robustness of equilibrium to incomplete information. Int J Game Theory 45, 839–857 (2016). https://doi.org/10.1007/s00182-015-0488-4
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DOI: https://doi.org/10.1007/s00182-015-0488-4
Keywords
- Incomplete information
- Robustness
- Bayesian Nash equilibrium
- \(\varepsilon \)-equilibrium
- Upper hemicontinuity
- Zero-sum games