<p>The concept that Brandenburger and Dekel (1987) call an "a posteriori equilibrium" is roughly the same as what Dekel and Siniscalchi call an "epistemic type structure for a complete information game" in which all types are rational and there is common belief in rationality. Therefore, Brandenburger and Dekel's Proposition 2.1, together with the remark that immediately follows the proof of Propoistion 2.1, is roughly the same as Theorem 1 in Dekel and Siniscalchi.  </p>

CC BY-SA 3.0
2015-01-24T02:02:09.790Z
2015-01-24T02:02:09.790Z
Tilman Borgers
<p>One of the fundamental results in epistemic game theory is that the solution concept of <em>correlated rationalizability</em> gives exactly those action profiles that are compatible with rationality and common belief in rationality. A precise statement and formulation of this result is given in </p>

<blockquote>
  <p>Tan, Tommy Chin-Chiu, and Sérgio Ribeiro da Costa Werlang. "The
  Bayesian foundations of solution concepts of games." <em>Journal of
  Economic Theory</em> 45.2 (1988): 370-391.</p>
</blockquote>

<p>as Theorem 5.2 and Theorem 5.3. An alternative reference often cited for this result (at least in the context of finite games, Tan &amp; Werlang allow for compact metric action spaces) is </p>

<blockquote>
  <p>Brandenburger, Adam, and Eddie Dekel. "Rationalizability and
  correlated equilibria." Econometrica: Journal of the Econometric
  Society (1987): 1391-1402.</p>
</blockquote>

<p>For example, the survey on epitemic game theory in the fourth volume of the handbook of game theory credits Brandenburger &amp; Dekel for this result (<a href="https://c92d0541-a-62cb3a1a-s-sites.googlegroups.com/site/eddiedekelsite/EpistemicGameTheory-130326.pdf?attachauth=ANoY7cps6Rt9t8BHAt5jqUXK4DlafRwpMzlsT_oMieCrXMlGJz1AMJwODmMVmtQaNtFjR8cs1mj_mXtrXoIZL6bzTgKNwwxm4spccFUheStoyQmTq0NtfpZ2hqoThby97f84NZdM-vEQWBTt_JP8tz-2iwDw4t0rlCqRkRloyiyMZRXErEIcE7SbEkXdaTUzx-9jp8Pn_d4I8O8ZTJ2UIg0ez-zuBj1OXFdQl-QReYlLo9-i4td0I6s%3D&amp;attredirects=0">online version</a>, see Theorem 1 there). I have actually seen many such references but was not able to locate the result in their paper. That paper contains 4 propositions and none of them corresponds to this result. The authors actually credit Tan &amp; Werlang and write "Tan and Werlang (1984) and Bernheim (1985) provide formal proofs of the equivalence between rationalizability and common knowledge of rationality." (Tan &amp; Werlang 1984 is the working paper version).</p>

<p><strong>What am I missing that everyone else gets?</strong></p>

CC BY-SA 3.0
2014-12-15T01:26:04.367Z
That's appears to be a case of Stigler's Law, http://en.wikipedia.org/wiki/Stigler%27s_law_of_eponymy
Alecos Papadopoulos
<mathematical-economics><game-theory>
Rationality and Common Belief in Rationality in Brandenburger & Dekel (1987)
Michael Greinecker
# Rationality and Common Belief in Rationality in Brandenburger & Dekel (1987)

(Asked by: username_0 on 2014-12-15)

One of the fundamental results in epistemic game theory is that the solution concept of *correlated rationalizability* gives exactly those action profiles that are compatible with rationality and common belief in rationality. A precise statement and formulation of this result is given in

> Tan, Tommy Chin-Chiu, and Sérgio Ribeiro da Costa Werlang. "The
> Bayesian foundations of solution concepts of games." *Journal of
> Economic Theory* 45.2 (1988): 370-391.

as Theorem 5.2 and Theorem 5.3. An alternative reference often cited for this result (at least in the context of finite games, Tan & Werlang allow for compact metric action spaces) is

> Brandenburger, Adam, and Eddie Dekel. "Rationalizability and
> correlated equilibria." Econometrica: Journal of the Econometric
> Society (1987): 1391-1402.

For example, the survey on epitemic game theory in the fourth volume of the handbook of game theory credits Brandenburger & Dekel for this result ([online version](https://c92d0541-a-62cb3a1a-s-sites.googlegroups.com/site/eddiedekelsite/EpistemicGameTheory-130326.pdf?attachauth=ANoY7cps6Rt9t8BHAt5jqUXK4DlafRwpMzlsT_oMieCrXMlGJz1AMJwODmMVmtQaNtFjR8cs1mj_mXtrXoIZL6bzTgKNwwxm4spccFUheStoyQmTq0NtfpZ2hqoThby97f84NZdM-vEQWBTt_JP8tz-2iwDw4t0rlCqRkRloyiyMZRXErEIcE7SbEkXdaTUzx-9jp8Pn_d4I8O8ZTJ2UIg0ez-zuBj1OXFdQl-QReYlLo9-i4td0I6s%3D&attredirects=0), see Theorem 1 there). I have actually seen many such references but was not able to locate the result in their paper. That paper contains 4 propositions and none of them corresponds to this result. The authors actually credit Tan & Werlang and write "Tan and Werlang (1984) and Bernheim (1985) provide formal proofs of the equivalence between rationalizability and common knowledge of rationality." (Tan & Werlang 1984 is the working paper version).

**What am I missing that everyone else gets?**

**Tags:** mathematical-economics, game-theory
> **Comments:**
> **username_1**: That's appears to be a case of Stigler's Law, http://en.wikipedia.org/wiki/Stigler%27s_law_of_eponymy  

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## 1 answers

### Answer by username_2 on 2015-01-24. Score: 4 | Accepted answer

The concept that Brandenburger and Dekel (1987) call an "a posteriori equilibrium" is roughly the same as what Dekel and Siniscalchi call an "epistemic type structure for a complete information game" in which all types are rational and there is common belief in rationality. Therefore, Brandenburger and Dekel's Proposition 2.1, together with the remark that immediately follows the proof of Propoistion 2.1, is roughly the same as Theorem 1 in Dekel and Siniscalchi.


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