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Proximate preferences and almost full revelation in the Crawford–Sobel game

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Crawford and Sobel (Econometrica 50(6):1431–1451, 1982) is a seminal contribution that introduced the study of costless signalling of privately held information by an expert to a decision maker. Among the chief reasons for its widespread application is the comparative statics they develop between the extent of strategically transmitted information and the degree of conflict in the two players’ preferences. This paper completes their analysis by establishing that in their general model, almost full revelation obtains as the two players’ preferences get arbitrarily close to each other.

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  1. See Gilligan and Krehbiel (1989), Krishna and Morgan (2001), Morgan and Stocken (2003), Benabou and Laroque (1992), Harris and Raviv (2008) among many other applications of the model. A significant theoretically motivated literature on strategic information transmission also exists which adds additional elements to the basic CS-model such as multi-dimensional type uncertainty, partial verifiability of actions, multiple experts reporting on the state, multiple principals, etc. For example, see Ambrus and Takahashi (2008) and the references therein or the recent survey by Sobel (2013).

  2. For example, this plays a critical role in discussing the relative merits of authority and delegation in Agastya et al. (2014).

  3. The analysis here is presented in terms of behavioral (pure) strategies whereas CS work with distributional strategies. This difference is inessential here. Furthermore, the definition of an equilibrium must specify players’ beliefs at all information sets, including out of the equilibrium path, as well as (1) and (2). Insofar as our concern is only in the characterization of the equilibrium outcome function (EOF), this is without loss of generality because, given a strategy profile \(( \sigma _{s} , \sigma _{r} )\) such that (1) and (2) hold, pick \(\hat{\theta }\) arbitrarily and let \(\hat{m} = \sigma _{s} ( \hat{\theta } )\). For any \(m \in \mathcal {M} \setminus R ( \sigma _{s} )\), which represents an unreached node in the candidate equilibrium \(( \sigma _{s} , \sigma _{r} )\), prescribe the beliefs of R at \(m\) to be the same as those at \(\hat{m}\) and redefine \(\sigma _{r} (m) = \sigma _{r} ( \hat{m})\). That is, R behaves at any unreached equilibrium message exactly as he does upon hearing \(\hat{m}\). Since the original incentive compatibility conditions prevent any type (other than \(\hat{\theta }\)) from mimicking the behavior of \(\hat{\theta }\), with the above prescribed beliefs, every type of S has an incentive to weakly report \(\sigma _{s} ( \theta )\) and makes \(( \sigma _{s} , \sigma _{r} )\) a perfect Bayesian equilibrium, in the sense of Fudenberg and Levine (1990).

  4. Equilibrium is after all obtained as the solution to a fixed-point problem. Therefore, the question of the existence of such an equilibrium reduces to asking for the continuity of a fixed-point mapping. Regularity is very close in spirit to the type of conditions that are imposed to ensure the continuity of fixed-point mappings with respect to some parameter. McLennan (2012) surveys the fixed-point theory from a perspective useful for Economics.

  5. Here, and elsewhere in the proof when \(a=0\) or \(a' =1\), \(x_{1} ( a,a' )\) and \(x_{2} ( a,a' )\) should be interpreted as the right and left derivatives respectively. Similarly for \(V_{1} ( a, \hat{a} ,a' )\) and \(V_{3} ( a, \hat{a} ,a' )\) later on in the proof.


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Correspondence to Parimal Kanti Bag.




(Claim 1) Throughout \(U ( \xi , \theta ) \equiv U^{s} ( \xi , \theta ,0 )\). We will first show that for \(a<a'\),Footnote 5

$$\begin{aligned} x_{1} ( a,a' ) +x_{2} ( a,a' )&\le 1. \end{aligned}$$

From the first-order condition that determines \(x ( a,a' )\) we have

$$\begin{aligned} \int \limits _{a}^{a'} U_{1} ( x ( a,a' ) , \theta ) \mathrm {d}F ( \theta ) \; \equiv \; 0. \end{aligned}$$

Setting \(\xi :=x ( a,a' )\) and partially differentiating the above with respect to \(a\) and \(a'\) gives:

$$\begin{aligned} x_{1} ( a,a' ) \int \limits _{a}^{a'} U_{11} ( \xi , \theta ) \mathrm {d}F ( \theta ) +f ( a' ) U_{1} ( \xi ,a' )&= 0 \\ x_{2} ( a,a' ) \int \limits _{a}^{a'} U_{11} ( \xi , \theta ) \mathrm {d}F ( \theta ) -f ( a ) U_{1} ( \xi ,a )&= 0. \end{aligned}$$

Summing the above, we have

$$\begin{aligned} 0&= \left( x_{1} ( a,a' ) +x_{2} ( a' ,a' ) \right) \int \limits _{a}^{a'} U_{11} ( \xi , \theta ) \mathrm {d}F ( \theta ) \\&\quad +f ( a' ) U_{1} ( \xi ,a' ) -f ( a ) U_{1} ( \xi ,a ) \\&= \left( x_{1} ( a,a' ) +x_{2} ( a' ,a' ) \right) \int \limits _{a}^{a'} U_{11} ( \xi , \theta ) \mathrm {d}F ( \theta ) \\&\quad + \hat{G} ( \xi ,a' ) - \hat{G} ( \xi ,a ) - \int \limits _{a}^{a'} U_{11} ( \xi , \theta ) \mathrm {d}F ( \theta ) \\&= \left( x_{1} ( a,a' ) +x_{2} ( a' ,a' ) -1 \right) \int \limits _{a}^{a'} U_{11} ( \xi , \theta ) \mathrm {d}F ( \theta ) + \hat{G} ( \xi ,a',0 ) - \hat{G} ( \xi ,a,0 ) \\&\le \left( x_{1} ( a,a' ) +x_{2} ( a' ,a' ) -1 \right) \int \limits _{a}^{a'} U_{11} ( \xi , \theta ) \mathrm {d}F ( \theta ) , \end{aligned}$$

where the inequality is from the hypothesis of the Claim, that \(\hat{G} ( \xi ,a,0 )\) is non-increasing in \(a\). The fact that \(U_{11} <0\) yields (9). Write

$$\begin{aligned} V^{i} ( \varvec{a} )&:= V \left( a_{i-1}^{*} ,a_{i}^{*} ,a_{i+1}^{*} ,0 \right) f\left( a^{*}_{i} \right) = \left( U_{1}^{r} \left( x^{i} ,a_{i} \right) -U_{1}^{r} \left( x^{i+1} ,a_{i} \right) \right) f \left( a^{*}_{i} \right) \end{aligned}$$

where \(x^{i} :=x ( a^{*}_{i-1} ,a^{*}_{i} )\) and for convenience, set \(n=N-1\). Then,

$$\begin{aligned} H ( \Pi _{N} ) ( \varvec{a}^{*}_{N} )&= \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \alpha _{1} &{}\quad \gamma _{1} &{}\quad &{}\quad \ldots &{} 0\\ \beta _{1} &{}\quad \alpha _{2} &{}\quad \ddots &{} &{}\quad \vdots \\ &{}\quad \beta _{2} &{}\quad \ddots &{}\quad \gamma _{n-2} &{}\quad 0\\ \vdots &{}\quad &{}\quad \ddots &{}\quad \alpha _{n-1} &{}\quad \gamma _{n-1}\\ 0 &{}\quad &{}\quad \cdots &{}\quad \beta _{n-1} &{}\quad \alpha _{n} \end{array} \right) \end{aligned}$$

where \(\alpha _{i} = \dfrac{\partial V^{i} ( \varvec{a}^{*}_{N} )}{\partial a_{i}}\) for \(i=1, \ldots ,n\), \(\beta _{i} = \frac{\partial V^{i} ( \varvec{a}^{*}_{N} )}{\partial a_{i+1}} \) and \(\gamma _{i} = \frac{\partial V^{i+1} ( \varvec{a}^{*}_{N} )}{\partial a_{i}} \) for \(i=1, \ldots ,n-1\). Of course, by symmetry of the Hessian, we have \(\beta _{i} = \gamma _{i}\). Therefore, for any vector \(\varvec{y} = ( \xi _{1} , \ldots , \xi _{n} ) \ne \mathbf {0}\), setting \(\beta _0= V_1( 0,a_1^*,a_2^*)f(a_1)\) and \(\beta _N= V_3 ( a_{n-1}^*,a_{n}^*,1)f(a_n)\) we have

$$\begin{aligned} \varvec{y}^{t} H \left( \varvec{a}^{*}_{N}\right) \varvec{y}&= \sum _{i=1}^{n} \alpha _{i} \xi _{i}^{2} +2 \sum _{i=1}^{n-1} \beta _{i}\xi _{i} \xi _{i+1}. \end{aligned}$$

Note that \(\beta _{i} >0\) (see Lemma 2 in CS for instance). Assume, for the moment, that

$$\begin{aligned} \alpha _{i}&\le - ( \beta _{i-1} + \beta _{i} ). \end{aligned}$$


$$\begin{aligned} \varvec{y}^{t} H \left( \varvec{a}^{*}_{N} \right) \varvec{y}&\le - \sum _{i=1}^{n-1} \beta _{i} \left( \xi _{i}^{2} + \xi _{i+1}^{2} -2 \xi _{i} \xi _{i+1}\right) - \beta _{0} \xi _{1}^{2} - \beta _{n} \xi _{n}^{2} \\&= - \sum _{i=1}^{n-1} \beta _{i} ( \xi _{i} - \xi _{i+1} )^{2} - \beta _{0} \xi _{1}^{2} - \beta _{n} \xi _{n}^{2} <0. \end{aligned}$$

In other words, \(H ( \varvec{a}^{*}_{N} )\) is negative definite, and hence invertible. So, it remains to show (10) to complete the proof. Using the notation \(x^{i} =x ( a_{i-1}^{*} ,a_{i}^{*} )\) and \(x^{i}_{j} =x_{j} ( a_{i-1}^{*} ,a_{i}^{*} )\), for \(j=1,2\),

$$\begin{aligned} \frac{1}{f(a^{*}_{i})} \times \frac{\partial V^{i}}{\partial a_{i-1}}&= x_{1}^{i} U_{1} \left( x^{i} ,a_{i}^{*} \right) \\ \frac{1}{f(a^{*}_{i})} \times \frac{\partial V^{i}}{\partial a_{i+1}}&= -x_{2}^{i+1} U_{1} \left( x^{i+1} ,a_{i}^{*}\right) \\ \frac{1}{f(a^{*}_{i})} \times \frac{\partial V^{i}}{\partial a_{i}}&= x_{2}^{i} U_{1} \left( x^{i} ,a_{i}^{*} \right) +U_{2} \left( x^{i} ,a_{i}^{*} \right) \\&\quad -x_{1}^{i+1} U_{1} \left( x^{i+1} ,a_{i}^{*} \right) -U_{2} \left( x^{i+1} ,a_{i}^{*} \right) . \end{aligned}$$

Therefore, for \(i=1, \ldots ,n\),

$$\begin{aligned} \frac{1}{f(a^{*}_{i})} \times \left( \frac{\partial V^{i}}{\partial a_{i-1}} + \frac{\partial V^{i}}{\partial a_{i}} + \frac{\partial V^{i}}{\partial a_{i+1}}\right)&= \left( x^{i}_{1} +x_{2}^{i}\right) U_{1} \left( x^{i} ,a_{i}^{*} \right) +U_{2} \left( x^{i} ,a_{i}^{*} \right) \\&\quad - \left( x_{1}^{i+1} +x_{1}^{i+1}\right) U_{1}\left( x^{i+1} ,a_{i}^{*} \right) \\&\quad -U_{2} \left( x^{i+1} ,a_{i}^{*}\right) . \end{aligned}$$

Since \(x^{i} <a^{*}_{i} <x^{i+1}\), for all \(i\), \(U_{1} ( x^{i} ,a_{i}^{*} ) >0>U_{1} ( x^{i+1} ,a_{i}^{*} )\). Using (9), we then have

$$\begin{aligned} \frac{\partial V^{i}}{\partial a_{i-1}} + \frac{\partial V^{i}}{\partial a_{i}} + \frac{\partial V^{i}}{\partial a_{i+1}}&\le {f(a^{*}_{i})} \times \left( G \left( x^{i} ,a_{i}^{*},0 \right) -G\left( x^{i+1} ,a_{i}^{*},0 \right) \right) \le 0, \nonumber \end{aligned}$$

which in turn completes the proof of (10) for \(i=1, \ldots ,n\).

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Agastya, M., Bag, P.K. & Chakraborty, I. Proximate preferences and almost full revelation in the Crawford–Sobel game. Econ Theory Bull 3, 201–212 (2015).

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