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A global game with heterogenous priors

Abstract

This paper relaxes the common prior assumption in the public and private information game of Morris and Shin (Eur Econ Rev 48(1):133–153, 2004). For the generalized game, where the agents’ prior expectations are heterogenous, it derives a sharp condition for the emergence of unique/multiple equilibria. This condition indicates that unique equilibria are played if players’ public disagreement is substantial. If disagreement is small, equilibrium multiplicity depends on the relative precisions of private signals and subjective priors. Extensions to environments with public signals of exogenous and endogenous quality show that prior heterogeneity, unlike heterogeneity in private information, provides a robust anchor for unique equilibria. Finally, irrespective of whether priors are common or not, we show that public signals can ensure equilibrium uniqueness, rather than multiplicity, if they are sufficiently precise.

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Fig. 1
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Notes

  1. In the “private information limit”, the private signal’s precision goes to infinity. A crucial consequence of this is that the importance of priors for the agent’s decisions vanishes.

  2. Izmalkov and Yildiz (2010) discuss prior heterogeneity in a game with a continuum of players. Izmalkov and Yildiz (2010), p. 25, focus on the formation of individual threshold strategies and avoid the “delicate” question of equilibrium multiplicity, which is our focus

  3. Steiner and Stewart (2008), Izmalkov and Yildiz (2010), and Mathevet (2014) abstract from public signals.

  4. In the context of the currency crises model of Morris and Shin (1998), attacking agents would short a country’s currency, and the central bank’s reserves \(\theta \) are either sufficient \((\theta >A)\) to defend the peg or not \((\theta <A)\). Alternatively, agents can run on a firm’s debt. The firm defaults if it has too few reserves.

  5. Raiffa and Schlaifer (2000), p. 250, for prior and posterior distributions of normally distributed variables which are used throughout the paper. Note also that we already use the (forthcoming) critical mass condition, (4), which requires that \(\theta ^*= A(\psi ^*,\theta ^*)\), and replace A with \(\theta ^*\) in (1) to obtain (2).

  6. That is, if (7) holds, then (6) never holds with equality for real-valued \(\theta ^{*}\)’s. Put differently, the polynomial which characterizes those values \(\theta ^*\), for which (6) holds with equality, has two complex roots.

  7. Indeed, if there are no private signals, the PIC reads \(\Phi \left( \sqrt{\alpha _p}(\theta ^*-\mu ^*)\right) =c\) and the CMC \(\Phi \left( \frac{1}{\sigma _{\mu }}(\mu ^*-E[\mu ])\right) =\theta ^*\), such that the uniqueness condition (7) becomes \(\sigma _{\mu }\ge \frac{1}{\sqrt{2\pi }}\).

  8. That is, we assume that agents trade stocks prior to the coordination game. These stocks are traded at a market price P and pay an unknown amount \(\theta \). This market price will, in equilibrium, aggregate dispersed private information and reveal the true fundamental \(\theta \) partially. Where the partial revelation is due to aggregate noise-trader activity, \(\sigma _{\varepsilon }\varepsilon \), \(\varepsilon \sim \mathcal {N}(0,1)\), on the asset’s supply side.

  9. Note that increases in the prior’s dispersion reduce the public signal’s precision \(\alpha _z=\frac{\alpha _x^2\alpha _{\psi }}{\sigma ^2_{\varepsilon }\alpha ^2}=\frac{\alpha _x^2}{\sigma _{\varepsilon }^2(\alpha _x+\alpha _p^2\sigma _{\mu }^2)}\); for intermediate values of \(\sigma _{\mu }\), it is therefore not necessarily true that increases in \(\sigma _{\mu }\) always favor uniqueness.

  10. Note that for \(y=\Phi ^{-1}(\theta ^{*})\), we have \(\frac{d\theta ^{*}}{dy}=\phi (y)\) and thus \(\frac{dy}{d\theta ^{*}}=\frac{1}{\phi (y)}=\frac{1}{\phi (\Phi ^{-1}(\theta ^{*}))}\).

  11. The existence of at least one solution is ensured. It follows from (22) that \(lim_{S\rightarrow \infty }\frac{\Phi (\theta ^*(S))^{-1}}{S}\) and \(lim_{S\rightarrow -\infty }\frac{\Phi (\theta ^*(S))^{-1}}{S}\) are constants. Rewriting \(Z(S)=\frac{\alpha }{\alpha _x}\psi ^*(S)-\frac{\alpha }{\alpha _x}\frac{1}{\sqrt{\alpha _{\psi }}}S\) as \(Z(S)=\frac{\alpha }{\alpha _x}S(\frac{\psi ^*(S)}{S}-\frac{\alpha }{\alpha _x}\frac{1}{\sqrt{\alpha _{\psi }}})\) and recalling \(\psi (\theta ^*(S))\) as given in (22), one can show that Z(S) varies with S between \(\infty \) and \(-\infty \).

  12. \(\sqrt{\frac{\alpha _x}{(\alpha _p+\alpha _z)^2}+\frac{\sigma _{\mu }^2}{1+\frac{\alpha _z}{\alpha _p}}}\le \sqrt{\frac{(\alpha _x+\alpha _z)^2}{\alpha _x\alpha _p^2}+\frac{\sigma _{\mu }^2(\alpha _x+\alpha _z)^2}{\alpha _x^2}}\) follows from the inequalities \(\frac{\alpha _x}{(\alpha _p+\alpha _z)^2}\le \frac{(\alpha _x+\alpha _z)^2}{\alpha _x\alpha _p^2}\) and \(\frac{\sigma _{\mu }^2}{1+\frac{\alpha _z}{\alpha _p}}\le \frac{\sigma _{\mu }^2(\alpha _x+\alpha _z)^2}{\alpha _x^2}\), which are easy to verify.

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Correspondence to Wolfgang Kuhle.

Additional information

I thank Sophie Bade, Nataliya Demchenko, David Frankel, Alia Gizatulina, Olga Gorelkina, Dominik Grafenhofer, Martin Hellwig, Paul Klemperer, Flavio Toxvaerd, Carl Christian von Weizsäcker, the referee, conference and seminar participants in Bonn, Stony Brook, Tel Aviv, and the 2014 ESEM Toulouse for helpful and encouraging discussions. First arxiv version December 2013.

Appendix: Proof of Propositions 4 and 5

Appendix: Proof of Propositions 4 and 5

We start by laying out the equations that describe equilibria. In turn, we characterize the equilibria described in Propositions 4 and 5 in two separate paragraphs.

$$\begin{aligned}&S=\Phi ^{-1}(A)+\sigma _{\varepsilon }\varepsilon , \quad \varepsilon \sim \mathcal {N}(0,1) \end{aligned}$$
(15)
$$\begin{aligned}&A= P(\psi \le \psi ^{*}(S)|\theta )=\theta \end{aligned}$$
(16)
$$\begin{aligned}&P(\theta \le \theta ^{*}|x,\mu ,S)=c. \end{aligned}$$
(17)

To calculate equilibria, we recall that agents act on \(x=\theta +\sigma _x\xi \), with \(\xi \sim \mathcal {N}(0,1)\) and \(\theta |\mu \sim \mathcal {N}(\mu ,\sigma _p^2)\), where the prior \(\mu \) is distributed over the population as \(\mu \sim \mathcal {N}(E[\mu ],\sigma _{\mu })\). Moreover, we define \(\psi =\frac{\alpha _x}{\alpha }x+\frac{\alpha _p}{\alpha }\mu \) with \(\alpha =\alpha _x+\alpha _p+\alpha _z\). The PIC (17) now writes as:

$$\begin{aligned} \Phi \left( \sqrt{\alpha }(\theta ^*-\psi ^*-\frac{\alpha _z}{\alpha }Z)\right) =c; \quad \alpha =\alpha _x+\alpha _p+\alpha _z. \end{aligned}$$
(18)

Again, (18) defines a critical \(\psi ^*(Z)\) such that agents attack if \(\psi \le \psi ^*\) and do not attack if \(\psi >\psi ^*\). To calculate the mass of attacking agents, we note that \(\psi |\theta \sim \mathcal {N}(\frac{\alpha _x}{\alpha }\theta +\frac{\alpha _p}{\alpha }E[\mu ],(\frac{\alpha ^2}{\alpha _x+\alpha ^2_p\sigma ^2_{\mu }})^{-1})\). Once we define \(\alpha _{\psi }=\frac{\alpha ^2}{\alpha _x+\alpha ^2_p\sigma ^2_{\mu }}\), the CMC (16) can be written as:

$$\begin{aligned} A=P(\psi <\psi ^*|\theta ^*)=\theta ^*\quad \Leftrightarrow \quad \Phi \left( \sqrt{\alpha _{\psi }}(\psi ^*-\frac{\alpha _x}{\alpha }\theta ^*-\frac{\alpha _p}{\alpha }E[\mu ])\right) =\theta ^*.\qquad \ \ \end{aligned}$$
(19)

Using this expression for the aggregate attack A in condition (19), we can return to the public signal S in (15) and write:

$$\begin{aligned} S=\sqrt{\alpha _{\psi }}(\psi ^*-\frac{\alpha _x}{\alpha }\theta -\frac{\alpha _p}{\alpha }E[\mu ])+\sigma _{\varepsilon }\varepsilon . \end{aligned}$$
(20)

where S in (20) is informationally equivalent to a signal

$$\begin{aligned} Z(S)=\frac{\alpha }{\alpha _x}\psi ^*(S)-\frac{\alpha }{\alpha _x}\frac{1}{\sqrt{\alpha _{\psi }}}S=\theta +\frac{\alpha _p}{\alpha _x}E[\mu ] -\sigma _{\varepsilon }\frac{\alpha }{\alpha _x}\frac{1}{\sqrt{\alpha _{\psi }}}\varepsilon . \end{aligned}$$
(21)

Regarding (21), we note that Z(S) contains two aspects (i) Z is a noisy public signal which reveals the true state of the economy \(\theta \) with precision \(\alpha _z=\frac{\alpha _x^2\alpha _{\psi }}{\sigma ^2_{\varepsilon }\alpha ^2}=\frac{\alpha _x^2}{\sigma _{\varepsilon }^2(\alpha _x+\alpha _p^2\sigma _{\mu }^2)}\) and (ii) the signal S allows agents to align their strategies \(\psi ^*(S)\). That is, for every given \(\bar{Z}\), there may be several S such that \(Z(S)=\bar{Z}\). That is, there is a potential source of equilibrium multiplicity, concerning S, to which we turn in Paragraph 2. For now, we take S as given and study the threshold equilibria \(\theta ^*(S),\psi ^*(S)\).

Proof of Proposition 4: multiplicity in thresholds \(\theta ^*\)

For every given signal S, we rewrite (19) as:

$$\begin{aligned} \psi ^*=\Phi ^{-1}(\theta ^*)\frac{1}{\sqrt{\alpha _{\psi }}}+\frac{\alpha _x}{\alpha }\theta ^*+\frac{\alpha _p}{\alpha }E[\mu ]. \end{aligned}$$
(22)

To obtain an equation in \(\theta ^*\) only, we substitute \(Z(S)=\frac{\alpha }{\alpha _x}\psi ^*(S)-\frac{\alpha }{\alpha _x}\frac{1}{\sqrt{\alpha _{\psi }}}S\) and (22) into (18). Rearranging then yields:

$$\begin{aligned} \Phi \left( \sqrt{\alpha }(\frac{\alpha _p}{\alpha }\theta ^* -\frac{1}{\sqrt{\alpha _{\psi }}}\frac{\alpha _x+\alpha _z}{\alpha _x}\Phi ^{-1}(\theta ^*) -\frac{(\alpha _x+\alpha _z)\alpha _p}{\alpha _x\alpha }E[\mu ] +\frac{\alpha _z}{\alpha _x}\frac{1}{\sqrt{\alpha _{\psi }}}S)\right) =c\nonumber \\ \end{aligned}$$
(23)

To derive the uniqueness condition, which ensures that there exists only one \(\theta ^*(S)\) for every given signal S, we differentiate (23) with respect to \(\theta ^*\):Footnote 10

$$\begin{aligned} \phi (\Phi (\theta ^*)^{-1})\le \frac{(\alpha _x+\alpha _z)}{\alpha _x}\sqrt{\left( \frac{\sqrt{\alpha _x}}{\alpha _p}\right) ^2+\sigma _{\mu }^2}, \end{aligned}$$
(24)

and hence, threshold equilibria are always unique iff \(\frac{1}{\sqrt{2\pi }}\le \frac{(\alpha _x+\alpha _z)}{\alpha _x}\sqrt{\left( \frac{\sqrt{\alpha _x}}{\alpha _p}\right) ^2+\sigma _{\mu }^2}\). Otherwise, if \(\frac{1}{\sqrt{2\pi }}\ge \frac{(\alpha _x+\alpha _z)}{\alpha _x}\sqrt{\left( \frac{\sqrt{\alpha _x}}{\alpha _p}\right) ^2+\sigma _{\mu }^2}\), there may exist up to three threshold equilibria

\(\theta ^*_1(S),\psi _1^*(S);\theta ^*_2(S),\psi _2^*(S);\theta ^*_3(S),\psi _3^*(S)\) for every given signal value S.

Proof of Proposition 5: multiplicity in signals \(S^*\)

To preclude multiple solutionsFootnote 11 \(S(\bar{Z})\) to the equation \(Z(S)=\bar{Z}\), where \(Z(S)=\frac{\alpha }{\alpha _x}\psi ^*(S)-\frac{\alpha }{\alpha _x}\frac{1}{\sqrt{\alpha _{\psi }}}S\), it will suffice to show that \(\frac{\partial Z(S)}{\partial S}_{|(21)}=\frac{\alpha }{\alpha _x}\frac{\partial \psi ^*}{\partial S}-\frac{\alpha }{\alpha _x}\frac{1}{\sqrt{\alpha _{\psi }}}\le 0\). To calculate the derivative \(\frac{\partial \psi (\theta ^*(S))}{\partial S}=\frac{\partial \psi ^*}{\partial \theta ^*}\frac{\partial \theta ^*}{\partial S}\), defined by (22) and (23), we differentiate (23) which yields \(\frac{\partial \theta ^*}{\partial S}=\frac{-\frac{\alpha _z}{\alpha _x}\frac{1}{\sqrt{\alpha _{\psi }}}}{\frac{\alpha _p}{\alpha } -\frac{1}{\sqrt{\alpha _{\psi }}}\frac{\alpha _x+\alpha _z}{\alpha _x}\frac{1}{\phi (\Phi (\theta ^*)^{-1})}}\) and (22), (which is a 1 : 1 mapping between \(\psi ^*\) and \(\theta ^*\)), to obtain \(\frac{\partial \psi ^*}{\partial \theta ^*}=\frac{1}{\sqrt{\alpha _{\psi }}\phi (\Phi (\theta ^*)^{-1})}+\frac{\alpha _x}{\alpha }\). Hence, we have

$$\begin{aligned} \frac{\partial Z(S)}{\partial S}= & {} \frac{\alpha }{\alpha _x}\frac{\partial \psi ^*}{\partial S}-\frac{\alpha }{\alpha _x}\frac{1}{\sqrt{\alpha _{\psi }}}=\frac{\alpha }{\alpha _x}\frac{\partial \psi ^*}{\partial \theta ^*}\frac{\partial \theta ^*}{\partial S}-\frac{\alpha }{\alpha _x}\frac{1}{\sqrt{\alpha _{\psi }}}\nonumber \\= & {} \underset{-}{\underbrace{-\frac{\alpha }{\alpha _x}\frac{1}{\sqrt{\alpha _{\psi }}}}} +\underset{+}{\underbrace{\left( \frac{1}{\sqrt{\alpha _{\psi }}\phi (\Phi (\theta ^*)^{-1})}+\frac{\alpha _x}{\alpha }\right) }} \underset{+/-}{\underbrace{\frac{-\frac{\alpha _z}{\alpha _x}\frac{1}{\sqrt{\alpha _{\psi }}}}{\frac{\alpha _p}{\alpha } -\frac{1}{\sqrt{\alpha _{\psi }}}\frac{\alpha _x+\alpha _z}{\alpha _x}\frac{1}{\phi (\Phi (\theta ^*)^{-1})}}}}\frac{\alpha }{\alpha _x}.\nonumber \\ \end{aligned}$$
(25)

Once we recall that \(\alpha _{\psi }=\frac{\alpha ^2}{\alpha _x+\alpha ^2_p\sigma ^2_{\mu }}\), rearranging (25) gives:

$$\begin{aligned} \frac{\partial Z(S)}{\partial S}= & {} -\frac{\sqrt{\alpha _x+\alpha ^2_p\sigma ^2_{\mu }}}{\alpha _x}+\left( \sqrt{\alpha _x+\alpha ^2_p\sigma ^2_{\mu }} +\alpha _x\phi (\Phi (\theta ^*)^{-1})\right) \nonumber \\&\quad \times \frac{-\frac{\alpha _z}{\alpha _x}}{\frac{\phi (\Phi (\theta ^*)^{-1})\alpha _p\alpha _x}{\sqrt{\alpha _x+\alpha ^2_p\sigma ^2_{\mu }}}-(\alpha _x+\alpha _z)}\nonumber \\= & {} \left[ -(\alpha _p+\alpha _z)\phi (\Phi (\theta ^*)^{-1})+\sqrt{\alpha _x+\alpha ^2_p\sigma ^2_{\mu }}\right] \frac{1}{\frac{\phi (\Phi (\theta ^*)^{-1})\alpha _p\alpha _x}{\sqrt{\alpha _x+\alpha ^2_p\sigma ^2_{\mu }}}-(\alpha _x+\alpha _z)} \nonumber \\ \end{aligned}$$
(26)

From (26), and the fact that \(\theta ^*\in (0,1)\), it follows that equilibria in signals are unique if

$$\begin{aligned} \frac{1}{\sqrt{2\pi }}\le \sqrt{\left( \frac{\sqrt{\alpha _x}}{\alpha _p+\alpha _z}\right) ^2+\frac{\sigma _{\mu }^2}{1+\frac{\alpha _z}{\alpha _p}}}, \quad \alpha _z=\frac{\alpha _x^2}{\sigma _{\varepsilon }^2(\alpha _x+\alpha _p^2\sigma _{\mu }^2)} \end{aligned}$$
(27)

and

$$\begin{aligned} \frac{1}{\sqrt{2\pi }}\le \sqrt{\frac{(\alpha _x+\alpha _z)^2}{\alpha _x\alpha _p^2}+\frac{\sigma _{\mu }^2(\alpha _x+\alpha _z)^2}{\alpha _x^2}}. \end{aligned}$$
(28)

That is, once (27) and (28) hold, we have \(\frac{\partial Z(S)}{\partial S}<0\), which ensures unique solutions \(S(\bar{Z})\) to the equation \(Z(S)=\bar{Z}\). Comparison shows that inequality (28) is less restrictive than (27).Footnote 12 Evaluation of (27) therefore yields: (i) In the limit where \(\sigma _{\mu }\rightarrow \infty \), equilibria in signals are unique. (ii) In the limit where \(\sigma _{\varepsilon }\rightarrow 0\) such that \(\alpha _{z}\rightarrow \infty \), there exist multiple equilibria in signals. (iii) Multiple equilibria are ensured in the limit where \(\alpha _{x}\rightarrow \infty \). (iv) Once \(\sigma ^2_{\mu }=0\) and \(\alpha _p=0\), condition (27) collapses into the standard uniqueness condition \(\sqrt{2\pi }\le \frac{\sqrt{\alpha _x}}{\alpha _z}\).

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Kuhle, W. A global game with heterogenous priors. Econ Theory Bull 4, 167–185 (2016). https://doi.org/10.1007/s40505-015-0075-7

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Keywords

  • Global games
  • Equilibrium selection
  • Heterogenous priors
  • Thin out effect

JEL Classification

  • D53
  • D83