Accessible Unlicensed Requires Authentication Published by De Gruyter January 17, 2018

Strategic Investment under Incomplete Information

Mohammad H. Dehghani

Abstract

This paper studies how hiding sunk cost of investment would affect investment strategies in a duopoly. The investment would improve profit. If this improvement is larger for the first mover than the second mover, this study finds a unique symmetric equilibrium for a subset of such cases. On the other hand, a larger improvement for the second mover results in a class of symmetric equilibria. For the first case, the surplus to sharing information increases with higher volatility of profit flow and lower uncertainty about the investment cost. For the second case, this surplus grows with both mentioned types of uncertainty.

JEL Classification: C7; D81; D82

A Appendix

A.1 Derivation of WiF(x;xjC,xiF)

Here, I explain how to derive WiF(x;xjC,xiF). I use the standard investment model as described by Dixit and Pindyck (1994). The Bellman equation for the follower before the leader invests is

(37)rWiF(x;xjC,xiF)dt=xD00dt+E[dWiF(x;xjC,xiF)].

Using Ito’s lemma, the Bellman equation leads to the following differential equation:

(38)rWiF(x;xjC,xiF)=xD00+μxddxWiF(x;xjC,xiF)+12σ2x2d2dx2WiF(x;xjC,xiF).

The general solution to the above equation is

(39)WiF(x;xjC,xiF)=xD00rμ+Axβ+Bxλ,

where λ is

(40)λ=12μσ212μσ22+2rσ2<0.

The value function needs to satisfy two boundary conditions. The first one is

(41)WiF(0;xjC,xiF)=0.

This condition is necessary to make sure that when x=0, the firm has no value. It follows from this condition that B=0. The second condition is necessary to make sure the value function for the follower is continuous at the threshold the leader invests.

(42)WiF(xjC;xjC,xiF)=ViF(xjC;xiF)

Note that in eq. (4), for x=xjC,

(43)xD00rμxjC(D00D01)rμxxjCβ=0,

and what remains is ViF(xjC;xiF). Hence, the function described for x<xjC is the solution to eq. (38).

A.2 Lemmas Related to Section 4

Proof of Lemma 4.1. In order to show that supFi=supFj, suppose it does not hold and without loss of generality assume supFi<supFj. Then it is optimal for firm i with the investment cost IU to invest at supFi. As x gets close to supFi, firm j knows that firm i will invest in any moment with probability one. Since it is optimal for firm i with investment cost IU to invest at supFi, it would be optimal for firm j with investment cost arbitrarily close to IU to invest at supFi or before. This contradicts the initial assumption that supFi<supFj.

Proof of Lemma 4.2. Suppose that in equilibrium xL(IL)<infFj. It follows from D10D00>D11D01 that xiL<xiF for any value of investment cost, and value of xL(IL) is less than any possible value of xjF. Also, from k=1, it follows that for firm i with the investment cost IL, investing as a first mover at xL(IL), is preferred to waiting for simultaneous investment at any threshold larger than xL(IL). Thus, there is no reason firm i with the investment cost IL to invest later than xL(IL) and it is optimal to invest at xL(IL) and it should be that infFi=xiL(IL). Since firm j knows that firm i invests at xL(IL) or before with zero probability, it is also optimal for firm j with the investment cost IL to invest at xL(IL). This contradicts with the assumption that xL(IL)<infFj. As a result infFjxL(IL) for j=1,2.

Note that it cannot be the case that infFi<infFjxL(IL), because firm i with the lowest investment cost prefers to invest at any threshold in (infFi,infFj), rather than infFi. Hence, infFi=infFj. In conclusion, if D10D00>D11D01 and k=1, in any equilibrium, infFi=infFjxL(IL).

Proof of Lemma 4.3. The first order derivative is f(y)=1β1yβ11. It is zero only at y=1 and the second order derivative is positive. Hence, for any y1, f(y)>f(1)=0.

Lemma A.1

The optimal value of xiI for firm i is not larger than xiS.

Proof of Lemma A.1. Since D10>D11, from eqs (10) and (9) it follows that xiL<xiS. Given that firm j has not invested yet, firm i anticipates that there are two possible outcomes after investing at xiI, firm j either invests at xiI or waits and invests after a delay. If firm i knew with certainty that firm j would not invest at xiI, then it would not be optimal to differ the investment after xiL is reached, and the optimal xiI could not be larger than xiL. Also, if firm i knew with certainty that firm j would invest immediately at xiI, then it would not be optimal to differ the investment after xiS is reached, and the optimal xiI could not be larger than xiS. The payoff keeps decreasing after the optimal threshold is reached. Since xiL<xiS, when firm i does not know which of these two possible outcomes will happen, it is never an optimal choice to delay the investment after xiS is reached.

A.3 Proof of Proposition 4.4.

It follows from D10D00>D11D01 that xiL<xiF. Adding this to the assumption infFjxiL guarantees that infFj<xiF. For any xiI[infFj,xiF], the payoff function is

WiI(x,xˆ;xiI)=11G(Ij(xˆ))[maxIj(xˆ),ILIj(xiI)WiF(x;xjI(Ij),xiF)dG(Ij)
(44)]+Ij(xiI)IUWiL(x;xiI,xF(Ij))dG(Ij).

The derivative of the above function with respect to xiI is

(45)ddxiIWiI(x,xˆ;xiI)=ddxiIEIj|xˆ[WiL(x;xiI,xF(Ij))]
dG(Ij(xiI)|xˆ)dxiIWiL(x;xiI,xF(Ij(xiI)))WiF(x;xjI(Ij(xiI)),xiF).

I use eqs (8), (9) and (10) to replace any Dninj with a combination of xiF, xiL and xiS. This gives more tractable and intuitive equations. Also remember that x and xˆ are smaller than any investment threshold. The derivative of WiI(x,xˆ;xiI) with respect to xiI at any xiI[infFj,xiF], is

ddxiIWiI(x,xˆ;xiI)=βIi1G(Ij(xˆ))xxiIβ{1G(Ij(xiI))1xiI1xiL
(46)}+G(Ij(xiI))Ij(xiI)fxiIxiF+xiIβ11xiL1xiSxiIxF(Ij(xiI))β11.

Setting the above derivative equal to zero, gives eq. (16).

If xiF<supFj, for any xiI in the interval [xiF,supFj], the payoff function is

WiI(x,xˆ;xiI)=11G(Ij(xˆ))[maxIj(xˆ),ILIj(xiF)WiF(x;xjI(Ij),xiF)dG(Ij)
(47)]+Ij(xiF)Ij(xiI)WiS(x;xjI(Ij))dG(Ij)+Ij(xiI)IUWiL(x;xiI,xF(Ij))dG(Ij)

and the derivative of WiI(x,xˆ;xiI) with respect to xiI is

ddxiIWiI(x,xˆ;xiI)=βIi1G(Ij(xˆ))xxiIβ{1G(Ij(xiI))1xiI1xiL
(48)}+G(Ij(xiI))Ij(xiI)xiIβ11xiL1xiSxiIxF(Ij(xiI))β11.

From Lemma 4.3 it follows that f(1)=0, as a result

(49)limxiIxiFddxiIWiI(x,xˆ;xiI)=limxiIxiFddxiIWiI(x,xˆ;xiI).

The value of xiL is less than xiS by definition and xiL<xiF as I showed before. Also from xjI(Ij)xF(Ij), by replacing Ij with Ij(xiI), it follows that xiI<xF(Ij(xiI)). Hence, value of the derivative in eq. (48) is negative for any xiI(xiF,supFj].

Because xiLinfFj, for xiI<infFjinfXF, the derivative ddxiIWiI(x,xˆ;xiI), which is equal to ddxiIEIj|xˆ[WiL(x;xiI,xF(Ij))], has a positive value. Also, choosing xiI>supFj is not any better than choosing supFj, because the other firm invests by the time supFj is reached.

The payoff function is continuous everywhere and differentiable inside Fj, the derivative is positive for xiI<infFj and negative for xiI>xiF. Hence, the best choice of xiI for firm i exists and it is in interval [infFj,xiF], where the derivative given by eq. (46) is equal to zero.

A.4 Proof of Proposition 4.5.

It follows directly from Proposition 4.4 that, if I1(x)=I2(x)=I(x) as characterized in eq. (22), then it is an equilibrium and I1(.)xF(.). Note that all the assumptions of Proposition 4.4 hold in this equilibrium.

To show the uniqueness, first I show that there is no equilibrium in which xiI>xiF for any firm with any investment cost. There is no equilibrium in which xF(Ij)<xjI(Ij) for any Ij in G, because it contradicts Lemma 4.2. Consider the only remaining case, an equilibrium in which xF(Ij)xjI(Ij) changes sign at some IjG. Let I˜j be the smallest zero of xF(Ij)xjI(Ij) and without loss of generality assume that I˜iI˜j. It follows from the assumption of this proposition that xiL<xiF, hence, Lemma 4.2 guarantees that xF(IL) is greater than infFj. As a result xF(Ij)xjI(Ij)>0 for any Ij less than I˜i, for j=1,2. Thus eq. (46) gives the value of left derivative of payoff function. For a firm i that invests at I˜i in equilibrium, it follows that xiF=xiI(I˜i)<xF(Ij(xiI(I˜i))). Thus, the derivative at x˜i=xiI(I˜i) from left side is

limxiIx˜iddxiIWiI(x,x^;xiI)=βIi1G(j(x^))(xx˜i)β{(1G(j(x˜i)))(1xiF1xiL)
(50)}+G(Ij(x˜i))Ij(x˜i)f1+x˜iβ11xiL1xiSx˜ixF(Ij(x˜i))β11

and has a negative value. Hence, there is some value of xiI less than x˜i that is preferred to x˜i and this contradicts with x˜i being chosen in equilibrium. As a result, in any equilibrium, it holds that xiIxiF, and any equilibrium must be a solution to problem eq. (18). Since the boundary conditions for the functions I1(x) and I2(x) are the same, there is no asymmetric solution that solves eq. (18). See Lambrecht and Perraudin (2003) for more details on a similar problem.

In conclusion, there is no equilibrium other than what is characterized by eq. (22) and always it holds that I1(.)xF(.).

A.5 Proof of Proposition 4.7.

To prove Proposition 4.7, first note that it follows from D11D00<D10D00D11D01 that xiFxiL<xiS.

Since infXFxiFxiL, any xiI smaller than infXF is also smaller than xiL. Hence, for any xiI<infXF, the derivative ddxiIWiI(x,xˆ;xiI), which is equal to ddxiIE[WiL(x;xiI,xF(Ij))], has a positive value and no xiI<infXF is an optimal choice for firm i’s investment threshold.

For any xiI[infXF,mininfFj,supXF], the payoff is

WiI(x,xˆ;xiI)=11G(Ij(xˆ))[WiS(x;xiI)maxIj(xˆ),ILxF1(xiI)dG(Ij)
(51)]+xF1(xiI)IUWiL(x;xiI,xF(Ij))dG(Ij),

and the derivative of the payoff function with respect to xiI is

(52)ddxiIWiI(x,xˆ;xiI)=βIi1G(Ij(xˆ))xxiIβ1xiI1G(xF1(xiI))xiLG(xF1(xiI))xiS.

If xiI<xiF, then eq. (52) is positive, because xiI<xiFxiL<xiS. If infFj<xiF, for any xiI<xiF in support of xjI, the payoff function is

(53)WiI(x,xˆ;xiI)=11G(Ij(xˆ))[maxIj(xˆ),ILIj(xiI)WiF(x;xjI(Ij),xiF)dG(Ij)
]+WiS(x;xiI)Ij(xiI)xF1(xiI)dG(Ij)+xF1(xiI)IUWiL(x;xiI,xF(Ij))dG(Ij),

and the derivative of WiI(x,xˆ;xiI) with respect to xiI is

ddxiIWiI(x,xˆ;xiI)=βIi1G(Ij(xˆ))xxiIβ[1G(Ij(xiI))xiI1G(xF1(xiI))xiL
(54)]G(xF1(xiI))G(Ij(xiI))xiS+G(Ij(xiI))Ij(xiI)fxiIxiF.

Lemma 4.3 guarantees that fxiIxiF is positive. From xF(Ij)xjI(Ij) it follows that G(xF1(xiI))G(Ij(xiI))0. Since xiI<xiFxiL<xiS, value of the derivative given by eq. (54), remains positive for values of xiI in interval [infFj,xiF). Thus for any xiI<xiF, the derivative is positive and optimal xiI cannot be smaller than xiF.

At this point I assume that FjXF. This assumption is equivalent to xF(IU)>infFj. I will later find the optimal xiI for the other case.

From Lemma A.1, it follows that optimal xiI cannot be larger than xiS. Since xiFsupFj, if supFj<xiS, firm i is indifferent between investment at any xiI[supFj,xiS], because this choice does not affect firm i’s investment threshold. Hence, the optimal xiI is interval [xF(IL),supFj].

If xiF<infFj, the derivative for any xiI[xiF,infFj] is given by eq. (52). For any xiI in [max{xiF,infFj},xF(IU)], the payoff function is

WiI(x,xˆ;xiI)=11G(Ij(xˆ))[maxIj(xˆ),ILIj(xiF)WiF(x;xjI(Ij),xiF)dG(Ij)
+Ij(xiF)Ij(xiI)WiS(x;xjI(Ij))dG(Ij)+WiS(x;xiI)Ij(xiI)xF1(xiI)dG(Ij)
(55)+]xF1(xiI)IUWiL(x;xiI,xF(Ij))dG(Ij),

and the derivative of WiI(x,xˆ;xiI) with respect to xiI is

(56)ddxiIWiI(x,xˆ;xiI)=
βIi1G(Ij(xˆ))xxiIβ1G(Ij(xiI))xiI1G(xF1(xiI))xiLG(xF1(xiI))G(Ij(xiI))xiS.

For any xiI in [xF(IU),supFj], the payoff function is

WiI(x,xˆ;xiI)=11G(Ij(xˆ))[maxIj(xˆ),ILIj(xiF)WiF(x;xjI(Ij),xiF)dG(Ij)
(57)]+Ij(xiF)Ij(xiI)WiS(x;xjI(Ij))dG(Ij)+WiS(x;xiI)Ij(xiI)IUdG(Ij),

and the derivative of WiI(x,xˆ;xiI) with respect to xiI is

(58)ddxiIWiI(x,xˆ;xiI)=βIi1G(Ij(xiI))1G(Ij(xˆ))xxiIβ1xiI1xiS.

Given firm j’s behavior, let Lj(xiI) be a function that equals to G(xF1(xiI)) for any xiIXF and is equal to one for xiI>xF(IU). For any xiIxF(IL), define

(59)gj(xiI)=1G(Ij(xiI))xiI1Lj(xiI)xiLLj(xiI)G(Ij(xiI))xiS.

Remember that for xiI<infXF, I defined Ij(xiI)=IL. Combining eqs (52), (56) and (58) together, for any xiI[xF(IL),supFj] that xiI>xiF, it follows that

(60)ddxiIWiI(x,xˆ;xiI)=βIi1G(Ij(xˆ))xxiIβgj(xiI).

The function gj(xiI) is continuous. From eq. (59), since 1G(Ij(xiI))[1Lj(xiI)][Lj(xiI)G(Ij(xiI))]=0, it follows that gj(xiI) is positive for any xiI[xF(IL),xiL]. From Assumption 4.6 and eq. (58) it follows that gj(.) is strictly decreasing in the interval [xiL,supFj], hence, if gj(xiI) is zero at any xiI in this interval, then that value of xiI is the optimal choice for firm i. Otherwise, gj(.) is positive in [xiL,supFj] and since the other firm definitely invests by the time supFj is reached, any xiIsupFj is optimal.

From Lemma A.1 it follows that the zero of gj(.) cannot be larger than xiS. Hence, if xiS<xF(IU), the optimal xiI solves eq. (24). From eq. (59) it follows that if xF(IU)xiS, then gj(.) is zero at xiI=xiS and is positive for values of xiI smaller than xiS.

In conclusion, if xiS<xF(IU), then the optimal xiI solves eq. (24), if xF(IU)xiS<supFj, then the optimal xiI is xiS, and if supFjxiS, then any xiIsupFj is optimal.

Consider the case that FjXF=. The derivative for any xiI[xiF,xF(IU)] is given by eq. (52). For any xiI[xF(IU),infFj], the expected payoff function is equal to WiS(x;xiI) and its derivative is

(61)ddxiIWiI(x,xˆ;xiI)=βIi1G(Ij(xˆ))xxiIβ1xiI1xiS.

For any xiI in Fj, the payoff function is

(62)WiI(x,xˆ;xiI)=
11G(Ij(xˆ))maxIj(xˆ),ILIj(xiI)WiS(x;xjI(Ij))dG(Ij)+WiS(x;xiI)Ij(xiI)IUdG(Ij)

and the derivative of WiI(x,xˆ;xiI) with respect to xiI is

(63)ddxiIWiI(x,xˆ;xiI)=βIi1G(Ij(xiI))1G(Ij(xˆ))xxiIβ1xiI1xiS.

If xiS<xF(IU), Equations (61) and (63) are negative and the optimal xiI is the zero of eq. (53) and solves

(64)1xiI=1G(xF1(xiI))xiL+G(xF1(xiI))xiS

From eqs (61) and (63) it follows that if xF(IU)xiS, then the derivative is zero at xiI=xiS and is positive for values of xiI smaller than xiS. If xiS<xF(IU), since for xiI<infXF, I defined Ij(xiI)=IL, then the zero of eq. (52) is the same as solution to eq. (24). If xF(IU)xiS<supFj, then the optimal xiI is xiS, and if supFjxiS, then any xiIsupFj is optimal.

A.6 Proof of Proposition 4.9.

First I show that for any xI(I,xˉ) defined by Definition 4.8, Assumption 4.6 holds. For any xI, from eq. (25) it follows that

(65)1G(I)xI2+G(xF1(xI))xF1(xI)1xL(I)1xS(I)G(I)1xI1xS(I)dIdxI=
1G(xF1(xI))xL(I)2xL(I)+G(xF1(xI))G(I)xS(I)2xS(I)dIdxI.

The derivative in Assumption 4.6 equals to the left side of the above equation. The right side of the above equation is negative for any xI, hence any solution to eq. (25) satisfies the condition in Assumption 4.6. Thus, it follows directly from Proposition 4.7 that for any xˉ{xF(IU),xS(IU)}, if firm j’s investment threshold is given by xI(.,xˉ), then it is optimal for firm i with investment cost I to invest at xI(I,xˉ). Hence, investment at xI(I,xˉ) is an equilibrium. Also it follows that xI(I,xˉ) is not smaller than xF(I).

A.7 Proof of Proposition 4.10.

To find the limit for the case that D10D00D11D01, note that as IU gets unboundedly large, IUα converges to zero, hence, from eq. (28) it follows that in the limit xI(I) solves

(66)IαxI=xF1(xI)αxL(I)+IαxF1(xI)αxS(I).

The solution to above equation is xI(I)=ββ1rμM2I.

For the case that D10D00>D11D01 and k=1, from eq. (27) it follows that the function xI(I) in the limit must solves

xI(I)1xL(I)1xI
(67)=αIfxIxF(I)+xIβ11xL(I)1xS(I)xIxF(I)β11.

The function xI(I)=ββ1rμM2I is a solution to the above. Note that xI(I)=ββ1rμM2.

Equation (27) is derived from the results of Proposition 4.5 which guarantees that xI(I)xF(I). Also eq. (27) is derived from the results of Proposition 4.9 which guarantees that xI(I)>xF(I). This inequalities will continue to exist in the limit as IU becomes unboundedly large. Hence, it holds that

(68)ββ1rμM1Iββ1rμD11D01I<ββ1rμM2I,

or equivalently M2<D11D01M1.

A.8 Proof of Lemma 5.1.

First I show that the sign of EIi,Ij[UiC(Ii,Ij)]EIi,Ij[UiI(Ii,Ij)] is not an explicit function of r and μ. From eqs (8), (9), (10) and (29), it follows that xiF, xiS, xiL and xiI are constant proportions of (rμ). These thresholds have no other explicit relationship with r and μ.

I show the same thing is true for xjiP. Let

(69)x21P=ββ1rμP21I2.

The threshold x21P is the smallest x that solves the equation V2L(x;x1F)V2F(x;x2F)=0. From this equation, it follows that

(70)x21PD10rμI2x1F(D10D11)rμx21Px1Fβx21PD01rμx2F(D11D01)rμI2x21Px2Fβ=0.

I use eq. (69) and substitute for x21P in the above. It follows that

ββ1D10D01P21I2I2ββ1D10D11D11D01I1D11D01P21.I2I1βI2β1D11D01P21β=0
ββ1D10D01P211ββ1D10D11P21D11D01P21.I2I1β11β1D11D01P21β=0
(71)D10D01P21β1βD10D11P21D11D01P21.I2I1β11βD11D01P21β=0.

Hence, P21 depends only on β and values of Dni,nj. The value of P21 is not an explicit function of r and μ.

The value functions WiF(.), WiL(.) and WiS(.) are expressed in eqs (4), (6) and (7). For values of x smaller than any threshold, they have the following form:

(72)wi(x)=xD00rμIixthresholdβ+kDkthresholdkrμxthresholdkβ.

Here, threshold can be any of the thresholds xiF, xiS, xiL, xiI and xjiP. Also Dk is in fact a term that depends only on values of Dni,nj.

In EIi,Ij[UiC(Ii,Ij)]EIi,Ij[UiI(Ii,Ij)], the term xD00rμ is canceled. Because this term is common among all value functions WiF(.), WiL(.) and WiS(.). The term thresholdkrμ is not an explicit function of r and μ, because all the thresholds are constant proportions of (rμ). The only place that the dependence on r and μ remains, is in the term xthresholdβ. Hence, EIi,Ij[UiC(Ii,Ij)]EIi,Ij[UiI(Ii,Ij)] can be written as A((rμ)x)β, where A is not an explicit function of r and μ. As a result, the sign of EIi,Ij[UiC(Ii,Ij)]EIi,Ij[UiI(Ii,Ij)] is not affected explicitly by r and μ.

Here I show that the sign of EIi,Ij[UiC(Ii,Ij)]EIi,Ij[UiI(Ii,Ij)] is independent of the lower bound of the distribution of investment cost, as long as parameter α remains the same. Pick any Pareto distribution characterized by IL and α. One can change the distribution by multiplying the random variable for investment cost by a positive b. In the new distribution, the lower bound of the support changes to bIL, and the parameter α remains unchanged. Note that as a result of this transformation, the investment thresholds will be multiplied by b as well. Let B be the value of EIi,Ij[UiC(Ii,Ij)]EIi,Ij[UiI(Ii,Ij)] under the original distribution where the lower bound of the support is IL. For the new distribution in which the lower bound of support is bIL, the value of EIi,Ij[UiC(Ii,Ij)]EIi,Ij[UiI(Ii,Ij)] is b(1β)B. This follows from the fact that the value functions WiF(.), WiL(.) and WiS(.) have the form given by (73). Hence, the sign of EIi,Ij[UiC(Ii,Ij)]EIi,Ij[UiI(Ii,Ij)] is the same for both of these distributions and the sign does not depend on the value of IL.

A.9 Proof of Lemma 5.2.

Within a family of distributions, the mean of distributions are the same. Take two distributions from the same family. The distribution that corresponds to α dominates the distribution that corresponds to α in the second order, if for any I

(73)IαIGα(I)dI<IαIGα(I)dI.

The lower bound for the support of a distribution, Iα, increases with larger α. If α>α, it follows that Iα>Iα. Hence, the inequality eq. (73) holds for IIα. To prove the lemma, it is sufficient to show that

(74)ddαIαIGα(I)dI<0,

for I>Iα. Note that

IαIGα(I)dI=IαI1IIααdI
=(IIα)+1α1Iα+1Iαα+1Iαα
=Iαα1Iα+1α1Iα+1Iαα
(75)=IIm+1α1Iα+1Iαα.

Hence,

ddαIαIGα(I)dI=ddα1α1Iα+1Iαα
(76)=1(α1)2Iα+1Iαα+1α11Iαα(ln(I))Iα+1+Iα+1α1dIααdα.

Note that

(77)dIααdα=ddαeαln(Iα)=Iααln(Iα)+αIαdIαdα.

Since Iα=Imα1α and dIαdα=Imα2, the above is equal to

(78)Iααln(Iα)+1α1.

Hence,

(79)ddαIαIGα(I)dI=Iα+1Iαα1α1ln(Iα)ln(I)+1α11α1.

For I>Iα, the above is negative. Hence, the distribution that corresponds to α dominates the distribution that corresponds to α in the second order, if α>α. This is true within any of the defined families of Pareto distributions.

Acknowledgements

I would like to thank Svetlana Boyarchenko for her continued advice and encouragement. I also thank Maxwell Stinchcombe and Thomas Wiseman for their insightful comments.

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Published Online: 2018-01-17

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