# 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.

## A Appendix

### A.1 Derivation of W i F ( x ; x j C , x i F )

Here, I explain how to derive

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

The general solution to the above equation is

where

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

This condition is necessary to make sure that when

Note that in eq. (4), for

and what remains is

### A.2 Lemmas Related to Section 4

**Proof of Lemma 4.1**. In order to show that

**Proof of Lemma 4.2**. Suppose that in equilibrium

Note that it cannot be the case that

**Proof of Lemma 4.3**. The first order derivative is

### Lemma A.1

The optimal value of

**Proof of Lemma A.1.** Since

### A.3 Proof of Proposition 4.4.

It follows from

The derivative of the above function with respect to

I use eqs (8), (9) and (10) to replace any

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

If

and the derivative of

From Lemma 4.3 it follows that

The value of

Because

The payoff function is continuous everywhere and differentiable inside

### A.4 Proof of Proposition 4.5.

It follows directly from Proposition 4.4 that, if

To show the uniqueness, first I show that there is no equilibrium in which

and has a negative value. Hence, there is some value of

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

### A.5 Proof of Proposition 4.7.

To prove Proposition 4.7, first note that it follows from

Since

For any

and the derivative of the payoff function with respect to

If

and the derivative of

Lemma 4.3 guarantees that

At this point I assume that

From Lemma A.1, it follows that optimal

If

and the derivative of

For any

and the derivative of

Given firm

Remember that for

The function

From Lemma A.1 it follows that the zero of

In conclusion, if

Consider the case that

For any

and the derivative of

If

From eqs (61) and (63) it follows that if

### A.6 Proof of Proposition 4.9.

First I show that for any

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

### A.7 Proof of Proposition 4.10.

To find the limit for the case that

The solution to above equation is

For the case that

The function

Equation (27) is derived from the results of Proposition 4.5 which guarantees that

or equivalently

### A.8 Proof of Lemma 5.1.

First I show that the sign of

I show the same thing is true for

The threshold

I use eq. (69) and substitute for

Hence,

The value functions

Here,

In

Here I show that the sign of

### 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

The lower bound for the support of a distribution,

for

Hence,

Note that

Since

Hence,

For

# 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.

### References

Boyarchenko, Svetlana, and Sergei Levendorskiĭ. 2014. “Preemption Games under Lévy Uncertainty.” *Games and Economic Behavior* 88: 354–380.10.1016/j.geb.2014.10.010Search in Google Scholar

Chevalier-Roignant, Benoît, Christoph M. Flath, Arnd Huchzermeier, and Lenos Trigeorgis. 2011. “Strategic Investment under Uncertainty: A Synthesis.” *European Journal of Operational Research* 215 (3):639–650.10.1016/j.ejor.2011.05.038Search in Google Scholar

Décamps, Jean-Paul, and Thomas Mariotti. 2004. “Investment Timing and Learning Externalities.” *Journal of Economic Theory* 118 (1): 80–102.10.1016/j.jet.2003.11.006Search in Google Scholar

Dixit, Avinash K., and Robert S. Pindyck. 1994. *Investment under Uncertainty*. Princeton.Search in Google Scholar

Fudenberg, Drew, and Jean Tirole. 1985. “Preemption and Rent Equalization in the Adoption of New Technology.” *The Review of Economic Studies* 52 (3):383–401.10.2307/2297660Search in Google Scholar

Fudenberg, Drew and Jean Tirole. 1991. *Game Theory*. MIT Press.Search in Google Scholar

Grenadier, Steven R., and Andrey Malenko. 2011. “Real Options Signaling Games with Applications to Corporate Finance.” *Review of Financial Studies* 24 (12):3993–4036.10.1093/rfs/hhr071Search in Google Scholar

Hahn, T. 2006. “The Cuba Library.” *Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment* 559 (1):273–277.10.1016/j.nima.2005.11.150Search in Google Scholar

Harris, Christopher, and John Vickers. 1985. “Perfect Equilibrium in a Model of a Race.” *The Review of Economic Studies* 52 (2):193–209.10.2307/2297616Search in Google Scholar

Harris, Christopher, and John Vickers. 1987. “Racing with Uncertainty.” *The Review of Economic Studies* 54 (1):1–21.10.2307/2297442Search in Google Scholar

Hopenhayn, Hugo A., and Francesco Squintani. 2011. “Preemption Games with Private Information.” *The Review of Economic Studies* 78 (2):667–692.10.1093/restud/rdq021Search in Google Scholar

Hsu, Yao-Wen, and Bart Lambrecht. 2007. “Preemptive Patenting under Uncertainty and Asymmetric Information.” *Annals of Operations Research* 151:5–28.10.1007/s10479-006-0125-5Search in Google Scholar

Huisman, Kuno J. M. 2001. *Technology Investment: A Game Theoretic Real Options Approach*. Kluwer Academic Publishers.Search in Google Scholar

Huisman, Kuno J. M., and Peter M. Kort. 2004. “Strategic Technology Adoption Taking into Account Future Technological Improvements: A Real Options Approach.” *European Journal of Operational Research* 159 (3):705–728.10.1016/S0377-2217(03)00421-1Search in Google Scholar

Lambrecht, Bart, and William Perraudin. 2003. “Real Options and Preemption under Incomplete Information.” *Journal of Economic Dynamics and Control* 27 (4):619–643.10.1016/S0165-1889(01)00064-1Search in Google Scholar

Maskin, Eric S., and Jean Tirole. 1988. A Theory of Dynamic Oligopoly, i: Overview and Quantity Competition with Large Fixed Costs.” *Econometrica* 56 (3):549–69.10.2307/1911700Search in Google Scholar

Mason, Robin, and Helen Weeds. 2010. “Investment, Uncertainty and Pre-emption.” *International Journal of Industrial Organization* 28 (3):278–287.10.1016/j.ijindorg.2009.09.004Search in Google Scholar

Pawlina, Grzegorz, and Peter M. Kort. 2006. “Real Options in an Asymmetric Duopoly: Who Benefits from Your Competitive Disadvantage?” *Journal of Economics and Management Strategy* 15 (1):1–35.10.1111/j.1530-9134.2006.00090.xSearch in Google Scholar

Reinganum, Jennifer F. 1981. “Market Structure and the Diffusion of New Technology.” *The Bell Journal of Economics* 12 (2):618–624.10.2307/3003576Search in Google Scholar

Thijssen, Jacco J. J., Kuno J. M. Huisman, and Peter M. Kort. 2006. “The Effects of Information on Strategic Investment and Welfare.” *Economic Theory* 28: 399–424.10.1007/s00199-005-0628-3Search in Google Scholar

Thijssen, Jacco J. J. 2010. “Preemption in a Real Option Game with a First Mover Advantage and Player-Specific Uncertainty.” *Journal of Economic Theory* 145 (6): 2448–2462.10.1016/j.jet.2010.10.002Search in Google Scholar

Weeds, Helen. 2002. “Strategic Delay in a Real Options Model of R&D Competition.” *The Review of Economic Studies* 69 (3): 729–747.10.1111/1467-937X.t01-1-00029Search in Google Scholar

**Published Online:**2018-01-17

© 2018 Walter de Gruyter GmbH, Berlin/Boston