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Preemption Games: Theory and Experiment Author(s): Steven T. Anderson, Daniel Friedman and Ryan Oprea Source: The American Economic Review, Vol. 100, No. 4 (SEPTEMBER 2010), pp. 1778-1803 Published by: American Economic Association Stable URL: Accessed: 04-02-2016 18:36 UTC

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American Economie Review 100 (September 2010): 1778-1803

http-J/www. aeaweb. or g/article s.php?doi=10.1257/aer. 100.4.1778

Preemption Games: Theory and Experiment

By Steven T. Anderson, Daniel Friedman, and Ryan Oprea*

El Mut?n, perhaps the world’s largest remaining iron ore deposit, was opened to private inves tors in the 1980s but, due to the high cost of developing the remote Bolivian site, there were no takers for two decades. In late 2005, spurred by rising commodity prices, the Brazilian company EBX finally seized the opportunity, preempting rivals based in China and India.1 Numerous similar examples can be found in the annals of mining and oil companies (Raymond F. Mikesell et al. 1971).

The strategic problem is as old as humanity: the value of a grove of figs fluctuates as the fruit

ripens (or is attacked by worms) and the first band of hunter-gatherers to seize it eats better than

neighboring bands. High tech firms today face similar issues when they introduce a new product. Delaying introduction might allow the product niche to expand, but new substitutes or changes in standards might shrink the niche. A rival could preempt the niche or it could vanish entirely due to a disruptive new technology (Clayton M. Christensen 1997). Similar considerations apply to retailers deciding when to open a “big box” outlet in a market too small to handle more than one

store, and also apply to academic researchers investigating a hot new topic. In this paper we study such situations both theoretically and empirically. We formalize them

as preemption games, using standard simplifications to put the strategic issues into sharp focus.2 In our games, the opportunity is available to a known number n + 1 of investors; it has a publicly observed value Vthat evolves according to geometric Brownian motion with known parameters; each investor has a privately known avoidable cost of investing; and the first mover preempts and obtains the entire value V.

Our model builds on an active literature reviewed in Marcel Boyer, Eric Gravel, and Pierre Lasserre (2004) that studies preemption contests for investments with option values. Unlike most of this literature (e.g., Steven R. Grenadier 2002; Helen Weeds 2002; Romain Bouis, Kuno J.M.

Huisman, and Peter M. Kort 2009), firms in our model are uncertain of their rivals’ costs. A trade-off for this added piece of realism is that unlike some of this literature but like much clas sic auction theory, preemption is complete; the winner takes all of the returns from investment. A separate literature considers preemption in very different environments related to R&D (e.g.,

* Anderson: National Minerals Information Center, US Geological Survey, Reston, VA 20192 (e-mail: Sanderson?; Friedman: Economics Department, University of California-Santa Cruz, 1156 High Street, Santa Cruz, CA 95064 (e-mail: [email protected]); Oprea: Economics Department, University of California-Santa Cruz, 1156 High Street, Santa Cruz, CA 95064 (e-mail: [email protected]). We are grateful to Jerome Bragdon and Todd Feldman for research assistance, to Nitai Farmer, Adam Freidin and James Pettit for programming support, and to the National Science Foundation (grant IIS-0527770) and the University of California for funding. We received helpful comments from Robert Jerrard, Vijay Krishna, Robert B. Wilson, James C. Cox, Donald Wittman, participants at the 2008 UCSB

Experimental and Behavioral Workshop, from seminar audiences at Cal Poly SLO, CUNY-Baruch, George Mason,

Houston/Rice, and Texas A&M Universities, and from three anonymous referees of this Review. All mistakes are our own.

1 Arai, Adriana, and Andrew J. Barden. 2006. “Bolivia Rules Out Brazil’s EBX for $1.1 Bin Project (Update2).”

Bloomberg, April 24. 2 Reality, of course, is always more complex. In the El Mut?n example, Bolivia’s newly elected government shut

down EBX’s operations in 2006, citing environmental and other concerns, and in late 2007 signed a 40-year concession contract with the Indian firm Jindal. Other parts of the story are more consistent with our model. The number of serious rivals was always reasonably clear. The true costs of the Brazilian and Indian firms (and their Chinese rival, Shandong)


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Jennifer F. Reinganum 1981; Drew Fudenberg and Jean Tirole 1985; Heidrun C. Hoppe and Ulrich Lehmann-Grube 2005) and to market entry (e.g., Dan Levin and James Peck 2003).

Bart M. Lambrecht and William R.M. Perraudin (2003) is our direct predecessor. Like us,

they draw on real options theory to investigate preemption of a stochastic investment opportunity by competitors whose costs are private information. Our theoretical results extend theirs by cov

ering more than two competitors, and by relaxing a restrictive technical assumption. Our model, unlike theirs, is explicitly rooted in auction theory as well as in real options theory. Indeed, spe cial cases of our model include Dutch auctions as well as deferral options.

Our theoretical contribution appears in Section I. We characterize the symmetric Bayesian Nash equilibrium of the preemption game with an arbitrary number of players. Players’ BNE

strategies take the form of a threshold value at which the opportunity is seized immediately. The

mapping from realized cost to equilibrium threshold is characterized in two different ways: by an ordinary differential equation and also by a recursion equation. We show that the auction and real option special cases lead to useful bounds on the BNE strategies.

Section II describes a laboratory experiment informed by the theory, using software cre ated expressly for the purpose. It presents the main treatments?Competition (triopoly) versus

Monopoly, and High versus Low Brownian parameters?and obtains four testable hypotheses. Section III explains other aspects of the laboratory implementation.

Section IV presents the results. The first three hypotheses fare quite well: the triopoly market structure leads to much lower markups (threshold value less cost) than the monopoly structure; the Brownian parameters have a major impact in the predicted direction in Monopoly but (again as predicted in BNE) have negligible impact in Competition; and the lowest cost investor indeed is far more likely to preempt than her rivals. The evidence is mixed on the last hypothesis: at the low cost end of the scale, investors’ markups indeed tend to decline in cost, but the predicted relationship breaks down at higher costs. However, these departures from prediction turn out to have very little impact on subjects’ actual earnings.

Following a concluding discussion, an Appendix collects mathematical derivations and the main proofs. Other Appendices available online provide additional mathematical details, discuss the lesser-known econometric techniques, report supplementary data analysis, and reproduce instructions to subjects. Our theoretical contribution incorporates and extends the PhD disserta tion of Anderson (2003). A companion paper, Oprea, Friedman, and Anderson (2009) describes a related laboratory experiment concerning the monopoly (n

= 0) case only.

I. Theoretical Results

This section analyzes two situations. In the first, called monopoly, a single investor / has sole access to an investment opportunity. In the second, called competition, two or more investors with private information concerning their own costs have access to the same opportunity, and the first to seize it renders it unavailable to the others.

were, in no small part, private information due to confidential subsidies arranged by their own governments as well as

confidential understandings with the Bolivian government. The firms faced the hazard that the Bolivian government

might renationalize El Mut?n, or declare it a protected national park, before any of them could seize the opportunity

(Barden and Arai 2006; Patrick J. McDonnell 2008. “Bolivia opening up huge iron ore deposit to mining.” Los Angeles Times, May 28. (accessed October 31, 2008).).

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1780 the american economic review september 2010

A. Monopoly

An investor / with discount rate3 p > 0 can launch a project whenever she chooses by sinking a given cost Ct > 0. The present value V of the project evolves via geometric Brownian motion with drift parameter a < p and volatility parameter a > 0:

(1) dV= aVdt + aVdz,

where z is the standard Wiener process. That is, the value follows a continuous time random walk in which the appreciation rate has mean a and standard deviation o per unit time. At times t > 0

prior to launching the project, the investor observes V(t). If she invests at time t, then she obtains

payoff [ V(t) ?

Ci]e~pt. The project is irreversible and generates no other payoffs. Thus, the task is to choose the investment time so as to maximize the expected payoff.

The solution goes back to Claude Henry (1974) and has been widely known since Robert McDonald and Daniel Siegel (1986); see Chapter 5 of Avinash K. Dixit and Robert S. Pindyck (1994) for a detailed exposition. The optimal policy takes the form: wait until V(t) hits the threshold

(2) VM(Q) = (1 + w)Ct,

then launch immediately. Note that the threshold is proportional to cost, and that the wait option premium w > 0 is an algebraic function of the volatility, drift and discount parameters <r, a and p.


(3) w =

j^j, where ? = \

– +

B. Competition

Now consider the case that each investor has n > 1 rivals. All investors / = 1,2,…,ai -f- 1 have access to the same investment opportunity, whose value V again evolves according to geo metric Brownian motion (1). Each investor / again knows her own cost Ch but doesn’t know the other investors’ costs Cj9j ^ i. She regards them as independent draws from a cumulative dis tribution function, H(C), with a positive continuous density function h(C) on support [CL, q/]. The first investor to launch, say at time tt > 0, obtains payoff [V(f/)

– C^e~p%i and the other

investors obtain zero payoff. All this is common knowledge. The resulting preemption game is denoted r(?9n,ti).

The preemption game has a unique symmetric Bayesian Nash Equilibrium (BNE). It is char acterized by an increasing function V*(C/) that maps the investor’s cost into a threshold value, above which she immediately invests. We now sketch the derivation and offer some intuition; the

Appendix spells out the derivations and the main proofs.

a + 2p J2. > 1.

3 Recall that the discount rate reflects pure time preference, the expiration hazard, and possibly risk aversion. As

explained in the next section, the laboratory experiment focuses on the expiration hazard, sometimes informally referred to as “preemption by Nature.”

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Using notation m, V0 and C defined below, the expected discounted payoff E[ V{t) can be written out as the following objective function:

(4) F(m\Chn) = \V*(m) –

?] V\m)

1 – H{m) 1 – H(C)

The choice variable in (4) is m G [CL, Cv], interpreted as the cost-type that the investor chooses as her potential “masquerade.”4

The first factor in the objective function (4) is simply the profit (or “markup”) [V*(m) ?

C,] obtained at the time of successful investment. The second factor, [V0/V*(m)]?, accounts for the time cost of delaying investment and the expiration hazard, given that V0 is the current value of the investment project. The Appendix shows that the monopolist’s value function consists of only these first two factors. It also notes that with competitors (n > 0), the restriction p > a can be relaxed and consequently (4) is valid for ? > 0, while of course (2-3) is valid only for ? > 1.

The third and final factor, [(1 –

H(m))/(l –

H(C))]n, is the probability that the n rivals all have higher costs (and therefore will not preempt), conditioned on the fact that none of them has

already invested. That conditioning is reflected in the denominator. Let V > VQbc the “highest peak” so far achieved by the random walk. Then C is the corresponding cost, i.e., V

= V*(C).

Since the preemption game is over as soon as the first investor moves, it turns out that the BNE threshold strategy is independent of V, C and V0 within the relevant range.

The key to obtaining the BNE is the best response (or “truthtelling”) property that investor i maximizes (4) at m

= Ct. The associated first-order condition can be expressed as the following

ordinary differential equation (ODE):

(5) V”(Q = nh(C.) [V(Q

– C,]V(Q

For reasons explained in the next subsection, we also impose the boundary condition


This boundary value problem has a unique solution V* that characterizes the symmetric BNE threshold for our preemption game. The Appendix shows that it can also be expressed as a con

ditional expectation and that it satisfies

(7) V*(C) = c + Jc [v\y)\ 1 – H(y)

Li-ff(c)j dy.

THEOREM 1: Let the cumulative distribution function H have a continuous density h with full support [CL, Cy], where 0 < CL < Cv < oc. Let it be common knowledge among all investors i = 1,…, n + 1 that Vs investment cost C, is an independent random variable with distribution H

and is observed only by investor i. Then

1. for any ? > 0, boundary value problem (5-6) has a unique solution V* : [CL, Cy] ?> R,

4 Of course, the threshold value itself is the natural choice variable but, as in auction theory, it turns out that m is

more convenient since V*(-) is invertible.

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2. a function V* satisfies the recursion equation (7) if and only if it solves the boundary problem (5-6), and

3. the premption game T[?,n,H] has a symmetric Bayesian-Nash equilibrium in which each investor i’s threshold is V* evaluated at realized cost Cv

To obtain V* numerically, one can use the Euler method of integrating the ODE (5) backward from the upper boundary value (6). Alternatively, one can take an initial approximation (such as

the auction solution V defined in the next subsection), substitute it for V* in the last expression in

(7) to obtain a better approximation, and iterate. The BNE threshold function V* is a fixed point of this mapping.

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