This webpage presents the bibliography on option games models.

Recall that option games comprise the combination of real options theory with game theoretic models.
Option games models addresses the optimal options exercise considering both exogenous uncertainty and competitors as rational "players".

The link below leads the reader to the bibliographical list (to another webpage). This list has a lot of references in option-games but of course is incomplete. I'll try to enlarge this list in the next website updates.

Below this link, there are some short bibliographical reviews of a selected literature.

Most contributions in option games literature come from real options researchers rather than game-theoreticians. Tools like stochastic processes and optimal control are more useful than fixed-point theorems and other related tools - see the webpage on Oligopoly under Uncertainty for a discussion.

However, another promising way to solve option-games models comes from two game-theoreticians, P.K. Dutta & A. Rustichini (1995, "(s, S) Equilibria in Stochastic Games", *Journal of Economic Theory*, vol.67, 1995, pp.1-39, who prove that the best response map satisfies a *strong monotonicity condition*, which is used to set the existence of Markov-Perfect Nash equilibria.

The third related school that can contribute to option games literature comes from researchers in optimal control, e.g., Basar & Olsder (1995), "Dynamic Non-Cooperative Game Theory""" and Dockner & Jorgensen & Van Long & Sorger (2000): "Differential Games in Economics and Management Science", mainly the ** stochastic differential games** branch. However, the bridge between option games and that branch in optimal control literature remains to be constructed. Therefore, this literature is not included here, but in the future I plan to include at least papers focused in stochastic differential games. The reader can find some related papers in the general bibliography webpage.

Good luck in your research!

For many applications, is necessary to take into account the results of more sophisticated models. Option games models have a large practical potential, but need more complex mathematical tools and a lot of research is still to be done, option games models are in the infancy.

There are five reviews, three from articles and two from books chapters. Moreover, in the ending of this page are suggested others articles related with option-games (with a promise to review in a future issue).

**Others Important Option Games Articles for Future
Review**

1) Dixit & Pindyck

Dixit, A.K. & R.S. Pindyck (1994): "Investment under
Uncertainty" *Princeton University Press*, Princeton, N.J., 1994

In chapter 9, section 3, they analyze the imperfect-competition case (duopoly) using option-game approach in continuous time. The case analized is for complete information. They don't analyze the case with incomplete information. By considering perpetual option, they get analytic solution for the thresholds and values of both leader and follower! The game is a preemption game, with two equilibria by permuting the roles of leader and follower of the two players.

In this model the authors apply other tool from the
stochastic calculus arsenal: *the expected first hitting time*, in order to calculate the
player's payoffs. The random time of option exercise makes the discount factor from this random time to the current date a random variable itself. The expected value of this stochastic discount factor is used for payoff calculation purposes.

2) Lambrecht & Perraudin (a)

Lambrecht, B. & W. Perraudin (1994): "Option Games"

Working Paper, *Cambridge University*, and *CEPR* (UK), August
1994, 17 pp.

In "Option Games", the authors analyze a
duopoly, with American put options as payoffs, where the exercise price
are costs of transactions known only by each player (incomplete
information) and the put-threshold is raised because there is an advantage
to act first (preemption game). They calculate the payoff using the
expected first time that the stochastic variable hits the threshold (this
is done in Dixit & Pindyck, too), for each player (minus the
intersection of players’ common scenarios).

The other agent threshold conditional distribution of probabilities is
updated using a method that is similar to "distributional strategies
approach", finding the optimal using standard Laplace maximization,
and derive the condition for the existence of the Nash equilibria.

They provide an example of the model for the "optimal" insider
trading, when there are more than one investor with the inside
information.

3) Lambrecht & Perraudin (b)

Lambrecht, B. & W. Perraudin (1996): "Real Option and
Preemption"

Working Paper, *Cambridge University*, *Birkbeck College*
(London) and *CEPR*, 35pp.

In "Real Option and Preemption", they focus
on preemption game, perpetual option, and incomplete information about the
other agent threshold. They simplify the problem considering that after
the first entry the other agent loses the investment opportunity. The
solution is analytic, and using the notation of our site, the threshold
for a perpetual option (**V***, where V is the operating project
value) is modified (see the Dixit & Pindyck book, p.142, eq.14, for
the threshold of the perpetual option to invest) incorporating the **"hazard
rate"** into the equation:

V* = I

Where** b** is the
positive (and higher than one) characteristic root of the differential
equation, **Y** is the other player (competitor), **h _{y}(V*)**
is the Y’s hazard rate. By adding the hazard rate in this equation,
the threshold decreases (in limit to the Marshallian level). If the hazard
rate is zero, the problem reduces to the traditional timing option
(without game). This solution is valid for preemption game, perpetual
option and for zero "follower payoff".

For the problem of the waiting game, finite option, and lower but non-zero "loser of the game" payoff - as in the petroleum drilling game, the SPE paper presented in this website, is not easy to adapt this elegant solution.

4) Trigeorgis

Trigeorgis, L. (1996): "Real Options - Managerial Flexibility and
Strategy in Resource Allocation" *MIT Press*, Cambridge, MA, 1996, 427 pp.

In the chapter 9 of his textbook, Trigeorgis also
studied the preemption game combined with option approach. The idea is
incorporate the preemption effect on the threshold, reducing it by adding
a positive value in the dividend yield parameter. This parameter has large
influence on the threshold (see a typical chart, for example, in Dixit &
Pindyck, p.157, Fig.5.6).

This additional dividend is a **"competitive dividend"**
which is lost for the owner of the real option, in analogy with the cash
dividend which is lost for the owner of a call option on a dividend-paying
stock (or the convenience yield which is lost by the owner of a future
contract of a commodity).

This is an exogenous way to model the competition, in contrast with the endogenous rational competition methods using typical game theory tools.

If in the __preemption game__ is added a __positive__ value on
the dividend yield, for the opposite timing game, the * war of attrition*,
one idea is

Trigeorgis also develops a game-tree analysis, similar
to the presented in the SPE option-game model, he also founds the
equilibria set of strategies by backward induction and using binomial
valuation.

He analyzes both cases, the preemption game and the waiting game. In this
last case (p.298), the problem is analogous of that here analyzed (there,
a R&D investment of a firm results in a cost benefit for who spend in
R&D, but also for the competitor). However, their analysis is more
simplified, for example not considering the incomplete information
subject, and so not updating the probabilities of the other player, using
the information obtained with each move.

5) Grenadier

Grenadier, S.R. (1996): "Strategic Exercise of Options: Development
Cascades and Overbuilding in Real Estate Markets" *Journal of Finance*, vol.51, no 5, December 1996, pp.1653-1679

Grenadier develops an option-game model for real estate
investment timing, again one preemption game.

Preemption by a competitor explains the exercise of the call option to
develop a real estate in declining demand. This paper considers the time
to build (ranging from 6 months to more than 5 years) effect on option
value.

He develops a two-builders subgame perfect equilibria, finding a pair of
symmetric Markovian exercise strategies. The option exercise is optimal
conditional on the other player exercise strategy. There is an infinite
set of equilibria exercise strategies, and he focused on the Pareto
optimal one.

The analysis of a sequential investment option, the "development
cascades", shows that the demand volatility reduces the median time
between investment starts.

1) Caplin, A. &
J. Leahy (1994): "Business as Usual, Market Crashes, and Wisdom After
the Fact" *American Economic Review*, vol.84, no 3, June 1994, pp.548-565

2) Smit, H.T.J.
& L.A. Ankum (1993): "A Real Options and Game-Theoretic Approach
to Corporate Investment Strategy under Competition" * Financial Management*, Autumn 1993, pp.241-250

3) Kulatilaka,
N. & E.C. Perotti (1997): "Strategic Growth Options" * Working Paper, Boston University & University of Amsterdam*,
April 1997, 22 pp. (forthcoming in Management Science).

4) Trigeorgis,
L. (1991): "Antecipated Competitive Entry and Early Preemptive
Investment in Deferrable Projects" * Journal of Economics and Business*, May 1991, vol.43, n.2,
pp.143-156

OBS: See also a light paper related to this subject, in Harvard Business
Review, Nov-Dec/97. The article title is "Strategy under Uncertainty"