I. Introduction: An Explosion of Frivolous Litigation? The aggregate quantity of litigation in the United States, both in comparison with other countries and over time, has been and remains the subject of much controversy. n1 But the often heard phrase "litigation explosion" is merely descriptive, being devoid of any particular normative content. A recent, related, and more specific debate concerns the nature or quality of litigation in specific areas, including but not limited to, medical malpractice, product liability, and securities fraud. n2 In particular, many legal and social commentators feel that America is and has been experiencing an explosion in so called strike lawsuits, also known as nuisance lawsuits or frivolous litigation.    An explosion in frivolous litigation is not normatively neutral. A perceived rise in frivolous lawsuits alleging securities fraud was a major impetus for the provisions imposing strict pleading requirements contained in the Private Securities Litigation Reform Act of 1995, which Congress enacted over President Clinton's veto. n3 The politics of American litigation reform and specific anti litigation campaigns are an interesting reflection of American culture, history, and society that is beyond the scope of this Article. n4    Whether there has been such a frivolous litigation explosion is a descriptive and historical question that is empirically challenging to resolve because nearly all lawsuits settle, n5 with many of the settlements involving confidentiality agreements. Everyone agrees there is a demand for more empirical research and work about civil procedure and litigation, n6 but unfortunately the supply of it is still rare. n7 But positive theoretical economics can provide insights into frivolous litigation in the form of implications from analytical, formal, rigorous, and systematic models. n8 What procedural or substantive reforms would reduce frivolous litigation is a normative question that raises difficult concerns involving procedural fairness, both outcome based and process based, in addition to questions regarding the nature and limits of substantive rights. But normative theoretical economics can help answer this question by identifying and comparing the various error and process costs of alternative reforms. n9    This Article develops a new theory of possibly frivolous litigation by focusing on a plaintiff's options to unilaterally abandon a lawsuit. n10 Federal Rule of Civil Procedure 41(a)(1)(i) and its various state law counterparts permit, under certain circumstances, a plaintiff to voluntarily dismiss her lawsuit without prejudice. n11    A. The Value of Litigation Abandonment Options: A Hypothetical Example The following hypothetical example illustrates the value of a plaintiff's option to abandon or drop litigation. Suppose that Portia sues Daphne. In addition, suppose that Portia's ex ante or initial expected probability of prevailing at trial is 1/2. Suppose also that, initially, the monetary judgment that Portia expects to win at trial is $ 1,000,000. Under these facts, Portia's actual initial expected judgment at trial is (1/2)($ 1,000,000), or $ 500,000. Suppose that Portia's total expected litigation cost of proceeding to a trial is $ 550,000. Portia's lawsuit will have a net expected value of $ 500,000 $ 550,000 = $ 50,000 0.    Now divide the lawsuit into two stages, discovery and trial, each of which costs Portia $ 275,000. In addition, suppose that discovery resolves all of the risks of Portia's litigation, so the posterior or ex post probability conditional upon discovery of Portia's prevailing in court is either 0 or 1. Portia would only proceed when she has a sure winner, and Portia would abandon a sure loser. Under those facts, the revised initial expected value or initial option value that the lawsuit has for Portia would be (1/2)($ 1,000,000 $ 275,000) $ 275,000 = (1/2)($ 725,000) $ 275,000 = $ 362,500 275,000 = $ 87,500 0. Notice that, holding fixed the other values of the parameters in this example, this lawsuit has initial positive option value as long as the monetary judgment from Portia's prevailing in court exceeds $ 825,000. Portia would initially file this Negative Expected Value ("NEV") lawsuit; however, after discovery, Portia would choose to drop this lawsuit if Portia and Daphne believe that Portia will lose at trial.    In order to illustrate in the simplest possible way the settlement value of this litigation, suppose that Daphne's litigation costs are the same as Portia's costs; that is, that Daphne's total expected litigation costs for proceeding to a trial are $ 550,000, with each of discovery and trial expected to cost Daphne $ 275,000. Finally, suppose that Daphne and Portia have equal bargaining strength. Then, after discovery, Daphne and Portia either learn that Portia has a sure loser in which case, Portia abandons the litigation or Daphne and Portia learn that Portia has a sure winner. If Portia has a sure winner, then Portia should accept any settlement amount that exceeds what she expects to get by proceeding to trial namely, $ 1,000,000 $ 275,000. In addition, Daphne should offer any settlement amount that is less than what she expects to lose by proceeding to trial namely, $ 1,000,000 + $ 275,000. Because Daphne and Portia have equal bargaining power, they agree to settle for $ 1,000,000, which is the midpoint of the range between $ 1,000,000 $ 275,000 and $ 1,000,000 + $ 275,000.    Now that we have figured out what Daphne and Portia would settle for after discovery but before trial if both learn that Portia has a winner, we can proceed to determine what happens before discovery. Upon Portia's commencing the litigation, both Daphne and Portia can reason, as we have above, that if they proceed and incur the costs of discovery namely, $ 275,000 each then, with an initial probability of 1/2, Portia will have a credible threat to proceed to trial. As a result, Daphne and Portia would settle for $ 1,000,000 after discovery. Thus, Portia should accept any settlement amount that exceeds what she initially expects to get by proceeding to discovery namely, (1/2)($ 1,000,000) $ 275,000. In addition, Daphne should offer any settlement amount that is less than what she initially expects to lose by proceeding to discovery namely, (1/2)($ 1,000,000) + $ 275,000. As before, because Daphne and Portia have equal bargaining power, they will settle for (1/2)($ 1,000,000), or $ 500,000, which is the midpoint of the range between $ 500,000 $ 275,000 and $ 500,000 + $ 275,000. Recall that when the plaintiff must pay for all her litigation costs up front that is, when the plaintiff does not have the option to abandon the litigation after discovery this is a NEV lawsuit. In other words, the plaintiff's threat for pursuing litigation is not credible, and the settlement value is zero.    B. Lawsuit Abandonment Options in Game Theoretic Litigation Models Appendix B of this Article presents a multi period algebraic options game theoretic model of litigation that generalizes the hypothetical numerical example above by incorporating general bargaining strengths, litigation costs, and probability beliefs on the part of the plaintiff and the defendant. Interested readers should look over the appendices of this Article before going forward. Appendix A provides an accessible, nontechnical, self contained, and user friendly primer about options for those unfamiliar with options. Appendix B contains a formal, mathematical game theoretic analysis of a plaintiff's options to unilaterally abandon a lawsuit. The rest of this Article is organized as follows. Part II of this Article places this Article's options approach to litigation, including quite possibly, frivolous litigation in the context of the literature of economic models about litigation in general and frivolous litigation in particular. This part of the Article explains that possibly frivolous lawsuits will be filed and settled when the values of a plaintiff's options to unilaterally abandon litigation exceed the costs of purchasing those litigation abandonment options by continuing the litigation. Part III of this Article addresses some of the limitations of this Article's abandonment options game theoretic model of litigation. n12 In particular, there is reason to believe that people have cognitive limitations in their abilities to reason backwards in sequential interactions. n13 Empirical and experimental evidence also exists that indicates that emotions affect how people make decisions. n14 Finally, recent psychological experiments indicate that decision makers often overvalue options and over invest in keeping options alive, even if those options present little intrinsic value. n15 Part IV of this Article briefly explains how and why many laws and judicial doctrines effectively preclude specific legal options.    II. An Economic Analysis of Litigation The application of microeconomics to litigation has a distinguished and relatively long history in the field of law and economics. n16 A rich, related literature analyzing civil procedure that utilizes microeconomics also exists. n17 Some legal practitioners utilize the powerful tools of single person decision theory and risk analysis to help facilitate the settlement of their clients' legal disputes. n18 But multiperson decision making theory or game theory describes a lawsuit more accurately than single person decision making theory does because of the interactive and strategic nature of litigation. Legal scholars have applied multiperson decision theory or game theory to analyze settlement negotiations in litigation. n19 Game theory's origins date back at least 2500 years and can be found in classic Chinese philosophical texts. n20 Multiperson decision theory, as game theory is more accurately and perhaps less frivolously described, is a branch of applied mathematics; n21 having numerous applications in biology, n22 economics, n23 management, n24 and politics. n25 It has become standard practice to apply game theory to analyze legal rules and institutions. n26 Proof of the acceptance of game theoretic reasoning in the legal scholar's toolkit is found in the five peer refereed journals about law and economics. n27 Finally, game theory played a crucial role in designing the Federal Communications Commission (FCC) auctions for assigning licenses to wavelengths for personal communication services, such as cell phones and wireless computer access services. Professor John McMillan provides an excellent account of this case study, which demonstrates the success of modern game theory, as applied to designing optimal regulatory policy. n28 Professor McMillan explains that the features of the auction format the FCC adopted essentially were those proposed by Professors Preston McAfee, Paul R. Milgrom, and Robert Wilson and experimentally tested by Professor Charles Plott. n29 As Professor McMillan stated, "When the theorists met the policy makers, concepts like Bayes Nash equilibrium, incentive compatibility constraints, and order statistic theorems came to be discussed in the corridors of power." n30        A. A General Comparison of the Expected Value and Options Models of Litigation Three path breaking models set the standard for the formal economic analysis of the settlement of litigation. n31 First, Professor William Landes explained why most criminal cases involve negotiated sentences instead of trial. n32 Second, Professor Richard Posner explained why the FTC and other administrative agencies settle most regulatory disputes via out of court settlements. n33 Third, Professor John Gould explained why most civil cases settle before trial. n34 The Landes Posner Gould (LPG) single person decision theory expected value approach to settlement of litigation culminated in Professor Steven Shavell's model comparing the incentives to sue and settle under alternative rules for allocating legal costs. n35    The standard approach in law and economics models to how people deal with risk involves assuming that legal decision makers maximize their expected utilities of wealth. This general assumption often then is reduced to assuming that legal decision makers maximize the net present discounted values of their expected wealth levels. In other words, neoclassical models assume that legal decision makers have as their utility function over wealth, the net present discounted value of wealth. This can be more accurately termed an expected value of wealth approach to risk.    An expected value approach to the risks in litigation would be appropriate if legal decision makers in litigation were locked into their initial decisions. What an expected value approach to risks ignores, however, are the opportunities to make future choices after learning more concerning the payoff relevant risks. In the lawsuit context, an expected value approach neither incorporates nor reflects the value of the flexibility provided by a plaintiff's option to unilaterally abandon litigation. n36 The values of a plaintiff's options to abandon litigation lead to qualitatively different implications concerning the incentives to sue, settle, or go to trial than under the usual expected value approach to lawsuits, and these option values can be quite large quantitatively.    Professor Bradford Cornell was the first scholar to develop some of the implications of the observation that plaintiffs have unilateral options to drop a lawsuit before incurring the cost of a full blown trial. n37 Professor Cornell showed that the option to drop a lawsuit increases a lawsuit's expected payoff, and, hence, the incentive to file a lawsuit. Professor Cornell's analysis extends the LPG model in which litigation decisions were based solely on the present discounted value of a lawsuit's costs and expected benefits by introducing an explicit options approach to litigation. n38 William J. Blanton applied Cornell's insights to evaluate the impact of changes in evidentiary rules on a plaintiff's incentive to file a lawsuit. n39 In particular, Blanton focuses on changes in the admissibility of expert scientific testimony resulting from the Supreme Court's decision in Daubert v. Merrell Dow Pharmaceuticals, Inc. n40 Blanton identified four principal ways in which any evidentiary, procedural, or substantive rule (or change in such a rule) can reduce the value of a plaintiff's option to unilaterally drop litigation by: (1) increasing the plaintiff's litigation costs, (2) front loading the plaintiff's litigation costs, (3) enhancing trial precision, and (4) obfuscating the plaintiff's ability to predict a trial outcome. n41 Frederick Dunbar and his colleagues provided options based approaches to nuisance lawsuits, plaintiffs' attorneys' behavior under contingent fee arrangements in securities litigation, securities litigation reform, and settlements in shareholder class actions. n42     I introduced an options model of contingency multipliers for attorney 's fees in public interest and civil rights litigation. n43 Professor Steven Shavell raised a set of related concerns in his affidavit for a civil rights case where attorney's fees were hotly contested. n44 Professor Shavell, however, did not frame his argument explicitly in terms of the language of an options approach to litigation. Also related are my proposals to incorporate options in teaching corporate law n45 and to use an options approach toward an understanding of why a firm could rationally choose to engage in predatory pricing. n46    Professor Lucian Bebchuk provided a theory of NEV lawsuits where threats to go to trial are credible due to divisibility over time of plaintiffs' litigation costs. n47 The options model of litigation in this Article differs from Professor Bebchuk's model because in his model litigants face certainty over expected trial outcomes and legal fees, while litigants in this Article's model face uncertainty over expected judgments, litigation costs, or both. The plaintiffs in the model of this Article have opportunities not only to learn about expected judgments and litigation costs during the litigation process, but also to drop litigation, conditional upon information they learn during the course of that litigation. The divisibility of legal costs also forms the basis for Professor William Landes's model of unitary versus sequential trials. n48 Professor Landes demonstrated that bifurcating liability and damages reduces expected litigation costs because no need to litigate damages arises if no liability exists. n49 In turn, such bifurcation increases the incentives to sue, increases the minimum acceptable settlement, and decreases the maximum settlement offer. n50    B. Comparing the Expected Value and Options Models of Frivolous Litigation    1. Defining "Frivolous Litigation" Both the positive and normative analyses of frivolous litigation depend upon one's definition of "frivolous litigation." As Professor Bone details, defining a frivolous lawsuit proves more complicated than one might initially think. n51    An obvious definition of a frivolous lawsuit is a case in which the plaintiff does not expect initially to prevail at trial. In other words, the plaintiff in a frivolous lawsuit is one who has suffered no legally recoverable damages because she either (1) suffered no harm, or (2) if she did suffer harm, she cannot recover it from the defendant under existing legal precedent. Her case lacks any legal merit because her expected judgment from proceeding to a trial is zero. Another, more inclusive definition of frivolous litigation also includes negative expected value lawsuits in other words, litigation in which the expected judgment is greater than zero but still remains less than the plaintiff's costs of proceeding to trial. Such negative expected value ("NEV") litigation appears to be irrational for plaintiffs to file and for defendants to settle.    A problem with both of the above definitions of frivolous litigation is that they include cases in which litigants are seeking to establish new legal theories that differ from existing legal precedent. Many people, including the author of this Article, believe that novel test cases in such legal areas as civil rights actions, e.g., actions regarding subconscious gender discrimination and unconscious racial discrimination, should not be considered frivolous litigation.     Professor Bone defines a frivolous lawsuit as one in which a plaintiff either (1) actually knows that the case completely, or virtually completely, lacks any merit under the legal theories being alleged; or (2) would have known that the case lacked merit under the legal theories being alleged, had the plaintiff conducted a reasonable investigation before filing. n52 This definition of frivolous litigation clearly differs from negative expected value litigation definitions in three ways. First, Bone's definition makes no comparison of expected judgments and litigation costs. Second, his definition does not include as frivolous those lawsuits described in the observation mentioned in the previous paragraph. Third, this definition includes a second component that has no analogues in the negative expected value definitions. In addition, as Professor Bone details, positive net expected value litigation explanations of frivolous litigation are unconvincing. n53 This is because positive net expected value litigation examples of frivolous litigation are due to either a plaintiff expecting substantial nonlegal benefits or courts making enough mistakes to imply that even a meritless lawsuit has a high likelihood of success at trial. While both of these scenarios are possible, neither is a sufficiently serious problem to warrant costly regulatory intervention.    2. Using Options Games to Develop a Definition of "Frivolous Litigation" A number of law and economics models address the dual questions of why plaintiffs file frivolous lawsuits and why defendants agree to settle frivolous lawsuits. n54 Existing models demonstrate that litigation deemed frivolous according to the definitions discussed above can still be worthwhile for the litigants to pursue if any of the following situations applies: the litigants possess different probability estimates of the plaintiff prevailing at trial; n55 a court makes a legal error; n56 the parties' litigation costs are incurred sequentially; n57 asymmetries exist between the litigants with respect to the size or timing of litigation costs; n58 plaintiffs have private information concerning their cases; n59 or plaintiffs have the ability to commit or pre commit to litigation. n60    The questions of why a plaintiff would choose to file a frivolous lawsuit and why a defendant would agree to settle a frivolous lawsuit are intellectually and practically troublesome. Asymmetric information game theoretic models answer both questions; however, as the phrase "asymmetric information" suggests, these models assume that just one side of the litigation in other words, either the plaintiff or the defendant realizes that the litigation is frivolous. n61 In other words, no mechanism exists for communication between the parties as to the truthful revelation of this private information besides litigation. The 2001 Nobel Prize in Economics recognized the pioneering research of Professors George Akerlof, Andrew Michael Spence, and Joseph E. Stiglitz, each of whom developed seminal concepts in the economics of symmetric information. n62 Such concepts as lemons, pooling, separation, and signaling games play crucial roles in asymmetric information game theoretic models of frivolous litigation. n63    This Article allows for the realistic possibilities that (1) initially, neither side of the litigation knows whether he or she is a participant in a frivolous lawsuit, and (2) perhaps more importantly, initially, neither side of the litigation knows for certain whether a court will hold that the lawsuit it frivolous. For example, medical malpractice plaintiffs may file lawsuits, in part, from a motivation to find out what really happened during a medical procedure that went awry. Even if a doctor seems not to be legally negligent, an empathetic jury might nonetheless find in favor of a sympathetic plaintiff.    The model in Appendix B of this Article discusses litigation that might possibly be frivolous. The adverb "possibly" reflects a realistic feature of litigation in the sense that, during its course or process, litigants and their attorneys will revise their expected costs and benefits of proceeding to a trial. In other words, parties and their lawyers will learn that a lawsuit is frivolous only after the lawsuit commences. Frivolous litigation is not rational for plaintiffs to file or for defendants to settle, if the litigation costs are incurred all at once up front or if the expected value of litigation does not change over the course of litigation. But litigation that later turns out to be frivolous can be initially rational for plaintiffs to file and for defendants to settle if litigation costs are incurred sequentially and if the expected value of litigation changes over the course of litigation.    The model in Appendix B of this Article develops a new theory regarding frivolous litigation that provides conditions under which litigation that may later turn out to be frivolous is initially credible for plaintiffs to file and for defendants to settle. n64 This novel theory of possibly frivolous litigation is based upon two central features of litigation. The first aspect of litigation to note is that, once a plaintiff makes the initial decision to file a lawsuit, Federal Rule of Civil Procedure 41(a)(1)(i) and its state counterparts provide the plaintiff with unilateral options to abandon that lawsuit under certain circumstances. n65 The second notable feature of litigation is that parties and their attorneys learn information concerning their litigation over the course of that litigation.    In the particular model presented in Appendix B, no asymmetric information exists. Therefore, one party is not learning things that the other party already knows. Instead, both parties in this Article's model learn information that both parties do not know, namely the judgment at trial or the costs of litigation. In Appendix B's model, the parties to litigation share common initial beliefs regarding the judgment at trial and the costs of litigation. Although the model does not restrict how parties revise their beliefs over time, the model does assume that both sides of the litigation modify those shared initial beliefs in the same manner, and hence, share the same beliefs in each period of litigation. In other words, litigants do not have to necessarily update their probability beliefs via Bayes's rule, but they do have to utilize a common rule for how they adjust probability beliefs over time. n66    For a decision maker, an important benefit of learning information is the opportunity to make choices based on that additional information. Such potentially valuable opportunities are precisely what decision makers gain from having options. n67 This central and fundamental insight underlies Professors C. Frederick Beckner's and Steven Salop's multistage decision model of sequential legal procedure, which computes the optimal standards of summary disposition (those minimizing the sum of information and error costs) and the optimal sequence of legal and factual issues that a court should address. n68 That insight also underlies Professor Landes's model regarding whether a court should hold separate trials for liability and damages, as opposed to a unified trial that considers both issues. n69 Professor Warren F. Schwartz also recognizes the value of options when he demonstrates that separating determinations of damages from determinations of liability could reduce litigation costs. n70    This Article fills a niche in the literature about possibly frivolous litigation. n71 This Article demonstrates how to harmoniously blend an options approach to lawsuits with a strategic approach to pretrial settlement bargaining. The analytical model in Appendix B of this Article both builds upon and combines two major literatures. The first literature is research about options, both in law in particular and, more generally, in strategic management. The second literature consists of game theoretic models of litigation. The model in Appendix B of this Article integrates these distinct literatures into a unified game theoretic options model of litigation. Strategic options models have only recently begun to appear in the financial and management literatures. n72 These models can become quite mathematically complicated rather quickly. n73    The model in Appendix B of this Article clarifies how and why options analysis explains when possibly frivolous litigation can be nonetheless credible for plaintiffs to threaten to file and for defendants to settle. Of course, not all threatened potentially frivolous lawsuits are going to be credible for plaintiffs to threaten to file and for defendants to settle. In fact, Appendix B of this Article proves that only lawsuits, including possibly frivolous ones, with positive net abandonment option values as to their expected costs, are credible for plaintiffs to threaten to file and for defendants to settle. n74 In other words, the gross abandonment option values for each stage of litigation must exceed the cost of that stage of litigation. The model in Appendix B of this Article demonstrates that lawsuits that have positive net expected values will also have positive net option values. Thus, any lawsuit with a positive expected value ("PEV") will be credible for a plaintiff to threaten to file (and actually to file) and for a defendant to settle. n75    All lawsuits, including NEV lawsuits, must have positive gross abandonment option value, because any random variable's abandonment option value is larger than or equal to its expected value. That conclusion logically follows from the notion that the abandonment option value of a random variable equals that random variable's expected value when all of the negative value realizations of that random variable are replaced by zero. Such a conceptualization of the abandonment option value of a random variable insightfully captures a pragmatic and valuable feature of abandonment options: namely, the flexibility to avoid negative outcome realizations of the underlying random variable. Thus, the abandonment option value of any random variable, including that of a plaintiff's expected judgment at litigation must be, non negative by definition.    Several economists have developed the related concept of an option value or quasi option value, both in the particular context of environmental preservation and in the more general setting of decision making under conditions of risk. n76 The relationship between these option values has caused some confusion in the literature. n77 By the phrase "the option value of a random variable," this Article simply means the expected value of that random variable, but with all of its negative value realizations replaced by zero. From this definition, it follows that, at every date, the option value of any random variable exceeds the expected value of any random variable.    The model in Appendix B of this Article provides four principal ways in which any evidentiary, procedural, or substantive rule (or change in such a rule) can increase the value of a plaintiff's litigation abandonment option, namely: (1) increasing the variance of trial judgment awards, n78 (2) increasing the divisibility of plaintiff's legal costs, n79 (3) back loading plaintiff's litigation costs, n80 and (4) decreasing plaintiff's total litigation costs. n81    However, the game theoretic options model of litigation in Appendix B of this Article differs from game theoretic expected value models of litigation in terms of its predictions. For example, in expected value game theoretic models of litigation involving risk neutral parties, a mean preserving increase or decrease in the variance of judgment at trial has no impact on the incentives to file, nor does it affect the Nash equilibrium settlement amounts. But in the game theoretic options model of litigation involving risk neutral parties, a mean preserving increase or decrease in the variance of judgment at trial increases or, respectively, decreases both the incentives to file and the Nash equilibrium settlement amounts. n82    The reason for this difference in the predictions of the game theoretic expected value and options models of litigation is that risk neutral litigants only care about expected values, and not variance, in the game theoretic expected value model of litigation. In contrast, in a game theoretic options model of litigation, the option values of the settlement amounts from litigation depend not only upon expected values, but also upon the variances of random variables even with risk neutral litigants.    A comparison of the English rule, which requires the losing party to pay the legal fees of both sides, and the American rule, which requires each side to pay for its own legal fees, also illustrates the difference between the predictions of the game theoretic expected value and options models of litigation. A large body of theoretical and empirical literature exists regarding the incentives to file a lawsuit under the American and English rules for allocating legal costs. n83 Almost all of this literature focuses exclusively on the average values or expected judgments of trials because risk neutral parties care only about means and not about variances, higher order moments, or any other characteristics of the distributions of trial outcomes. Professor Shavell proved that a plaintiff is more likely to file a lawsuit under the American rule than under the English rule if the plaintiff does not expect to prevail because the plaintiff expects to pay only for the plaintiff's litigation costs rather than both sides' litigation costs. n84 Appendix B of this Article proves that, under certain hypotheses, the filing of possibly frivolous litigation is more likely under the English rule than under the American rule if the probability of the plaintiff prevailing at trial is sufficiently large or if the plaintiff's expected aggregate litigation costs sufficiently exceed the defendant's expected aggregate litigation costs. n85        3. Applying These Models to the Discovery Context and Beyond While Appendix B of this Article develops a multi period options model of lawsuits, this general approach to litigation can be illustrated by making the simplifying assumption that litigation consists only of two stages, namely discovery and trial. Because modern liberal rules of pleading allow the survival of a fairly broad class of claims, a plaintiff's lawyers can engage in discovery if a plaintiff files a lawsuit and survives a defendant's motion to dismiss for failure to state a claim. n86 Before discovery, the plaintiff's attorney has only filed suit and initial motions; therefore, legal fees up to then are usually small in comparison to the sizable amounts charged during discovery. In fact, an empirical survey of attorneys found that about 50% of the aggregate costs of litigation are discovery costs. n87 A plaintiff can avoid incurring those significant discovery costs by dropping the litigation.    But, even if a plaintiff decides to have her attorney engage in discovery, a plaintiff is not locked into proceeding to a trial. In fact, discovery provides information and opportunities to update beliefs as to the probability of the plaintiff's winning at trial. Federal and state rules governing discovery confer upon parties the legal rights to obtain information from other parties before trial via document requests, interrogatories, and the deposition of witnesses. n88 But both the attorney client privilege and the work product doctrine limit the information another party can discover. n89 The discovery process provides a plaintiff's attorney with the opportunity to conduct research into a case; the attorney can develop it further if it looks promising in terms of an expected judgment or a settlement, or he can recommend that a plaintiff drop the case if it does not look promising. For simplicity, assume that discovery completely resolves the uncertainty over the actual merits of a case. Then, after discovery, both sides of the case will know the probability of the plaintiff's prevailing at trial is either zero or one. Therefore, in the second period, a plaintiff will be willing to incur the sizable and irreversible costs of trial if she learns that she has a sure winner, while a plaintiff will drop the case unilaterally if she learns that she has a sure loser.    Many sophisticated game theoretic models of discovery, n90 discovery rules, n91 and efficient discovery exist. n92 Discovery generates benefits and costs that differ significantly between plaintiffs and defendants. n93 A defendant's cost of complying with a plaintiff's discovery requests for non privileged, relevant documents can be quite substantial, whether the defendant is a corporation, doctor, or even just another individual. n94 A clear potential for discovery abuse exists because of the externality involved where plaintiffs receive the informational benefits of discovery but defendants bear its costs. n95 So, even if the discovery request will likely produce benefits that exceed its costs, one party receives the benefits from, while another party bears the costs of, discovery. n96 Thus, even if a discovery request is socially desirable, in the sense that its benefits exceed its costs, it can provide a small plaintiff an advantage over a large defendant, as Professors David Rosenberg and Steven Shavell demonstrate in their analysis of NEV lawsuits. n97 Although both sides to a lawsuit can make discovery requests, a plaintiff does not incur much cost in complying with discovery requests when she lacks "truckloads of documents." n98 The cost of complying with discovery requests illustrates how litigation abandonment options may create problems akin to a strategy of raising rival's costs in the context of business competition and the game theoretic industrial organization literature. n99    More generally, a lawsuit consists not just of discovery and trial stages. Several other stages exist, as well. The plaintiff's lawyer files a complaint; the defendant's lawyer files a pre answer motion to dismiss the plaintiff's complaint, e.g., under the Federal Rules of Civil Procedure, a motion to dismiss "for failure to state a claim upon which relief can be granted;" n100 the defendant's lawyer answers the compliant by making admissions, n101 making denials, n102 raising affirmative defenses, n103 or filing counterclaims or cross claims; n104 the lawyers file third party complaints; n105 the lawyers amend or supplement their pleadings; n106 the lawyers make any required automatic disclosures; n107 the lawyers conduct, object to, and respond to discovery requests for the production of documents; n108 the lawyers send and the parties must sign answers to interrogatories; n109 the lawyers take oral depositions; n110 the lawyers request and comply with court orders for independent physical or mental medical examinations; n111 the lawyers promulgate and respond to requests for admissions; n112 the lawyers file and respond to motions for directed verdict; n113 the lawyers proceed to trial by, among other things, conducting opening arguments, examining and cross examining witnesses, n114 presenting non testimonial evidence, and making closing arguments; the lawyers file and respond to motions for judgment as a matter of law before a verdict (also known under some state rules of civil procedure as motions for summary judgment); n115 the lawyers file and respond to motions for judgment as a matter of law after the verdict (also known under some state rules of civil procedure as motions for j.n.o.v., which stands for judgment non obstante veredicto); n116 the lawyers file and respond to motions for a new trial; n117 and, finally, the lawyers file and respond to motions to alter or amend a judgment. n118 Thus, litigation is a multistage process that provides plaintiffs not just a single option but instead, a sequence of abandonment options analogous to those found in sequential investment.    4. Properties of Litigation Abandonment Options Litigation abandonment options have several interesting features. First, plaintiffs do not pay litigation abandonment option premiums to defendants but instead, to plaintiffs' attorneys. If plaintiffs are not paying clients, but instead are suing under contingency fee arrangements or attorney fee award statutes, then plaintiffs' attorneys incur litigation abandonment option premiums up front. Second, defendants provide these litigation abandonment options to plaintiffs by virtue of their activity choices and the relevant substantive and procedural laws. Thus, changes in either procedural or substantive law can alter the value of litigation abandonment options. Third, plaintiffs' litigation abandonment options are similar to the options that a natural resources company, oil refinery, pharmaceutical company, petrochemical firm, or, in fact, any business that is engaged in research and development ("R&D") has to abandon product or process innovation. n119 But there is a very important difference between a plaintiff's litigation abandonment options and those in R &D. On the one hand, lawsuits are wasteful from the joint perspective of plaintiffs, defendants, and perhaps, society as a whole if the costs imposed upon a court, a judge, and jury (if any) exceed the precedent and process values from adjudication of the litigation. In litigation, the plaintiff and the defendant will both lose if they make investments in a lawsuit, as opposed to resolving their differences via some alternative dispute resolution method. On the other hand, a corporation engaging in R&D, as well as its employees, its equity owners, its debt holders, its current and future customers, the surrounding community, and possibly other third parties, all stand to gain from the development and sale of a new product. Thus, even though litigation abandonment options are similar to other familiar examples of options, litigation abandonment options differ from other existing options in several important ways.    III. Limitations of Strategic Litigation Option Analysis This part of the Article appraises limitations of an options game theoretic approach to litigation. Some of these limitations in the particular context of litigation are the result of general behavioral limitations on game theoretic analysis. n120 First, there are cognitive limitations as to how people conceptualize, frame, make, process, and understand choices over time. n121 Second, traditional, or non psychological, game theoretic models assume that people do not experience any emotions or feelings. Both of these limitations are particularly serious in litigation settings because most litigation is quite time consuming and emotionally draining, if not protracted and contentious. It is possible that, even if litigants themselves are myopic and overly emotional, their lawyers might be more farsighted and less emotional. Unfortunately, however, lawyers may exacerbate cognitive and emotional issues, due to conflicts of interest and repeat play considerations, such as those involving developing a reputation for being tough or playing hardball in pretrial settlement negotiations. n122 This part of the Article considers these limitations in turn and explores possible responses to such limitations.    A. Cognitive Limitations of Strategic Options Analysis The standard procedure for solving dynamic games of complete information utilizes a technique known as "backward induction." n123 This method for calculating an equilibrium solution to an extensive form game of perfect information starts by determining the optimal choice for the player who moves last. It continues by then determining the optimal course of action for the player who moves penultimately, and so forth, until determining the optimal decision for the player who moves first. An alternative way to understand backward induction focuses on the sequential rationality of players' strategies. The requirement of sequential rationality relates to another intuitive notion: that of credibility of threats. Professor Bebchuk systematically applied the credibility constraint in his approach to NEV lawsuits. n124     As Professor Bebchuk noted, backward induction arguments have become standard in studying multi period strategic environments. n125 The history of backward induction arguments dates back at least to Zermelo's demonstration that in chess, either white or black can ensure itself a draw regardless of how the other side plays. n126 Later, the philosopher Kierkegaard said, "It is quite true what Philosophy says: that Life must be understood backwards. But that makes one forget the other saying: that it must be lived forwards." n127 Similarly, backward induction arguments presume that decision makers have the computational ability to, and in fact do, correctly forecast all of the future choices that are to be made in a game. The longer or more complex a game, the more descriptively problematic is the assumption of rational expectations about strategic decisions. n128    Numerous experiments demonstrate that people are quite limited in their abilities to perform backwards induction for even relatively simple game situations. n129 The inconsistency between empirical experimental play results and backward induction based solutions for a famous game called "the centipede game" illustrates the predictive limitations of using backward induction arguments for sufficiently lengthy games. n130 Even in only two stage or three stage sequential bargaining experimental games, subjects actually play very differently from backward induction based equilibrium solutions for those games. n131 One way to resolve these and related backward induction paradoxes is to posit a small degree of uncertainty into the players' knowledge of what motivates other players. n132     The game theoretic analysis of litigation abandonment options in Appendix B of this Article utilizes backward induction arguments to analyze lawsuits, despite the fact that the above concerns are disturbing and convincing. Utilizing backward induction arguments to analyze lawsuits is appropriate because litigants have financial and psychological incentives to be sequentially rational. Indeed, litigants are more likely to be sequentially rational than are experimental subjects, who may face artificial time constraints and might lack the motivations of greed and other emotional responses frequently found in litigation. n133 Also, even if the litigants themselves fail to be sequentially rational due to, for example, cognitive difficulties, they hire lawyers who provide not only legal knowledge and expertise, but also negotiating experience and professionalism. Presumably, part of being a professional is being unwilling to make and carry out incredible threats. In a sense, then, litigation involves professionals who have reasons to be sequentially rational. Of course, both defendants' and plaintiffs' attorneys are often repeat players and their behavior might be rational across cases as opposed to within any given case. A final defense is the often made hand waving argument that market reputation and competition discipline lawyers who fail to be sequentially rational. In other words, lawyers who fail to be sequentially rational by making and carrying out incredible threats will become known for doing so and lose business to lawyers who do not do so. They will lose business because carrying out an incredible threat means by definition that carrying out such a threat hurts the plaintiff.    B. Emotional and Psychological Factors in Strategic Options Analysis Almost all formal economic models of litigation focus primarily on the monetary incentives to sue, settle, or proceed to a trial. Professors Huang's & Wu's psychological game theoretic models of litigation provide an exception. Their models demonstrate how emotions such as anger, outrage, and shock can prevent or delay settlement in litigation by changing the incentives of parties to sue, to settle, or to go to trial. n134 In addition, the United States Supreme Court described and endorsed the wide ranging rights of parties to control and participate in their litigation based upon a psychological theory of process based value to precluding feelings of unjust treatment. n135 Empirical and experimental psychological research demonstrates that people are more likely to accept an adverse outcome and to believe that an adjudicatory process is fair if they have the opportunity to personally participate in that process, that is, have the adjudicator hear their stories. n136 Emotional considerations usually predominate in particular legal areas, including, but not necessarily limited to battery, child custody, criminal offenses, defamation, divorce, false imprisonment, intentional infliction of emotional distress, invasion of privacy, medical malpractice, products liability (especially involving bodily injury), and worker's compensation. These areas often involve hot emotions because they involve physical or emotional harms or invasions.    Whether a lawsuit has a positive or negative expected value to a plaintiff, a lawsuit always has net negative expected value to a defendant (ignoring the filing of counterclaims) because of a defendant's litigation costs. Indeed, avoiding such costs is often the rationale for settlement. In reality, it is not just legal costs, but also the opportunity costs, such as the prospective harm to a defendant's reputation, that might lead a defendant to settle a lawsuit by effectively purchasing the plaintiff's litigation continuation options. An often used pejorative term is that of "vexatious litigation." In a well known quotation from a securities fraud lawsuit, the Chief Justice of the Supreme Court, William Rhenquist, spoke of the danger of "vexatious litigation" that could result from the prosecution of "a complaint which by objective standards may have very little chance of success at trial" because, among other reasons, "the very pendency of the lawsuit may frustrate or delay normal business activity of the defendant which is totally unrelated to the lawsuit." n137 Exactly what constitutes a vexatious lawsuit is debatable in the same manner as precisely what constitutes a frivolous lawsuit. But, certainly, the heated emotional considerations that motivate a plaintiff to use a lawsuit to harass a defendant provide an example of a nonmonetary aspect of vexatious litigation.    A recent set of psychological experiments indicates that decision makers generally overvalue their options and exhibit a willingness to invest greater effort and larger sums of money to keep options viable, even when such options have little intrinsic value. n138 The tendencies uncovered experimentally were robust with regard to decision makers' experiences, information about outcomes, and saliency about option costs. n139 In other words, options may offer subjective values exceeding their decision theoretic value for two psychological reasons. First, people sometimes derive pleasure from merely having the right to choose. n140 This phenomenon is perhaps related to a desire for or illusion of control. n141 Second, people sometimes experience loss aversion and a type of endowment effect for options. n142 This phenomenon relates to the phenomenon of litigants experiencing framing effects as described by prospect theory, causing frivolous litigation and lack of settlement during pretrial bargaining. n143     Finally, the emotional and psychological costs of exercising options in general may prevent people from exercising options that are monetarily inexpensive. For example, most prospective law students probably undervalue the value of beginning a legal education because they ignore their options to drop out. In other words, a student can decide to abandon legal education after the first class, day, week, semester, or year of law school (or in fact anytime before graduation). However, many law students refuse to become legal drop outs because doing so is too embarrassing or costly in terms of their psyche or ego. As with most options to abandon some course of action, people may feel a compulsion to not be quitters in their own eyes or in those of certain observers, such as family members, friends, or even political constituencies, in the case of politicians who do not want a reputation for being inconsistent. The option to modify a course of action might be less emotionally or psychologically costly than the option to abandon or discontinue a course of action (even though abandonment is a particular form of modification). Thus, a student might not drop out of law school, but might instead choose to change her course of study or legal career path.    In conclusion, litigants experience both cognitive limitations as well as emotional and psychological factors that are not present in strategic options analysis of litigation. There are some reasons to believe that lawyers might experience less of those cognitive limitations and emotional and psychological factors in their decision making, as compared to their clients. But clearly it would be helpful to conduct further empirical research concerning how lawyers actually behave and theoretical research about how to incorporate cognitive limitations as well as emotional and psychological factors into strategic options analysis of litigation.    IV. Other Legal Applications of Options Analysis This part of the Article focuses on applying options analysis to regulate other litigation options in civil procedure, preclusion law, constitutional law, and family law. Constraints of space and time permit only a brief glimpse of the full potential of these legal applications. Although all the possible applications below only pertain to civil actions, numerous legal options in the areas of criminal law and procedure, such as options provided by prosecutorial discretion and plea bargaining also exist.    A. Other Litigation Options: An Overview Thus far, this Article has analyzed lawsuits from the perspective of the options that a plaintiff has to unilaterally abandon or drop litigation. Other litigation options also arise naturally in litigation and civil procedure. For example, some states allow for the practice of additur, n144 in which a court denies a plaintiff's motion for a new trial, conditional on a defendant accepting more liability than a jury awarded. From an options perspective, additur involves a court presenting a defendant with an option to accept more liability than a jury award in exchange for a plaintiff not being permitted to exercise her option to file and to later abandon a new trial.    In the medical malpractice area, Professor Jeffrey O'Connell, along with several co authors, has proposed a reform plan under which a physician has the option to make a plaintiff an early offer to pay for economic losses in the form of medical expenses and lost wages. n145 In exchange for accepting such an offer, a plaintiff relinquishes her option to file and later to abandon litigation for damages to compensate for non economic harms, unless that plaintiff can prove that the physician was guilty of gross criminal negligence. Applying options theory provides a qualitative, if not quantitative, analysis of both a physician's option to make such early offers and the forgone value of a patient's option to abandon litigation seeking damages for pain and suffering.     Finally, an options perspective to litigation can provide not only descriptive or positive analysis, but also prescriptive or normative analysis of litigation related behavior. For example, options theory suggests that a rational and far sighted manufacturer should factor into the price of her product a per unit amount for covering the option values of products liability cases, which are larger than merely expected litigation costs or damage awards from defending or settling products liability cases. n146 But if only a single manufacturer in a reasonably competitive market raises her prices to cover the option values, instead of expected values of product liability, then it might price itself out of business relative to its competitors in the short run. Another example of prescriptive or normative analysis comes from realizing that, all other things equal, the deterrence impact of settlements or trials based upon the option values of litigation exceeds the deterrence impact of settlements or trials based upon their expected monetary damage awards. n147 This relative comparison applies equally forcefully to the deterrence of harms from accidents, contract breaches, governmental takings of private property, and nuisances. n148 A final example is to analyze how litigation abandonment options affect Professors George Priest's and Benjamin Klein's selective litigation hypothesis that a nonrandom sample of cases filed will result in trial. n149        B. Precluding Legal Options This Article demonstrates that many procedural and substantive legal rules provide options that are valuable because they provide flexibility. It is well known, conversely, that inflexibility can be advantageous in strategic bargaining. n150 Economists and game theorists often speak of people utilizing (pre)commitment devices to improve their bargaining position; for example, automobile salespeople, one member of a couple, and employers often claim their hands are tied. An options perspective about law also reveals that numerous legal doctrines and rules increase the price of, if not preclude, certain other legal options. For example, both the Model Code of Professional Responsibility and the Model Rules of Professional Conduct preclude certain behavior that would otherwise be available to lawyers as options. For another example, both state and federal sovereigns have developed a number of self limitation doctrines and statutes that prevent them from exercising their full adjudicatory, constitutional authority over non local cases. n151 Most prominent among these subconstitutional restrictions on geographic forum selection are the laws of venue and forum non conveniens. n152    1. Applying Options Theory to Res Judicata and Collateral Estoppel Another example of this concept is found in the judicially created doctrines of collateral estoppel and res judicata. Collateral estoppel, also known as issue preclusion, prevents the same issues from being relitigated in subsequent lawsuits. For the doctrine of collateral estoppel to preclude an issue, that issue must be identical, actually litigated, decided, and necessary for the court's judgment in a prior lawsuit. n153 The standard first year law school civil procedure casebook's explanation of the reasons behind collateral estoppel and res judicata is achieving finality or the repose of judgments. n154 Another traditional rationale for both collateral estoppel and res judicata is judicial economy. n155 Finally, decisional consistency is often cited as an additional benefit of collateral estoppel and res judicata. n156 But collateral estoppel and res judicata both also influence the settlement values of litigation. Professor Bruce L. Hay argues that collateral estoppel and res judicata both function to better align the settlement values of lawsuits with their merits. n157 The few economic analyses of collateral estoppel and res judicata do not specifically analyze these judicial doctrines as explicitly proscribing future litigation options. n158 The perspective of this Article suggests analyzing the preclusion rules of collateral estoppel and res judicata specifically from an options perspective, which is beyond the scope of this Article.    2. Applying Options Theory to the Proposal and Ratification of Constitutional Amendments The political value of precluding some amendment options also helps to explain the supermajoritarian requirements for the proposal and ratification of constitutional amendments. n159 The United States Constitution requires that at least two thirds of both houses of Congress vote to propose a constitutional amendment. n160 Alternatively, two thirds of the state legislatures must petition Congress to call a constitutional convention. n161 The second method of proposing a constitutional amendment never has been utilized. In addition, the United States Constitution requires three quarters of the state legislatures or state conventions to ratify a Constitutional amendment. n162 The only amendment ratified by state conventions was the Twenty first Amendment, which repealed Prohibition. n163 Although over eleven thousand constitutional amendments have been introduced in Congress since 1793, only thirty three of these have received the requisite two thirds vote of Congress to be submitted to the states for ratification. n164 Of those, six never were ratified, including, most notably, the Equal Rights Amendment proposed in 1972 and, most recently, the D.C. Voting Rights Amendment proposed in 1978. n165 The constitutional amendment process "guards equally against that extreme facility, which would render the Constitution too mutable; and that extreme difficulty, which might perpetuate its discovered faults." n166    3. Applying Options Theory to Judicial Decision Making A similar concern about precluding judicial and legislative options helps explain why the current United States Supreme Court engages in judicial minimalism. n167 Deciding a particular case not only decides that case on its merits, but it also affects future activity, behavior, and cases via precedent and the resulting effects on incentives. In addition, the principles of analogical reasoning and the demands of logical consistency mean that any judicial decision might constrain or preclude future, related judicial or legislative options to make decisions about related legal issues.    4. Applying Options Theory to Family Law Finally, as Professors Dixit and Pindyck suggested in their book:    Marriage entails significant costs of courtship, and divorce has its own monetary and emotional costs. Happiness or misery within the marriage can be only imperfectly forecast in advance, and continues to fluctuate stochastically even after the event. Therefore waiting for a better match has an option value. n168 This Article suggests analyzing family law statutes in terms of how they regulate the options to marry or divorce. For example, the family law statutes of many states require couples to wait for a specified period of time after the application for a marriage certificate before its issuance. n169 Symmetrically, some of these states also stipulate that a couple may not divorce until after the passage of a mandatory waiting period, which usually exceeds the mandatory prenuptial waiting period. n170 One can understand both types of family law statutes as raising the waiting time or non monetary price of, if not precluding, certain marriage or divorce options. The debate over whether a state will legally recognize the marriage of gay and lesbian couples effectively concerns whether a state will preclude legal marriage options and the attendant legal rights (themselves options) that follow. In fact, legal rights in general are options their owners may choose not to exercise because of either low payoffs or high costs of exercising them.    V. Conclusions This Article introduces a new options game theoretic model of possibly frivolous litigation. This novel theory is a hybrid approach that combines an options approach to litigation incentives and game theoretic models of pretrial settlement negotiations. This Article derives a set of necessary and sufficient conditions for the credible threatening and filing of possibly frivolous litigation based on whether the option values from abandoning litigation exceed the cost of those litigation abandonment options. Similarly, defendants only settle those possibly frivolous lawsuits with initial positive net (of the cost of litigation) abandonment option values. This Article also considers limitations of strategic litigation options analysis. Finally, this Article very briefly introduces other applications of options analysis to law by pointing out other litigation options and preclusions of legal options in various legal areas.    The technical details of this new options game theoretic model of possibly frivolous litigation are presented in Appendix B of this Article. But the intuition of this new model is captured by the numerical example presented in the Introduction. The procedural fact that both Federal Rule of Civil Procedure 41(a)(1)(i) and its state counterparts provide a plaintiff with the option to unilaterally and voluntarily abandon her lawsuit under certain circumstances without prejudice provides plaintiffs litigation abandonment options with value. If those positive option values exceed a plaintiff's expected litigation costs, then a plaintiff will credibly threaten to and will file a lawsuit. This Article develops a formal mathematical theory of how to value a plaintiff's litigation abandonment options.        Appendix A: An Options Primer What do buying a house, having children, and recalling the governor of California have in common? These three seemingly unrelated experiences all involve not only sequential decision making, n171 but also exercising or preserving various options. House hunting may involve looking over many houses that differ along numerous dimensions. n172 These differences mean that it can be difficult for a buyer to compare houses in order to determine an optimal stopping rule for house shopping. n173 Passing on a particular house preserves options to buy other houses, but risks losing an option to buy that particular house later. Multiple potential buyers might express interest in a particular house and end up bidding against each other. A bidding contest over a house means that a potential buyer has fewer negotiating options because she may feel she has to make her initial bid her best offer, instead of engaging in a series of negotiating rounds. Deciding to bid on, and then possibly losing bids on, houses can become an emotional roller coaster. n174 The purchase of a home is part of the American dream, but for most Americans, their home is their most expensive purchase (at least, until then). Thus, most (at least first time) home buyers finance part of the price of their purchase by taking out a mortgage. Virtually every home mortgage grants a homeowner the option to pay off the mortgage early without any penalties for prepayment. n175    Whether and when to have children, as well as how many to have, are various options that people face. Before (and even after) a child is conceived, there are numerous options regarding birth control. Conception can be assisted by reproductive and genetic technologies. n176 Once a child is conceived, many options exist regarding prenatal care, whether to carry the child to full term, and methods of delivery. But, while a woman is pregnant, she does not have the option to become pregnant again until after her first pregnancy concludes. In addition, numerous adoption options exist. n177 After a child is born, parents have fewer options in terms of alternative joint activities or purchases. Finally, numerous child rearing options exist. Of course, children have options to have their own children.    California's 2003 gubernatorial recall election was authorized by the California Constitution provision that provides California voters with options to recall their elected officials. n178 Some social observers and political commentators feared that California's recall election set a dangerous precedent because it could lead to voters exercising their options to hold recall elections of any elected officials who make unpopular decisions. n179 As a result, elected officials may come to engage in perpetual campaigning, and elections might degenerate into no more than contests of personality or popularity. But, fifteen states, the District of Columbia, Guam, and the Virgin Islands already have laws that provide their voters with options to recall elected state officials, and thirty six states have laws that provide their voters with options to recall various local officials. n180 In fact, a Gallup Poll conducted in 1987 found that 67% of a nationwide sample of 1009 people supported amending the United States Constitution to provide for the recall of members of Congress, and 55% of that same sample supported a constitutional amendment providing for the recall of the President. n181 Only one third of those polled opposed the idea of recalling nationally elected officials. n182    Another common feature of buying a house, having children, and recalling the governor of California is risk. In fact, virtually every decision involves an element of risk. Attorneys, their clients, elected officials, judges, jurors, legislators, litigants, negotiators, regulators, and voters face various risks, including those arising from appellate, contractual, electoral, financial, judicial, legislative, regulatory, statutory, and technological sources. It is increasingly critical for such decision makers to respond effectively to such risks. Just as omnipresent as risks are the methods by which individuals, organizations, and institutions cope or deal with risks, including diversifying, hedging, insuring, and learning. In a sense, (payoff relevant) information can be thought of as the reduction of risk or the negative of risk.    One particular method of handling risks is by utilizing options. An option provides its holder with a right, as opposed to an obligation, to choose some action in the future. The word option "comes from the medieval French and is derived from the Latin optio, optare, meaning to choose, to wish, to desire." n183 Options are valuable from a decision theoretic perspective when there are unresolved risks because they provide the flexibility not to be locked into an irreversible course of action. In other words, options have no value if there is no risk and decisions are reversible. n184 After all, risks involve not only dangers, but also opportunities. n185 Options allow those facing risky environments to profit from the upside potential of the risks they face, while truncating losses from the downside possibility. Options thus offer asymmetric, kinked, or non linear payoffs because options permit actors to make future decisions after learning relevant information concerning the risks they face.    A few legal scholars already have begun to apply options analysis to study legal rules and institutions. n186 But financial economists and management scholars have been studying options theory and its applications in the practice of financial engineering and management science for over a quarter of a century. n187 Options are classified as financial or real. n188 Financial options are contracts that give their holders the right, but not an obligation, to buy (or sell) a particular quantity of some underlying financial asset at a certain price on (or before) a certain date. n189 Examples of underlying financial instruments on which options are written include bonds, stocks, commodities (such as corn, soybeans, wheat, gold, or silver) futures contracts, foreign currencies, or stock indices. A vast literature addresses the financial theory, institutional details, pricing models, regulation, and valuation of various financial options. n190    Because options concepts, ideas, and terminology may not be familiar to some readers, this appendix offers a brief overview about options in general. n191 A call option provides its owner with the right, but not an obligation, to buy a specified quantity of some underlying item at some price called the strike or exercise price. n192 A put option provides its owner with the right, but not an obligation, to sell a specified quantity of an underlying item at some price called the strike or exercise price. n193 An option's price is called its premium in order to avoid confusion with exercise or strike prices. n194 An option is at the money if the current price of the item equals the strike price. A call option is out of the money (respectively, in the money) if the current price of the underlying item the option is written on is less (respectively, greater) than its exercise price. Similarly, a put option is out of the money (respectively, in the money) if the current price of the underlying item the option is written on is greater (respectively, less) than its exercise price.    The intrinsic value of a call (respectively, put) option is the difference between the current price of the underlying item on which the option is written and the strike price (respectively, the difference between the strike price and the current price of the underlying item that the option is written on). Even an option that is currently out of the money has a positive (although possibly, very small) value because of its time value. The time value of an option will be positive (although possibly, very small) as long as the option has not yet expired because, in the remaining time before its expiration, an option may finish in the money. It is, of course, true, symmetrically, that an option may finish out of the money. But, because options do not require their owners to buy or sell the underlying items on which the options are written, rational option holders will simply choose not to exercise options that are out of the money. The flexible nature of an option explains why, intuitively, an option always has a value that is nonnegative, gross of the option premium, because the holder of an option can decide not to exercise the option.    Financial options permit decision makers to hedge financial types of risk such as those arising from fluctuations in stock prices, interest rates, or currency rates. n195 Financial options are a type of state contingent securities. Professor Kenneth J. Arrow, a recipient of the 1972 Nobel Prize in Economics, introduced the concept and theory of state contingent securities in a paper that he presented in 1952. n196 Many of today's exotic financial derivatives that Wall Street investment banks create, such as catastrophe bonds (whose payoffs are linked to natural disasters such as earthquakes and hurricanes), n197 utilize Professor Arrow's work. n198 Professor Robert J. Shiller has proposed the creation of even more contingent securities markets to hedge aggregate income risks, home price risks, income distribution inequality risks, intergenerational risks, international risks, and livelihood risks. n199 Professor Stephen A. Ross proved that, under certain assumptions, trading simple financial call and put options written on a single index of existing securities can realize any possible desired pattern of payoffs across contingencies and over time. n200    Financial options have revolutionized modern financial markets by facilitating the reallocation of underlying financial market risks. The 1997 Nobel Prize in Economics recognized the path breaking financial option pricing models of Professors Fisher Black, Robert C. Merton, and Myron S. Scholes. n201 Widely publicized, huge losses from trading in financial options by well known corporations such as Barrings Bank, Dell Computer, Gibson Greetings, and Procter & Gamble; n202 municipalities such as Orange County, California; n203 and hedge funds such as Long Term Capital Management (LTCM) n204 illustrated the potential dangers from speculation in financial options. n205 Recently, many commentators have questioned whether incentive compensation in the form of executive stock options contributed to the series of corporate fraud and mismanagement scandals, and whether companies should expense their executive stock options. n206    Real options involve decisions concerning activities whose risks have not been completely reduced to financial assets or financial commodities. n207 Real options are so named to differentiate them from financial options such as well known executive stock options. n208 The phrase "real options" in corporate finance refers to the options that managers have to add value to an organization by adjusting its production plans; n209 alter capacity, output levels, or scale of operations; break up, divide, or partition investment opportunities; defer before (further) investing; switch inputs, outputs, or production methods; and grow from a pilot project. n210 Indeed, any dynamic investment opportunity presents a sequence of real options. n211 For example, business deal making negotiations entail numerous real options. n212 Multinational corporations face numerous real options in making foreign investments. n213    In describing a generic decision making setting, the phrase "real option" implies that options theory is applicable to analyzing the sequential choices that are inherent in such a dynamic and uncertain environment. Real options have a fascinating history. n214 Familiar (and perhaps, some unfamiliar) examples of real options include the options to: abandon, perhaps temporarily (i.e. mothball) a project; n215 become delinquent in property tax payments; n216 breach a contract and pay liquidated or expectation damages; n217 build or develop real estate property versus delaying construction; n218 continue with education; n219 declare bankruptcy; n220 delay a project; n221 dissolve a business arrangement, marital or corporate union, merger, partnership, or any other form of ongoing or steady relationship; n222 drill, develop, or start production from oil wells; n223 engage in venture capital start up investing; n224 exchange one asset for another; n225 heat new construction with electricity, heating oil, or natural gas; n226 lease airplanes, assets, copiers, power plants, real estate, satellites, trucks, or zoo animals; n227 maintain academic employment under faculty tenure; n228 make a movie from a script and follow up sequels if the original movie is a box office success; n229 purchase assets, items, or properties; n230 threaten to employ fewer workers if a firm has a flexible production technology; n231 throw away food obtained from an all you care to eat buffet or freely dispose of items generally; try predatory pricing or to leverage monopoly power in one market into monopoly power in another market; n232 and utilize (export or import) quota licenses. n233    Real options theory applies financial option pricing models to derive qualitative, if not (yet) quantitative, estimates of real option values. n234 Many quantitative financial and real option pricing models assume that the underlying risks evolve over time according to a particular stochastic process, known as geometric Brownian motion with drift. n235 Although there is no reason that such a distributional assumption would accurately describe litigation risks in general, other quantitative option pricing models, such as the binomial or two state option pricing model, may approximately describe a particular lawsuit. n236 More generally, qualitative, as opposed to quantitative, financial and real option valuation models apply to litigation abandonment options because qualitative option valuation models make no distributional assumptions regarding the stochastic process of underlying risks. n237 Nonetheless, qualitative option valuation models provide upper and lower bounds for option values. n238        Appendix B: A Game Theoretic Options Model of Litigation This appendix introduces a game theoretic options model of possibly frivolous litigation under the assumption that defendants and plaintiffs maximize their expected net wealth and initially have incomplete but common knowledge regarding all information concerning their litigation. This appendix adopts these quite strong assumptions to focus attention on the additional and novel insights provided by viewing litigation from an options theoretic perspective. This Article demonstrates that options analysis generates different conclusions and implications from those of expected value analysis under an identical set of assumptions. Litigation costs are stochastic processes in the game theoretic options model in this appendix. More generally, litigants might choose the levels of litigation expenditures as endogenous variables as opposed to facing litigation costs that are exogenously distributed random variables. It is left for another day to model endogenous litigation expenditures in a strategic options analysis of litigation.    A. Notation The following notation is used in the formal model. Denote the plaintiff's total litigation costs by P. Denote the defendant's total litigation costs by D. Divide the number of stages in pretrial bargaining by the index t = 1, ... , n. All money values at periods t 1 are denominated in terms of their present discounted values at t = 1 (using a common discount rate or factor). Let I t represent the plaintiff's litigation costs at stage t. Thus, by definition, P = summ 1 [su'n']I t . Let C t represent the defendant's litigation costs at stage t. Then, by definition, D = summ 1 [su'n']C t . Let P t denote the plaintiff's remaining litigation costs after stage t. Then, by definition, P t = summ t [su'n']I k . Let E 1 (P t ) denote the initial expected present value of plaintiff's remaining litigation costs once stage t is reached. Let D t denote the defendant's remaining litigation costs once stage t is reached. Then, by definition, D t = summ t [su'n']C k . Let E 1 (D t ) denote the initial expected present value of defendant's remaining litigation costs once stage t is reached. Let [alpha] denote the relative bargaining strength of the plaintiff; so that, 0 lteq [alpha] lteq 1.    At stage t, the size of the dollar amount of judgment expected at trial is J t . At stage t, the subjective probability of the plaintiff prevailing at trial is denoted p t . At stage t, the expected value of the judgment expected at trial is defined as x t = p t J t . Denote the initial net present discounted expected value of the settlement if the litigation settles at stage t by E 1 (s t ). Solving recursively via backwards induction, E 1 (s t ) = x 1 + [alpha] E 1 (D t ) (1 [alpha] )E 1 (P t ). Let OV 1 (s t ) be the initial abandonment option value of the settlement at stage t. Finally, let E 1 (I t ) be the initial present expected value of the plaintiff's litigation cost at stage t.    The following game theoretic options model of lawsuits assumes that [J t ], [I t ], and [C t ] are stochastic processes, whose distributions are agreed upon and common knowledge among the litigants and their attorneys. Recall that the litigants are assumed to be risk neutral, to share a common discount rate, and to face no effective wealth constraints. Finally, J t , I t , and C t are assumed to be independent random variables at each t.    B. Necessary and Sufficient Conditions for the Credible Filing of Litigation This characterization of the incentives to file lawsuits illustrates the power of backwards induction.    Proposition 1: A necessary and sufficient condition for a lawsuit to be filed is that the initial value of all the abandonment options exceed the initial value of their expected costs or premiums. In other words, for all t = 1, ... , n; these inequalities hold at date 1:    OV 1 (s t ) gteq E 1 (I t )    Proof: (a) Necessity: If for any t lteq n, OV 1 (s t ) E 1 (I t ); then both parties expect at stage 1 that the plaintiff will not continue the lawsuit at stage t. Thus, reasoning backwards, both parties expect at stage 1 that the plaintiff's threat at stage t to continue the lawsuit is not credible.    (b) Sufficiency: Conversely, if for all t lteq n, OV 1 (s t ) gteq E 1 (I t ); then both parties expect at stage 1 that the plaintiff will at each stage t be able to credibly threaten to continue the lawsuit for its abandonment option value at that stage.    It is straightforward to show that any Positive Expected Value (PEV) lawsuit will always satisfy the above condition.     Proposition 2: If a lawsuit has PEV, then all of the abandonment options will have initial values that exceed their initial expected cost.    Proof: A PEV lawsuit by definition satisfies x 1 gteq E 1 (P). Because P = summ 1 [su'n']I t [su'i'] and I t gteq 0 for all t, it follows that x 1 gteq E 1 (I t ) + E 1 (P t ) for all t. So, [alpha] x 1 gteq [alpha] E 1 (I t ) and (1 [alpha] )x 1 gteq (1 [alpha] )E 1 (I t )+(1 [alpha] )E 1 (P t ) for all t: Adding these last two inequalities together results in the inequality, x 1 (1 [alpha] )E 1 (P t ) gteq E 1 (I t ) for all t. This implies that x 1 + [alpha] E 1 (D t ) (1 [alpha] )E 1 (P t ) gteq E 1 (I t ) for all t because [alpha] E 1 (D t ) gteq 0. But, E 1 (s t ) = x 1 + [alpha] E 1 (D t ) (1 [alpha] )E 1 (P t ) by definition. So, for all t; E 1 (s t ) gteq E 1 (I t ). Finally, by the definition of abandonment option value, we conclude that for all t: OV 1 (s t ) gteq E 1 (s t ) gteq E 1 (I t ).    Thus, a lawsuit having PEV is a sufficient, but not necessary condition for a lawsuit to be credibly filed by a plaintiff. The last step in the proof of the above corollary, namely that for all t, OV 1 (s t ) gteq E 1 (s t ), is merely an instance of the more general proposition that at every date, the abandonment option value of a random variable is greater than its expected value. This is true because the abandonment option value of a random variable can be thought of as being equal to its expected value with all of its negative value realizations replaced by zero.    C. Litigation Abandonment Option Values and Equilibrium Settlement Amounts In expected value litigation models, the parties compare their deterministic cash outflows from the costs of litigation with their probability weighted expected monetary payoffs to litigation. If the parties have the same expected values for trial, they will settle rather than go to court, in order to save on trial costs (even if they are risk neutral) or because they are risk averse (even if trial costs are zero). Parties only go to trial if they have sufficiently different beliefs about the probability that the plaintiff will prevail at trial or about the size of the judgment if the plaintiff should win at trial. In expected value litigation models, different expected values for the outcome of trial are necessary, but not sufficient for trial. n239 Settlement occurs if and only if there is a range of mutually acceptable settlement amounts. This interval will be non empty if and only if the difference between the plaintiff's expected gain and the defendant's expected loss from going to trial is less than the sum of their litigation costs. The parties will settle immediately at an amount in the range of mutually acceptable settlement amounts. The precise settlement amount in that range is determined by the values of the parties' relative bargaining strengths. A similar immediate settlement result holds true in this game theoretic options model of lawsuits, the difference being the value of the settlement amount. n240    Proposition 3: If the parties to litigation share the same initial common probability beliefs [p t ] and have common knowledge over [J t ], [I t ], and [C t ], then both parties will agree to settle the litigation in period 1 for the Nash equilibrium amount S* = OV 1 (s 1 ) gteq E 1 (s 1 ).    Proof: If the litigants share common prior probability beliefs regarding the distributions of the relevant random variables, then they also will agree on the values of OV 1 (s t ) and E 1 (I t ) and the inequality conditions in Proposition 1 being satisfied for all t lteq n. Thus, they will agree to settle immediately to avoid incurring litigation costs. In other words, the defendant will effectively agree to buy the plaintiff's initial abandonment option for its value OV 1 (s 1 ), which is at least as large as the net present discounted expected value of the lawsuit, E 1 (s 1 ).    In this game theoretic options model, all lawsuits are settled immediately in the case of homogeneous probability beliefs [p t ] between the plaintiff and defendant and common knowledge regarding the distributions of [J t ], [I t ], and [C t ]. As with settlement in game theoretic expected value models, the settlement amount in this game theoretic options model is constructed iteratively period by period from the last period backwards. If a lawsuit were to be credibly filed and not settled immediately due to differing beliefs [p t ] or lack of common knowledge over the distributions of [J t ], [I t ], and [C t ], it might settle nonetheless at some later period, e.g., after discovery, due to convergence of probability beliefs [p t ] or common knowledge about [J t ], [I t ], and [C t ]. In particular, optimism or self serving biases can prevent immediate settlement, as is the case in expected value litigation models. n241 Finally, notice that higher than expected realized litigation costs may cause a plaintiff unilaterally to drop her lawsuit, because the actual premium of the litigation abandonment option at that stage exceeds its value.    D. Qualitative Comparative Statics or Sensitivity Analysis A game theoretic options model of litigation has different implications for how various policies or shifts in the underlying legal random variables change the incentives to file litigation and the size of Nash equilibrium settlement amounts than those that are predicted by game theoretic expected value models of litigation. Economists utilize the phrase "comparative statics analysis" to refer to a comparison of how equilibrium behavior differs for different parameter values. n242 Another way to think of a comparative statics result is that it analyzes how sensitive behavior endogenously determined in equilibrium is to changes in exogenous variables. Thus, comparative statics results are forms of sensitivity analysis. In this Article's game theoretic options model of litigation, many of these comparative statics results are driven by the fact that the option value of a random variable increases with its variance because of the option to avoid downside risk, while a random variable's expected value does not necessarily increase with its variance.    The first comparative statics result concerns the awarding of punitive damages, n243 a practice in certain areas of the law, such as treble damages in antitrust, n244 punitive multiples in certain tort actions, n245 or willful contract breach. Punitive damages increase the incentive to file lawsuits because such damages increase the amount of expected judgments. But, above and beyond the mean increasing effect on judgments, punitive damages also increase the variance of judgments, and hence, they not only increase the net present discounted values, but also the option values, of settlement.    Proposition 4: Holding all other variables fixed, punitive damages increase the incentives to file lawsuits and equilibrium settlement amounts more than just a variance preserving increase in judgments by the same factor as the punitive multiple does.    Proof: All other things being equal, punitive damages increase the variance of x t for all t and thus increase OV 1 (s t ) for all t. Thus, the necessary and sufficient conditions for filing lawsuits are more likely to hold with punitive damages than without punitive damages. In addition, the necessary and sufficient conditions for filing lawsuits are more likely to hold with punitive damages than with merely a variance preserving increase in judgments by the same factor as the punitive multiple.    The other side of the above result concerns the frequently suggested policy of capping the damages that juries can award. Although these proposals usually lament both the unpredictability and seemingly random nature of jury awards, the argument behind these reforms focuses on the absolute magnitude of the punitive component of jury awards. A game theoretic options model of litigation makes clear that not only the size of expected punitive damages, but also the variance of punitive damages affects the incentives to sue and settle. This is because, above and beyond the mean decreasing effect on judgments, damage caps also decrease the variance of judgments. Hence, they not only decrease the net present values but also the option values of settlement.    Proposition 5: Holding all other variables fixed, damage caps decrease the incentives to file lawsuits and equilibrium settlement amounts more that just a variance preserving decrease in judgments by the same factor as the damage caps.    Proof: All other things being equal, damage caps decrease the variance of x t for all t and thus decrease OV 1 (s t ) for all t. Therefore, the necessary and sufficient conditions for filing lawsuits are less likely to hold than with damage caps than without damage caps. In addition, the necessary and sufficient conditions for filing lawsuits are less likely to hold with damage caps than with merely a variance preserving decrease in judgments by the same factor as the damage caps.    The above two results concerning effects on incentives to file lawsuits of substantive or procedural reforms are special cases of the next general comparative statics result about how the abandonment option value of a lawsuit changes as the variance of the trial judgment award changes.    Proposition 6: Holding all other variables fixed, an increase (respectively, decrease) in the variance of the trial judgment award increases (respectively, decreases) the incentives to file lawsuits and equilibrium settlement amounts.    Proof: All other things being equal, higher (respectively, lower) variance in the trial judgment award increases (respectively, decreases) OV 1 (s t ) for all t. Thus, the necessary and sufficient conditions for filing a lawsuit are more (respectively, less) likely to hold with changes in the variance of the trial judgment award than without such changes.    The next result provides a set of conditions under which the English rule for allocating legal costs, under which the party losing at trial is required to also pay for the winning party's legal costs, increases the incentive to file lawsuits and equilibrium settlement amounts. Professor Cornell observed that even risk neutral plaintiffs are better off under the English rule for allocating litigation costs than under the American rule for allocating litigation costs, under which each side of litigation pays for its own legal costs. This is due to the increased variance of trial outcomes, when taking into account paying for legal costs under the English rule for allocating litigation costs as compared to the American rule for allocating litigation costs. n246 But a lawsuit having an expected judgment of x under the American rule becomes a lawsuit having an expected judgment of x+pP (1 p)D under the English rule. Thus, shifting from the American rule to the English rule not only increases the variance of, but also changes the mean of, trial outcomes. Hence, whether the English rule increases or decreases the likelihood of litigation and settlement amounts compared to the American rule depends respectively on whether the English rule's variance increasing effect on the plaintiff's abandonment option value outweighs its mean decreasing effect on the plaintiff's abandonment option value or vice versa. As the next result makes clear, a sufficient condition for the English rule's variance increasing effect on the plaintiff's abandonment option value to outweigh its mean decreasing effect on the plaintiff's abandonment option value is that for all t, p t E t (P) gteq (1 p t )E t (D).    Proposition 7: If for all t, p t E t (P) gteq (1 p t )E t (D), then the English rule increases the likelihood of filing a lawsuit and the settlement amount compared to the American rule. n247    Proof: Litigation under the English rule instead of the American rule changes both the mean and variance of the litigation process. At each period t, the expected judgment changes from x t to x t +p t E t (P) (1 p t )E t (D), where the conditional expected values E t (P) and E t (D) are conditional on the realized values of litigation costs up to stage t for the plaintiff and defendant, respectively. By hypothesis, for all t, x t +p t E t (P) (1 p t )E t (D) gteq x t . Thus, the English rule does not decrease the expected judgment compared with the American rule. The English rule also increases the variance in expected judgment compared with the American rule. Each of these effects on the mean and variance of judgment increases OV 1 (s t ). But the English rule does not change the plaintiff's litigation costs at any given stage because they are incurred sequentially. Thus, the necessary and sufficient condition for filing a lawsuit is more likely to hold under the English rule for allocating litigation costs than under the American rule for allocating litigation costs.    The next proposition explains how the option value of a lawsuit changes as the variance of the defendant's litigation costs changes, all other things being equal. More (respectively, less) risk over the defendant's legal costs at any given stage increases (respectively, decreases) a plaintiff's incentive to file a lawsuit because of the increased (respectively, decreased) savings in defendant's avoided legal costs from settling before that stage.    Proposition 8: Holding all other variables fixed, increasing (respectively, decreasing) the variance of defendant's litigation costs at any stage k lteq n, increases (respectively, decreases) the incentives to file lawsuits and equilibrium settlement amounts.    Proof: Holding all other variables fixed, more (respectively, less) variance in the defendant's litigation costs at stage k lteq n increases (respectively, decreases) OV 1 (s t ) for all t lteq k. Thus, the necessary and sufficient conditions for filing a lawsuit are more (respectively, less) likely to hold with changes in the variance of the defendant's litigation costs than without changes in the variance of the defendant's litigation costs.    Because changing the variance of the plaintiff's litigation costs also generally changes the mean of the plaintiff's litigation costs; changing the variance of the plaintiff's litigation costs affects both sides of the inequalities from the necessary and sufficient conditions for the (credible) filing of a lawsuit. To isolate the impact of changing the variance of the plaintiff's litigation costs upon the option value of lawsuit, the next proposition analyzes how the option value of a lawsuit changes as the variance of plaintiff's litigation costs changes in a mean preserving manner.    Proposition 9: Holding all other variables fixed, a mean preserving increase (respectively, decrease) in the variance of plaintiff's litigation costs at any stage k lteq n, increases (respectively, decreases) the incentives to file lawsuits and equilibrium settlement amounts.     Proof: All other things being equal, higher (respectively, lower) variance in the plaintiff's litigation costs at stage k increases (respectively, decreases) OV 1 (s t ) for all t lteq k. Thus, the necessary and sufficient condition for filing a lawsuit is more likely to hold with mean preserving changes in the variance of the plaintiff's litigation costs than without mean preserving changes in the variance of the plaintiff's litigation costs.    The next proposition analytically demonstrates how this appendix's model of initially NEV lawsuits due to non negative values of litigation abandonment options generalizes Bebchuk's model of NEV litigation.    Proposition 10: The set of parameter values for which initially NEV lawsuits are brought is larger than in Bebchuk's model of NEV litigation. n248 The difference between the set of parameter values for which NEV lawsuits are filed in a game theoretic options model and Bebchuk's game theoretic expected value model of litigation is a function of the difference between OV 1 (s t ) and E 1 (s t ), which in turn depends on the ability to subdivide the litigation into stages and the opportunities to learn more information.    Proof: Bebchuk's conditions for the filing of a lawsuit can be thought of as t, E 1 (s t ) I t . Bebchuk's model describes the situation of a lawsuit in which the values of all of the variables are known with certainty by the litigants. Under symmetric uncertainty, Bebchuk's conditions become E 1 (s t ) E 1 (I t ). Because the lawsuit can be dropped t, OV 1 (s t ) gteq 0 and moreover OV 1 (s t ) gteq E 1 (s t ). Thus, whenever E 1 (s t ) E 1 (I t ), OV 1 (s t ) E 1 (I t ) also holds. But, OV 1 (s t ) E 1 (I t ) can hold even though E 1 (I t ) E 1 (s t ).    Interpreting a comparison between the relative sizes of the set of parameter values in Bebchuk's non stochastic model that satisfy the necessary and sufficient conditions for the credible filing of NEV lawsuits with that of the set of parameter values in this appendix's stochastic litigation abandonment options model that satisfy the necessary and sufficient conditions for the credible filing of initially NEV lawsuits requires a bit of care. When any non stochastic model is embedded in a stochastic model involving the same variables as in the non stochastic model, the whole parameter space of the non stochastic model is only a single point in the parameter space of the stochastic model. In other words, for most economically and legally relevant choices of topologies and measures, the entire parameter space of the non stochastic model will only be a small or negligible set in the parameter space of the stochastic model. n249 Thus, any proper subset of the parameter space of the non stochastic model is a fortiori a small and negligible proper subset in the parameter space of the stochastic model. It thus comes as no surprise that a stochastic game theoretic options model of NEV litigation generalizes Bebchuk's non stochastic game theoretic expected value model of NEV litigation because any stochastic model generalizes any non stochastic model involving the same variables in the sense that stochastic random variables generalize non stochastic random variables.    The next two results analyze the impact of changes in a plaintiff's litigation costs on that plaintiff's incentive to file litigation and the resulting equilibrium settlement amount. Increased or greater divisibility of a plaintiff's legal costs only bolstered the credibility of a plaintiff's threats to continue a lawsuit in Bebchuk's nonstochastic model n250 and Cornell's non game theoretic model. n251 A similar proposition holds in this Article's game theoretic stochastic model.     Proposition 11: A finer partition of a plaintiff's legal costs can only bolster the credibility of that plaintiff's threats to continue a lawsuit and therefore increase equilibrium settlement amounts.    Proof: Let a finer partition of the plaintiff's legal costs be formed by at least subdividing some stage k into two substages: i and j. By construction, the plaintiff's legal costs in stage k can be decomposed into two component legal costs in stage i and stage j: I k = I i + I j . If the plaintiff initially had credible threats for continuing the lawsuit through to trial, then by proposition 1, option values of settlement at each stage are larger than the initial expected premia of those continuation options. In other words, for all t lteq n; these inequalities hold at date 1: OV 1 (s t ) gteq E 1 (I t ). In particular, at stage 1 it is expected that at stage k, OV 1 (s k ) gteq E 1 (I k ) = E 1 (I i ) + E 1 (I j ). By definition of the random variables s t , s k = s i = s j because there is no intermediate bargaining between stages k and k+1. Thus, OV 1 (s i ) = OV 1 (s k ) gteq E 1 (I k ) E 1 (I i ) and OV 1 (s j ) = OV 1 (s k ) gteq E 1 (I k ) E 1 (I j ). By proposition 1, this means that all of the plaintiff's threats for continuing the lawsuit through to trial remain credible.    If the plaintiff initially did not have a credible threat at stage k for continuing the lawsuit through to trial, then by proposition 1, OV 1 (s k ) E 1 (I k ) = E 1 (I i ) + E 1 (I j ). As noted above, s k = s i = s j because there is no intermediate bargaining between stages k and k+1. It is now possible that both OV 1 (s k ) gteq E 1 (I i ) and OV 1 (s k ) gteq E 1 (I j ). Of course, that is not guaranteed because it is also possible that OV 1 (s k ) E 1 (I i ) yet OV 1 (s k ) gteq E 1 (I j ), or OV 1 (s k ) gteq E 1 I i ) yet OV 1 (s k ) E 1 I j ), or OV 1 (s k ) E 1 I i ) and OV 1 (s k ) E 1 I j ). If any one of these three possibilities holds, then the plaintiff is initially expected not to have a credible threat at stage i, j, or both for continuing the lawsuit through to trial.    In Cornell's non game theoretic model, front loading a plaintiff's legal costs, meaning increasing that plaintiff's expected litigation costs at earlier stages while preserving the plaintiff's total expected litigation costs, reduced that plaintiff's litigation abandonment option value. n252 A similar proposition holds in this Article's game theoretic stochastic model.     Proposition 12: Holding all other variables fixed, front loading a plaintiff's litigation costs decreases the incentives to file lawsuits and equilibrium settlement amounts.    Proof: All other things being fixed, front loading a plaintiff's litigation costs increases E 1 (I t ) for initial values of t = 1, 2, ... . Thus, the necessary and sufficient conditions for initially filing lawsuits are less likely to hold when a plaintiff's litigation costs are front loaded than when a plaintiff's litigation costs are not front loaded.    Conversely, back loading a plaintiff's legal costs, meaning decreasing that plaintiff's expected litigation costs at earlier stages while preserving the plaintiff's total expected litigation costs, increases that plaintiff's litigation abandonment option value.    Proposition 13: Holding all other variables fixed, back loading a plaintiff's litigation costs increases the incentives to file lawsuits and equilibrium settlement amounts.    Proof: All other things being held equal, back loading a plaintiff's litigation costs decreases E 1 (I t ) for initial values of t = 1, 2, ... . Thus, the necessary and sufficient conditions for initially filing lawsuits are more likely to hold when a plaintiff's litigation costs are back loaded than when a plaintiff's litigation costs are not back loaded.    In Cornell's non game theoretic model, increasing a plaintiff's total legal costs reduced that plaintiff's litigation abandonment option value. n253 A similar proposition holds in this Article's game theoretic stochastic model.    Proposition 14: Holding all other variables fixed, increasing a plaintiff's total expected litigation costs decreases that plaintiff's incentives to file lawsuits and equilibrium settlement amount.    Proof: All other things being held fixed, increasing a plaintiff's total expected litigation costs increases E 1 (I t ) for some value(s) of t. Thus, at least one of the necessary and sufficient conditions for initially filing lawsuits is less likely to hold when a plaintiff 's total expected litigation costs increase than when a plaintiff's total expected litigation costs stay constant.    Conversely, decreasing a plaintiff's total legal costs increases that plaintiff's litigation abandonment option value.    Proposition 15: Holding all other variables fixed, decreasing a plaintiff's total expected litigation costs increases that plaintiff's incentives to file lawsuits and equilibrium settlement amount.    Proof: All other things being equal, decreasing a plaintiff's total expected litigation costs decreases E 1 (I t ) for some value(s) of t. Thus, at least one of the necessary and sufficient conditions for initially filing lawsuits is more likely to hold when a plaintiff's total expected litigation costs decrease than when a plaintiff's total expected litigation costs do not change.