A note on the welfare and policy implications of a two-period real option game under imperfect information

Wang, Congcong; Wang, Yuhan; Chen, Shanshan; Luckraz, Shravan; Pansera, Bruno Antonio Article A note on the welfare and policy implications of a twoperiod real option game under imperfect information Games Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Wang, Congcong; Wang, Yuhan; Chen, Shanshan; Luckraz, Shravan; Pansera, Bruno Antonio (2025) : A note on the welfare and policy implications of a two-period real option game under imperfect information, Games, ISSN 2073-4336, MDPI, Basel, Vol. 16, Iss. 1, pp. 1-7, https://doi.org/10.3390/g16010007 This Version is available at: https://hdl.handle.net/10419/330121 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Academic Editor: Ulrich Berger Received: 3 December 2024 Revised: 18 January 2025 Accepted: 24 January 2025 Published: 3 February 2025 Citation: Wang, C., Wang, Y., Chen, S., Luckraz, S., & Pansera, B. A. (2025). A Note on the Welfare and Policy Implications of a Two-Period Real Option Game Under Imperfect Information. Games,16(1), 7. https://doi.org/10.3390/g16010007 Copyright: © 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/ licenses/by/4.0/). Article A Note on the Welfare and Policy Implications of a Two-Period Real Option Game Under Imperfect Information Congcong Wang 1,† , Yuhan Wang 1,† , Shanshan Chen 1,† , Shravan Luckraz 2, * ,† and Bruno Antonio Pansera 3, * ,† 1School of Finance, Zhejiang University of Finance and Economics, Hangzhou 310012, China 2School of Economics and CeDeX China, University of Nottingham Ningbo China, Ningbo 315104, China 3Department of Law, Economics and Human Sciences & Decisions_Lab, University Mediterranea of Reggio Calabria, Via dell’Universitá 25, I-89124 Reggio Calabria, Italy *Correspondence: [email protected](S.L.); [email protected] (B.A.P.) †These authors contributed equally to this work. Abstract: We show that the discrete real option game model proposed in the recent literature can be extended to the case of imperfect information. As a result, the model can cover a wider range of applications. However, we also observe that the effectiveness of implementing the subsidy is affected by the imperfect informational structure. Keywords: welfare; policy implications; two-period real option game; imperfect information 1. Introduction In a recent study, Wang et al. (2023) analyzed the welfare implications of a discrete real option game where firms make investment decisions sequentially. One of their key findings was that, under equilibrium conditions where no firm invests in the first period, the government can introduce a subsidy in a timely manner to induce at least one firm to invest early. However, while their analysis assumes a sequential decision-making process with a designated leader firm moving first, many real-world scenarios involve firms operating under imperfect information. In such contexts, firms lack knowledge of whether competitors have already invested when making their decisions. This is particularly relevant in applications like Public–Private Partnerships (PPPs), where informational asymmetries are prevalent. In this paper, we relax the assumption of perfect information and reconsider the real option game under imperfect information. We explore the resulting welfare implications and evaluate the effectiveness of policy interventions in this more general informational framework. Our findings reveal that while the equilibrium outcomes remain unchanged, the shift to an imperfect informational structure significantly impacts the effectiveness of subsidies. This work builds on a rich literature examining real options and strategic decision making under uncertainty. For example, Lambrecht and Perraudin (2003) integrated partial information and anticipatory behavior into a model of competitive investment decisions, showing how optimal strategies range between the zero-NPV threshold and a monopolist’s preferred strategy. Similarly, Weeds (2002) investigated irreversible investments in competitive R&D under uncertain profits within a patent system, highlighting how lack of cooperation leads to delays in investment due to fears of initiating a patent race. Huisman and Kort (2004) explored dynamic markets where firms compete in adopting new technologies, revealing how the likelihood of technological innovation affects strategic behaviors—shifting from preemption games to wars of attrition. Miltersen and Schwart Games 2025,16, 7 https://doi.org/10.3390/g16010007 Games 2025,16, 7 2 of 7 (2004) extended the real options framework to include game-theoretic dynamics for evaluating patent-protected R&D projects under competition. Pawlina and Kort (2006) examined asymmetric investment costs in a duopoly, identifying three distinct equilibrium strategies. Murto and Keppo (2002) developed a game model where multiple firms compete for a single investment opportunity, demonstrating the existence of Nash equilibria based on varying assumptions about firms’ knowledge of competitors’ project evaluations. Our work also relates to Smit and Ankum (1993), who adopted a microeconomic approach to analyze competing business investments using real options and game theory. They forecasted cash inflows from operations that use economic rents or excess profit (see also Smit,2003;Smit & Trigeorgis,2006), while Grenadier (1996) applied a game equilibrium model to housing development, investigating how market demand and asset values shape investment capacity. Martzoukos and Zacharias (2009) highlighted strategic decision making in R&D, incorporating spillover effects and pricing dynamics, and McGahan (1993) examined sectoral strategies integrating real options and game theory with an emphasis on collaborative versus competitive R&D. Kulatilaka (1993) provided a flexible framework for evaluating investment decisions under uncertainty, such as switching between operating modes in a dual-fuel industrial boiler project. Myers (1977) linked corporate debt behavior to real option values, identifying how risky loans can constrain investment strategies and reduce firm value. Tondji (2016) examined welfare outcomes in R&D-intensive markets using Cournot and Bertrand competition models, while Paddock et al. (1988) extended financial options theory to evaluate claims on real assets like offshore oil leases. In light of this extensive literature, our study offers new insights into the low participation rates observed in PPP projects. As noted by Wang et al. (2023), participation in such initiatives in China remains as low as 10–20% despite the introduction of subsidies. Our model identifies two primary reasons for this phenomenon. First, firms face strong incentives to delay investment due to uncertain future profits. Second, competition under imperfect information deters entry, as firms cannot anticipate whether they will face a competitor in later stages of the game. This contrasts with the sequential move case, where followers have full visibility of the leader’s actions. The remainder of the paper is structured as follows: Section 2introduces the extended version of the Wang et al. model and presents our main results and Section 3concludes. 2. A 2-Period 2-Stage Real Option Game We consider the (Wang et al.,2023) model in an environment with imperfect information. The game is played over two periods, t= 0, 1. At t= 0, firms make their investment decisions simultaneously. As in (Wang et al.,2023), at t= 1, demand follows a simple binomial process and depending on the investment decisions the market structure could be a duopoly or monopoly at t= 0 or t= 1 or in both periods. The game is played under perfect information across the period, but imperfect information within each period. The cost of investment is denoted by I . As in (Wang et al.,2023) the demand and profit functions for firm iis given as follows: pi=a−bqi+θqj,i=1, 2 i=j,a,b>0 (1) πi=Yt(piqi−cqi)(2) where Yt , for t= 0, 1, is an exogenous shock, which can be on the supply side or on the demand side and each firm faces a marginal cost of c . As in (Wang et al.,2023), we assume that Y0is deterministic while Y1is stochastic with the possible realization of either a good Games 2025,16, 7 3 of 7 state denoted by G or a bad state denoted by B . More formally, the stochastic process is as follows: Y1=(uY0if G dY0if B(3) where 0 ≤d< 1 <u . Let the probability p denote the probability that state G is realized. As in (Wang et al.,2023), if we let µ≡pu + ( 1 −p)d , the expected value of Y1 is given by E(Y1)=µY0 . We denote by r the risk free interest rate and we let R≡ 1 +r . As in (Wang et al.,2023), we denote the duopoly symmetric profit of a firm in period t by D(1, 1)Yt so that from the above we have: D(1, 1)≡ (1−θ) b(1+θ)a−c 2−θ2!. (4) Furthermore, we denote the monopoly profit of the firm in period t by D(1, 0)Yt so that we have: D(1, 0)≡1 b (a−c)2 4!. (5) When no firm invests, each firm obtains 0 profits. This is denoted by D(0, 0)Yt and finally, when only one firm invests, the firm that did not invest obtains 0 profits, which we denote by D(0, 1)Yt. Thus, we have: D(0, 0)=D(0, 1)=0 (6) Following (Wang et al.,2023), we have D(1, 0)>D(1, 1) and the following inequalities hold: D(1, 0)>D(1, 1)>D(0, 0)=D(0, 1)=0 (7) and D(1, 0)−D(0, 0)>D(1, 1)−D(0, 1)(8) We represent the above 2-period game as an extensive game and its complete description is given in the game tree below. We now find the perfect Nash Equilibrium subgame. As in (Wang et al.,2023), we use the following expressions in the presentation of our results. Let E≡D(1, 1)Y0R+(1−p)d R−p and F≡D(1, 0)Y0R+(1−p)d R−p . We assume that the magnitude of the positive (negative) shock if the good (bad) states realizes is sufficiently high (low), that is: D(1, 1)dY0<E(9) and D(1, 1)uY0>F(10) As a result of the above assumption, we have the following: D(1, 0)dY0<E<F<D(1, 1)uY0(11) In Figure 1, the game is represented as a game in extensive form. Each has five information sets (1.1–1.5) for player 1 and information sets (2.1–2.2) for player 2. A strategy for the k player is a mapping sk from {{1.i}}5 i=1→{I,NI} , where k= 1, 2. Let S1 denote the set of all strategies of player 1 while S2 denote the set of all strategies of player 2. Moreover, let S=S1×S1 denote the set of all strategy profiles of the game. We say that some profile s≡(s1;s1)∈S is a perfect Nash Equilibrium (SPNE) subgame of the game Games 2025,16, 7 4 of 7 if its restriction to each subgame is a Nash Equilibrium of that subgame. Consider the following profiles: s∗,s∗∗,s∗∗∗, where: s∗≡(s∗ 1;s∗ 2), (12) s∗ 1=s∗ 2=(I,I,NI,I,NI), (13) s∗∗ ≡(s∗∗ k;s∗∗ l), (14) s∗∗ k=(I,I,NI,I,NI), (15) s∗∗ l=(NI,I,NI,I,NI), where k=1, 2 and k=l(16) s∗∗∗ ≡(s∗∗ 1;s∗∗ 2), (17) s∗∗∗ 1=s∗∗∗ 2=(NI,I,NI,I,NI), (18) Our main result characterizes the set of equilibria of the game described by Figures 1and 2. Figure 1. The 2-period, 2-stage, 2-player game. Figure 2. The game tree for period 0. Games 2025,16, 7 5 of 7 Proposition 1. The subgame perfect Nash equilibria of the game are given as follows: s=     s∗if D(1, 0)dY0<I ≤ E s∗∗ if E <I ≤ F s∗∗∗ if F <I ≤ D(1, 1)uY0 (19) Proof. From inequalities D(1, 0)dY0<E , F<D(1, 1)uY0 , I ≤ D(1, 1)uY0 and D(1, 0)dY0<I , we know that each firm will invest in the G state and will not invest in the B state in period 1 under all three strategies. Using backward induction, we can then proceed to period 0. The reduced period 0 game tree is given in Figure 3. Figure 3. The general case. Therefore, we consider the following three cases of the proposition. Case (i): D(1, 0)dY0<I ≤ E. In Case (i), solving the game using backward induction again, it can be shown that the inequalities ensure that we have the equilibrium path shown in Figure 4. Figure 4. The diagram for Case 1. Case (ii) E<I ≤ F. In Case (ii), solving the game using backward induction again, it can be shown that the inequalities ensure that we have the equilibrium path shown in Figure 5. Figure 5. The diagram for Case 2. Case (iii): F<I ≤ D(1, 1)uY0. In Case (iii), solving the game using backward induction again, it can be shown that the inequalities ensure that we have the equilibrium path shown in Figure 6. Games 2025,16, 7 6 of 7 Figure 6. The diagram for Case 3. This completes the proof. Proposition 1shows that the main proposition of (Wang et al.,2023) remains robust to the imperfect information structure. Thus, under imperfect information if the cost of investment is sufficiently low, both firms would exercise their options in the first period, whereas if it is sufficiently high, then both firms would not exercise their options in the first period. Moreover, there exists a range of values of the investment costs for which only one firm invests in period 0. We can also establish the robustness of propositions 2 and 3 of (Wang et al.,2023) as follows. Proposition 2. Under an imperfect information structure, the thresholds investment levels E and F are increasing in d and p while decreasing in R. Proposition 3. Suppose that the information structure is imperfect, F<I ≤ D(1, 1)uY0 holds and hR+qd R−1iY0WM>I . Then, welfare improves if the government subsidizes at least one firm in the initial period. In this case, the subsidy is given by S=I − F+ϵ , where ϵ is positive. As a result, a subgame perfect Nash Equilibrium in which exactly one firm invests in period 0 can be achieved. Remark 1. The policymaker cannot implement the subsidy without resorting to some arbitrary rule that discriminates between the two firms. This is in contrast to (Wang et al.,2023), where, at the beginning of stage 1in period 0, the policymaker could announce that it would subsidize the leader if it invests and then implement the subsidy if the leader has invested. Under simultaneous moves, the designations “leader” and “follower” no longer exist, and therefore, although the policymaker could still implement a subsidy that induces exactly one firm to invest, it can only do so by arbitrarily discriminating between firm 1and firm 2. 3. Conclusions We have shown that the previous result remains unchanged under this new informational structure. However, we found that the effectiveness of the implementation of the policy is reduced for the simultaneous move case. Author Contributions: Conceptualization, C.W., Y.W., S.C., S.L. and B.A.P.; validation, C.W., Y.W., S.C., S.L. and B.A.P.; formal analysis, C.W., Y.W., S.C., S.L. and B.A.P.; investigation, C.W., Y.W., S.C., S.L. and B.A.P.; writing—original draft preparation, C.W., Y.W., S.C., S.L. and B.A.P.; writing—review and editing, C.W., Y.W., S.C., S.L. and B.A.P.; supervision, S.L. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the Zhejiang Provincial Natural Science Foundation of China under Grant No. LMS25G030001. Data Availability Statement: No new data were created or analyzed in this study. Data sharing is not applicable to this article. Games 2025,16, 7 7 of 7 Acknowledgments: The authors express their gratitude to the three anonymous reviewers and the academic editor for their useful comments, suggestions, and observations aimed at improving the paper. Conflicts of Interest: The authors declare no conflicts of interest. References Grenadier, S. (1996). The strategic exercise of options: Development cascades and overbuilding in real estate markets. Journal of Finance, 51(5), 1653–1679. [CrossRef] Huisman, K., & Kort, P. (2004). Strategic Technology Adoption taking into account Future Technological Improvements: A Real Options Approach. European Journal of Operational Research,159, 705–728. [CrossRef] Kulatilaka, N. (1993). The value of flexibility: The case of a dual-fuel industrial steam boiler. Financial Management,22(3), 271–280. [CrossRef] Lambrecht, B., & Perraudin, W. (2003). Real Options and Preemption under Incomplete Information. Journal of Economic Dynamic and Control,27, 619–643. [CrossRef] Martzoukos, S., & Zacharias, E. (2009). Real option games with R&D and learning spillovers. MPRA. McGahan, A. M. (1993). The incentive not to invest: Capacity commitments in the compact disc introduction. Research and Technology Innovation, Management and Policy,5(4), 177–197. Miltersen, K. R., & Schwart, E. S. (2004). R&D Investments with Competitive Interactions. Review of Finance,8(3), 355–401. Murto, P., & Keppo, J. (2002). A Game Model of Irreversible Investment under Uncertainty. International Game Theory Review,4, 127–140. [CrossRef] Myers, S. C. (1977). Determinants of Corporate Borrowing. Journal of Financial Economics,5(2), 147–175. [CrossRef] Paddock, J. L., Siegel, D. R., & Smith, J. L. (1988). Option valuation of claims on real assets : The case of offshore petroleum leases. The Quarterly Journal of Economics,103(3), 479–508. [CrossRef] Pawlina, G., & Kort, P. (2006). Real Options in an Asymmetric Duopoly: Who Benefits from your Competitive Disadvantage? Journal of Economics & Management Strategy,15, 1–35. Smit, H. T. J. (2003). Infrastructure investment as a real options game: The case of European airport expansion. Financial Management, 32(4), 27–57. [CrossRef] Smit, H. T. J., & Ankum, L. A. (1993). A Real Option and Game-Theoretic Approach to Corporate Investment Strategy Under Competition. Financial Management,22(3), 241–250. [CrossRef] Smit, H. T. J., & Trigeorgis, L. (2006). Real options and games: Competition, alliances and other applications of valuation and strategy. Review of Financial Economics,15(2), 95–112. [CrossRef] Tondji, J. B. (2016). Welfare analysis of cournot and bertrand competition with(out) investment in R&D. MPRA. Wang, C., Chen, S., Wang, Y., Luckraz, S., & Pansera, B. A. (2023). On the welfare and policy implications of a two-period real option game. Socio-Economic Planning Sciences,90, 101717. [CrossRef] Weeds, H. (2002). Strategic delay in a real options model of R&D competition. The Review of Economic Studies,69(3), 729–747. Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.