Zhen Liu

Publications in Peer-Reviewed Journals

"Games with Incomplete Information when Players Are Partially Aware of Others' Signals", Journal of Mathematical Economics, (2016), pp. 58-70. Available at


Empirical and experimental findings suggest that players may underestimate others' private information in incomplete-information games. We modify standard epistemic assumptions of static incomplete-information games to allow partial signal-awareness. That is, players can be unaware of some of the signals available to others. When learning to play, players form conjectures that underrepresent the sophistication of others' strategy profile. The resulting solution concept is equilibrium with partial signal-awareness (EPSA). We illustrate EPSA by various examples and prove an existence result and an equivalence result between it and Bayesian Nash equilibrium. We then compare EPSA and analogy-based expectation equilibrium (ABEE) (Jehiel and Koessler, 2008) and prove an equivalence result between them for analogy-based expectation games that satisfy the condition of comparative-informativeness. We also discuss how they differ.

"The Dirty Face Problem with Unawareness", The B.E. Journals of Theoretical Economics, Vol. 8: Iss. 1 (Topics), Article 28. Available at http://www.bepress.com/bejte/vol8/iss1/art28  


By revisiting the classic dirty face problem we highlight the notion of ``unawareness'' (a simpler state space) and compare it with ``impreciseness'' (a coarser information partition). The outcomes are derived with various information structures. That allows us to 1) demonstrate a model of differing interpretations of public information using asymmetric awareness; 2) compare and contrast the impact of unawareness and impreciseness on learning; 3) analyze the value of awareness and the value of preciseness; and 4) demonstrate the effect of unawareness and impreciseness on information aggregation through indirect aggregation.

Working Papers

"Consumer Unawareness and Competitive Strategies" with Chun Qiu
Download at SSRN


Many purchase decisions rely on complex information. In reality, some consumers may not be well informed and unaware of their lack of information, a situation termed consumer unawareness. In a stylized model of market entry, this paper investigates how consumer unawareness of preference-product match affects firms' strategies in product, pricing, and promotion. It shows that for entry to occur, the level of unawareness must be intermediate. When entry occurs, firms offer differentiated products to avoid head-on competition and use mixed pricing strategies. As the level of unawareness decreases, both the incumbent and entrant become less competitive in pricing. This paper also analyzes how firms strategically promote consumer awareness through advertising. While the promotion of the entrant helps increase its demand after entry, the promotion of the incumbent can serve as either a barrier or an invitation to entry. Fearing the former, the entrant may not enter, or enter with limited promotion; benefiting from the latter, the entrant may enter and free ride on the incumbent's promotion. As a result of the former situation, a relatively high degree of consumer unawareness is maintained despite competition. This paper also identifies the incumbent's brand equity and promotion costs as critical factors that drive firms' strategic choices. Finally, this paper presents economic and managerial implications of the findings.

"Information Acquisition in the Era of Fair Disclosure: An Application of Asymmetric Awareness"
Version 0216, Version 0308, Version 1007
(Previously titled "Fair Disclosure and Investor Asymmetric Awareness in Stock Markets.")


As the cost of financial information dissemination continues to decline, investors, firms, and regulators are gradually adopting the principle of fair disclosure, which requires no preferential public disclosure. We use a simple model to examine the impact of this change on information acquisition with two alternative assumptions: (1) Investors have symmetric awareness about the underlying uncertainties, and (2) this awareness is asymmetric among them. Under the first assumption, the change reduces information asymmetry among investors and induces acquisition of high-quality information. Under the second assumption, however, the reduction of information asymmetry may be limited, and information acquisition is less efficient. Specifically, investors with high awareness may either acquire high-quality information at a higher cost or not acquire it; investors with low awareness only acquire low-quality information. The loss in overall information quality is greater when awareness asymmetry is moderate than when it is high or low; this causes information asymmetry between the insiders and outside investors as a whole. These results offer explanations for intriguing empirical findings regarding the effect of a recent accounting regulation (Regulation Fair Disclosure).

"Social Security Literacy and Retirement Well-Being?" (in progress), with Hugo Benitez-silva and Berna Demiralp


We build upon the growing literature on financial literacy, which studies the prevalence of lack of knowledge about various financial issues, and propose to analyze how much people know about the Social Security rules using both a small pilot survey we have already conducted, and a follow-up and extended survey, hopefully funded by this proposal. We then propose to assess the consequences of the apparent prevalence of lack of information by individuals about the rules governing the Social Security system using a realistic and empirically-based life-cycle model of retirement behavior under uncertainty. We will investigate the individual's retirement and savings decisions under incomplete information and unawareness, in which a portion of the population does not know some or all of the rules of the system. We will compare the outcomes in these cases to the outcome under full information, computing the average welfare gain resulting from the acquisition of information regarding the Social Security system. Our analysis can illuminate the need for policies that foster knowledge of the system, which can improve welfare among Americans of all ages.

"An Extended No Bet Theorem"


Given two players holding a common prior and distinct information partitions, the No Bet theorem (Sebenius and Geanakoplos, 1982) says that when at a state it is common knowledge that one's conditional expectation is no less than a certain number but the other one's is not greater than it, their conditional expectations must be the same. The extended theorem generalizes the result by taking away the separating number which is sometimes not available in practice. We also generalize the extended theorem to the case where priors are heterogeneous. As an application, we show that if the ranking of each player's expectation, or just the identity of the highest/lowest one is common knowledge, players must agree in the logic of the extended theorem, but may not agree under the original No Bet theorem.

A Short Survey on the Applications of Unawareness