# Shih En Lu's Papers

**Cheap Talk**

Monotonic Cheap Talk, September 2016 (available upon request)

This paper studies monotonic equilibria in a multi-sender version of Crawford and Sobel's (1982) cheap talk model, i.e. pure-strategy equilibria where senders' strategies are weakly monotonic in the state and where the receiver's strategy is strictly monotonic in the senders' messages. Monotonic equilibria have interval form, are bounded away from full revelation, and are straightforward to compute; they are closely related to the set of coordination-free equilibria identified by Lu (2016). When senders can be ranked according to bias: (i) in monotonic equilibria, senders most biased toward larger actions are informative when the receiver's desired action is smallest, and vice versa; and (ii) monotonic equilibria can be made collusion-proof in a strong sense by appropriately placing the receiver's off-path actions. If assumed alone, weak monotonicity of sender strategies generally has only a weak implication for the realized state-to-action function in a pure-strategy equilibrium. Strict monotonicity of the receiver's strategy is motivated by the possibility of misunderstanding.

Coordination-Free Equilibria in Cheap Talk Games

Online Appendix

Journal of Economic Theory 168, 2017: 177-208.

This paper characterizes generic equilibrium play in a multi-sender version of Crawford and Sobel's (1982) cheap talk model, when robustness to a broad class of beliefs about noise in the senders' observation of the state is required. Just like in the one-sender model, information transmission is partial, equilibria have an interval form, and they can be computed through a generalized version of Crawford and Sobel's forward solution procedure. Fixing the senders' biases, full revelation is not achievable even as the state space becomes large. Intuitive welfare predictions, such as the desirability of consulting senders with small and opposite biases, follow.

Almost Fully Revealing Cheap Talk with Imperfectly Informed Senders (with Attila Ambrus)

Formerly "Robust Almost Fully Revealing Equilibria in Multi-Sender Cheap Talk"

Games and Economic Behavior 88, 2014: 174-189.

We show that in multi-sender communication games where senders imperfectly observe the state, if the state space is large enough, then there can exist equilibria arbitrarily close to full revelation of the state as the noise in the senders' observations gets small. In the case of replacement noise, where the senders observe the true state with high probability, we show this under mild assumptions, for both unbounded and large bounded state spaces. In the case of continuous noise, where senders observe a signal distributed continuously over a small interval around the true state, we establish this for unbounded state spaces. The results imply that when there are multiple experts from whom to solicit information, if the state space is large, then even when the state is observed imperfectly, there are communication equilibria that are strictly better for the principal than delegating the decision right to one of the experts.

**Bargaining**

Legislative Bargaining with Long Finite Horizons (with Attila Ambrus), March 2016

Institutional rules provide natural deadlines for negotiations in legislative bargaining. In the continuous-time bargaining model framework of Ambrus and Lu (2015), we show that as the time horizon of legislative bargaining increases, equilibrium payoffs with deadline converge to a stationary equilibrium payoffs of the infinite-horizon bargaining game. We provide a characterization of these limit payoffs, and show that under a K-majority rule, the payoffs of the K legislators with the lowest relative recognition probabilities have to be equal to each other when positive. Hence, by varying recognition probabilities, possible limit equilibrium payoffs are constrained to a lower-dimensional subset of the set of all possible allocations. This result contrasts with Kalandrakis' (2006) finding that in the infinite-horizon Baron and Ferejohn (1989) framework, for any discount factor, any division of the surplus can be achieved as a stationary equilibrium payoff through some choice of recognition probabilities.

Self-Control and Bargaining

Formerly "Quasi-Hyperbolic Discounting and the Dual-Self Model in Rubinstein-Ståhl Type Bargaining"

Journal of Economic Theory 165, 2016: 390-413.

This paper examines a bargaining game with alternating proposals where sophisticated quasi-hyperbolic discounters negotiate over an infinite stream of payoffs. In Markov perfect equilibrium, payoffs are almost always unique, and a small advantage in self-control can result in a large advantage in payoff. In subgame-perfect equilibrium, a multiplicity of payoffs and delay can arise, despite the complete information setting. Markov perfect equilibria are the best subgame-perfect equilibria for the agent with more self-control, and the worst for the agent with less self-control. Naïveté can help a player by increasing their reservation value.

Related: Models of Limited Self-Control: Comparison and Implications for Bargaining

Economics Letters 145, 2016: 186-191.

This paper compares two models of limited intertemporal self-control: the linear-cost version of Fudenberg and Levine's dual-self model (2006) and the quasi-hyperbolic discounting model. The main distinction between the two frameworks can be formulated as whether agents care about future self-control costs: dual selves do, while quasi-hyperbolic discounters do not. The dual-self model is applied to a bargaining game with alternating proposals where players negotiate over an infinite stream of payoffs, and it is shown that, in subgame-perfect equilibrium, the first proposer's payoff is unique and agreement is immediate. By contrast, Lu (2016) shows that with quasi-hyperbolic discounters, a multiplicity of payoffs and delay can arise in equilibrium.

A Continuous-Time Model of Multilateral Bargaining (with Attila Ambrus)

American Economic Journal: Microeconomics 7(1), 2015: 208-249.

We propose a finite-horizon continuous-time framework for coalitional bargaining, in which players can make offers at random discrete times. In our model: (i) expected payoffs in Markov perfect equilibrium (MPE) are unique, generating sharp predictions and facilitating comparative statics; (ii) MPE are the only subgame perfect Nash equilibria (SPNE) that can be approximated by SPNE of nearby discrete-time bargaining models. We investigate the limit MPE payoffs as the time horizon goes to infinity and players get infinitely patient. In convex games, we establish that the set of these limit payoffs achievable by varying recognition rates is exactly the core of the characteristic function.

**Miscellaneous**

Matching, Marriage and Children: Differences across Sexual Orientations (with Douglas W. Allen)

Supplementary Appendix

Review of Economics of the Household 15, 2017: 527-547.

Please note: this paper has no direct implications about the desirability of same-sex marriage, an issue about which the authors have different opinions.

There are many differences in behavior across couples of different sexual orientations --- some well known, others not. We propose a model which explains differences in expected matching behavior, marriage rates, non-child-friendly activities, and fertility, based on different costs of procreation and complementarities between marriage and children. The model predicts that the biological traits of same-sex couples, unlike those of heterosexual couples, should not be correlated --- holding constant other household production characteristics. In addition, the model predicts that heterosexuals have a higher probability of having children and getting married, and that childless heterosexuals are less likely to engage in behaviors not complementary with children than childless gays and lesbians. Using two nationally representative probability samples that self-identify sexual orientation, these predictions are confirmed.