Bruno — Research

Persuading Part of an Audience

I propose a cheap-talk model in which the sender can use private messages and only cares about persuading a subset of her audience. For example, a candidate only needs to persuade a majority of the electorate in order to win an election. I find that senders can gain credibility by speaking truthfully to some receivers while lying to others. In general settings, the model admits information transmission in equilibrium for some prior beliefs. The sender can approximate her preferred outcome when the fraction of the audience she needs to persuade is sufficiently small. I characterize the sender-optimal equilibrium and the benefit of not having to persuade your whole audience in separable environments. I also analyze different applications and verify that the results are robust to some perturbations of the model, including non-transparent motives as in Crawford and Sobel (1982), and full commitment as in Kamenica and Gentzkow (2011). example.

Example — targeted poplitical campaigns

Suppose a politician a politician (she) running for office. The state of the world equals either 0 or 1. Each voter (he) will vote for the politician if and only if his expectation about the state is greater than \(1/2\). The voters’ common prior expectation about the state lies in \((1/3,1/2)\), so that nobody is willing to vote for the politician ex ante. Suppose that the politician knows the true state and can talk privately with each voter. I claim that, if there are sufficiently many voters, then there exists an equilibrium in which she wins the election for sure regardless of the state.

This is possible because the politician only needs to persuade half plus one of the electorate in order to win the election. She can attain this goal by using the following strategy. If the state is indeed \(1\), then she will let every single voter know this fact. If the state is \(0\), then she will randomly choose half plus one of the voters and tell them that the state is \(1\), despite the fact that it is not.

A voter that receives a message saying that the state equals \(1\) knows that this could be a lie. However, he also knows that he would be more likely to receive this message if it was actually true. Hence, the message conveys some information. When the population is large enough, it conveys sufficient information to overturn prior beliefs arbitrarily close to \(1/3\). In that case, every voter who receives this message prefers to vote for the politician.

In the paper studies a general cheap-talk model in which an informed sender can engage in private conversation with many receivers and cares about the behavior of some but not all of them. My main finding is that having to persuade only part of an audience significantly facilitates information transmission and increases persuasion power.

Pricing Algorithms and Tacit Collusion

There is an increasing tendency for firms to use pricing algorithms that speedily react to market conditions, such as the ones used by major airlines and online retailers like Amazon. I consider a dynamic model in which firms commit to pricing algorithms in the short run. Over time, their algorithms can be revealed to their competitors and firms can revise them, if they so wish. I show how pricing algorithms not only facilitate collusion but inevitably lead to it. To be precise, within certain parameter ranges, in any equilibrium of the dynamic game with algorithmic pricing, the joint profits of the firms are close to those of a monopolist.

Discerning Solution Concepts

(with Nail Kashaev) — New impoved version! Revision Submitted to the Journal of Business & Economic Statistics

The empirical analysis of discrete complete-information games has relied on behavioral restrictions in the form of solution concepts, such as Nash equilibrium. Choosing the right solution concept is crucial not just for the identification of payoff parameters, but also for the validity and informativeness of counterfactual exercises and policy implications. We say that a solution concept is discernible if it is possible to determine whether it generated the observed data on the players’ behavior and covariates. We propose a set of conditions that make it possible to discern solution concepts. In particular, our conditions are sufficient to tell whether the players’ choices emerged from Nash equilibria. We can also discern between rationalizable behavior, maxmin behavior, and collusive behavior. Finally, we identify the correlation structure of unobserved shocks in our model using a novel approach. example.

Example — identifying equilibrium conditions

Firms \(\ i=1,2\ \) decide whether to enter a market \(\ (y_i=1)\ \) or not \(\ (y_i=0)\). Profits are as in Figure 1, where \(\ \beta_0 =(\beta_{0j})\ \) is a vector of unknown structural parameters, \(\ \mathbf{x} = (\mathbf{x}_i)\ \) is a vector of observable continuous firm characteristics with large support, and \(\ \mathbf{e} = (\mathbf{e}_i)\ \) is a vector of unobserved i.i.d. standard normal random errors independent of \(\ \mathbf{x}\).

	\(0\)	\(1\)
\(0\)	\(0,\ 0\)	\(0,\ \beta_{02}\mathbf{x}_2 - e_2\)
\(1\)	\(\beta_{01}\mathbf{x}_1 - \mathbf{e}_1,\ 0\)	\(\beta_{01}\mathbf{x}_1 - \beta_{03} - \mathbf{e}_1,\ \beta_{01}\mathbf{x}_2 - \beta_{03} - \mathbf{e}_2\)

Fig 1 — Profit matrix

Suppose the econometrician observes the joint distribution of \(\ \mathbf{x}\ \) and entry decisions. Assuming that firms are profit maximizers and this fact is mutual knowledge, could she tell from the data whether firm's entry choices always constitute Nash equilibria?

The answer is “yes, she could!”

First, the large support of \(\ \mathbf{x}\ \) and the profit-maximization assumption identify \(\ \beta_0\ \) at infinity. For example, as \(\ \mathbf{x}_1\to-\infty\), firm \(\ 1\ \) stays out of the market almost surely, and thus \[ \Pr\left(\mathbf{y}_2=1 \big|\mathbf{x}_1\right) = \Pr\left(\mathbf{e}_2 \leq \beta_{02}\mathbf{x}_2 \big| \mathbf{x}_1\right). \] Because the left-hand-side probability is observed, this equation identifies \(\ \beta_{02}\).

Second, we have to assume that \(\ \mathbf{x}\ \) only affects behavior through payoffs. Then, we can show that the facts that the covariates have continuous support and the error terms’ distribution belongs to the exponential-family imply that the disribution of choices conditional on both \(\ \mathbf{x}\ \) and \(\ \mathbf{e}\ \) is identified.

Therefore, the econometrician can recover both the profit functions and the firm's behavior as a function of both observed and unobserved heterogeneity. This allows her to determine whether, how often and when firm's choices are in equilibrium. In our paper, Nail and I generalize this approach to identify a wide class of behavioral and structural assumptions for a general class of discrete semi-parametric games of complete information.

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A bailout-crisis game

(with Ruilin Zhou and Bruno Sultanum)

We study the optimal design of a liability sharing arrangement between two agents as an infinitely repeated game. Each period the active agent can take a costly, unobservable action to try to avert a crisis. If a crisis occurs, both agents decide how much to contribute to mitigate it. When the cost of the avoidace action is too high the first-best is not achievable. In that case, at every contrained Pareto efficient PPE of the repeated game, the active agent “shirks” infinitely often, and when a crisis happens, the active agent is “bailed out” infinitely often. The frequencies of crises and bailouts are endogenously determined at equilibrium.

Interdependent Choices

Interdependent-choice equilibrium (ICE) is an extension of correlated equilibrium in which the mediator is able to choose the timing of her signals, and observe the actions taken by the players. It characterizes all the outcomes that can be implemented in single shot interactions without repetition, side payments, binding contracts or any other form of delegation. example.

Example — cooperaton in a prisoners’ dilemma

Two suspects of a crime are offered a sentence reduction in exchange for a confession. While making his statement, each prisoners chooses whether to cooperate by remaining silent \(\ (\mathrm{C})\), or defect by signing a full confession \(\ (\mathrm{D})\). Preferences are as in Figure 1.

	\(\mathrm{C}\)	\(\mathrm{D}\)
\(\mathrm{C}\)	\(1,\ 1\)	\(-k,\ 1+g\)
\(\mathrm{D}\)	\(1+g,\ -k\)	\(0,\ 0\)

Fig 1 — Prisoners’ payoffs, whith \(\ g,k>0\)

The prisoners need not choose simultaneously nor independently. There may be different ways to coordinate thier choices. For instance, they could hire the same lawyer to schedule and be present in all the interactions with the DA and instruct her as follows:

Uniformly randomize the order of our meetings. If the DA offers a deal, recommend that we reject it, unless one of us has already confessed, in which case recommend that we also confess. Other than those recommendations, do not provide us with any additional information.

The resulting situation corresponds to the extensive form game in Figure 2. As long as \(\ g\leq 1,\) the strategies represented with red arrows are a SPNE in which both prisoners cooperate. This is possible because, along the equilibrium path, each prisoner is concerned that if he confesses then his accomplice will learn of his defection and will punish him by also confessing.

The prisoners can sustain cooperation in equilibrium, despite it being a single-shot interaction, and without using side payments, signing contracts or using any other form of binding commitment.

In the rest of the paper, I find a canonincal class of mechanisms that characterizes the set of all outcome distributions that can be implemented without commitment, not just for the prisoners’ dilemma, but for any finite strategic-form game. The set of implementable outcomes admits a parsimonious representation in terms of a fnite set of affine inequalitites.

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Ordinal Dominance and Risk Aversion

(with Bulat Gafarov) Economic Theory Bulletin ⋅ 2015 ⋅ 3(2):287–298

We extend Börgers (1993) by showing that pure and mixed dominance are equivalent when agents are sufficiently risk averse. This implies that, in the absence of cardinal information, rationalizability is equivalent to the iterated removal of strategies that are dominated by a pure strategy. example.

Example — a fair bet

A rational agent chooses between betting on an event \(\ E\), betting against \(\ E\), and not betting at all. Her preferences are represented by the vNM indexes in Figure 1.

	\(E\)	not \(E\)
bet on \(E\)	\(2\)	\(0\)
bet against \(E\)	\(0\)	\(2\)
do not bet	\(\gamma\)	\(\gamma\)

Fig 1 — Payoff matrix with \(\gamma\in(0,2)\)

Note that the ordinal ranking of state-action pairs remains unchanged as long as \(\ 0 < \gamma < 2\). Also, not betting is not strictly dominated by any pure action, and, if \(\ \gamma \geq 1\), it is also not strictly dominated by any mixed action. However, if \(\ \gamma <1\), then mixing uniformly between betting for and against \( E\) strictly dominates not betting.

Here, \(\ \gamma\ \) measures the degree of risk aversion of the agent. Hence, dominance by pure actions coincides with dominance by mixed actions if the agent is sufficiently risk averse, and there exists a sufficiently risk-averse utility function which is compatible with the given ordinal preferences.

In the rest of the paper, Bulat and I show that these two observations continue to hold for a large class of decision problems under uncertainty with ordinal preferences.

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An Algorithm for Multiplayer Dynamic Games

(with Bruno Sultanum) New draft in progress.