There are two primary ways of influencing the actions of strategic agents: through providing incentives and through influencing beliefs. The former is the domain of traditional mechanism design, and involves the promise of payments or goods contingent on behavior. The latter is the domain of information design, or persuasion, and involves the selective provision of information pertaining to the payoffs and costs of various actions. There are striking similarities and parallels between the two worlds, both in terms of the domains in which they are studied — for example in auctions [Emek et al., 2014; Dughmi et al., 2014; Daskalakis et al., 2016] and routing [Bhaskar et al., 2016] — as well in the mathematical models and techniques used to characterize and compute optimal policies (e.g. [Dughmi and Xu, 2016, 2017; Kolotilin et al., 2017]). Combining the approaches and techniques of mechanism design and persuasion leads to a more powerful toolkit for the design of economic systems, and this paper takes a step in that direction.
We work with two models of persuasion: the Bayesian Persuasion model of Kamenica and Gentzkow , and the multi-receiver private Bayesian persuasion model of Arieli and Babichenko  (further developed in Babichenko and Barman  and Dughmi and Xu ) which we generalize to allow externalities. In the spirit of mechanism design and principal-agent problems, we generalize both models by permitting payments, which serve as additional incentive for the receiver(s) of information to behave in accordance with the wishes of the sender (principal).
We then explore these models through the lens of Lagrangian duality, much in the spirit of the literature applying duality to auction theory and Bayesian mechanism design. In particular, we vary constraints on the payments (arbitrary, nonnegative, budget balanced) and the information/reward structure (symmetric vs asymmetric actions), and derive canonical and/or tractable optimal policies through duality.
The Persuasion Models.
In the Bayesian persuasion model, there is a receiver who must select one of a number of actions, and a sender looking to influence the receiver’s choice in order to maximize her own expected payoff. We adopt the perspective of the sender. A state of nature, drawn from a common knowledge prior distribution, determines the payoff of each action to each of the sender and the receiver. The sender has an informational advantage over the receiver: access to the realization of the state of nature. The problem facing the sender is that of computing and committing to the optimal signaling scheme: a randomized map from states of nature to signals. Once the state is drawn by nature, the signaling scheme is invoked and the corresponding signal is sent to the receiver; she then updates her prior belief and chooses the action maximizing her expected payoff. The multi-receiver private Bayesian persuasion model generalizes the previous model to multiple receivers. There is still a common knowledge prior distribution over states of nature, and a single sender with an informational advantage. We restrict attention to the special case of two actions for each receiver. The state of nature now determines a set function for the sender and a set function for each of the receivers: each set function maps the set of receivers taking action to a payoff. A signaling scheme now is a randomized map from states of nature to a signal for each receiver.
In both models, a simple revelation principle style argument shows that it suffices to restrict attention to schemes which are direct and persuasive (see e.g. Kamenica and Gentzkow ; Arieli and Babichenko ). A direct scheme is one in which signals correspond to action recommendations. Such a scheme is persuasive if it is a Bayes-Nash equilibrium for each receiver to follow the recommendation.
We augment each model by allowing a special form of payment contract. In addition to committing to a direct signaling scheme, the sender also commits to a payment for each action . If the signaling scheme recommends action , and the receiver follows the recommendation, she is then paid by the sender (or pays the sender if ). If the receiver deviates from the scheme’s recommendation, no payment is exchanged. Since payments are exchanged only when the receiver follows the recommendation, nonnegative payments can be viewed as augmenting the “persuasiveness” of the signaling scheme. Negative payments are less natural, although their consideration will be technically instructive.
We distinguish three classes of payment contracts: unrestricted (allowing arbitrary positive and negative payments), nonnegative, and budget balanced. For the latter, the sender’s expected payment should be zero over states of nature and randomness in the signaling scheme, assuming the receiver(s) follow the recommendations of the scheme. Independent to our work, Li  also considers adding payments to Bayesian persuasion. They analyze a special case of Bayesian persuasion with two states of nature, and examine how adding a payment contract influences the optimal policy in that scenario. Our approach diverges from this work by considering more general settings of Bayesian persuasion and various classes of payment contracts.
Duality as a Unifying Lens.
Persuasion and auction design share striking parallels. Indeed, both are economic design problems in which the outputs — recommendation(s) in the case of persuasion and allocations of goods in the case of an auction — are subjected to incentive constraints which at the surface appear quite similar in the two settings. This is made explicit by Dughmi and Xu , who draw an analogy between persuasion and single-item auctions: actions are analogous to bidders, and recommending an action is analogous to allocating the item. Through this analogy, they were able to leverage techniques from auction theory — in particular Border’s theorem [Border, 1991] — to characterize and compute optimal signaling schemes for Bayesian persuasion when action payoffs are i.i.d. This analogy is imperfect, however, as illustrated by the impossibility result of Dughmi and Xu  for independent non-identical action payoffs, contrasting the tractability of single-item auctions with independent bidders.
Despite being imperfect, however, this similarity is suggestive: if a Border’s theorem based approach of optimization of interim rules can be applied to persuasion, why not the “virtual value” approach of Myerson  as well? Myerson’s approach can be viewed through the more general lens of Lagrangian duality, in particular as a consequence of Lagrangifying the incentive constraints. The duality-based approach has been applied to much more general mechanism design settings, producing (often approximate) generalizations of Myerson’s virtual-value characterization which have led to simple and approximately optimal mechanisms in a number of multi-parameter settings (e.g. Cai et al. ; Eden et al. [2017a, b]; Brustle et al. ; Fu et al. ; Liu and Psomas ; Cai and Zhao ; Haghpanah and Hartline , and classic results such as Rochet and Choné ). It is therefore natural that we embark on the same exploration for persuasion, as well as for models (such as ours) which combine the approaches of persuasion and mechanism design. As a best case scenario, we can hope for “simple” characterizations of optimal or near-optimal schemes, akin to those derived from duality in mechanism design. Particularly attractive are “canonical” characterizations which depend minimally on the details of the instance at hand.
In Section 3, we apply Lagrangian duality to Bayesian persuasion, and derive some elementary properties of the primal/dual pair which enable our results to follow. Our first main result is in Section 5, and concerns (single-receiver) Bayesian persuasion with a prior distribution which is symmetric across actions111We refer the reader to Dughmi ; Dughmi and Xu  for some natural examples of persuasion with symmetric actions.
, and no payments are allowed. We show that a single dual variable naturally interpolates between two extreme schemes: at one extreme (equals zero) we get the (non-persuasive) scheme which always recommends the sender’s ex-post preferred action, at the other extreme ( very large) we get the (persuasive) scheme which recommends the receiver’s ex-post preferred action. Intermediate values of yield schemes which point-wise optimize the sender payoff plus times the receiver’s payoff. Moreover, there is a threshold below which the induced scheme is non-persuasive, and above which the scheme is persuasive. This induces the sender-optimal persuasive signaling scheme. This characterization is detail-free, in the sense that it reduces the prior distribution to a relative weighting of receiver to sender payoff, in particular the minimum necessary for persuasiveness. Furthermore, this optimal scheme is Pareto efficient in a strong (ex-post) sense: for every state , no outcome Pareto dominates the one picked by the scheme.
In Section 6, we use duality to characterize Bayesian persuasion schemes with payments. When arbitrary payments are allowed and the prior is symmetric, the optimal signaling scheme is canonical and does not depend on the prior: it always (i.e. in every state of nature) recommends the action that maximizes the sender utility plus times the receiver’s utility. Payments accompanying this scheme are computed easily via a simple payment identity. Our main result in this section is a dichotomy for Bayesian persuasion with a symmetric prior, but non-negative payments: the optimal scheme is either the same as the aforementioned arbitrary-payment scheme (in the event that non-negative payments are needed), or else is the optimal no-payment signaling scheme. Finally, with only two actions and an arbitrary prior, we show that when arbitrary payments are allowed, the optimal scheme always recommends the action maximizing sender utility plus twice receiver utility. Again, all of our optimal schemes are ex-post Pareto efficient. We note that the strongest positive results of Dughmi and Xu  (exact polynomial time solvability) hold in the setting of i.i.d. actions. Our results therefore extend and simplify theirs, while lending further insight.
In Section 7, we turn our attention to a multi-receiver private persuasion model with externalities. Again, we employ duality to analyze the optimal scheme in this setting. Our first main result shows that, when we allow budget balanced payments, there exists an optimal scheme which is “simple” in the following sense: It always recommends an action maximizing a weighted sum of sender utility and the receivers’ marginal utility from following the recommendation of the scheme. The relative weighting is determined by a single dual variable. This characterization is interesting when contrasted with the optimal no-payment scheme, which is not as simple in general, and we show is sometimes strictly outperformed (in terms of sender expected utility) by the optimal budget-balanced scheme. Our second main result is a generalization of an algorithmic result of Dughmi and Xu  to multi-receiver persuasion with positive externalities: when no payments are allowed, and sender and receiver utility functions lie in some cone of set functions , we use duality to exhibit a polynomial time reduction from optimal signaling to the optimization problem for set functions in . Due to the lack of space, details of this last result are in Appendix B.
Bayesian persuasion is a game between a sender, also termed as the principal, and a receiver, also termed as the agent. There is also a set of possible states of nature . The true state of nature is drawn from a prior distribution , known to both the sender and the receiver. The receiver has a set of possible actions to pick. Based on the action picked by the receiver and the true state of nature , the sender and the receiver gain payoffs222 We use the words “payoff" and “reward" interchangeably in this paper., denoted by and respectively. In the Bayesian persuasion game, the sender commits to a signaling scheme , where in general is a mapping from to distributions over possible signals, . Then the sender observes , and sends a signal to the receiver, where for the observed . Given signal , the receiver updates her belief about the state of nature , and selects an action that maximizes her expected payoff under this posterior distribution. In this paper, without loss of generality, and by applying the revelation principle [Kamenica and Gentzkow, 2011], we focus on signaling schemes for which . We use the notation
to denote the probability that the sender recommends actionconditioned on the state of nature being . A signaling scheme is said to implement , if it samples signal given the state of nature .
Persuasive signaling and optimal Bayesian persuasion.
A signaling scheme is persuasive if the receiver is best off following the sender’s recommendation, i.e. following the sender’s recommendation maximizes the receiver’s expected payoff under the receiver’s posterior belief about the state of the nature (conditioned on the received signal). In the optimal Bayesian persuasion problem, we wish to find, over all persuasive schemes, the one that maximizes the sender’s expected payoff.
As typical in information structure design, we frequently think of each action as having a “type” depending on the state of nature.333 We refer the reader to Dughmi and Xu  for a list examples of Bayesian persuasions and how types are defined in those. That is,
is a vector in, for some parameter . Action has type , which completely determines the sender and receiver payoffs should action be selected, independent of . More clearly, there exist pairs of payoffs such that when the receiver selects action with , is the payoff to the sender and is the payoff to the receiver. Note that and , and for universality we stick to this notation (see Appendix C). Distributions in this setting may be independent, if is a product distribution, where each is a distribution over . Distributions in this setting may also be symmetric, if is invariant under all permutations. Distributions that are both independent and symmetric are i.i.d.
Bayesian persuasion with payments.
In this paper, we introduce a natural model of payments into the Bayesian persuasion problem. In addition to recommending an action , the sender is allowed to “incentivize” the receiver to take the recommendation with an additional payment . Mathematically, it also makes sense to consider when payments are allowed to be negative (which corresponds to the sender “charging” the receiver in order to follow the recommendation). This certainly could be relevant for practice (e.g. if the sender/receiver can commit to contracts), but the non-negative payment model is clearly more natural. We study four different payment models: zero payments (i.e. classic Bayesian persuasion), non-negative payments (i.e. when the sender cannot charge the receiver), budget-balanced payments (i.e. the payment of the sender is zero in expectation) and general payments (i.e. payments are arbitrary real numbers). To unify the notation throughout the paper, we use to denote the feasible set of payments, which plays the role of a different polytope for each relevant payment model (note that in all four models is closed and convex). A signaling scheme is said to implement if given the observed state of nature , it samples and pays the receiver a (randomized) payment . Throughout this paper, it will be much more convenient to focus on specifying the expected payments for following the recommendation of action , for any scheme. We similarly define the implementability of .444 It is also easy to see how to implement payments from and an implementation of : for a given state , sample a payment whenever signal is recommended.
Payment identity and optimal payments.
We conclude with an observation about Bayesian persuasion with payments. Similar to auction design, there is a “payment identity” capturing which payments will make an implementable signaling scheme persuasive. In contrast to auctions, however, it is easy to see that every signaling scheme can be made persuasive with sufficiently high payments.
Let be any distribution over states of nature, and be any signaling scheme (not necessarily persuasive). Then there exist thresholds such that is persuasive if and only if for all .
Let be the receiver’s expected utility by taking action when the scheme recommends action . Define . Therefore, for all , and any scheme that pays is certainly persuasive when recommending . Moreover, if , then there exists some s.t. , and the scheme is not persuasive. ∎
We refer to the guaranteed in Observation 1 as the optimal payments for , and as the optimal non-negative payments for .
Bayesian persuasion with multiple agents, binary actions and externalities.
In this paper, we formalize a model with externalities, naturally extending the model introduced recently in Arieli and Babichenko  and Dughmi and Xu  for private persuasion with no externalities. Consider the general setup of Bayesian persuasion with one sender and receivers, where each receiver can take either action or action . Using the revelation principle for Bayesian persuasion, we can restrict our attention to direct signaling schemes, where a direct (randomized) scheme can be thought of as a mapping from to distributions over subsets of , indicating to which receivers action is recommended. We denote such a scheme by for every . Given the subset of receivers taking action , let be the payoff of receiver and be sender’s payoff. We further assume that is a monotone set function. We also allow payments of the form , where is the payment agent receives by following action , if it is recommended. We occasionally use the notation , which are called marginal utilities in this paper. For our computational results, we further assume positive externalities, i.e. for every agent , can only increase if agent switches from action to .
3 Lagrangian Duality and Bayesian persuasion
In this section, we create a unified analysis toolbox for various Bayesian persuasion problems through the lens of LP duality. Specifically, we use Lagrangian duals to reveal the structures of the optimal signaling, á la successful instances of a similar technique in Bayesian mechanism design [Cai et al., 2016]
. As a prerequisite, we heavily use linear programming techniques introduced inKamenica and Gentzkow , and further improved in Dughmi and Xu , and build a bedrock for the analyses in future sections.
3.1 LP for general Bayesian persuasion with payments
Linear programming formulations of Bayesian persuasion without payments have been introduced in numerous prior works (e.g. Dughmi and Xu ). We now modify the program slightly to capture payments, by observing that whenever action is recommended the receiver gets additional payoff and the sender loses a payoff . The following program solves the sender’s optimization problem.
The first set of constraints, also called persuasiveness constraints, are similar to Incentive Compatibility (IC) constraints from auction design, and ensure that the receiver is best off (in expectation) by following the recommendation and paying the payment. The rest ensure that the scheme is in fact a valid distribution over recommendations, and the payments are feasible. While not yet a linear program, the above program can be made linear by a simple change of variables to expected payments :555 Recall that .
where we abuse the notation by using to denote the feasible polytope of average payments .
3.2 The partial Lagrangian dual
One of the main tools we use in this paper is taking the partial Lagrangians of the linear programs of the Bayesian persuasion problem. Basically, similar to Cai et al. , we do not take a “complete dual” of the LP formulation, instead “Lagrangifying” only the persuasiveness constraints, and leave all feasibility constraints in the primal.666 There are significant conceptual differences between “incentive constraints” in auction design and “persuasiveness constraints”. Some of these differences were briefly discussed in Section 1; we will highlight them further in future technical sections. To take the partial Lagrangians, we apply the method of Lagrangian multipliers by (a) introducing dual variables for every pair of actions with , (b) multiplying the persuasiveness constraints with duals , and (c) moving the persuasiveness constraints to the objective. By rearranging the terms we have the following observation.
Assigning dual variables to the persuasiveness constraint guaranteeing that the receiver prefers to take action over action when is recommended gives the following partial Lagrangian.
We conclude by reminding the reader that the optimal signaling schemes and payments are solutions to the following min-max programs by applying strong duality ( denotes the appropriate dual feasible polytope):
Definition 2 (Dual-adjusted receiver payoff).
For any assignment of dual variables , define .
[Strong Duality for Bayesian Persuasion] There exist dual variables such that the optimal signaling scheme, for every state of nature , recommends the action maximizing the dual-adjusted receiver payoff, i.e. . Moreover, if , then when action is recommended, the receiver is indifferent between following the recommendation and taking action instead (Complementary Slackness).777 We will not actually make use of complementary slackness in this paper, but include it here for completeness.
This is is an immediate consequence of strong duality, plus some observations about the structure of the Bayesian Persuasion LP. Observe first that indeed . Now, consider the RHS of Equation 2. For given dual variables , observe that the right-most term (depending on ) does not depend on at all. Observe that the remaining term () is trivial to optimize over all such that for all : simply recommend the action maximizing for all . Therefore, when are the optimal dual variables, the optimal signaling scheme must recommend the action maximizing on all states of nature. Complementary slackness follows immediately from strong duality. ∎
3.3 Exploiting symmetries
When viewing actions by their types (recall this means that we view states of nature as a profile , with each ), some of our results consider symmetric settings, where for any permutation . It is well-known that symmetric LPs admit symmetric solutions (e.g. Herr and Bödi ), and this fact has indeed been exploited in prior work on signaling and auctions [Dughmi and Xu, 2016; Cai et al., 2012]. The proof of this is straight-forward, but we sketch it below (and provide a full proof in Appendix A). First, we state clearly what we mean by symmetric solutions.
We say that a signaling scheme is symmetric if for all and for all permutations , we have for all . We say that a dual solution is symmetric if for all .
Let be a symmetric instance of Bayesian Persuasion, with any of the four referenced constraints on payments (none, non-negative, budget-balanced, or arbitrary). Then there exists an optimal symmetric primal and an optimal symmetric dual for .
The proof is essentially a formalization of the following intuition. First, if a scheme is optimal and persuasive, and is symmetric, then we can first “relabel” the actions according to any permutation before implementing , and it will still be optimal and persuasive. Second, any distribution over optimal, persuasive schemes is still optimal and persuasive. And third, if we consider the scheme that randomly samples a permutation with which to relabel the actions before implementing , then this scheme is symmetric. A full proof appears in Appendix A. ∎
With Proposition 2 in hand, we can now draw further conclusions regarding the format of the Lagrangian function in the special case that is symmetric:
When is symmetric, there exists a constant such that the optimal scheme, on every state of nature , selects an action maximizing . Moreover, for the same and some constant , the Lagrangian takes the following form:
Consider an optimal and symmetric dual solution, which is guaranteed to exist by Proposition 2, in which for all . Then we get the following simplified form for :
Where we have defined . It is now easy to see that the signaling scheme maximizing necessarily on every state of nature recommends an action maximizing . ∎
4 Illustrative Examples
Here, we work through two illustrative examples to highlight the role of different payment models in Bayesian persuasion. Our examples shed some insights on the payment choices. We formalize these insights in the later sections.
4.1 A tempting but false argument
Initially, it may seem that the optimal scheme with arbitrary payments must recommend an action in for all (and pay the optimal payments). In fact, the following argument seems to confirm this intuition. Consider an optimal persuasive signaling scheme and assume (towards a contradiction) that there is some state of nature where does not recommend an element of . Let denote the action recommended, and denote an element in the argmax. Modify by moving all the probability mass from to , i.e. recommend action in state , and increase by . Let be the new signaling scheme. The following two observations are immediate. First, the sender’s payoff under is strictly larger than the sender’s payoff under . This is because the sender pays an additional , and gets additional payoff ; the total change in sender payoff is , by the hypothesis about action . Second, the receiver’s payoff under is equal to the receiver’s payoff under . The receiver gets an additional in payment, and gets additional payoff . Note that one of these terms might be negative.
Therefore, the sender’s payoff in is strictly improved, while the receiver is just as happy, i.e., it seems like we have reached a contradiction to the optimality of . The catch is that is not necessarily persuasive! Since payments in Bayesian persuasion are quite different than payments in (say) auctions, we find it educational to present here a concrete example that confirms that the scheme described is suboptimal.888The proofs of our theorems of course provide confirmation themselves, but since the intuition is especially jarring for those unfamiliar with Bayesian persuasion, the example should be instructive.
Consider the following. There are two actions, and . Each action gives the receiver payoff or , uniform and iid. Identically for the sender payoff. In other words, there are four possible action types and , encoding the receiver and sender payoffs respectively. We will compare the following two schemes: (1) Recommend a uniformly random action in , and pay the optimal payments (the scheme suggested above), (2) Recommend a uniformly random action that maximizes , and pay the optimal payments (the optimal scheme, as suggested by Proposition 4).
Towards analyzing the first scheme, observe that conditioned on action having type , it is recommended with probability : it is always recommended unless has type , in which case it is recommended with probability . Conditioned on action having type or , it is recommended with probability , and conditioned on having type it is recommended with probability . The expected receiver payoff for taking the recommendation, conditioned on action being recommended, is . The expected receiver payoff for taking action when was recommended is . Therefore, the optimal payments are for each action. The sender’s expected payoff when the recommendation is followed is also , so the total sender utility (accounting for payments) is .
Let’s analyze the second scheme. The difference between the two schemes is that the second one tie-breaks in favor of over . So, conditioned on action having type , it’s still recommended with probability . Conditioned on having type , it’s recommended with probability , type is recommended with probability , and type is recommended with probability . The expected receiver payoff, conditioned on action being recommended, is now . The expected payoff for taking action when was recommended is then . This means that the optimal payments are for each action. The sender’s expected payoff when the recommendation is followed is . The sender’s total utility is , indeed better than .
4.2 Contrasting payment schemes
The following example demonstrates how negative payments can be quite counterintuitive in Bayesian persuasion, and how various constraints on payments differ. Consider an instance with one sender and one receiver that has two possible actions 0 and 1. Suppose that , that the two states are equiprobable, and that the payoff functions of the sender and the receiver satisfy: and . The payoffs of the sender and the receiver always sum up to zero, so this is a zero-sum persuasion game.
The optimal signaling without payments is to not reveal any information to the receiver. To see this, observe that the receiver must get payoff at least , as they can guarantee this by ignoring the sender’s signal and taking action . Thus, the sender cannot get utility strictly bigger than , which they can guarantee by recommending action . This scheme is in fact optimal even when the sender is allowed to use non-negative payments. To see this, observe that the receiver’s utility is again at least : they can always take action regardless of the recommendation, and be paid a non-negative amount. Again, the sender cannot get utility strictly bigger than .
Next, consider the following scheme with negative payments, as suggested by Proposition 4: always recommend action and charge . This scheme is persuasive, because the receiver’s expected payoff for taking action is . This gives the sender expected utility equal to .
Finally, consider the following budget balanced scheme, recommended by Proposition 7. When , recommend action with probability . When , recommend action deterministically. The receiver payoff for taking action , conditioned on it being recommended, is . The receiver payoff for taking action , conditioned on being recommended, is . Therefore, the sender must pay to make this persuasive, for an expected total payment of . Conditioned on action being recommended, the receiver payoff for taking action is , while her payoff for taking action is . The sender can charge and remain persuasive, for a total expected charge of . The total sender payoff (not counting payments) is then . Accounting for payments, the sender pays , and charges , for an additional payoff of . The sender’s total utility is . If , we recover the optimal scheme with possibly negative payments. If , then the total expected payment is , and the scheme is budget balanced. In this case, the expected sender utility is , strictly bigger than the optimal expected utility with non-negative payments.
The above example highlights the following concepts. First, the purpose of negative payments is to get extra payoff out of “strictly persuasive” schemes. Second, depending on the context, negative payments might not always be well motivated. For example, if the setup is such that a sender is making a non-binding recommendation to a receiver, then in the above examples the receiver would simply choose not to engage in the recommendation and always take action . On the other hand, if the sender acts as a “gatekeeper” to the actions, e.g. because the actions are which event to attend that the sender is hosting or which fund to invest in that the sender manages, then negative payments are well motivated. Third, all three payment methods considered in this paper are distinct.
5 Symmetric Actions and No Payments
Here, we consider the standard symmetric setting without payments. We show how to derive structure on the optimal scheme by making use of duality. In any symmetric instance of Bayesian persuasion, there exists an optimal dual solution such that for all by applying Proposition 2. Now, by plugging directly into Corollary 1 and observing that for all , we get that the Lagrangian for the optimal dual takes the following form:
We immediately conclude that the optimal scheme recommends, for every state , an action maximizing . With a little more work, we can conclude something stronger about the exact value of .
Definition 4 (-scaled welfare maximizer).
For a given multiplier , define to be the scheme with if , and if . In other words, recommends a uniformly random action in .
Let be the smallest such that the scheme is persuasive for . Then is the optimal scheme for .
The proof will follow from the following. We claim that the persuasiveness of is monotone increasing in (larger is more persuasive), while the sender payoff is monotone decreasing in . Together this immediately concludes that the optimal persuasive scheme is . We’ll first need a technical lemma.
Let , be symmetric schemes and let be symmetric. Let the expected receiver payoff for accepting recommendation for be at least as large as the expected receiver payoff for accepting recommendation for . Then if is persuasive, so is .999 Note that this does not hold generally, and absolutely requires the symmetry assumptions.
First, observe that by symmetry, the expected payoff for any action conditioned on that action being recommended is the same. Moreover, the expected payoff for any action conditioned on action being recommended is also the same (for all ). Therefore, the scheme is persuasive as long as for all actions, the expected payoff conditioned on being recommended exceeds the expected payoff conditioned on not being recommended. This holds if and only if the expected payoff conditioned on being recommended exceeds the unconditional expected payoff. As the unconditional expected payoff for an action is independent of the payoff scheme (denote it by ), we conclude that a signaling scheme is persuasive if and only if the expected receiver payoff for following the recommendation exceeds . The lemma immediately follows. ∎
If is persuasive, then is persuasive, for all .
First, observe that is symmetric for all . Further observe that the receiver’s expected payoff for following the recommendation is monotone in : On every state of nature , the recommended action maximizes . The proof now immediately follows by Lemma 1. ∎
The sender’s expected payoff when the receiver follows is monotone non-increasing in .
Simply observe that on every state of nature , the recommended action maximizes . As increases, the sender payoff for the recommended action decreases. ∎
The proof of Proposition 3 now immediately follows. ∎
6 Optimal Single Agent Signaling with Payments
In this section we study the single receiver Bayesian persuasion game with payments. While we consider our “main results” to be the case where payments are constrained to be non-negative, it’s instructive to study general (positive or negative) payments. We characterize the optimal signaling scheme with payments in the general setting, drawing similar conclusions to Cai et al.  for optimal auctions. This is not a main result, but may be of independent interest. An easy corollary of this characterization, however, immediately allows us to claim something interesting in the case of two actions. The optimal scheme, for all states , recommends the action maximizing (paying the optimal payments), and this holds for any distribution.
A deeper application of this characterization lets us characterize the optimal scheme for a single receiver with symmetric actions. In this setting, we show that the optimal scheme recommends the action maximizing (paying the optimal payments).101010 Further recall that the sender/receiver payoffs for action are completely determined by action ’s type. So this can also be phrased as recommending a uniformly random action with type , where maximizes over all present types . When payments are non-negative, we prove that the optimal scheme is always either the optimal scheme without payments at all, or the optimal scheme with arbitrary payments.
6.1 The general setting with payments
Recall Equation 1, where we had the Langrangian function for an arbitrary polytope :
Also recall that for every choice of the Lagrange multipliers, is an upper bound to the performance of the optimal persuasive scheme. Strong duality further implies that this bound is tight for some choice of the Lagrange multipliers.
Observe that if payments are allowed to be arbitrary, then is unbounded whenever the coefficient for is non-zero for any (as we can simply set to be or . Therefore, we certainly have in the optimal dual, for all actions . This means that for each action , the dual variables form a distribution over actions other than . The simplified Lagrangian becomes
For every choice of Lagrange multipliers , the scheme that maximizes recommends, for every state of nature , the action that maximizes .
The optimal persuasive scheme recommends, at each state of nature , the action that maximizes , where is the optimal Lagrange multiplier.
Observation 3 provides a general framework to reason about optimal signaling schemes with arbitrary payments. We repeat now a connection to optimal auction design: In optimal auction design, there are some cases where the optimal dual is “canonical,” and doesn’t depend on the input distribution (e.g. single-dimensional) [Myerson, 1981]. In such settings, one can identify simple structure of the optimal mechanism. The case is similar in signaling: some cases admit a canonical optimal dual that doesn’t depend on the input distribution. In these cases, we obtain simple characterizations of the optimal scheme.
6.2 Two actions, arbitrary payments
In the case of only two actions, Observation 3 immediately allows us to derive the simple structure of the optimal signaling scheme.
When , for every distribution , the optimal persuasive scheme with possibly negative payments always recommends the action that maximizes (and pays the optimal payments).
Observe that there are only two Lagrange multipliers, and , and are both equal to in the optimal dual (by Observation 3). Therefore, the Lagrangian can be further simplified:
Observe that the term does not depend on the action selected at all. So in order to maximize , the scheme must recommend the action maximizing for every state of nature . ∎
6.3 Symmetric actions
Here, we draw conclusions for the symmetric setting with payments. Again getting initial traction from a canonical form for the optimal dual. Recall by Corollary 1, the Lagrangian in this setting is given by:
When arbitrary payments are allowed, i.e. , then the multiplier of the payment variable must be equal to zero for all . Otherwise the Lagrangian would be unbounded. This immediately implies that for the optimal dual, we have , and the Lagrangian becomes
The proof of the following proposition then immediately follows.
In the single sender, single receiver setting with symmetric actions and arbitrary payments, the optimal scheme recommends, on every state of nature , the action that maximizes (and pays the optimal payments).
Non-negative payments: a dichotomy
When payments are restricted to be non-negative, it is no longer the case that is pinned down completely (in particular, must certainly be , or else setting would result in an unbounded Lagrangian, but it is indeed possible to have ). Our next result shows that the optimal scheme in this scenario is essentially either the optimal scheme without any payments, or the optimal scheme for arbitrary payments (because the payments are already non-negative)
In the single sender, single receiver setting with independent and identically distributed actions and non-negative payments, the optimal scheme is either (1) the optimal no-payment scheme, or (2) recommends the action that maximizes (and pays the optimal non-negative payments).
Let be the guaranteed by Corollary 1. Then there is an optimal scheme that maximizes over all feasible .
If , then is strictly negative, and hence the multiplier of each payment variable is strictly negative. Therefore, every scheme that maximizes must have for all actions . Hence, the scheme is in fact feasible and persuasive for the no-payments case, and must be the optimal scheme without payments (as every scheme without payments is also feasible for non-negative payments, and is optimal among all schemes with non-negative payments).
If , then we immediately observe that this is exactly the same Lagrangian as for arbitrary payments, and therefore the second part of the proposition follows. ∎
7 Optimal Multi-agent Signaling with Externalities and Payments
In this section, we study Bayesian persuasion for multiple receivers with binary actions and general externalities through the lens of duality. Our model is a natural extension of the model recently introduced in Arieli and Babichenko , and further developed in Dughmi and Xu . We refer the reader to Section 2 for notation and definitions. Missing proofs can be found in Appendix A. In Appendix B we study optimal signaling with positive externalities: when no payments are allowed, and sender and receiver utility functions lie in some cone of set functions , we use duality to exhibit a polynomial time reduction from optimal signaling to the optimization problem for set functions in .
7.1 Linear programming formulation
Similar to all other Bayesian persuasion problems, one can formulate finding the optimal signaling scheme with payments as a linear program, but this time with exponentially many variables (again, is the set of feasible payments):
The first two sets of constraints in this LP are essentially persuasion constraints, i.e. if a receiver is in the recommended set she is better off picking action and if a receiver is not in the recommended set she is better off picking action . For notation brevity, we rewrite the persuasiveness constraints as:
where . Note that for .
7.2 Budget balanced payments under externalities
As explored in Section 6, adding monetary payments to the signaling problem, whether payments are positive or negative, is a natural idea to boost the performance of signaling schemes. It is expected that by adding arbitrary payments, one can increase the expected sender’s utility, as we have seen in Section 4.2; however, it is not clear what happens if we allow restricted payments, e.g. payments that are budget-balanced. A general and mathematically interesting setting to study this question is the general multi-agent signaling with externalities. We seek to find a simple characterization for the optimal scheme, and understand its properties. In the same setting, we also study arbitrary payments and show how one can modify the previous budget balanced scheme to find the optimal scheme with arbitrary payments. This model is again mathematically interesting, but perhaps not as natural as the budget-balanced.
Here is the surprising upshot of the story: with the help of Lagrangian duality we can design signaling schemes with payments that have a simple form (in contrast to the optimal scheme without payments), are budget-balanced in expectation (i.e. zero total payment in expectation), and sender’s expected payoff 111111 When the scheme is budget-balanced, there is no difference between expected utility and expected payoff of the sender. is no smaller than that of the optimal signaling without payments. Also, as we explained in Section 4.2, somewhat surprisingly, it is possible that the sender’s expected utility strictly increases by a budget-balanced scheme.
A budget-balanced signaling scheme for the multi-agent binary-actions with externalities setting is a pair of an allocation rule and a payment rule satisfying:
where is the payment to receiver in state , when the receiver takes action .
Proposition 7 (Optimal budget-balanced signaling with externalities).
In multi-agent binary-action with externalities setting, there exists a persuasive signaling scheme such that:
It is budget-balanced as in Definition 5 (but may charge negative payments),
It maximizes the “total virtual payoff". That is, there exists a parameter such that for every state of nature the scheme recommends action to a subset of agents that maximizes , where ,
The expected total payoff of the sender is no smaller than the expected total payoff of the sender under any budget-balance persuasive scheme (with or without payments).
One can show a similar characterization for the optimal signaling scheme with arbitrary payments. In fact, because the payments are now unrestricted, the corresponding dual constraints are going to be equalities. This fact results in the following generalization of Proposition 4 for the special case of .
Proposition 8 (Optimal signaling with externalities and arbitrary payments).
In the multi-agent binary-action with externalities setting, the optimal signaling scheme with arbitrary payments maximizes the total payoff, i.e. for every state the scheme recommends action to a subset of agents that maximizes , where .
We augment Bayesian persuasion by introducing payments, in single and multi-receiver settings. We obtain a number of results, all enabled via Lagrangian duality. For symmetric, single-receiver persuasion with no payments we show that a detail-free, ex-post Pareto optimal scheme is optimal. In the same setting, if arbitrary payments are allowed, the optimal scheme does not even depend on the prior: it always recommends the action that maximizes the sender utility plus times the receiver utility, where is the number of actions. When payments are restricted to be non-negative we prove a dichotomy: the optimal scheme is either the arbitrary-payment signaling scheme or the optimal no-payment scheme. When there are multiple receivers with binary actions and externalities, we prove that a simple scheme is optimal under a budget balanced constraint on the payments. Finally, in the same setting, when no payments are allowed, when the sender and receiver utility functions lie in some cone of set functions , we use duality to give a polynomial time reduction from optimal signaling to the optimization problem for set functions in .
Our work focuses on characterizing optimal schemes in the simplest cases beyond the tools of prior work. Notably, our tools gain the most traction where there is a “canonical” optimal dual (i.e. when the optimal Lagrangian multipliers are independent of the underlying distribution). Compare this to single-dimensional settings in auction design where the payment identity/monotonicity implies a canonical optimal dual as well Myerson .
The clear direction for future work is to consider more general settings, possibly through the lens of “simple versus optimal” persuasion schemes (aligned with similar work in mechanism design). That is, rather than targeting optimal dual solutions (which are likely unwieldy too far beyond the canonical settings studied her), perhaps approximately optimal dual solutions will yield tractable insight for proving approximation results. Results of this form are limited without making use of duality, but do exist. For example, Dughmi and Xu  provides an even simpler signaling scheme for a single receiver with a symmetric prior that guarantees a -approximation. Another general direction for future work is to further explore the interplay between payments and persuasion.
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Appendix A Missing proofs.
Proof of Proposition 2.
Now, consider any optimal, persuasive signaling scheme , and any optimal dual . For a permutation , let denote the state of nature , denote the prices . Let denote the dual variables with . Finally, let denote the signaling scheme that on state of nature , recommends action with probability .
Now we just want to confirm that and form another optimal primal/dual pair for any permutation (because is symmetric). Let’s first compute the probability that recommends action . By symmetry of , the states and occur with the same probability. On state , recommends action with probability . On state , recommends action with probability (this is just chasing through the definition of ). Therefore, because and occur with the same probability, recommends action with the same probability recommends action