# Prophet Inequalities for Bayesian Persuasion

IJCAI 2020, pp. 175-181, 2020.

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Abstract:

We study an information-structure design problem (i.e., a Bayesian persuasion problem) in an online scenario. Inspired by the classic gambler's problem, consider a set of candidates who arrive sequentially and are evaluated by one agent (the sender). This agent learns the value from hiring the candidate to herself as well as the value to ...More

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Introduction

- In many settings an informed agent wants to use private information in order to persuade other agents to take some action that he would benefit from.
- The authors design online signaling schemes that are good for the sender, i.e., the goal is to find direct and persuasive schemes that maximize the sender’s expected utility of the hired candidate.
- Let xij be the probability to signal HIRE upon arrival of a candidate of type j in round i.

Highlights

- In many settings an informed agent wants to use private information in order to persuade other agents to take some action that he would benefit from
- Our goal is to provide good online signaling schemes that maximize the expected utility of the sender
- We study online Bayesian persuasion inspired by prophet inequalities for the classic gambler’s problem
- We show that a simple signaling scheme designed by Dughmi and Xu [2016] for the offline case can be applied online and that it provides a (1−1/e)-approximation
- We provide an online signaling scheme that provides a 1/2-approximation and show that this is the best possible guarantee
- We design online signaling schemes that are good for the sender, i.e., our goal is to find direct and persuasive schemes that maximize the sender’s expected utility of the hired candidate

Results

- R will take an action independent of the signal of S and accept the candidate in round 1, since it yields the higher expected value.
- The unique persuasive scheme in the online scenario for S is to signal HIRE in the first round, which has utility 0 for S.
- The optimal value of LP (4) is an upper bound on the expected utility for S in any offline persuasive signaling scheme.
- Suppose xij be the ex-post probability that a HIRE signal is issued for candidate j from Di. The authors show that the vector x defined in this way is a feasible solution for LP (4).
- The constraint j xijρij ≥ ρE · j xij is fulfilled for all i ∈ [n], since the scheme is persuasive and does not allow a profitable deviation upon a HIRE signal in round i to the outside option.
- On the other hand, R obeys the signal and does not hire, the expected utility from the remaining scheme in rounds i + 1, .
- To obtain persuasiveness it suffices to provide R with an expected value of ρE conditioned on a HIRE signal.
- There is a persuasive signaling scheme in the online setting with k hires satisfying k-SSQ that yields a
- The optimal value constitutes an upper bound for the expected utility for S in the optimal offline scheme – if the authors set xij to the ex-post probability of hiring candidate i of type j in the offline scheme, the vector x is feasible for the LP and the objective function value is the expected utility of S in the offline scheme.
- The exact same calculations as in Theorem 3 show that conditioned on a HIRE signal in round i, the expected value of the candidate for R is at least ρE.

Conclusion

- The authors circumvent this problem due to the following: (1) The authors' scheme will ensure at most a single receiver gets a HIRE-signal in every round.
- The utility of receiver Rt is additive over candidates hired by Rt. The proof of the following result is deferred to the full version of this paper.
- With public signals S faces the feasibility problem discussed above, i.e., multiple receivers might get an incentive to hire the same candidate.

Summary

- In many settings an informed agent wants to use private information in order to persuade other agents to take some action that he would benefit from.
- The authors design online signaling schemes that are good for the sender, i.e., the goal is to find direct and persuasive schemes that maximize the sender’s expected utility of the hired candidate.
- Let xij be the probability to signal HIRE upon arrival of a candidate of type j in round i.
- R will take an action independent of the signal of S and accept the candidate in round 1, since it yields the higher expected value.
- The unique persuasive scheme in the online scenario for S is to signal HIRE in the first round, which has utility 0 for S.
- The optimal value of LP (4) is an upper bound on the expected utility for S in any offline persuasive signaling scheme.
- Suppose xij be the ex-post probability that a HIRE signal is issued for candidate j from Di. The authors show that the vector x defined in this way is a feasible solution for LP (4).
- The constraint j xijρij ≥ ρE · j xij is fulfilled for all i ∈ [n], since the scheme is persuasive and does not allow a profitable deviation upon a HIRE signal in round i to the outside option.
- On the other hand, R obeys the signal and does not hire, the expected utility from the remaining scheme in rounds i + 1, .
- To obtain persuasiveness it suffices to provide R with an expected value of ρE conditioned on a HIRE signal.
- There is a persuasive signaling scheme in the online setting with k hires satisfying k-SSQ that yields a
- The optimal value constitutes an upper bound for the expected utility for S in the optimal offline scheme – if the authors set xij to the ex-post probability of hiring candidate i of type j in the offline scheme, the vector x is feasible for the LP and the objective function value is the expected utility of S in the offline scheme.
- The exact same calculations as in Theorem 3 show that conditioned on a HIRE signal in round i, the expected value of the candidate for R is at least ρE.
- The authors circumvent this problem due to the following: (1) The authors' scheme will ensure at most a single receiver gets a HIRE-signal in every round.
- The utility of receiver Rt is additive over candidates hired by Rt. The proof of the following result is deferred to the full version of this paper.
- With public signals S faces the feasibility problem discussed above, i.e., multiple receivers might get an incentive to hire the same candidate.

Related work

- Our work extends the study of prophet inequalities which was introduced by Krengel and Sucheston [1977]. More recently, the problem was the focus of a lot of research which showed improvements for special cases and introduced combinatorial variants of the original problem [Dutting et al, 2017; Kleinberg and Weinberg, 2019; Alaei, 2014; Chawla et al, 2010; Correa et al, 2017; Correa et al, 2019; Esfandiari et al, 2015]. The study of Bayesian persuasion was initiated by Aumann and Maschler [1966] and came back into focus more recently following the work of Kamenica and Gentzkow [2011]. A plethora of authors worked on variants of persuasion [Celli et al, 2019; Arieli and Babichenko, 2019; Ely, 2017; Ely et al, 2015; Au, 2015; Dughmi and Xu, 2017], including combinations of persuasion with online learning and multi-armed bandit problems (see, e.g., [Kremer et al, 2014; Frazier et al, 2014; Mansour et al, 2015] and subsequent work). For a recent survey on related algorithmic work see Dughmi [2017].

An online model of Bayesian persuasion closely related to our work was studied in our recent work [Hahn et al, 2019]. We study a similar round-wise persuasion game with very different assumptions regarding the a-priori knowledge of sender and receiver. The scenario is inspired by the secretary problem – candidate values are unknown and adversarially chosen but candidates arrive in uniform random order. This uncertainty also has consequences for the definition of persuasiveness. In contrast, in this paper we assume candidate values are drawn independently from known distributions, which allows to compute expected utilities and to apply standard notions of persuasiveness. Moreover, we use significantly different techniques for analysis.

Funding

- Niklas Hahn and Martin Hoefer are funded by the GermanIsrael Foundation grant I-1419-118.4/2017
- Rann Smorodinsky is funded by the joint United States-Israel Binational Science Foundation and National Science Foundation grant 2016734, German-Israel Foundation grant I-1419118.4/2017, Israel Ministry of Science and Technology grant 19400214, Technion VPR grants, and the Bernard M

Reference

- [Alaei, 2014] Saeed Alaei. Bayesian combinatorial auctions: Expanding single buyer mechanisms to many buyers. SIAM J. Comput., 43(2):930–972, 2014.
- [Alonso and Camara, 2016] Ricardo Alonso and Odilon Camara. Persuading voters. Amer. Econ. Rev., 106(11):3590–3605, 2016.
- [Arieli and Babichenko, 2019] Itai Arieli and Yakov Babichenko. Private bayesian persuasion. J. Econ. Theory, 182:185–217, 2019.
- [Au, 2015] Pak Hung Au. Dynamic information disclosure. RAND J. Econ., 46(4):791–823, 2015.
- [Aumann and Maschler, 1966] Robert Aumann and Michael Maschler. Game theoretic aspects of gradual disarmament.
- Report of the US Arms Control and Disarmament Agency, 80:1–55, 1966.
- [Badanidiyuru et al., 2018] Ashwinkumar Badanidiyuru, Kshipra Bhawalkar, and Haifeng Xu. Targeting and signaling in ad auctions. In Proc. 29th Symp. Disc. Algorithms (SODA), pages 2545–2563, 2018.
- [Celli et al., 2019] Andrea Celli, Stefano Coniglio, and Nicola Gatti. Bayesian persuasion with sequential games. CoRR, abs/1908.00877, 2019.
- [Chawla et al., 2010] Shuchi Chawla, Jason Hartline, David Malec, and Balasubramanian Sivan. Multi-parameter mechanism design and sequential posted pricing. In Proc. 42nd Symp. Theory Comput. (STOC), pages 311–320, 2010.
- [Correa et al., 2017] Jose Correa, Patricio Foncea, Ruben Hoeksma, Tim Oosterwijk, and Tjark Vredeveld. Posted price mechanisms for a random stream of customers. In Proc. 18th Conf. Econ. Comput. (EC), pages 169–186, 2017.
- [Correa et al., 2019] Jose Correa, Paul Dutting, Felix Fischer, and Kevin Schewior. Prophet inequalities for I.I.D. random variables from an unknown distribution. In Proc. 20th Conf. Econ. Comput. (EC), pages 3–17, 2019.
- [Dughmi and Xu, 2016] Shaddin Dughmi and Haifeng Xu. Algorithmic Bayesian persuasion. In Proc. 48th Symp. Theory Comput. (STOC), pages 412–425, 2016.
- [Dughmi and Xu, 2017] Shaddin Dughmi and Haifeng Xu. Algorithmic persuasion with no externalities. In Proc. 18th Conf. Econ. Comput. (EC), pages 351–368, 2017.
- [Dughmi, 2017] Shaddin Dughmi. Algorithmic information structure design: a survey. SIGecom Exchanges, 15(2):2– 24, 2017.
- [Dutting et al., 2017] Paul Dutting, Michal Feldman, Thomas Kesselheim, and Brendan Lucier. Prophet inequalities made easy: Stochastic optimization by pricing non-stochastic inputs. In Proc. 27th Symp. Found. Comput. Sci. (FOCS), pages 540–551, 2017.
- [Ely et al., 2015] Jeffrey Ely, Alexander Frankel, and Emir Kamenica. Suspense and surprise. J. Political Econ., 123(1):215–260, 2015.
- [Ely, 2017] Jeffrey Ely. Beeps. Amer. Econ. Rev., 107(1):31– 53, 2017.
- [Emek et al., 2012] Yuval Emek, Michal Feldman, Iftah Gamzu, Renato Paes Leme, and Moshe Tennenholtz. Signaling schemes for revenue maximization. In Proc. 13th Conf. Econ. Comput. (EC), pages 514–531, 2012.
- [Esfandiari et al., 2015] Hossein Esfandiari, MohammadTaghi Hajiaghayi, Vahid Liaghat, and Morteza Monemizadeh. Prophet secretary. In Proc. 23rd European Symp. Algorithms (ESA), pages 496–508, 2015.
- [Frazier et al., 2014] Peter Frazier, David Kempe, Jon Kleinberg, and Robert Kleinberg. Incentivizing exploration. In Proc. 15th Conf. Econ. Comput. (EC), pages 5–22, 2014.
- [Goldstein and Leitner, 2018] Itay Goldstein and Yaron Leitner. Stress tests and information disclosure. J. Econ. Theory, 177:34–69, 2018.
- [Hahn et al., 2019] Niklas Hahn, Martin Hoefer, and Rann Smorodinsky. The secretary recommendation problem. CoRR, abs/1907.04252, 2019.
- [Kamenica and Gentzkow, 2011] Emir Kamenica and Matthew Gentzkow. Bayesian persuasion. Amer. Econ. Rev., 101(6):2590–2615, 2011.
- [Kleinberg and Weinberg, 2019] Robert Kleinberg and Matthew Weinberg. Matroid prophet inequalities and applications to multi-dimensional mechanism design. Games Econom. Behav., 113:97–115, 2019.
- [Kolotilin, 2015] Anton Kolotilin. Experimental design to persuade. Games Econom. Behav., 90:215 – 226, 2015.
- [Kremer et al., 2014] Ilan Kremer, Yishay Mansour, and Motty Perry. Implementing the wisdom of the crowd. J. Political Econ., 122:988–1012, 2014.
- [Krengel and Sucheston, 1977] Ulrich Krengel and Louis Sucheston. Semiamarts and finite values. Bull. Amer. Math. Soc, 83:745–747, 1977.
- [Mansour et al., 2015] Yishay Mansour, Aleksandrs Slivkins, and Vasilis Syrgkanis. Bayesian incentivecompatible bandit exploration. In Proc. 16th Conf. Econ. Comput. (EC), pages 565–582, 2015.
- [Rabinovich et al., 2015] Zinovi Rabinovich, Albert Xin Jiang, Manish Jain, and Haifeng Xu. Information disclosure as a means to security. In Proc. 14th Conf. Auton. Agents and Multi-Agent Syst. (AAMAS), pages 645–653, 2015.
- [Xu et al., 2015] Haifeng Xu, Zinovi Rabinovich, Shaddin Dughmi, and Milind Tambe. Exploring information asymmetry in two-stage security games. In Proc. 29th Conf. Artif. Intell. (AAAI), pages 1057–1063, 2015.
- [Xu et al., 2016] Haifeng Xu, Rupert Freeman, Vincent Conitzer, Shaddin Dughmi, and Milind Tambe. Signaling in Bayesian Stackelberg games. In Proc. 15th Conf. Auton. Agents and Multi-Agent Syst. (AAMAS), pages 150–158, 2016.

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