Abstract
Search for equilibria in games is a hard problem and many games do not have a pure Nash equilibrium (PNE). Incentive mechanisms have been shown to secure a PNE in certain families of games. The present study utilizes the similarity between Asymmetric Distributed Constraints Optimization Problems (ADCOPs) and games, to construct search algorithms for finding outcomes and incentives that secure a pure Nash equilibrium in Boolean games. The set of values returned by the search algorithm for a chosen incentive mechanism is termed an incentive plan. The two incentive mechanisms that are used by the present study are taxation and side-payments. Both are described and their performance on PNE search in Boolean games is evaluated. An incentive plan for the taxation mechanism consists of the values of imposed tax, while an incentive plan for the side payments mechanism consists of values for a set of transfer functions. The distributed search algorithms address two different requirements. One is to find an incentive plan that enables to secure a PNE. The other requirement is that the algorithms return a PNE that satisfies some global objective, such as a PNE that maximizes social welfare. The Boolean game is first transformed into an ADCOP. Then, a designated distributed search algorithm is applied to find the desired outcome. Two distributed search algorithms are described, incorporating k-ary constraints as well as soft constraints that relate to global objectives. The new and innovative algorithm - Concurrent Asymmetric Branch and Bound - is found experimentally to be much faster than the former algorithm. An extensive experimental evaluation on several types of social-network-based Boolean games is presented. The degree of intervention in the game is found to be small for both incentive mechanisms. In other words, the overall tax or the total amount of side-payments is a small fraction of the general utility. The density of the networks has a substantial effect on the solution quality as well as on the computational effort to find them.
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Notes
\({{\sum }_{2}^{p}} = {\textit {NP}}^{\textit {NP}}\) is the class of all the languages that can be recognized in polynomial time by a nondeterministic Turing machine equipped with NP oracles [43].
Throughout the paper the symbol \(\overline {a}\) is used to denote the negation of a (i.e., ¬a). This is clearer to the eye for formulas that include the negation of longer literals.
Coordination games [8].
References
Anshelevich, E., Dasgupta, A., Kleinberg, J., Tardos, E., Wexler, T., Roughgarden, T. (2008). The price of stability for network design with fair cost allocation. SIAM Journal on Computing, 38(4), 1602–1623.
Barabási, A.-L., & Albert, R. (1999). Emergence of scaling in random networks. Science, 286(5439), 509–512.
Bonzon, E., Lagasquie-Schiex, M.-C., Lang, J. (2009). Dependencies between players in boolean games. Int. J. Approx. Reasoning, 50(6), 899–914.
Bonzon, E., Lagasquie-Schiex, M.-C., Lang, J., Zanuttini, B. (2006). Boolean games revisited. In: 17th Euro. Conf. Artif. Intell. - ECAI, pp. 265–269, Riva del Garda, Italy.
Bonzon, E., Lagasquie-Schiex, M.-C., Lang, J., Zanuttini, B. (2009). Compact preference representation and Boolean games. Autonomous Agents and Multi-Agent Systems, 18, 1–35.
Boutilier, C., Brafman, R.I., Hoos, H.H., Poole, D. (2003). CP-Nets: A tool for representing and reasoning with conditional Ceteris Paribus preference statements. Journal of Artificial Intelligence Research, 21, 135–191.
De Clercq, S., Bauters, K., Schockaert, S., Mihaylov, M., Nowė, A., De Cock, M. (2017). Exact and heuristic methods for solving boolean games. Autonomous Agents and Multi-Agent Systems, 31(1), 66–106.
Cooper, R. (1999). Coordination games. Cambridge: Cambridge University Press.
Cordes, J.J. (1999). Horizontal equity. In: The encyclopedia of taxation and tax policy. Urban Institute Press.
De Clercq, S., Bauters, K., Schockaert, S., De Cock, M., Nowé, A. (2014). Using answer set programming for solving Boolean games. In: 14th Intern. Conf. Princip. Knowl. Represent. Reason. (KR), Vienna, Austria.
De Clercq, S., Bauters, K., Schockaert, S., Mihaylov, M., De Cock, M., Nowé, A. (2014). Decentralized computation of pareto optimal pure Nash equilibria of Boolean games with privacy concerns. In: ICAART, pp. 50–59.
Dunne, P.E., & van der Hoek, W. (2004). Representation and complexity in boolean games. In: European workshop on logics in artificial intelligence, pp. 347–359. Springer.
Dunne, P.E., van der Hoek, W., Kraus, S., Wooldridge, M. (2008). Cooperative Boolean games. In: 7th Intern. Conf. Auton. Agents Multi Agent Sys. (AAMAS-08), pp. 1015–1022, Estoril, Portugal.
Dunne, P.E., & Wooldridge, M. (2012). Towards tractable Boolean games. In: 11th Intern. Conf. Auton. Agents Multi Agent Sys. (AAMAS-2012), pages 939–946, Valencia, Spain.
Erdȯs, P., & Rényi, A. (1959). On random graphs. Publicationes Mathematicae Debrecen, 6, 290–297.
Gamson, W.A. (1961). A theory of coalition formation. American Sociological Review, 26, 373–382.
Gershman, A., Grubshtein, A., Meisels, A., Rokach, L., Zivan, R. (2008). Scheduling meetings by agents. In: Proc. 7th Intern. Conf. on Pract. & Theo. Automated Timetabling (PATAT 2008), Montreal.
Grant, J., Kraus, S., Wooldridge, M., Zuckerman, I. (2014). Manipulating games by sharing information. Studia Logica, 102(2), 267–295.
Grinshpoun, T., Grubshtein, A., Zivan, R., Netzer, A., Meisels, A. (2013). Asymmetric distributed constraint optimization problems. Journal of Artificial Intelligence Research, 47, 613–647.
Grubshtein, A., & Meisels, A. (2012). A distributed cooperative approach for optimizing a family of network games. In: Intelligent distributed computing V - proceedings of the 5th international symposium on intelligent distributed computing - IDC 2011, Delft, The Netherlands - October 2011, volume 382 of Studies in Computational Intelligence, pp. 49–62. Springer.
Grubshtein, A., & Meisels, A. (2012). Finding a nash equilibrium by asynchronous backtracking. In: Principles and practice of constraint programming - 18th international conference, CP 2012, quėbec City, QC, Canada, October 8-12, 2012. Proceedings, pp. 925–940.
Harrenstein, P., Turrini, P., Wooldridge, M. (2014). Hard and soft equilibria in Boolean games. In: 13th Intern. Conf. Auton. Agents Multi Agent Sys. (AAMAS-2014), pages 845–852, Paris, France.
Harrenstein, P., van der Hoek, W., Meyer, J.-J., Witteveen, C. (2001). Boolean games. In: TARK, pp. 287–298.
Hirayama, K., & Yokoo, M. (1997). Distributed partial constraint satisfaction problem. In: CP, pp. 222–236.
Jackson, M.O., & Wilkie, S. (2005). Endogenous games and mechanisms: Side payments among players. The Review of Economic Studies, 72(2), 543–566.
Kun, J., Powers, B., Reyzin, L. (2013). Anti-coordination games and stable graph colorings. In: SAGT, pp. 122–133.
Lėautė, T., & Faltings, B. (2011). Distributed constraint optimization under stochastic uncertainty. In: Proceedings of the 25th AAAI conference on artificial intelligence, AAAI San Francisco, California, USA, August 7-11, 2011,.
Levit, V., Grinshpoun, T., Meisels, A. (2013). Boolean games for charging electric vehicles. In: IAT, pp 86–93.
Levit, V., Grinshpoun, T., Meisels, A., Bazzan, A.L.C. (2013). Taxation search in boolean games. In: 12th Intern. Conf. Auton. Agents Multi-Agent Sys. (AAMAS-13), pp. 183–190, Saint Paul, MN, USA.
Levit, V., Komarovsky, Z., Grinshpoun, T., Meisels, A. (2015). Tradeoffs between incentive mechanisms in boolean games. In: 24th Intern. Joint Conf. on Artif. Intell. (IJCAI-2015), pp 68–74, Buenos Aires, Argentina.
Litov, O., & Meisels, A. (2017). Forward bounding on pseudo-trees for dcops and adcops. Artificial Intelligence, 252, 83–99.
Lynch, N.A. (1997). Distributed algorithms. Morgan kaufmann Series.
Mavronicolas, M., Monien, B., Wagner, K.W. (2007). Weighted Boolean formula games. In: WINE, pp. 469–481.
Meisels, A., Razgon, I., Kaplansky, E., Zivan, R. (2002). Comparing performance of distributed constraints processing algorithms. In: Proc. AAMAS-2002 workshop on distributed constraint reasoning DCR, pp. 86–93, Bologna.
Meisels, A. (2007). Distributed search by constrained agents: Algorithms, performance, Communication. Springer Verlag.
Modi, J., & Veloso, M. (2004). Multiagent meeting scheduling with rescheduling. In: Proc. 5th workshop on distributed constraints reasoning DCR-04, Toronto.
Modi, P.J., Shen, W., Tambe, M., Yokoo, M. (2005). ADOPT: Asynchronous distributed constraints optimization with quality guarantees. Artificial Intelligence, 161, 11–2:49–180.
Monderer, D., & Tennenholtz, M. (2004). k-implementation. Journal of Artificial Intelligence Research, 21, 37–62.
Netzer, A., Grubshtein, A., Meisels, A. (2012). Concurrent forward bounding for distributed constraint optimization problems. Artif Intell., 193, 186–216.
Nguyen T.-V.-A., & Lallouet, A. (2014). A complete solver for constraint games. In: Principles and practice of constraint programming - 20th international conference, CP 2014, Lyon, France, September 8-12, 2014. Proceedings, pp. 58–74.
Nisan, N., Roughgarden, T, Tardos, E., Vazirani, V.V. (2007). Algorithmic game theory. Cambridge: Cambridge University Press.
Osborne, M., & Rubinstein, A. (1994). A course in game theory. The MIT Press.
Papadimitriou, C.H. (1994). Computational complexity. Addison-Wesley.
Porter, Ryan, Nudelman, Eugene, Shoham, Yoav. (2008). Simple search methods for finding a nash equilibrium. Games and Economic Behavior, 63(2), 642–662.
Roughgarden, T. (2005). Selfish routing and the price of anarchy. The MIT press.
Sauro, Luigi, & Villata, Serena. (2013). Dependency in cooperative Boolean games. Journal of Logic and Computation, 23(2), 425–444.
Turrini, P. (2013). Endogenous Boolean games. In: 24th Intern. Joint Conf. on Artif. Intell. (IJCAI-2013), pp. 390–396, Beijing, China.
Wooldridge, M., Endriss, U., Kraus, S., Lang, J. (2013). Incentive engineering for Boolean games. Artificial Intelligence, 195, 418–439.
Zhang, W., Xing, Z., Wang, G., Wittenburg, L. (2005). Distributed stochastic search and distributed breakout: properties, comparishon and applications to constraints optimization problems in sensor networks. Artificial Intelligence, 161, 1–2:55–88.
Zivan, R., & Meisels, A. (2005). Dynamic ordering for asynchronous backtracking on discsps. In: CP-2005, pages 32–46, Sigtes (Barcelona), Spain.
Zivan, R., & Meisels, A. (2006). Concurrent search for distributed csps. Artificial Intelligence, 170, 440–461.
Zivan, R., & Meisels, A. (2006). Message delay and disCSP search algorithms. Annals of Mathematics and Artificial Intelligence (AMAI), 46, 415–439.
Zivan, R., Glinton, R., Sycara, K.P. (2009). Distributed constraint optimization for large teams of mobile sensing agents. In: Proceedings of the 2009 IEEE/WIC/ACM international conference on intelligent agent technology, IAT 2009, Milan, Italy, 15-18 September 2009, pp. 347–354.
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Levit, V., Komarovsky, Z., Grinshpoun, T. et al. Incentive-based search for equilibria in boolean games. Constraints 24, 288–319 (2019). https://doi.org/10.1007/s10601-019-09304-y
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DOI: https://doi.org/10.1007/s10601-019-09304-y