Computational Social Choice (ComSoc) has delivered impactful improvements in several real world settings ranging from optimizing kidney exchanges Dickerson et al. (2012) to devising mechanisms which to assign students to schools and/or courses more fair and efficient manners Budish and Cantillon (2012)
. ComSoc has also had impact in a number of other disciplines within computer science including recommender systems, data mining, machine learning, and preference handlingRossi et al. (2011); Chevaleyre et al. (2008). From its earliest days, much theoretical work in ComSoc has centered on worst case assumptions Bartholdi, III et al. (1989). Indeed, in the last 10 years, there has been a groundswell of such research which shows little signs of slowing Conitzer et al. (2007); Faliszewski and Procaccia (2010); Faliszewski et al. (2010).
Within ComSoc, much work focuses on manipulative or strategic behavior, which may take many different forms including manipulation and control of election and aggregation functions Brandt et al. (2016). Often, advanced algorithmic techniques such as fixed parameter tractability to move beyond these worst case assumptions Conitzer (2010); Faliszewski and Procaccia (2010). Approximation algorithms have played an important part, helping to determine the winner of some hard to compute voting rules Caragiannis et al. (2014); Skowron et al. (2013). Approximation has been used in other areas of social choice including mechanism design, often to achieve good results when the “worst case” is too hard Procaccia and Tennenholtz (2009). Additional algorithmic work has centered on average case complexity (which typically suppose very uniform sampling of instances) Procaccia and Rosenschein (2007) and/or attempting to understand the parameters which make an aggregation or choice rule hard to manipulate Conitzer and Sandholm (2006); Xia and Conitzer (2008).
In one of the papers that founded the field, Bartholdi, III et al. (1989)
warned against exclusively focusing on worst case assumptions stating, “The existence of effective heuristics would weaken any practical import of our idea. It would be very interesting to find such heuristics.” For the last several years we have championed the use of real world data in ComSocMattei and Walsh (2013) and are happy to see more and more researchers working in this area (e.g., Goldman and Procaccia (2014); Tal et al. (2015)).111For a more comprehensive listing with links to over 80 papers and a wealth of tools and resources, please see www.preflib.org. We see an increased focus on experimentation, heuristics, and verifying theory and models through properly incentivized data collection and experimentation as key research direction for ComSoc. Some of the most impactful work in ComSoc has come from the development of theory that is specifically informed by real world data and/or practical application that is then rigorously tested (e.g., Budish and Cantillon (2012); Dickerson et al. (2012)).
Contribution. In this short note we detail a study of tournaments Laslier (1997) using real world and generated data. We show that, despite the NP-completeness of the Tournament Fixing Problem (TFP) Aziz et al. (2014), enumerating all the seedings for which a particular player can win is quickly solvable for a large number of generated and real world instances. Additionally, we show that the popular Condorcet Random (CR) model used to generate synthetic tournaments (i) does not match real world data and (ii) is drawing from a fundamentally different distribution than real world tournaments. The statistical and modeling methodologies we use for this research may be of independent interest to empirical researchers in social choice.
2 Preliminaries: Tournaments
Whether or not teams can advance through a knockout tournament tree in order to claim a championship is a question on the minds of many during the FIFA World Cup, ATP Tennis Tournaments, NCAA Basketball Tournaments, and numerous soccer leagues around the world. The scheduling of the tournament, the seeding which dictates whom will play whom, and its manipulation in order to maximize a particular team’s chance of winning is a well studied problem in ComSoc and other areas Hazon et al. (2008); Vassilevska Williams (2010); Vu and Shoham (2011); Horen and Riezman (1985); Kim et al. (2016); Kim and Vassilevska Williams (2015).
Following Aziz et al. (2014), we are given a set of players and a deterministic pairwise comparisons for all players in . For every in , if then we say that player beats player in a head to head competition; this means that in a pairwise comparison. In a balanced knockout tournament we have that . Given a set of players , a balanced knockout tournament is a balanced binary tree with leaf nodes and a draw . There are multiple isomorphic (ordered) assignments of agents in to the leaf nodes, we represent these as a single unordered draw . Observe that there are assignments to leaf nodes but only draws.
A knockout tournament is the selection procedure where each pair of sibling leaf nodes competes against each other. The winner of this competition proceeds up the tree into the next round; the winner of the knockout tournament is the player that reaches the root note. In this study we want to understand the computational properties of the Tournament Fixing Problem (TFP).
Tournament Fixing Problem (TFP):
Instance: A set of players , a deterministic pairwise comparision matrix , and a disginuished player .
Question: Does there exist a draw for the players in where is the winner of )?
It was recently proven that even if a manipulator knows the outcome of each pairwise matchup, the problem of finding a seeding for which a particular team will win is NP-hard Aziz et al. (2014), thus concluding a long line of inquiry into this problem. More recent results have shown that, for a number of natural model restrictions, the TFP is easily solvable Brandt et al. (2016). Note that the code developed for the next section can be easily generalized to the case where we do not enforce the balanced constraint on .
One popular model for generating for experiment is the Condorcet Random (CR) model, introduced by Young (1988). The model has one tuning parameter
which gives the probability that a lower ranked team will defeat (upset) a higher ranked team in a head-to-head matchup. In general,. A Uniform Random Tournament has . The manipulation of tournament seedings has been studied using both random models Stanton and Vassilevska Williams (2011); Kim et al. (2016); Kim and Vassilevska Williams (2015) and a mix of real data and random models Hazon et al. (2008). A more sophisticated model of random tournaments was developed by Russell and van Beek (2011) – upset probabilities were fitted to historical data, and varied according to the difference in the ranking of the teams (surprisingly for their tennis data, the upset probability remained relatively invariant to the difference in player rankings).
To explore how theory lines up with practice, we looked at two questions:
Does the NP-completeness of knockout tournament seeding manipulations tell us about the complexity of manipulation in practice?
Are random models which are popularly used in ComSoc supported by real world data?
To answer these questions we use data from the 2009-2014 English Premier League and German Bundesliga, along with lifetime head-to-head statistics of the top 16 tennis players on the ATP world tour to create 16 team pairwise tournament datasets.
3 Tournament Fixing in Practice
In order to convert the datasets into deterministic pairwise tournaments we used two different strategies. Using the lifetime head to head ATP tour results we get a set of weighted pairwise comparisons that we can use as input to understand tournaments. Using all data available up until Feb. 1, 2014 provides something that is not a tournament graph. There are several ties and several players who have never played each other. Given the matchup matrix, we extracted a tournament graph by saying one player beats another if their historical average is . This does not create a tournament graph, hence we award all ties, including if two players have never met, to the higher ranked player. This results in Rafael Nadal being a Condorcet winner and he is thus removed from the following analysis. For the soccer data, we deterministically assigned as the pairwise winner the team which had scored more total goals in the home-and-away format for a particular year (ties broken by away goals).
We implemented a simple constraint program in miniZinc Nethercote et al. (2007) to search for a seeding that ensures a given team wins. The miniZinc model which was then solved using a modified version of GeCode (http://www.gecode.org). The modifications to GeCode served to make the counting of all solutions faster by removing all diagnostic printing to STDOUT and implementing some other small caching optimizations. All experiments were run on a system running Debian 6.0.10 with a 2.0 GHz Intel Xeon E5405 CPU and 4 GB of RAM; GeCode was restricted to using 4 cores.
For all the tournaments in our experiments we have 16 teams which means that the entire search space is possible seedings. Table 1 shows the results for each of the players in the ATP World Top 16 including the total number of seedings won (and percentage of the overall total number of seedings), the number of choice points (nodes of the search tree) explored to find the first (resp. all) seedings, and and time (minutes) explored to to find all seedings for which a player can win a tournament. Table 2 gives summary statistics for the number of choice points (nodes of the search tree) and time (minutes) explored to to find the first (resp. all) seedings for which a team can win a knockout tournament for all teams across all 13 datasets.
Despite this being a NP-hard problem, it was extremely easy in every dataset to find a winning seeding for any given team (or prove that none existed). Exhaustively, counting all the winning seedings took more time but even this was achievable. Only the Bundseliga 2011 experiment came anywhere close to exploring even a small fraction of the total possible seedings.
The practical upshot of these computational results is that real world tournaments exhibit a lot of structure which is possible to leverage for practical computation. While the simple techniques we employed may not scale to the possible seedings in the 128 player Wimbledon Tournament, they can be used for more modestly sized tournaments. The low number of choice points explored for these instances may indicate that there is practically exploitable structure in larger tournaments; an interesting avenue for future research.
4 Verifying Real World Tournament Models
We turn to a second fundamental question. The CR model has been used to derive theoretical bounds on the complexity of manipulating the seeding in knockout tournaments Stanton and Vassilevska Williams (2011). But does it adequately model real world tournaments? In the soccer datasets we take the teams goals scored over the total as the pairwise probability, while we use the life time head-to-head average to determine probabilities in the tennis data. To test the modeling power of CR, we generated pairwise probabilities with . We then computed, for each of the real world datasets and all values of in steps of
, the probabilities that teams would win a knockout tournament using a simple sampling procedure which converged to the actual probability quickly; uniformly sampling over all possible seedings and updating the probability estimates for each team for a particular seeding, the probability that every team wins the tournament can be computed efficientlyMattei et al. (2015); Vu et al. (2013, 2009).
Our hypothesis is that the probability of winning a knockout tournament for a team in the real world data is a random variable drawn from the same distribution as the CR model. We approach this question in two parts. (1) Using a Kolmogorov-Smirnov (KS) tests withas our significance threshold Clauset et al. (2009), we can determine if the data is drawn from the same type of distribution, i.e. a normal distribution or a heavy tailed distribution such as a log-normal or power law distribution. (2) Then for a candidate pair of samples, we determine if the fitting parameters of the distribution are similar.
The KS test compares the distance between the cumulative distribution (CDF) of two empirical samples to determine whether or not two different samples are drawn from the same distribution. Figure 1(A) shows the CDF of the 2014 Bundesliga league data along with several settings of . Table 3 gives the minimum and maximum values of
, per dataset, for which we can say that the probability distribution of a team winning a knockout tournament according to the CR model is likely drawn from the same distribution as the respective real world dataset (KS test,). We can reject CR models with values of outside these ranges; as these models are not likely to emerge from the same distribution as the real world datasets. We also provide average upset probability for each datafile to compare with the results of Russell and van Beek (2011).
Examining our results, we find no support for the Uniform Random Tournament model. Likewise, setting or
generates data which is drawn from a different distribution than most real world datasets we survey. The tennis data seems to be an outlier here, supporting a very low value of, likely due to Rafael Nadal, who has a winning lifetime record against all other players in the ATP top 16 as of Feb. 1, 2014.
As we cannot reject all models given by CR outright we must look more closely at the underlying distribution and attempt to fit the empirical data to a likely distribution. For this we will dive more deeply into the 2014 Bundesliga League data, as the range for is similar to the average of across all datasets and the average upset probability for 2014 yields a model which is a good match for the underlying data. The 2014 Bundesliga data has an average upset probability of and a best fit probability according to the KS test of .
We must first identify what kind of probability distribution the samples are drawn from in order to tell if they are the same or different. At first glance, the winning probabilities appear to be drawn from a power law or some other heavy tailed distribution distribution such as a log-normal Mitzenmacher (2004); Clauset et al. (2009); Limpert et al. (2001)
. The study of heavy tailed distributions in empirical data is a rich topic that touches a number of disciplines including physics, computer science, literature, transportation science, geology, biology as these distributions describe a number of natural phenomena such as the spread of diseases, the association of nodes in scale free networks, the connections of neurons in the brain, the distribution of wealth amongst citizens, city sizes, and other interesting phenomenaAlstott et al. (2014); Clauset et al. (2009); Limpert et al. (2001). Recently, more sophisticated methods of determining if an empirical distribution follows a particular heavy tailed distribution have been developed, consequently showing strong evidence that distributions once thought power laws (e.g., node connections on the internet and wealth distribution) are likely not explained by a power law distribution Clauset et al. (2009) but rather by log-normal distributions Limpert et al. (2001). The current standard for fitting heavy tailed distributions in physics and other fields (and the one we will employ) involves the use of robust statistical packages to estimate the fitting parameters then testing the fitted model for basic plausibility through the use of a likelihood ratio test Alstott et al. (2014). This process will help us decide which distribution is the strongest fit for our data as well as provide us with the actual fitting parameters to compare the real world and generated data.
Figure 1 (B) shows the results of fitting the 2014 Bundesliga League data to a power law for a random variable of the form as well as the fit for a log-normal distribution with median
and multiplicative standard deviation. Using a likelihood ratio test we find that the log-normal is a significantly better fit for the data than the power law distribution (, ). This makes intuitive sense in this context as each matchup can be seen as a (somewhat) independent random variable, and the product of multiple positive random variables gives a log-normal distribution Limpert et al. (2001). The fit parameters for the 2014 Bundesliga League data are and while for the best fitting CR model with is and . While those two distributions are similar, it implies that perhaps a more nuanced, multi-parameter model is needed to capture the matchup probabilities for tournaments.
5 Discussion and Future Directions
In order to transform into a more impactful area we must demonstrate the effectiveness of our methods in real settings, and let these real settings drive our research. Kagel and Roth (1995) describe the journey for experimental economics: evolving from theory to simulated or repurposed data to full fledged laboratory and field experiments. This progression enabled a “conversation” between the experimentalists and the theoreticians which enabled the field to expand, evolve, and have the impact that it does today in a variety of contexts.
Our case study of tournaments shows that the need to verify our models with data is an important and interesting future direction for ComSoc. Working to verify these models can point the way to new domain restrictions or necessary model generalizations. There has been more data driven research, like the kind presented here, thanks to PrefLib Mattei and Walsh (2013) and other initiatives, e.g., Goldman and Procaccia (2014); Tal et al. (2015) ; we hope this trend continues. This research complements the existing research on axiomatic characterizations, worst case complexity, and algorithmic considerations (see, e.g., Davies et al. (2011); Skowron et al. (2013); Goldsmith et al. (2014)). While there are some issues with using repurposed data and discarding context (see, e.g., the discussion by John Langford of Microsoft Research about the UCI Machine Learning Research Repository at http://hunch.net/?p=159) it is a start towards a more nuanced discussion about mechanisms and preferences.
Results about preference or domain restrictions can lead to elegant algorithmic results, but these restrictions should be complemented by some evidence that the restrictions are applicable. For example, in the over 300 complete, strict order datasets in PrefLib Mattei and Walsh (2013), none are single-peaked, a popular profile restriction in voting Faliszewski and Procaccia (2010). Untested normative or generative data models can sometimes lead us astray; if we use self reported data from surveys, or hold out data which does not fit our preconceived model, we may introduce bias and thus draw conclusions that are spurious Regenwetter et al. (2006); Popova et al. (2013). Our study of tournaments makes unrealistic assumptions about model completion in order to produce deterministic tournaments. However, even these simple rounding rules yield instances that are much simpler than the worst case results would imply.
Perhaps one way forward for ComSoc is incorporating even more ideas from experimental economics Kagel and Roth (1995) including the use of human subjects experiments on Mechanical Turk Mason and Suri (2012); Crump et al. (2013) like those performed by, e.g., Mao and Suri (2013) and Mao et al. (2013). Work with human subjects can lead to a more refined view of strategic behavior and inform more interesting and realistic models on which we can base good, tested theory Thompson et al. (2013); Tal et al. (2015).
We would like to thank Haris Aziz for help collecting data and insightful comments. Data61/CSIRO (formerly known as NICTA) is funded by the Australian Government through the Department of Communications and the ARC through the ICT Centre of Excellence Program.
- Alstott et al. (2014) Alstott, J., Bullmore, E., Plenz, D., 2014. powerlaw: A Python package for analysis of heavy-tailed distributions. PloS one 9 (1), e85777.
Aziz et al. (2014)
Aziz, H., Gaspers, S., Mackenzie, S., Mattei, N., Stursberg, P., Walsh, T., 2014. Fixing a balanced knockout tournament. In: Proceedings of the 28th AAAI Conference on Artificial Intelligence (AAAI).
- Bartholdi, III et al. (1989) Bartholdi, III, J., Tovey, C., Trick, M., 1989. The computational difficulty of manipulating an election. Social Choice and Welfare 6 (3), 227–241.
- Brandt et al. (2016) Brandt, F., Conitzer, V., Endriss, U., Lang, J., Procaccia, A. D. (Eds.), 2016. Handbook of Computational Social Choice. Cambridge University Press.
- Budish and Cantillon (2012) Budish, E., Cantillon, E., 2012. The multi-unit assignment problem: Theory and evidence from course allocation at Harvard. The American Economic Review 102 (5), 2237–2271.
- Caragiannis et al. (2014) Caragiannis, I., Kaklamanis, C., Karanikolas, N., Procaccia, A. D., 2014. Socially desirable approximations for Dodgson?s voting rule. ACM Transactions on Algorithms (TALG) 10 (2), 6.
- Chevaleyre et al. (2008) Chevaleyre, Y., Endriss, U., Lang, J., Maudet, N., 2008. Preference handling in combinatorial domains: From AI to social choice. AI Magazine 29 (4), 37–46.
- Clauset et al. (2009) Clauset, A., Shalizi, C. R., Newman, M. J., 2009. Power-law distributions in empirical data. SIAM review 51 (4), 661–703.
- Conitzer (2010) Conitzer, V., 2010. Making decisions based on the preferences of multiple agents. Communications of the ACM 53 (3), 84–94.
- Conitzer and Sandholm (2006) Conitzer, V., Sandholm, T., 2006. Nonexistence of voting rules that are usually hard to manipulate. In: Proceedings of the 21st AAAI Conference on Artificial Intelligence (AAAI). pp. 627–634.
- Conitzer et al. (2007) Conitzer, V., Sandholm, T., Lang, J., 2007. When are elections with few candidates hard to manipulate? Journal of the ACM 54 (3), 14.
- Crump et al. (2013) Crump, M. J. C., McDonnell, J. V., Gureckis, T. M., 2013. Evaluating Amazon’s Mechanical Turk as a tool for experimental behavioral research. PloS one 8 (3), e57410.
- Davies et al. (2011) Davies, J., Katsirelos, G., Narodytska, N., Walsh, T., 2011. Complexity of and algorithms for Borda manipulation. In: Proceedings of the 25th AAAI Conference on Artificial Intelligence (AAAI). pp. 657–662.
- Dickerson et al. (2012) Dickerson, J. P., Procaccia, A. D., Sandholm, T., 2012. Optimizing kidney exchange with transplant chains: Theory and reality. In: Proceedings of the 11th International Joint Conference on Autonomous Agents and Multi-Agent Systems (AAMAS). pp. 711–718.
- Faliszewski et al. (2010) Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L. A., 2010. Using complexity to protect elections. Journal of the ACM 53 (11), 74–82.
- Faliszewski and Procaccia (2010) Faliszewski, P., Procaccia, A. D., 2010. AI’s war on manipulation: Are we winning? AI Magazine 31 (4), 53–64.
- Goldman and Procaccia (2014) Goldman, J., Procaccia, A. D., 2014. Spliddit: Unleashing fair division algorithms. jacm 13 (2), 41–46.
- Goldsmith et al. (2014) Goldsmith, J., Lang, J., Mattei, N., Perny, P., 2014. Voting with rank dependent scoring rules. In: Proceedings of the 28th AAAI Conference on Artificial Intelligence (AAAI).
- Hazon et al. (2008) Hazon, N., Dunne, P. E., Kraus, S., Wooldridge, M., 2008. How to rig elections and competitions. In: Proceedings of the 2nd International Workshop on Computational Social Choice (COMSOC).
- Horen and Riezman (1985) Horen, J., Riezman, R., 1985. Comparing draws for single elimination tournaments. Operations Research 33 (2), 249–262.
- Kagel and Roth (1995) Kagel, J. H., Roth, A. E., 1995. The Handbook of Experimental Economics. Princeton University.
- Kim et al. (2016) Kim, M. P., Suksompong, W., Vassilevska Williams, V., 2016. Who can win a single-elimination tournament? In: Proceedings of the 30th AAAI Conference on Artificial Intelligence (AAAI). pp. 516–522.
- Kim and Vassilevska Williams (2015) Kim, M. P., Vassilevska Williams, V., 2015. Fixing tournaments for kings, chokers, and more. In: Proceedings of the 24th International Joint Conference on Artificial Intelligence (IJCAI). pp. 561–567.
- Laslier (1997) Laslier, J. F., 1997. Tournament Solutions and Majority Voting. Springer-Verlag, Berlin.
- Limpert et al. (2001) Limpert, E., Stahel, W. A., Abbt, M., 2001. Log-normal distributions across the sciences: Keys and clues. BioScience 51 (5).
- Mao et al. (2013) Mao, A., Procaccia, A. D., Chen, Y., 2013. Better human computation through principled voting. In: Proceedings of the 27th AAAI Conference on Artificial Intelligence (AAAI).
- Mao and Suri (2013) Mao, A., Suri, S., 2013. How, when, and why to do online behavioral experiments. In: Proceedings of the 9th International Workshop on Internet and Network Economics (WINE). Tutorial Forum.
- Mason and Suri (2012) Mason, W., Suri, S., 2012. Conducting behavioral research on Amazon’s Mechanical Turk. Behavior Research Methods 44 (1), 1–23.
- Mattei et al. (2015) Mattei, N., Goldsmith, J., Klapper, A., Mundhenk, M., 2015. On the complexity of bribery and manipulation in tournaments with uncertain information. Journal of Applied Logic 13 (4), 557–581.
- Mattei and Walsh (2013) Mattei, N., Walsh, T., 2013. Preflib: A library for preferences, http://www.preflib.org. In: Proceedings of the 3rd International Conference on Algorithmic Decision Theory (ADT).
- Maurer (1980) Maurer, S. B., 1980. The king chicken theorems. Mathematics Magazine, 67–80.
- Mitzenmacher (2004) Mitzenmacher, M., 2004. A brief history of generative models for power law and lognormal distributions. Internet Mathematics 1 (2), 226–251.
- Nethercote et al. (2007) Nethercote, N., Stuckey, P. J., Becket, R., Brand, S., Duck, G. J., Tack, G., 2007. Minizinc: Towards a standard cp modelling language. In: Proceedings of the 13th International Conference on Principles and Practice of Constraint Programming (CP). Springer, pp. 529–543.
- Popova et al. (2013) Popova, A., Regenwetter, M., Mattei, N., 2013. A behavioral perspective on social choice. Annals of Mathematics and Artificial Intelligence 68 (1–3), 135–160.
- Procaccia and Rosenschein (2007) Procaccia, A. D., Rosenschein, J. S., 2007. Junta distributions and the average-case complexity of manipulating elections. Journal of Artificial Intelligence Research 28, 157–181.
- Procaccia and Tennenholtz (2009) Procaccia, A. D., Tennenholtz, M., 2009. Approximate mechanism design without money. In: Proceedings of the 10th ACM Conference on Electronic Commerce (ACM-EC). pp. 177–186.
- Regenwetter et al. (2006) Regenwetter, M., Grofman, B., Marley, A. A. J., Tsetlin, I. M., 2006. Behavioral Social Choice: Probabilistic Models, Statistical Inference, and Applications. Cambridge Univ. Press.
- Rossi et al. (2011) Rossi, F., Venable, K., Walsh, T., 2011. A Short Introduction to Preferences: Between Artificial Intelligence and Social Choice. Synthesis Lectures on Artificial Intelligence and Machine Learning 5 (4), 1–102.
- Russell and van Beek (2011) Russell, T., van Beek, P., 2011. An empirical study of seeding manipulations and their prevention. In: Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI). pp. 350–356.
- Skowron et al. (2013) Skowron, P., Faliszewski, P., Slinko, A., 2013. Achieving fully proportional representation is easy in practice. In: Proceedings of the 12th International Joint Conference on Autonomous Agents and Multi-Agent Systems (AAMAS). pp. 399–406.
- Stanton and Vassilevska Williams (2011) Stanton, I., Vassilevska Williams, V., 2011. Manipulating stochastically generated single-elimination tournaments for nearly all players. In: Proceedings of the 7th International Workshop on Internet and Network Economics (WINE). pp. 326–337.
- Tal et al. (2015) Tal, M., Meir, R., Gal, Y., 2015. A study of human behavior in online voting. In: Proceedings of the 14th International Joint Conference on Autonomous Agents and Multi-Agent Systems (AAMAS). pp. 665–673.
- Thompson et al. (2013) Thompson, D. R. M., Lev, O., Leyton-Brown, K., Rosenschein, J., 2013. Empirical analysis of plurality election equilibria. In: Proceedings of the 12th International Joint Conference on Autonomous Agents and Multi-Agent Systems (AAMAS). pp. 391–398.
- Vassilevska Williams (2010) Vassilevska Williams, V., 2010. Fixing a tournament. In: Proceedings of the 24th AAAI Conference on Artificial Intelligence (AAAI). AAAI Press, pp. 895–900.
- Vu et al. (2009) Vu, T., Altman, A., Shoham, Y., 2009. On the complexity of schedule control problems for knockout tournaments. In: Proceedings of the 8th International Joint Conference on Autonomous Agents and Multi-Agent Systems (AAMAS). pp. 225–232.
- Vu et al. (2013) Vu, T., Hazon, N., Altman, A., Kraus, S., Shoham, Y., Wooldridge, M., 2013. On the complexity of schedule control problems for knock-out tournaments.
- Vu and Shoham (2011) Vu, T., Shoham, Y., 2011. Fair seeding in knockout tournaments. ACM Transactions on Intelligent Systems Technology 3 (1), 1–17.
- Xia and Conitzer (2008) Xia, L., Conitzer, V., 2008. Generalized scoring rules and the frequency of coalitional manipulability. In: Proceedings of the 9th ACM Conference on Electronic Commerce (ACM-EC). pp. 109–118.
- Young (1988) Young, H. P., 1988. Condorcet’s theory of voting. The American Political Science Review 82 (4), 1231 – 1244.