The Complexity of Online Manipulation of Sequential Elections

Most work on manipulation assumes that all preferences are known to the manipulators. However, in many settings elections are open and sequential, and manipulators may know the already cast votes but may not know the future votes. We introduce a framework, in which manipulators can see the past votes but not the future ones, to model online coalitional manipulation of sequential elections, and we show that in this setting manipulation can be extremely complex even for election systems with simple winner problems. Yet we also show that for some of the most important election systems such manipulation is simple in certain settings. This suggests that when using sequential voting, one should pay great attention to the details of the setting in choosing one's voting rule. Among the highlights of our classifications are: We show that, depending on the size of the manipulative coalition, the online manipulation problem can be complete for each level of the polynomial hierarchy or even for PSPACE. We obtain the most dramatic contrast to date between the nonunique-winner and unique-winner models: Online weighted manipulation for plurality is in P in the nonunique-winner model, yet is coNP-hard (constructive case) and NP-hard (destructive case) in the unique-winner model. And we obtain what to the best of our knowledge are the first P^NP[1]-completeness and P^NP-completeness results in the field of computational social choice, in particular proving such completeness for, respectively, the complexity of 3-candidate and 4-candidate (and unlimited-candidate) online weighted coalition manipulation of veto elections.Comment: 24 page

Complexity of Stability

Graph parameters such as the clique number, the chromatic number, and the independence number are central in many areas, ranging from computer networks to linguistics to computational neuroscience to social networks. In particular, the chromatic number of a graph (i.e., the smallest number of colors needed to color all vertices such that no two adjacent vertices are of the same color) can be applied in solving practical tasks as diverse as pattern matching, scheduling jobs to machines, allocating registers in compiler optimization, and even solving Sudoku puzzles. Typically, however, the underlying graphs are subject to (often minor) changes. To make these applications of graph parameters robust, it is important to know which graphs are stable for them in the sense that adding or deleting single edges or vertices does not change them. We initiate the study of stability of graphs for such parameters in terms of their computational complexity. We show that, for various central graph parameters, the problem of determining whether or not a given graph is stable is complete for \Theta_2^p, a well-known complexity class in the second level of the polynomial hierarchy, which is also known as "parallel access to NP.

Strategic Voting and Social Networks

With the ever increasing ubiquity of social networks in our everyday lives, comes an increasing urgency for us to understand their impact on human behavior. Social networks quantify the ways in which we communicate with each other, and therefore shape the flow of information through the community. It is this same flow of information that we utilize to make sound, strategic decisions. This thesis focuses on one particular type of decisions: voting. When a community engages in voting, it is soliciting the opinions of its members, who present it in the form of a ballot. The community may then choose a course of action based on the submitted ballots. Individual voters, however, are under no obligation to submit sincere ballots that accurately reflects their opinions; they may instead submit a strategic ballot in hopes of affecting the election's outcome to their advantage. This thesis examines the interplay between social network structure and strategic voting behavior. In particular, we will explore how social network structure affects the flow of information through a population, and thereby affect the strategic behavior of voters, and ultimately, the outcomes of elections. We will begin by considering how network structure affects information propagation. This work builds upon the rich body of literature called opinion dynamics by proposing a model for skeptical agents --- agents that distrust other agents for holding opinions that differ too wildly from their own. We show that network structure is one of several factors that affects the degree of penetration that radical opinions can achieve through the community. Next, we propose a model for strategic voting in social networks, where voters are self-interested and rational, but may only use the limited information available through their social network contacts to formulate strategic ballots. In particular, we study the ``Echo Chamber Effect'', the tendency for humans to favor connections with similar people, and show that it leads to the election of less suitable candidates. We also extend this voter model by using boundedly-rational heuristics to scale up our simulations to larger populations. We propose a general framework for voting agents embedded in social networks, and show that our heuristic models can demonstrate a variation of the ``Micromega Law'' which relates the popularity of smaller parties to the size of the population. Finally, we examine another avenue for strategic behavior: choosing when to cast your vote. We propose a type of voting mechanism called ``Sticker Voting'', where voters cast ballots by placing stickers on their favored alternatives, thereby publicly and irrevocably declaring their support. We present a complete analysis of several simple instances of the Sticker Voting game and discuss how our results reflect human voting behavior

The Complexity of Online Manipulation of Sequential Elections

Most work on manipulation assumes that all preferences are known to the manipulators. However, in many settings elections are open and sequential, and manipulators may know the already cast votes but may not know the future votes. We introduce a framework, in which manipulators can see the past votes but not the future ones, to model online coalitional manipulation of sequential elections, and we show that in this setting manipulation can be extremely complex even for election systems with simple winner problems. Yet we also show that for some of the most important election systems such manipulation is simple in certain settings. This suggests that when using sequential voting, one should pay great attention to the details of the setting in choosing one's voting rule. Among the highlights of our classifications are: We show that, depending on the size of the manipulative coalition, the online manipulation problem can be complete for each level of the polynomial hierarchy or even for PSPACE. We obtain the most dramatic contrast to date between the nonunique-winner and unique-winner models: Online weighted manipulation for plurality is in P in the nonunique-winner model, yet is coNP-hard (constructive case) and NP-hard (destructive case) in the unique-winner model. And we obtain what to the best of our knowledge are the first P^NP[1]-completeness and P^NP-completeness results in the field of computational social choice, in particular proving such completeness for, respectively, the complexity of 3-candidate and 4-candidate (and unlimited-candidate) online weighted coalition manipulation of veto elections

The Complexity of Online Manipulation of Sequential Elections

Most work on manipulation assumes that all preferences are known to the manipulators. However, in many settings elections are open and sequential, and manipulators may know the already cast votes but may not know the future votes. We introduce a framework, in which manipulators can see the past votes but not the future ones, to model online coalitional manipulation of sequential elections, and we show that in this setting manipulation can be extremely complex even for election systems with simple winner problems. Yet we also show that for some of the most important election systems such manipulation is simple in certain settings. This suggests that when using sequential voting, one should pay great attention to the details of the setting in choosing one's voting rule. Among the highlights of our classifications are: We show that, depending on the size of the manipulative coalition, the online manipulation problem can be complete for each level of the polynomial hierarchy or even for PSPACE. And we obtain the most dramatic contrast to date between the nonunique-winner and unique-winner models: Online weighted manipulation for plurality is in P in the nonunique-winner model, yet is coNP-hard (constructive case) and NP-hard (destructive case) in the unique-winner model