We focus on the case where both the leader and the followers are entitled to mixed strategies. We show that the problem is NP-hard and not in APX unless P=NP, and that the same holds for deciding whether, in an equilibrium, one of the leader's actions will be played with a strictly positive probability. We also illustrate that, in the general case, pessimistic LFNEs may not be finite.

We propose different compact (and exact) nonconvex (mixed-integer) nonlinear programming formulations for the optimistic case, a faster implicit-enumeration algorithm which is suitable when the leader is only entitled to pure strategies, and a black box (heuristic) algorithm for the both the optimistic and pessimistic cases. Computational experiments are reported and illustrated.

We conclude by sketching an exact (up to finite precision) branch-and-bound algorithm for the pessimistic case based either on Farkas's lemma for the case where the followers are restricted to pure strategies, or on a result by Balas on the convex hull of a disjunction of polyhedra for the general case in mixed strategies.

This is joint work with Nicola Gatti and Nicola Basilico.

When: March 9, 2016 at 10.30

Where: Aula Fausto Saleri - 6th floor Dipartimento di Matematica, Building 14, Politecnico di Milano

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Title: A note on comparative ambiguity aversion and justifiability

Aula Fausto Saleri - 6th floor Dipartimento di Matematica

April 22, 2015 at 14.30

A note on comparative ambiguity aversion and justifiability |

The program:

“Equilibria and Stability in a Model of Political and Social Conflict. The Symmetric Case”

Conflicts are ubiquitous in social and political environments. Two are the common characteristics of these, otherwise extremely different, conflicts: very often they came as a surprise and their evolution through time is usually very heterogenous, sometimes they quickly disappear sometimes they degenerate in catastrophe. If we consider World War I, one hundred years later, historians still wonder how such a catastrophic war came almost out of nowhere and why what could have been a manageable crisis turned into all-out-war. In the end, the historians’ conclusion is that there are no sure solutions, because contexts change: what will turn off conflicts in one setting will spread the conflict as if it is a wildfire. From a theoretical point of view, we believe that these phenomena can be explained in two different and complementary ways: either using comparative statics showing how equilibria properties change suddenly as parameters (contexts) change, or with multiple equilibria and out-of equilibrium dynamics to emphasize the complex and cyclic behavior of conflicts. In this paper, we study the result of symmetric strategic interaction between two parties. Both parties are able to mobilize a certain effort to pursue their goals, that are partially diverging. The target of their political action can be the government, as well as public policies or private objects, but they have all in common a conflict situation, which starts from a particular historical situation and which can evolve in several different ways, so that, depending on the results that are achieved during the evolutions, the starting situation and the actions of the involved actors, it can break up, it can stabilize at a particular level, it can grow up and become a mass movement or it can ebb and flow periodically. We present a model in which the efforts of both the opponents are described as the outcome of strategic decisions, and the evolution is governed by several factors. Hence, the initial conditions describing the historical premises of the political (or social, economical) situation can define the final outcome of the conflict, which eventually could be independent of the other aspects.

“Personal Information on Social Networks: Incentives for Data Release and Fair Monetization”

Recent years have seen the explosion of online business models based on the exploitation of users’ personal data collected on social networks. The potential benefits of this ecosystem, however, are currently undermined by tussles around privacy and the uncontrolled use of this data. In this talk, we propose and analyze new game-theoretic models that allow us to discuss economics and incentives challenges arising in a personal data ecosystem where users have control over their data. In the first part of the talk, we present a user-centric public good model, in which users have full control of their personal information and strategically choose whether they want to reveal some data and with which precision. We analyze the users’ incentives to obfuscate their data and to reveal truthfully the precision of the data disclosed. We then propose simple ways in which an analyst collecting the data can increase the aggregate precision level by slightly modifying the game. In the second part of the talk, we focus on the economic value of personal data. In a social network, the value of data revealed depends on data revealed by other users. To analyze the impact of the social network on this value, we propose a cooperative game theoretic model on a graph. We then propose a fair mechanism based on the Shapley value to quantify the value of the personal information of each user depending on his social interactions, and we analyze the impact on the system of our mechanism.

"Efficient algorithms for the verification problem of sequential equilibrium and quasi-perfect equilibrium"

We study the computational problem (aka verification problem) of certifying that a given solution is a sequential equilibrium and/or a quasi-perfect equilibrium. While the verification problem of Nash equilibrium is easy, only results of NP–hardness are known for Nash equilibrium refinements for sequential games. We show that, when the assessment (i.e., a pair defining agents’ behavioral strategies and beliefs) is given in input, there exists an efficient algorithm with any number of agents. Furthermore, we show that, when only the behavioral strategy profile is given in input, there exists an efficient algorithm with two–agent games (with more agents the problem is already known to be NP–hard). The algorithm can be also adopted to derive the agents’ consistent (in the sense of Kreps and Wilson) beliefs from their strategies. In addition, a simple variation of this algorithm can be adopted for the verification problem of the quasi–perfect equilibrium in two–agent games (also in this case, with more agents the problem is already known to be NP–hard).

Equilibria and Stability in a Model of Political and Social Conflict. |

Personal information on social networks |

Efficient algorithms for the verification problem |

Speaker: Josep Freixas

Title: Tactival vote in committes. Applications to law, teaching, business and sports.

Abstract: In (j,k)-games each player chooses amongst j ordered options and there are k possible outcomes.

In this talk, we consider the case where players are assumed to prefer some outcomes to others, and note that when k>2 the players have an incentive to vote strategically. In doing so, we combine the theory of cooperative game theory with social choice theory, especially the theory of single-peaked preferences.

We define the concept of a (j,k)-game with preferences and what it means for it to be manipulable by a player. We also consider Nash equilibriums with pure strategies for these games and find conditions that guarantee their existence.

Sala Seminari, DEIB

February 5, 2015, 11:00-12:00

Title: Tactival vote in committes. Applications to law, teaching, business and sports.

Abstract: In (j,k)-games each player chooses amongst j ordered options and there are k possible outcomes.

In this talk, we consider the case where players are assumed to prefer some outcomes to others, and note that when k>2 the players have an incentive to vote strategically. In doing so, we combine the theory of cooperative game theory with social choice theory, especially the theory of single-peaked preferences.

We define the concept of a (j,k)-game with preferences and what it means for it to be manipulable by a player. We also consider Nash equilibriums with pure strategies for these games and find conditions that guarantee their existence.

Sala Seminari, DEIB

February 5, 2015, 11:00-12:00

Tactical vote in committes.pdf |

Title: Power indices: a measure of centrality in networks

Abstract: A cooperative game describes a situation in which all players can freely interact with each other, i.e. all coalitions are feasible. However, this is not the case in many real world situations. A typical situation is when the restriction of communication possibilities between players is modelled by a network. In this scenario power indices have been used to evaluate the centrality of nodes (the players of the cooperative game). In the talk, an overview of solution concepts for cooperative games with restricted communications will be provided and applications to centrality analysis in social and biological networks will be discussed.

Sala Seminari, DEIB

December 10 2014, 12:00-13:00

Power indices: a measure of centrality in networks.pdf |