Partial Order Games

We introduce a non-cooperative game model in which players’ decision nodes are partially ordered by a dependence relation, which directly captures informational dependencies in the game. In saying that a decision node v is dependent on decision nodes v1,…,vk, we mean that the information available to a strategy making a choice at v is precisely the choices that were made at v1,…,vk. Although partial order games are no more expressive than extensive form games of imperfect information (we show that any partial order game can be reduced to a strategically equivalent extensive form game of imperfect information, though possibly at the cost of an exponential blowup in the size of the game), they provide a more natural and compact representation for many strategic settings of interest. After introducing the game model, we investigate the relationship to extensive form games of imperfect information, the problem of computing Nash equilibria, and conditions that enable backwards induction in this new model.

Do supermajority rules really deter extremism? the role of electoral competition

This study investigates how supermajority rules in a legislature affect electoral competition. We construct an extensive-form game wherein parties choose policy platforms in an election. Post election, the policy is determined based on a legislative voting rule. At symmetric equilibrium, supermajority rules induce divergence of policy platforms if and only if the parties are sufficiently attached to their preferred platform. Thus, supermajority rules may not always lead to moderate policies once electoral competition is considered.

A Game-Theoretic Account of Responsibility Allocation

When designing or analyzing multi-agent systems, a fundamental problem is responsibility ascription: to specify which agents are responsible for the joint outcome of their behaviors and to which extent. We model strategic multi-agent interaction as an extensive form game of imperfect information and define notions of forward (prospective) and backward (retrospective) responsibility. Forward responsibility identifies the responsibility of a group of agents for an outcome along all possible plays, whereas backward responsibility identifies the responsibility along a given play. We further distinguish between strategic and causal backward responsibility, where the former captures the epistemic knowledge of players along a play, while the latter formalizes which players – possibly unknowingly – caused the outcome. A formal connection between forward and backward notions is established in the case of perfect recall. We further ascribe quantitative responsibility through cooperative game theory. We show through a number of examples that our approach encompasses several prior formal accounts of responsibility attribution.

Monty Hall three door ’anomaly’ revisited: a note on deferment in an extensive form game

The Monty Hall game is one of the most discussed decision problems, but where a convincing behavioral explanation of the systematic deviations from probability theory is still lacking. Most people not changing their initial choice, when this is beneficial under information updating, demands further explanation. Not only trust and the incentive of interestingly prolonging the game for the audience can explain this kind of behavior, but the strategic setting can be modeled more sophisticatedly. When aiming to increase the odds of winning, while Monty’s incentives are unknown, then not to switch doors can be considered as the most secure strategy and avoids a sure loss when Monty’s guiding aim is not to give away the prize. Understanding and modeling the Monty Hall game can be regarded as an ideal teaching example for fundamental statistic understandings.

The stability of multi-level governments

This paper studies the stability of a multi-level government. We analyze an extensive form game played between two politicians leading two levels of government. We characterize the conditions that render such government structures stable. We also show that if leaders care about electoral rents and the preferences of the constituencies at different levels are misaligned, then the decentralized government structure may be unsustainable. This result is puzzling because, from a normative perspective, the optimality of decentralized decisions via a multi-level government structure is relevant precisely when different territorial constituencies exhibit preference heterogeneity.

When to Follow the Tip: Security Games with Strategic Informants

Although security games have attracted intensive research attention over the past years, few existing works consider how information from local communities would affect the game. In this paper, we introduce a new player -- a strategic informant, who can observe and report upcoming attacks -- to the defender-attacker security game setting. Characterized by a private type, the informant has his utility structure that leads to his strategic behaviors. We model the game as a 3-player extensive-form game and propose a novel solution concept of Strong Stackelberg-perfect Bayesian equilibrium. To compute the optimal defender strategy, we first show that although the informant can have infinitely many types in general, the optimal defense plan can only include a finite (exponential) number of different patrol strategies. We then prove that there exists a defense plan with only a linear number of patrol strategies that achieve the optimal defender's utility, which significantly reduces the computational burden and allows us to solve the game in polynomial time using linear programming. Finally, we conduct extensive experiments to show the effect of the strategic informant and demonstrate the effectiveness of our algorithm.

Modeling competitive situations

This chapter studies how competitive situations are conventionally modeled in noncooperative game theory. It uses two sorts or forms of models: the so-called extensive form game and the normal or strategic form game. An extensive form representation of a noncooperative game is composed of the following list of items: a list of players; a game tree; an assignment of decision nodes to players or to nature; lists of actions available at each decision node and a correspondence between immediate successors of each decision node and available actions; information sets; an assignment of payoffs for each player to terminal nodes; and probability assessments over the initial nodes and over the actions at any node that is assigned to nature. There is no single way to proceed in general from a normal form game to a corresponding extensive form game. In one obvious extensive form, the players all choose complete strategies simultaneously, but often other extensive forms could be constructed from a given normal form.

Secure Degrees of Freedom in Networks with User Misbehavior

We investigate the secure degrees of freedom (s.d.o.f.) of three new channel models: broadcast channel with combating helpers, interference channel with selfish users, and multiple access wiretap channel with deviating users. The goal of introducing these channel models is to investigate various malicious interactions that arise in networks, including active adversaries. That is in contrast with the common assumption in the literature that the users follow a certain protocol altruistically and transmit both message-carrying and cooperative jamming signals in an optimum manner. In the first model, over a classical broadcast channel with confidential messages (BCCM), there are two helpers, each associated with one of the receivers. In the second model, over a classical interference channel with confidential messages (ICCM), there is a helper and users are selfish. By casting each problem as an extensive-form game and applying recursive real interference alignment, we show that, for the first model, the combating intentions of the helpers are neutralized and the full s.d.o.f. is retained; for the second model, selfishness precludes secure communication and no s.d.o.f. is achieved. In the third model, we consider the multiple access wiretap channel (MAC-WTC), where multiple legitimate users wish to have secure communication with a legitimate receiver in the presence of an eavesdropper. We consider the case when a subset of users deviate from the optimum protocol that attains the exact s.d.o.f. of this channel. We consider two kinds of deviation: when some of the users stop transmitting cooperative jamming signals, and when a user starts sending intentional jamming signals. For the first scenario, we investigate possible responses of the remaining users to counteract such deviation. For the second scenario, we use an extensive-form game formulation for the interactions of the deviating and well-behaving users. We prove that a deviating user can drive the s.d.o.f. to zero; however, the remaining users can exploit its intentional jamming signals as cooperative jamming signals against the eavesdropper and achieve an optimum s.d.o.f.