Leadership in Singleton Congestion Games: What is Hard and What is Easy

We study the problem of computing Stackelberg equilibria Stackelberg games whose underlying structure is in congestion games, focusing on the case where each player can choose a single resource (a.k.a. singleton congestion games) and one of them acts as leader. In particular, we address the cases where the players either have the same action spaces (i.e., the set of resources they can choose is the same for all of them) or different ones, and where their costs are either monotonic functions of the resource congestion or not. We show that, in the case where the players have different action spaces, the cost the leader incurs in a Stackelberg equilibrium cannot be approximated in polynomial time up to within any polynomial factor in the size of the game unless P = NP, independently of the cost functions being monotonic or not. We show that a similar result also holds when the players have nonmonotonic cost functions, even if their action spaces are the same. Differently, we prove that the case with identical action spaces and monotonic cost functions is easy, and propose polynomial-time algorithm for it. We also improve an algorithm for the computation of a socially optimal equilibrium in singleton congestion games with the same action spaces without leadership, and extend it to the computation of a Stackelberg equilibrium for the case where the leader is restricted to pure strategies. For the cases in which the problem of finding an equilibrium is hard, we show how, in the optimistic setting where the followers break ties in favor of the leader, the problem can be formulated via mixed-integer linear programming techniques, which computational experiments show to scale quite well

Quasi-Perfect Stackelberg Equilibrium

Equilibrium refinements are important in extensive-form (i.e., tree-form) games, where they amend weaknesses of the Nash equilibrium concept by requiring sequential rationality and other beneficial properties. One of the most attractive refinement concepts is quasi-perfect equilibrium. While quasi-perfection has been studied in extensive-form games, it is poorly understood in Stackelberg settings---that is, settings where a leader can commit to a strategy---which are important for modeling, for example, security games. In this paper, we introduce the axiomatic definition of quasi-perfect Stackelberg equilibrium. We develop a broad class of game perturbation schemes that lead to them in the limit. Our class of perturbation schemes strictly generalizes prior perturbation schemes introduced for the computation of (non-Stackelberg) quasi-perfect equilibria. Based on our perturbation schemes, we develop a branch-and-bound algorithm for computing a quasi-perfect Stackelberg equilibrium. It leverages a perturbed variant of the linear program for computing a Stackelberg extensive-form correlated equilibrium. Experiments show that our algorithm can be used to find an approximate quasi-perfect Stackelberg equilibrium in games with thousands of nodes

Computing a Pessimistic Stackelberg Equilibrium with Multiple Followers: The Mixed-Pure Case

The search problem of computing a Stackelberg (or leader-follower)equilibrium (also referred to as an optimal strategy to commit to) has been widely investigated in the scientific literature in, almost exclusively, the single-follower setting. Although the optimistic and pessimistic versions of the problem, i.e., those where the single follower breaks any ties among multiple equilibria either in favour or against the leader, are solved with different methodologies, both cases allow for efficient, polynomial-time algorithms based on linear programming. The situation is different with multiple followers, where results are only sporadic and depend strictly on the nature of the followers' game. In this paper, we investigate the setting of a normal-form game with a single leader and multiple followers who, after observing the leader's commitment, play a Nash equilibrium. When both leader and followers are allowed to play mixed strategies, the corresponding search problem, both in the optimistic and pessimistic versions, is known to be inapproximable in polynomial time to within any multiplicative polynomial factor unless $$\textsf {P}=\textsf {NP}$$. Exact algorithms are known only for the optimistic case. We focus on the case where the followers play pure strategies—a restriction that applies to a number of real-world scenarios and which, in principle, makes the problem easier—under the assumption of pessimism (the optimistic version of the problem can be straightforwardly solved in polynomial time). After casting this search problem (with followers playing pure strategies) as a pessimistic bilevel programming problem, we show that, with two followers, the problem is NP-hard and, with three or more followers, it cannot be approximated in polynomial time to within any multiplicative factor which is polynomial in the size of the normal-form game, nor, assuming utilities in [0, 1], to within any constant additive loss stricly smaller than 1 unless $$\textsf {P}=\textsf {NP}$$. This shows that, differently from what happens in the optimistic version, hardness and inapproximability in the pessimistic problem are not due to the adoption of mixed strategies. We then show that the problem admits, in the general case, a supremum but not a maximum, and we propose a single-level mathematical programming reformulation which asks for the maximization of a nonconcave quadratic function over an unbounded nonconvex feasible region defined by linear and quadratic constraints. Since, due to admitting a supremum but not a maximum, only a restricted version of this formulation can be solved to optimality with state-of-the-art methods, we propose an exact ad hoc algorithm (which we also embed within a branch-and-bound scheme) capable of computing the supremum of the problem and, for cases where there is no leader's strategy where such value is attained, also an $$\alpha$$-approximate strategy where $$\alpha > 0$$ is an arbitrary additive loss (at most as large as the supremum). We conclude the paper by evaluating the scalability of our algorithms via computational experiments on a well-established testbed of game instances

The Leverage Self-Delusion: Perceived Wealth and Cognitive Sophistication

By means of a laboratory experiment, we show that, contrary to standard consumer theory, financially equivalent balance sheet profiles may be perceived as non fungible in a controlled frictionless environment with no probabilistic attributes. A large majority of subjects indeed have a bias in the perception of wealth, such that balance sheet composition matters: for a given net worth with values of assets and debt that are financially certain and risk-free, a greater asset-debt ratio implies greater perceived wealth. The predominance of this bias is explained by low cognitive sophistication and great inattention. Moreover, biased subjects are less patient, less debt averse, more likely to increase spending out of unexpected gains and report greater propensities to consume. A standard optimal consumption choice model, enriched with a rational but inattentive agent à la Gabaix (2014, 2019), aligns our key experimental findings

The Leverage Self-Delusion: Perceived Wealth and Cognitive Sophistication

By means of a laboratory experiment, we show that, contrary to standard consumer theory, financially equivalent balance sheet profiles may be perceived as non fungible in a controlled frictionless environment with no probabilistic attributes. A large majority of subjects indeed have a bias in the perception of wealth, such that balance sheet composition matters: for a given net worth with values of assets and debt that are financially certain and risk-free, a greater asset-debt ratio implies greater perceived wealth. The predominance of this bias is explained by low cognitive sophistication and great inattention. Moreover, biased subjects are less patient, less debt averse, more likely to increase spending out of unexpected gains and report greater propensities to consume. A standard optimal consumption choice model, enriched with a rational but inattentive agent à la Gabaix (2014, 2019), aligns our key experimental findings