## Contributed Session Fri.1.MA 005

#### Friday, 10:30 - 12:00 h, Room: MA 005

**Cluster 8: Game theory** [...]

### New models and solution concepts II

**Chair: Yin Chen**

**Friday, 10:30 - 10:55 h, Room: MA 005, Talk 1**

**Ming Hu**

Existence, uniqueness, and computation of robust Nash equilibrium in a class of multi-leader-follower games

**Coauthor: Masao Fukushima**

**Abstract:**

The multi-leader-follower game can be looked on as a generalization of the Nash equilibrium problem (NEP), which contains several leaders and followers. On the other hand, in such real-world problems, uncertainty normally exists and sometimes cannot simply be ignored. To handle mathematical programming problems with uncertainty, the robust optimization (RO) technique assumes that the uncertain data belong to some sets, and the objective function is minimized with respect to the worst case scenario. In this paper, we focus on a class of multi-leader-follower games under uncertainty with some special structure. We particularly assume that the follower's problem only contains equality constraints. By means of the RO technique, we first formulate the game as the robust Nash equilibrium problem, and then the generalized variational inequality (GVI) problem. We then establish some results on the existence and uniqueness of a robust leader-follower Nash equilibrium. We also apply the forward-backward splitting method to solve the GVI formulation of the problem and present some numerical examples to illustrate the behavior of robust Nash equilibria.

**Friday, 11:00 - 11:25 h, Room: MA 005, Talk 2**

**Silvia Schwarze**

Equilibria in generalized Nash games with applications to games on polyhedra

**Coauthors: Justo Puerto, Anita Schöbel**

**Abstract:**

In generalized Nash equilibrium (GNE) games, a player’s strategy set depends on the strategy decisions of the competitors. In particular, we consider games on polyhedra, where the strategy space is represented by a polyhedron. We investigate best-reply improvement paths in games on polyhedra and prove the finiteness of such paths for special cases. In particular, under the assumption of a potential game, we prove existence of equilibria for strictly convex payoffs.

In addition, we study multiobjective characterizations of equilibria for general (nonpolyhedral) GNE games for the case of monotone payoffs. We show that nondominated points in the decision space are equilibria. Moreover, the equivalence of the sets of equilibria and nondominated points is ensured by establishing an additional restriction on the feasible strategy sets, leading to the new definition of comprehensive sets. As a result, multiobjective optimization techniques carry over to GNE games with monotone payoffs. In addition, we discuss the relation to efficient solutions in the payoff space. Applying those results to games on polyhedra, we yield linear programming formulations for finding equilibria.

**Friday, 11:30 - 11:55 h, Room: MA 005, Talk 3**

**Yin Chen**

Computing perfect equilibria of finite *n*-person games in normal form with an interior-point path-following method

**Coauthor: Chuangyin Dang**

**Abstract:**

For any given sufficiently small positive number *ε*, we show that the imposition of a minimum probability *ε* on each pure strategy in a Nash equilibrium leads to an *ε*-perfect equilibrium of a finite *n*-person game in normal form. To compute such an *ε*-perfect equilibrium, we introduce a homotopy parameter to combine a weighted logarithmic barrier term with each player's payoff function and devise a new game. When the parameter varies from 0 to 1, the new game deforms from a trivial game to the original game. With the help of a perturbation term, it is proved that there exists a smooth interior-point path that starts from an unique Nash equilibrium of the trivial game and leads to an *ε*-perfect equilibrium of the original game at its limit. A predictor-corrector method is presented to follow the path. As an application of this result, we derive a scheme to compute a perfect equilibrium. Numerical results show that the scheme is effective and efficient.