### 1 — 08:30 — Learning in Multi-Leader Single-Follower Stackelberg Games

In this work we want to tackle the problem of learning in multi leader single follower (MLSF) Stackelberg games. These games are less studied in the literature than other formulations, even though they are very relevant in a multitude of applications, such as security games, strategic bidding in markets or electricity production.

The game we study couples all the leaders between each others and with the follower, through their objective functions. The complexity arises from this coupling, which we encode using a reaction function. This function represents how the other players react to the actions proposed by the leaders. In a standard Stackelberg game, the reaction function is the optimal response; in this presentation, we consider different possibilities, aimed at hiding information between the players. In particular each leader will only observe the reaction function output, and it will have to learn its strategy through that.

The learning algorithm is based on the concepts of multi-agent reinforcement learning, and the structure of the algorithm relies on the game properties, since the goal is to have an interpretable and provably convergent algorithm. In particular we also focus on how much information needs to be shared between agents in order to prove the convergence.

### 2 — 09:00 — Nash Equilibrium of Code-Share Mechanism under Airline Frequency Competition Game with Second Order Cone Programming

In the modern aviation industry, strategically setting flight frequencies between airports is critical for increasing market shares, thus playing a pivotal role in the competitive dynamics among airlines. With the growth of the aviation market, the nature of commercial relationships between airlines has evolved from mere competition to the formation of alliances and collaborations. Among various cooperative strategies, the code-sharing agreement emerges as a particularly essential approach.

Code-sharing serves as a mechanism that allows airlines to sell seats on flights they do not operationally control, fostering participation through commercial agreements. This practice is frequently utilized in hub airports to improve connectivity between long-haul or high-frequency short-haul flights. Statistics indicate that more than half of U.S. airlines have embraced code-sharing strategies, with a notable prevalence on routes neighboring popular hub airports.

Despite the prevalence of codeshare in the airline industry, it is worth noting that there is no comprehensive quantitative model that addresses both codeshare strategy and flight frequency competition. Given its importance, this study attempts to develop a frequency competition model that incorporates code-sharing strategies in a game-theoretic context and uses second-order cones to characterize some of these constraints. To find the Nash Equilibrium for this problem, we apply the nature of the optimality conditions of second-order cone programming to build a KKT model containing primal, dual, and complementary slackness constraints. We hope to utilize the proposed model to perform calculations on available data to determine the optimal frequency for each segment and provide strategic information for decisions related to code sharing, suitable partners, and profit sharing among airlines.

### 3 — 09:30 — Computing Quasi-Proper Equilibria: From a Characterization to a Differentiable Path-Following Method

As a different paradigm from Myerson's properness for rationality on strategy perturbation, the concept of quasi-proper equilibrium was formulated by van Damme for finite extensive-form games with perfect recall. As for quasi-perfect equilibrium, the admissibility of quasi-proper equilibrium gains itself an advantage over Myerson's proper equilibrium. Nevertheless, the formulation contains insufficient information on how to find such an equilibrium. To address this issue, this paper develops an equivalent definition of quasi-proper equilibrium through the introduction of an auxiliary behavioral strategy profile into the formation of an epsilon-quasi-proper equilibrium. To show its computational effectiveness, we illustrate with simple examples how one can utilize the definition to analytically find all quasi-proper equilibria. The exploitation of the definition leads to a differentiable path-following method to compute quasi-proper equilibria. In the development of the method, we incorporate with an extra variable ranging between zero and one a logarithmic-barrier term into the payoff function of a player at each information set and constitute a logarithmic-barrier extensive-form game. At each information set in this barrier game, a player solves against any given behavioral strategies of other players and the payer herself at other information sets a strictly convex optimization problem. An application of the optimality conditions to the barrier game together with the equilibrium condition yields a polynomial equilibrium system. With this equilibrium system, we establish the existence of a smooth path that starts from an arbitrary totally mixed behavioral strategy profile and ends at a proper equilibrium as the extra variable vanishes. Numerical results further confirm the effectiveness and efficiency of the method.

### 4 — 10:00 — Tight Bounds for the Price of Fairness

A central decision maker (CDM), who seeks an efficient allocation of scarce resources among a finite number of players, often has to incorporate fairness criteria to avoid unfair outcomes. Indeed, the Price of Fairness (POF), a term coined in Bertsimas et al (2011), refers to the efficiency loss due to the incorporation of fairness criteria into the allocation method. Quantifying the POF would help the CDM strike an appropriate balance between efficiency and fairness. In this paper we improve upon existing results in the literature, by providing tight bounds for the POF for the proportional fairness criterion for any $n$, when the maximum achievable utilities of the players are equal or are not equal. Further, while Bertsimas et al (2011) have already derived a tight bound for the max-min fairness criterion for the case that all players have equal maximum achievable utilities, we also provide a tight bound in scenarios where these utilities are not equal. Finally, we investigate the sensitivity of our bounds and bounds of Bertsimas et al (2011) for the POF to the variability of the maximum achievable utilities, and identify certain patterns of the maximum achievable utilities for which the POF attains its maximum of 1-1/n.