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This study is focused on periodic Fisher markets where items with time-dependent and stochastic values are regularly replenished and buyers aim to maximize their utilities by spending budgets on these items. Traditional approaches of finding a market equilibrium in the single-period Fisher market rely on complete information about buyers' utility functions and budgets. However, it is impractical to consistently enforce buyers to disclose this private information in a periodic setting. We introduce a distributed auction algorithm, online proportional response, wherein buyers update bids solely based on the randomly fluctuating values of items in each period. The market then allocates items based on the bids provided by the buyers. Utilizing the known Shmyrev convex program that characterizes market equilibrium of a Fisher market, two performance metrics are proposed: the fairness regret is the cumulative difference in the objective value of a stochastic Shmyrev convex program between an online algorithm and an offline optimum, and the individual buyer's regret gauges the deviation in terms of utility for each buyer between the online algorithm and the offline optimum. Our algorithm attains a problem-dependent upper bound contingent on the number of items and buyers under stationary inputs in fairness regret. Additionally, we conduct analysis of regret under various non-stationary stochastic input models to demonstrate the algorithm's efficiency across diverse scenarios. The online proportional response algorithm addresses privacy concerns by allowing buyers to update bids without revealing sensitive information and ensures decentralized decision-making, fostering autonomy and potential improvements in buyer satisfaction. Furthermore, our algorithm is universally applicable to many worlds and shows the robust performance guarantees.

An election campaign among two or more parties can be viewed as a game of two or more players, each of which has its own candidates as the pure strategies. People, as voters, comprise supporters for each party, and a candidate brings utility for the supporters of each party. Each party nominates exactly one of its candidates to compete against the other party's. A candidate is assumed to win the election with greater or equal odds if it brings more utility for all the people. The payoff of each player is the expected utility that its supporters get. The game is egoistic if every candidate benefits its party's supporters more than any candidate from a competing party does. In this paper, we first prove that it is NP-complete to determine whether an election game in a succinct representation, which is called the general form, has a pure-strategy Nash equilibrium even if it is egoistic. Next, we propose a fixed-parameter tractable algorithm to compute a pure-strategy Nash equilibrium of an egoistic election game and show that a naive constant time algorithm leads to a (1+e)-approximate pure-strategy Nash equilibrium when the winning probability is computed by a softmax function. Finally, perhaps surprisingly, we show that the price of anarchy for egoistic election games is upper bounded by the number of parties. Our results suggest that an election becomes unpredictable in terms of stability and efficiency when more than two parties are involved, and, to some extent, also provides supporting arguments for why the two-party system is prevalent in democratic countries., Comment: A previous version appeared at the 6th Games, Agents, and Incentives Workshop (GAIW-24). The current version has been submitted to SAGT 2024

In this paper, we propose a simple and intuitive model to investigate the efficiency of the two-party election system, especially regarding the nomination process. Each of the two parties has its own candidates, and each of them brings utilities for the people including the supporters and non-supporters. In an election, each party nominates exactly one of its candidates to compete against the other party's. The candidate wins the election with higher odds if he or she brings more utility for all the people. We model such competition as a "two-party election game" such that each party is a player with two or more pure strategies corresponding to its potential candidates, and the payoff of each party is a mixed utility from a selected pair of competing candidates. By looking into the three models, namely, the linear link, Bradley-Terry, and the softmax models, which differ in how to formulate a candidate's winning odds against the competing candidate, we show that the two-party election game may neither have any pure Nash equilibrium nor a bounded price of anarchy. Nevertheless, by considering the conventional "egoism", which states that any candidate benefits his/her party's supporters more than any candidate from the competing party does, we prove that the two-party election game in both the linear link model and the softmax model always has pure Nash equilibria, and furthermore, the price of anarchy is constantly bounded.

From a perspective of designing or engineering for opinion formation games in social networks, the "opinion maximization (or minimization)" problem has been studied mainly for designing subset selecting algorithms. We define a two-player zero-sum Stackelberg game of competitive opinion optimization by letting the player under study as the leader minimize the sum of expressed opinions by doing so-called "internal opinion design", knowing that the other adversarial player as the follower is to maximize the same objective by also conducting her own internal opinion design. We furthermore consider multiagent learning, specifically using the Optimistic Gradient Descent Ascent, and analyze its convergence to equilibria in the simultaneous version of competitive opinion optimization.