Browsing by Subject "Nash equilibrium"

Now showing 1 - 20 of 30

Open Access
Analysis of a decentralized supply chain under partial cooperation
(2005) Güllü, R.; Van Houtum G. J.; Sargut F. Z.; Erkip, N.
In this article, we analyze a decentralized supply chain consisting of a supplier and two independent retailers. In each order cycle, retailers place their orders at the supplier to minimize inventory-related expected costs at the end of their respective response times. There are two types of lead times involved. At the end of the supplier lead time, retailers are given an opportunity to readjust their initial orders (without changing the total order size), so that both retailers can improve their expected costs at the end of respective retailer lead times (the time it takes for items to be shipped from the supplier to the retailers). Because of the possibility of cooperation at the end of supplier lead time, each retailer will consider the other's order-up-to level in making the ordering decision. Under mild conditions, we prove the existence of a unique Nash equilibrium for the retailer order-up-to levels, and show that they can be obtained by solving a set of newsboy-like equations. We also present computational analysis that provides valuable managerial insight for design and operation of decentralized systems under the possibility of partial cooperation.
Open Access
Binary signaling under subjective priors and costs as a game
(Institute of Electrical and Electronics Engineers Inc., 2019) Sarıtaş, S.; Gezici, Sinan; Yüksel, S.; Teel, A. R.; Egerstedt, M.
Many decentralized and networked control problems involve decision makers which have either misaligned criteria or subjective priors. In the context of such a setup, in this paper we consider binary signaling problems in which the decision makers (the transmitter and the receiver) have subjective priors and/or misaligned objective functions. Depending on the commitment nature of the transmitter to his policies, we formulate the binary signaling problem as a Bayesian game under either Nash or Stackelberg equilibrium concepts and establish equilibrium solutions and their properties. In addition, the effects of subjective priors and costs on Nash and Stackelberg equilibria are analyzed. It is shown that there can be informative or non-informative equilibria in the binary signaling game under the Stackelberg assumption, but there always exists an equilibrium. However, apart from the informative and non-informative equilibria cases, under certain conditions, there does not exist a Nash equilibrium when the receiver is restricted to use deterministic policies. For the corresponding team setup, however, an equilibrium typically always exists and is always informative. Furthermore, we investigate the effects of small perturbations in priors and costs on equilibrium values around the team setup (with identical costs and priors), and show that the Stackelberg equilibrium behavior is not robust to small perturbations whereas the Nash equilibrium is.
Open Access
Consensus as a Nash equilibrium of a dynamic game
(IEEE, 2016) Niazi, Muhammad Umar B.; Özgüler, Arif Bülent; Yıldız, Aykut
Consensus formation in a social network is modeled by a dynamic game of a prescribed duration played by members of the network. Each member independently minimizes a cost function that represents his/her motive. An integral cost function penalizes a member's differences of opinion from the others as well as from his/her own initial opinion, weighted by influence and stubbornness parameters. Each member uses its rate of change of opinion as a control input. This defines a dynamic non-cooperative game that turns out to have a unique Nash equilibrium. Analytic explicit expressions are derived for the opinion trajectory of each member for two representative cases obtained by suitable assumptions on the graph topology of the network. These trajectories are then examined under different assumptions on the relative sizes of the influence and stubbornness parameters that appear in the cost functions.
Open Access
Foraging motion of swarms as nash equilibria of differential games
(2016-09) Yıldız, Aykut
The question of whether foraging swarms can form as a result of a non-cooperative game played by individuals is shown here to have an affirmative answer. A dynamic (or, differential) game played by N agents in one-dimensional motion is introduced and models, for instance, a foraging ant colony. Each agent controls its velocity to minimize its total work done in a finite time interval. The agents in the game start from a set of initial positions and migrate towards a target foraging location. Such swarm games are shown to have unique Nash equilibra under two different foraging location specifications and both equilibria display many features of a foraging swarm behavior observed in biological swarms. Explicit expressions are derived for pairwise distances between individuals of the swarm, swarm size, and swarm center location during foraging. Foraging swarms in one-dimensional motion with four different information structures are studied. These are complete and partial information structures, hierarchical leadership and one leader structures. In the complete information structure, every agent observes its distance to every other agent and makes use of this information in its effort optimization. In partial information structure, the agents know the position of only its neighboring agents. In the hierarchical leadership structure, the agents look only forward and measures its distance to the agents ahead. In single leader structure, the agents know the position of only leader. In all cases, a Nash equilibrium exists under some realistic assumptions on the sizes of the weighing parameters in the cost functions. The consequences of having a “passive” leader in a swarm are also investigated. We model foraging swarms with leader and followers again as non-cooperative, multi-agent differential games. We consider two types of leadership structures, namely, hierarchical leadership and a single leader structure. In both games, the type of leadership is assumed to be passive since a leader is singled out only due to its rank in the initial queue. We identify the realistic assumptions under which a unique Nash equilibrium exists in each game and derive the properties of the Nash equilibriums in detail. It is shown that having a passive leader economizes in the total information exchange at the expense of aggregation stability in a swarm.
Open Access
Foraging motion of swarms with leaders as Nash equilibria
(Elsevier Ltd, 2016) Yıldız, A.; Özgüler, A. B.
The consequences of having a leader in a swarm are investigated using differential game theory. We model foraging swarms with leader and followers as a non-cooperative, multi-agent differential game. The agents in the game start from a set of initial positions and migrate towards a target. The agents are assumed to have no desire, partial desire or full desire to reach the target. We consider two types of leadership structures, namely hierarchical leadership and a single leader. In both games, the type of leadership is assumed to be passive. We identify the realistic assumptions under which a unique Nash equilibrium exists in each game and derive the properties of the Nash solutions in detail. It is shown that having a passive leader economizes in the total information exchange at the expense of aggregation stability in a swarm. It turns out that, the leader is able to organize the non-identical followers into harmony under missing information.
Open Access
Foraging swarms as Nash equilibria of dynamic games
(IEEE, 2014) Özgüler, A. B.; Yildiz, A.
The question of whether foraging swarms can form as a result of a noncooperative game played by individuals is shown here to have an affirmative answer. A dynamic game played by N agents in 1-D motion is introduced and models, for instance, a foraging ant colony. Each agent controls its velocity to minimize its total work done in a finite time interval. The game is shown to have a unique Nash equilibrium under two different foraging location specifications, and both equilibria display many features of a foraging swarm behavior observed in biological swarms. Explicit expressions are derived for pairwise distances between individuals of the swarm, swarm size, and swarm center location during foraging.
Open Access
A game theoretic approach to channel switching in the presence of jamming
(IEEE, 2021-10-14) Bozkurt, Berk; Sezer, A. D.; Gezici, Sinan; Girici, T.
In this letter, a channel switching problem is investigated in the presence of jamming based on a game theoretic approach. First, a convex formulation of the optimal channel switching problem is proposed for a given jamming strategy. Then, considering a fixed channel switching strategy, an explicit solution of the optimal jammer power allocation problem is obtained. Consequently, a game theoretic formulation is proposed and the existence of a pure-strategy Nash equilibrium is shown for the proposed channel switching game between the transmitter and the jammer.
Open Access
Hypothesis testing under subjective priors and costs as a signaling game
(IEEE, 2019) Sarıtaş, S.; Gezici, Sinan; Yüksel, S.
Many communication, sensor network, and networked control problems involve agents (decision makers) which have either misaligned objective functions or subjective probabilistic models. In the context of such setups, we consider binary signaling problems in which the decision makers (the transmitter and the receiver) have subjective priors and/or misaligned objective functions. Depending on the commitment nature of the transmitter to his policies, we formulate the binary signaling problem as a Bayesian game under either Nash or Stackelberg equilibrium concepts and establish equilibrium solutions and their properties. We show that there can be informative or non-informative equilibria in the binary signaling game under the Stackelberg and Nash assumptions, and derive the conditions under which an informative equilibrium exists for the Stackelberg and Nash setups. For the corresponding team setup, however, an equilibrium typically always exists and is always informative. Furthermore, we investigate the effects of small perturbations in priors and costs on equilibrium values around the team setup (with identical costs and priors), and show that the Stackelberg equilibrium behavior is not robust to small perturbations whereas the Nash equilibrium is.
Open Access
Implementation via code of rights
(2008) Yıldız, Kemal
Implementation of a social choice rule can be thought of as a design of power (re)distribution in the society whose ”equilibrium outcomes” coincide with the alternatives chosen by the social choice rule at any preference profile of the society. In this paper, we introduce a new societal framework for implementation which takes the power distribution in the society, represented by a code of rights, as its point of departure. We examine and identify how implementation via code of rights (referred to as gamma implementation) is related to classical Nash implementation via mechanism. We characterize gamma implementability when the state space on which the rights structure is to be specified consists of the alternatives from which a social choice is to be made. We show that any social choice rule is gamma implementable if it satisfies pivotal oligarchic monotonicity condition that we introduce. Moreover, pivotal oligarchic monotonicity condition combined with Pareto optimality is sufficient for a non-empty valued social choice rule to be gamma implementable. Finally we revisit liberal’s paradox of A.K. Sen, which turns out to fit very well into the gamma implementation framework.
Open Access
Implementation via rights structures
(Academic Press, 2018) Koray, Semih; Yildiz, K.
Implementation of socially desirable alternatives can be thought of as a way to design power distribution in a society such that the equilibrium outcomes coincide with the alternatives chosen at each preference profile. In this paper, we introduce a new institutional framework for implementation, which takes power distribution in a society as its point of departure. We use the notion of a rights structure, introduced by Sertel (2001), to formalize the power distribution in a society. We formulate and characterize implementability via rights structures under different specifications, which require having well-defined convergence dynamics and being consistent with farsighted behavior. We identify how implementation via rights structures is related to implementation via mechanisms. In the presence of at least three agents, we find the class of rights structures, implementability via which is equivalent to Nash and strong Nash implementability. We also introduce a strategic counterpart of implementation via rights structures in terms of deviation-constrained mechanisms.
Open Access
Inventory control under substitutable demand: A stochastic game application
(John Wiley & Sons, 2002) Avsşr, Z. M.; Baykal-Gürsoy, M.
Substitutable product inventory problem is analyzed using the concepts of stochastic game theory. It is assumed that there are two substitutable products that are sold by different retailers and the demand for each product is random. Game theoretic nature of this problem is the result of substitution between products. Since retailers compete for the substitutable demand, ordering decision of each retailer depends on the ordering decision of the other retailer. Under the discounted payoff criterion, this problem is formulated as a two‐person nonzero‐sum stochastic game. In the case of linear ordering cost, it is shown that there exists a Nash equilibrium characterized by a pair of stationary base stock strategies for the infinite horizon problem. This is the unique Nash equilibrium within the class of stationary base stock strategies.
Open Access
Learning the optimum as a Nash equilibrium
(Elsevier BV, 2000) Özyıldırım, S.; Alemdar, N. M.
This paper shows the computational benefits of a game theoretic approach to optimization of high dimensional control problems. A dynamic noncooperative game framework is adopted to partition the control space and to search the optimum as the equilibrium of a k-person dynamic game played by k-parallel genetic algorithms. When there are multiple inputs, we delegate control authority over a set of control variables exclusively to one player so that k artificially intelligent players explore and communicate to learn the global optimum as the Nash equilibrium. In the case of a single input, each player's decision authority becomes active on exclusive sets of dates so that k GAs construct the optimal control trajectory as the equilibrium of evolving best-to-date responses. Sample problems are provided to demonstrate the gains in computational speed and accuracy. © 2000 Elsevier Science B.V.
Open Access
Market entry decisions: effects of absolute and relative confidence
(Hogrefe Publishing, 2008) Bolger, F.; Pulford, B. D.; Colman, A. M.
In a market entry game, the number of entrants usually approaches game-theoretic equilibrium quickly, but in real-world markets business start-ups typically exceed market capacity, resulting in chronically high failure rates and suboptimal industry profits. Excessive entry has been attributed to overconfidence arising when expected payoffs depend partly on skill. In an experimental test of this hypothesis, 96 participants played 24 rounds of a market entry game, with expected payoffs dependent partly on skill on half the rounds, after their confidence was manipulated and measured. The results provide direct support for the hypothesis that high levels of confidence are largely responsible for excessive entry, and they suggest that absolute confidence, independent of interpersonal comparison, rather than confidence about one's abilities relative to others, drives excessive entry decisions when skill is involved.
Open Access
Nash equilibria for exchangeable team against team games and their mean field limit
(IEEE, 2023-07-03) Sanjari, S.; Saldı, Naci; Yüksel, S.
We study stochastic mean-field games among finite number of teams each with large finite as well as infinite numbers of decision makers (DMs). We establish the existence of a Nash equilibrium (NE) and show that a NE exhibits exchangeability in the finite DM regime and symmetry in the infinite one. We establish the existence of a randomized NE that is exchangeable (not necessarily symmetric) among DMs within each team for a general class of exchangeable stochastic games. As the number of DMs within each team drives to infinity that is for the mean-field games among teams), using a de Finetti representation theorem, we establish the existence of a randomized NE that is symmetric (i.e., identical) among DMs within each team and also independently randomized. Finally, we establish that a NE for a class of mean-field games among teams (which is symmetric) constitutes an approximate NE for the corresponding pre-limit game among teams with mean-field interaction and large but finite number of DMs.
Open Access
Nash equilibria for exchangeable team-against-team games, their mean-field limit, and the role of common randomness
(Society for Industrial and Applied Mathematics, 2024-05-16) Sanjari, Sina; Saldı, Naci; Yüksel, Serdar
We study stochastic exchangeable games among a finite number of teams consisting of a large but finite number of decision makers as well as their mean-field limit with infinite number of decision makers in each team. For this class of games within static and dynamic settings, we introduce sets of randomized policies under various decentralized information structures with pri- vately independent or common randomness for decision makers within each team. (i) For a general class of exchangeable stochastic games with a finite number of decision makers, we first establish the existence of a Nash equilibrium under randomized policies (with common randomness) within each team that are exchangeable (but not necessarily symmetric, i.e., identical) among decision makers within each team. (ii) As the number of decision makers within each team goes to infinity (that is, for the mean-field limit game among teams), we show that a Nash equilibrium exists under randomized policies within each team that are independently randomized and symmetric among decision makers within each team (that is, there is no common randomness). (iii) Finally, we establish that a Nash equilibrium for a class of mean-field games among teams under independently randomized symmetric policies constitutes an approximate Nash equilibrium for the corresponding prelimit (exchangeable) game among teams with finite but large numbers of decision makers. (iv) We thus establish a rigor- ous connection between agent-based-modeling and team-against-team games, via the representative agents defining the game played in equilibrium, and we furthermore show that common randomness is not necessary for large team-against-team games, unlike the case with small-sized ones.
Open Access
Nash equilibria in claim based estate division problems
(2014) İnel, Abdulkadir
Estate division game is an allocation of an estate between players based on a rule. In this thesis, we consider estate division games and study the necessary and sufficient conditions for division rules under which Nash equilibria induce equal division. Ashlagi, Karag¨ozoˇglu, Klaus (2012) introduce classes of properties for division rules and show that they are sufficient for all Nash equilibria to induce equal division. In this study, we propose a different property, namely conditional full compensation, and prove that it is also a sufficient condition for division rules in order for all Nash equilibria outcomes under these rules to be equal division. We, then, show that under any rule satisfying claims boundedness and conditional equal division lower bound, equal division is a Nash equilibrium outcome. Finally, we prove that letting at least one player get more than the difference between the whole estate and the sum of other players’ claims is a necessary condition for all Nash equilibria to induce equal division.
Open Access
A novel characterization of nash-implementable social choice rules via neutrality
(2020-07) Mammadov, Aghaheybat
In this thesis, we study Nash implementability of social choice rules in relation with the neutrality notion. Several works in the literature provide a characterization of Nash-implementable social choice rules. However, they do not explicitly show the degree of neutrality property in the Nash equilibrium concept which is also existing in Nash-implementable rules. In this study, we deﬁne a weak version of the neutrality condition critical neutrality which is associated with the critical domain of a social choice rule. The critical neutrality notion when conjoined with Maskin monotonicity turns out to be equivalent to Nash implementability. Moreover, we propose an algorithm to obtain a maximal domain of preference proﬁles on which a speciﬁed social choice rule is Nash-implementable, by utilizing critical neutrality as a tool. The main result of the thesis is in support of the view that implementability on the full domain of preference proﬁles is highly related with implementability on the critical domain in Nash equilibrium and possibly, in other solution concepts.
Open Access
Opinion dynamics of stubborn agents under the presence of a troll as differential game
(Scientific and Technical Research Council of Turkey - TUBITAK,Turkiye Bilimsel ve Teknik Arastirma Kurumu, 2021-11-30) Yıldız, A.; Özgüler, Arif Bülent
The question of whether opinions of stubborn agents result in Nash equilibrium under the presence of troll is investigated in this study. The opinion dynamics is modelled as a differential game played by n agents during a finite time horizon. Two types of agents, ordinary agents and troll, are considered in this game. Troll is treated as a malicious stubborn content maker who disagrees with every other agent. On the other hand, ordinary agents maintain cooperative communication with other ordinary agents and they disagree with the troll. Under this scenario, explicit expressions of opinion trajectories are obtained by applying Pontryagin’s principle on the cost function. This approach provides insight into the social networks that comprise a troll in addition to ordinary agents
Open Access
Organizational refinements of Nash equilibrium
(Springer, 2021-10) Kamihigashi, T.; Keskin, K.; Sağlam, Çağrı
Strong Nash equilibrium (see Aumann, 1959) and coalition-proof Nash equilibrium (see Bernheim et al., 1987) rely on the idea that players are allowed to form coalitions and make joint deviations. Both of these notions consider cases in which any coalition can be formed. Accordingly, there may arise “conflicts of interest” that prevent a player from choosing an action that simultaneously meets the requirements of two coalitions to which he or she belongs. Here, we address this observation by studying an organizational framework such that the coalitional structure is (i) motivated by real-life examples where players cannot form some coalitions and (ii) formulated in such a way that no conflicts of interest remain. We define an organization as an ordered collection of partitions of the player set such that any partition is coarser than the partitions that precede it. For any given organization, we introduce the notion of organizational Nash equilibrium. We analyze the existence of equilibrium in a subclass of games with strategic complementarities and illustrate how the proposed notion refines the set of Nash equilibria in some examples of normal form games
Open Access
Partially informed agents can form a swarm in a nash equilibrium
(Institute of Electrical and Electronics Engineers, 2015) Yildiz, A.; Ozguler, A. B.
Foraging swarms in one-dimensional motion with incomplete position information are studied in the context of a noncooperative differential game. In this game, the swarming individuals act with partial information as it is assumed that each agent knows the positions of only the adjacent ones. It is shown that a Nash equilibrium solution that exhibits many features of a foraging swarm such as movement coordination, self-organization, stability, and formation control exists. © 1963-2012 IEEE.