Around rationality of cycles

We prove certain results comparing rationality of algebraic cycles over the function field of a quadric and over the base field. These results have already been obtained by Alexander Vishik in the case of characteristic 0, which allowed him to work with algebraic cobordism theory. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2.     

Brauer characters and rationality

A finite group $G$ has no non-trivial rational-valued irreducible $p$ -Brauer characters if and only if $G$ has no non-trivial rational elements of order not divisible by $p$ .     

Bounded rationality in multiobjective games

In this paper, we will establish the connections between bounded rationality and multiobjective games. We obtain some new results for robustness to ?$?$-equilibria and structural stability of multiobjective games and generalized multiobjective games.     

The castelnuovo criterion of rationality

A new proof is provided for the rationality criterion for algebraic surfaces over an arbitrary base field, usingl-adic cohomologies.Translated from Matematicheskie Zametki, Vol.11, No. 1, pp. 27–32, January, 1972.     

Extended Chebyshev spaces in rationality

On a closed bounded interval, a given Extended Chebyshev space can be defined by means of generalised derivatives associated with systems of weight functions. Only recently we could identify all such systems, describing an iterative process to build them. In the present work, we interpret the first step of this process as the construction of rational spaces based on Extended Chebyshev spaces. This construction establishes an interesting symmetry between all Extended Chebyshev spaces “good for design” (i.e., all those which contain constants and which possess blossoms) and the rational spaces based on them (Extended Chebyshev spaces in rationality). In particular, this symmetry results in a very simple relation between the corresponding blossoms. A special case is obtained when considering polynomial spaces as examples of Extended Chebyshev spaces. The classical rational spaces then appear as examples of Extended Chebyshev spaces good for design, that is, possessing blossoms. This offers interesting new insights on the famous so-called rational Bézier curves.     

Multicriteria procedural rationality on SISDEMA

This paper considers the application, under the multicriteria procedural rationality paradigm (MPRP), of two environmental selection tools included in the priorization module (PRIOR) of the SISDEMA decisional system: preference structures and stability intervals. These tools, apart from being used in analysing the robustness of the model, allow us to obtain some critical points of the environmental decision-making process (DMP) which will be employed in the negotiation and consensus building process between the actors involved in the resolution of the problem.     

Frobenius groups and retract rationality

Let k$k$ be any field, G$G$ be a finite group acting on the rational function field k(_xg:g∈G)$k(x_g:g\in G)$ by h⋅_xg=_xhg$h\cdot x_g=x_{hg}$ for any h,g∈G$h,g\in G$. Define k(G)=k(_xg:g∈G)^G$k\left(G\right)=k{(x_g:g\in G)}^G$. Noether’s problem asks whether k(G)$k\left(G\right)$ is rational (= purely transcendental) over k$k$. A weaker notion, retract rationality introduced by Saltman, is also very useful for the study of Noether’s problem. We prove that, if G$G$ is a Frobenius group with abelian Frobenius kernel, then k(G)$k\left(G\right)$ is retract k$k$-rational for any field k$k$ satisfying some mild conditions. As an application, we show that, for any algebraic number field k$k$, for any Frobenius group G$G$ with Frobenius complement isomorphic to S_L2(_F5)$S{L}_2\left({\mathbb{F}}_5\right)$, there is a Galois extension field K$K$ over k$k$ whose Galois group is isomorphic to G$G$, i.e. the inverse Galois problem is valid for the pair (G,k)$(G,k)$. The same result is true for any non-solvable Frobenius group if k(_ζ8)$k\left({\zeta }_8\right)$ is a cyclic extension of k$k$.     

Bounded rationality and correlated equilibria

We study an interactive framework that explicitly allows for nonrational behavior. We do not place any restrictions on how players’ behavior deviates from rationality, but rather, on players’ higher-order beliefs about the frequency of such deviations. We assume that there exists a probability p such that all players believe, with at least probability p, that their opponents play rationally. This, together with the assumption of a common prior, leads to what we call the set of p-rational outcomes, which we define and characterize for arbitrary probability p. We then show that this set varies continuously in p and converges to the set of correlated equilibria as p approaches 1, thus establishing robustness of the correlated equilibrium concept to relaxing rationality and common knowledge of rationality. The p-rational outcomes are easy to compute, also for games of incomplete information. Importantly, they can be applied to observed frequencies of play for arbitrary normal-form games to derive a measure of rationality \(\overline{p}\) that bounds from below the probability with which any given player chooses actions consistent with payoff maximization and common knowledge of payoff maximization.     

Stable rationality of certain PGLn-quotients

Summary Stable rationality of the field of matrix invariants M _n ×M _n ) ^PGL _n is proved forn=5 andn=7. In combination with existing results this shows that (V) ^PGL _n is stably rational wheneverV is an almost free representation ofPGL _n andn divides 420=2²·3·5·7.Oblatum 1-VII-1989 & 15-VI-1990 & 19-VII-1990Partially supported by the DFGResearch associate of the NFWO     

Speculative dynamics with bounded rationality learning

We study a linear model for a future market characterized by the presence of different classes of traders. In the market there are three classes of traders: rational traders, feedback traders and fundamentalist traders. Each class of traders is described by a trading strategy and by an information set about the fundamental. The analysis is developed under bounded rationality, rational traders forming expectations do not know the “true” model but believe in a misspecified model. The convergence of the learning activity to the Rational Expectations Equilibria of the model is analyzed. Two different learning mechanisms are studied: the Ordinary Least Squares algorithm and the Least Mean Squares algorithm. The main goal of the study is to analyze how the presence of different classes of traders in the market affects the robustness of the Rational Expectations Equilibria of the model with respect to bounded rationality learning. Moreover we verify the claim that bubbles and erratic behavior in the stock price dynamics may arise because of learning non-convergence to Rational Expectations Equilibria. The results show that if the Ordinary Least Squares algorithm is used by the agents to update beliefs, convergence to one of the two Rational Expectations Equilibria of the model is ensured only if there are positive feedback traders in the market. On the contrary, the Least Mean Squares algorithm guarantees convergence to the Rational Expectations Equilibria given an appropriate initial belief.     

Nonlinear delay monopoly with bounded rationality

The purpose of this paper is to study the dynamics of a monopolistic firm in a continuous-time framework. The firm is assumed to be boundedly rational and to experience time delays in obtaining and implementing information on output. The dynamic adjustment process is based on the gradient of the expected profit. The paper is divided into three parts: we examine delay effects on dynamics caused by one-time delay and two-time delays in the first two parts. Global dynamics and analytical results on local dynamics are numerically confirmed in the third part. Four main results are demonstrated. First, the stability switch from stability to instability occurs only once in the case of a single delay. Second, the alternation of stability and instability can continue if two time delays are involved. Third, the occurence of Hopf bifurcation is analytically shown if stability is lost. Finally, in a bifurcation process, there are a period-doubling cascade to chaos and a period-halving cascade to the equilibrium point in the case of two time delays if the difference between the two delays is large.     

关于非凸的有限理性的稳定性

关于有限理性方面的文献, 大多数都是在满足凸性条件下研究有限理性的相关性质, 在一定程度上限制了其应用范围. 应用Ekeland变分原理, 减弱了有限理性模型的假设条件, 考虑在不满足凸性条件下的有限理性模型的稳定性问题. 具体给出了非凸的Ky Fan点问题解的稳定性, 非凸非紧的Ky Fan点问题解的稳定性, 非凸向量值函数Ky Fan点解的稳定性和非凸非紧向量值函数Ky Fan点解的稳定性. 作为应用, 还给出了非凸的n人非合作博弈有限理性模型解的稳定性和非凸的多目标博弈有限理性模型解的稳定性.     

Repeated congestion games with bounded rationality

We consider a repeated congestion game with imperfect monitoring. At each stage, each player chooses to use some facilities and pays a cost that increases with the congestion. Two versions of the model are examined: a public monitoring setting where agents observe the cost of each available facility, and a private monitoring one where players observe only the cost of the facilities they use. A partial folk theorem holds: a Pareto-optimal outcome may result from selfish behavior and be sustained by a belief-free equilibrium of the repeated game. We prove this result assuming that players use strategies of bounded complexity and we estimate the strategic complexity needed to achieve efficiency. It is shown that, under some conditions on the number of players and the structure of the game, this complexity is very small even under private monitoring. The case of network routing games is examined in detail.