Linear algebraic structure of zero-determinant strategies in repeated games

Zero-determinant (ZD) strategies, a recently found novel class of strategies in repeated games, has attracted much attention in evolutionary game theory. A ZD strategy unilaterally enforces a linear relation between average payoffs of players. Although existence and evolutional stability of ZD strategies have been studied in simple games, their mathematical properties have not been well-known yet. For example, what happens when more than one players employ ZD strategies have not been clarified. In this paper, we provide a general framework for investigating situations where more than one players employ ZD strategies in terms of linear algebra. First, we theoretically prove that a set of linear relations of average payoffs enforced by ZD strategies always has solutions, which implies that incompatible linear relations are impossible. Second, we prove that linear payoff relations are independent of each other under some conditions. These results hold for general games with public monitoring including perfect-monitoring games. Furthermore, we provide a simple example of a two-player game in which one player can simultaneously enforce two linear relations, that is, simultaneously control her and her opponent's average payoffs. All of these results elucidate general mathematical properties of ZD strategies.Comment: 19 pages, 2 figure

Comparative Study of Cyanobacterial and E. coli RNA Polymerases: Misincorporation, Abortive Transcription, and Dependence on Divalent Cations

If Mg2+ ion is replaced by Mn2+ ion, RNA polymerase tends to misincorporate noncognate nucleotide, which is thought to be one of the reasons for the toxicity of Mn2+ ion. Therefore, most cells have Mn2+ ion at low intracellular concentrations, but cyanobacteria need the ion at a millimolar concentration to maintain photosynthetic machinery. To analyse the mechanism for resistance against the abundant Mn2+ ion, we compared the properties of cyanobacterial and E. coli RNA polymerases. The cyanobacterial enzyme showed a lower level of abortive transcription and less misincorporation than the E. coli enzyme. Moreover, the cyanobacterial enzyme showed a slower rate of the whole elongation by an order of magnitude, paused more frequently, and cleaved its transcript faster in the absence of NTPs. In conclusion, cyanobacterial RNA polymerase maintains the fidelity of transcription against Mn2+ ion by deliberate incorporation of a nucleotide at the cost of the elongation rate. The cyanobacterial and the E. coli enzymes showed different sensitivities to Mg2+ ion, and the physiological role of the difference is also discussed

Competitiveness and subsidy or tax policy for new technology adoption in duopoly

We consider a problem of subsidy or tax policy for new technology adoption by duopolistic firms. The technology is developed in and transferred by a foreign country to the domestic country. It is free but each firm must expend some fixed set-up cost for education of its staff to adopt and use it. Assuming that each firm maximizes the weighted average of absolute and relative profits, we examine the relationship between competitiveness and subsidy or tax policies for technology adoption, and show that when firm behavior is not competitive (the weight on the relative profit is small), the optimal policy of the government may be taxation; when firm behavior is competitive (the weight on the relative profit is large), the optimal policy is subsidization or inaction and not taxation. However, if firm behavior is extremely competitive (close to perfect competition), taxation case re-emerges

Subsidy for New Technology Adoption in Duopoly with Differentiated Goods under Absolute and Relative Profit Maximization

Abstract. We present an analysis about subsidy policy for adoption of new technology in duopoly with differentiated goods under absolute and relative profit maximization. Technology itself is free, however, firms must expend fixed set-up costs to adopt new technology. There are various cases about optimal policies depending on the level of the set-up cost and whether the goods of the firms are substitutes or complements. In particular, under relative profit maximization there is a case such that the social welfare is maximized when one firm adopts new technology, but no firm adopts new technology without subsidy. Then, the government should give a subsidy to only one firm. It is a discriminatory policy. The government gives a chance to receive a subsidy to only one firm.Keywords. Subsidy for new technology adoption, Absolute and relative profit maximization, Duopoly.JEL. D43, L13

On a strictly convex and strictly sub-additive cost function with positive fixed cost

We investigate the existence of a strictly convex and strictly sub-additive cost function with positive fixed cost. If there is a positive fixed cost, any cost function can not be super-additive, and concavity (including linearity) of cost function implies strict sub-additivity. Then, does there exist a strictly convex and strictly sub-additive cost function? We will present such a cost function. It is close to a linear function although it is strictly convex

Royalty and license fee under oligopoly with or without entry of innovator: Two-step auction

When an outside innovating firm has a cost-reducing technology, it can sell licenses of its technology to incumbent firms, or enter the market and at the same time sell licenses, or enter the market without license. We examine the definitions of license fees in such situations under oligopoly with three firms, one outside innovating firm and two incumbent firms, considering threat by entry of the innovating firm using a two-step auction. Also we suppose that the innovating firm sells its licenses using a combination of royalty per output and a fixed license fee