A Generalized Diagonal Wythoff Nim

In this paper we study a family of 2-pile Take Away games, that we denote by Generalized Diagonal Wythoff Nim (GDWN). The story begins with 2-pile Nim whose sets of options and $P$-positions are $\{\{0,t\}\mid t\in \N\}$ and \{(t,t)\mid t\in \M \} respectively. If we to 2-pile Nim adjoin the main-\emph{diagonal} $\{(t,t)\mid t\in \N\}$ as options, the new game is Wythoff Nim. It is well-known that the $P$-positions of this game lie on two 'beams' originating at the origin with slopes $\Phi= \frac{1+\sqrt{5}}{2}>1$ and $\frac{1}{\Phi} < 1$. Hence one may think of this as if, in the process of going from Nim to Wythoff Nim, the set of $P$-positions has \emph{split} and landed some distance off the main diagonal. This geometrical observation has motivated us to ask the following intuitive question. Does this splitting of the set of $P$-positions continue in some meaningful way if we, to the game of Wythoff Nim, adjoin some new \emph{generalized diagonal} move, that is a move of the form $\{pt, qt\}$, where $0 < p < q$ are fixed positive integers and $t > 0$? Does the answer perhaps depend on the specific values of $p$ and $q$? We state three conjectures of which the weakest form is: $\lim_{t\in \N}\frac{b_t}{a_t}$ exists, and equals $\Phi$, if and only if $(p, q)$ is a certain \emph{non-splitting pair}, and where $\{\{a_t, b_t\}\}$ represents the set of $P$-positions of the new game. Then we prove this conjecture for the special case $(p,q) = (1,2)$ (a \emph{splitting pair}). We prove the other direction whenever $q / p < \Phi$. In the Appendix, a variety of experimental data is included, aiming to point out some directions for future work on GDWN games.Comment: 38 pages, 34 figure

The ⋆\star-operator and Invariant Subtraction Games

We study 2-player impartial games, so called \emph{invariant subtraction games}, of the type, given a set of allowed moves the players take turn in moving one single piece on a large Chess board towards the position $\boldsymbol 0$. Here, invariance means that each allowed move is available inside the whole board. Then we define a new game, $\star$ of the old game, by taking the $P$-positions, except $\boldsymbol 0$, as moves in the new game. One such game is \W^\star= (Wythoff Nim)$^\star$, where the moves are defined by complementary Beatty sequences with irrational moduli. Here we give a polynomial time algorithm for infinitely many $P$-positions of \W^\star. A repeated application of $\star$ turns out to give especially nice properties for a certain subfamily of the invariant subtraction games, the \emph{permutation games}, which we introduce here. We also introduce the family of \emph{ornament games}, whose $P$-positions define complementary Beatty sequences with rational moduli---hence related to A. S. Fraenkel's `variant' Rat- and Mouse games---and give closed forms for the moves of such games. We also prove that ($k$-pile Nim)$^{\star\star}$ = $k$-pile Nim.Comment: 30 pages, 5 figure

The Role of Sovereign Wealth Funds in Global Managament of Excess Foreign Exchange Reserves

This paper finds evidence that for many countries Sovereign Wealth Funds are the alternative vehicle for management of excess foreign exchange reserves. These funds can be seen as a substitutes for monetary authorities as well as institutional innovations on global financial markets. Sovereign Wealth Funds offer to countries various economic and financial benefits. They facilitate saving intergenerational transfer of proceeds from nonrenewable resources and help reduce cyclical volatility driven by changes in commodity export prices. These state-run funds help to reduce the opportunity cost of reserves holdings due to greater portfolio diversification of reserve-assets and allow countries to accumulate large capital inflow without negative consequences such as exchange rate appreciations, price distortions, liquidity expansion, domestic asset bubbles, financial sector imbalances and inflations. Sovereign Wealth Funds can support domestic economy during the crises as a investors of last resort and stabilize international financial markets by supplying liquidity and reducing market volatility. Sovereign Wealth Funds are likely to continue growing and increase their relative importance in global financial markets.Dzięki inwestowaniu w szeroką gamę aktywów na rynkach międzynarodowych państwowe fundusze majątkowe zmniejszają lub wręcz eliminują koszty alternatywne związane z utrzymywaniem rezerw. Fundusze te ułatwiają międzypokoleniowy transfer środków pochodzących z eksploatacji zasobów nieodnawialnych jak również mogą być wykorzystywane do wspierania gospodarki podczas kryzysów kiedy to jako inwestorzy ostatniej instancji zapewniają płynność zarówno sektora finansowego jak i pozostałych gałęzi gospodarki. Państwowe fundusze majątkowe postrzegane są jako narzędzie wspierające stabilność makroekonomiczną gospodarki oraz forma zabezpieczenia przyszłego dobrobytu ekonomicznego kraju. Podmioty te wnoszą ponadto istotny wkład w funkcjonowanie gospodarki światowej. Jako długoterminowi, pasywni inwestorzy, którzy nie stosują w swoich strategiach inwestycyjnych dźwigni, państwowe fundusze majątkowe wywierać mogą stabilizujący wpływ na międzynarodowe rynki finansowe zwiększając ich płynność oraz obniżając wahania rynkowe. Wnioski wyciągnięte w artykule wskazują, że w najbliższym latach możliwy jest dalszy rozwój rynku państwowych funduszy majątkowych i wzrost ich znaczenia na międzynarodowych rynkach finansowych

Sovereign Wealth Funds - new players on global financial markets

The objective of this paper is analysis of Sovereign Wealth Funds, which are becoming increasingly important players in the international monetary and financial system. Those funds are attracting growing attention not only due to last investment activities in brand-name global firms, but also due to lack of transparency and information about themselves. The article consists of two part. In the first part of the paper based on the latest literature the author presents definitions of Sovereign Wealth Funds and main factors responsible for the rise and growth of those funds. The second part of the paper deals with investment characteristics made by the largest of them. The main conclusion of this paper is that empirical analysis do not prove the thesis that investment made by SWF's has a political background. The latest available data suggest that those funds avoid investing in sensitive sectors like defense, aerospace, high technology and transportation