Simplified three player Kuhn poker

We study a very small three player poker game (one-third street Kuhn poker), and a simplified version of the game that is interesting because it has three distinct equilibrium solutions. For one-third street Kuhn poker, we are able to find all of the equilibrium solutions analytically. For large enough pot size, $P$, there is a degree of freedom in the solution that allows one player to transfer profit between the other two players without changing their own profit. This has potentially interesting consequences in repeated play of the game. We also show that in a simplified version of the game with $P>5$, there is one equilibrium solution if $5 < P < P^* \equiv (5+\sqrt{73})/2$, and three distinct equilibrium solutions if $P > P^*$. This may be the simplest non-trivial multiplayer poker game with more than one distinct equilibrium solution and provides us with a test case for theories of dynamic strategy adjustment over multiple realisations of the game. We then study a third order system of ordinary differential equations that models the dynamics of three players who try to maximise their expectation by continuously varying their betting frequencies. We find that the dynamics of this system are oscillatory, with two distinct types of solution. We then study a difference equation model, based on repeated play of the game, in which each player continually updates their estimates of the other players' betting frequencies. We find that the dynamics are noisy, but basically oscillatory for short enough estimation periods and slow enough frequency adjustments, but that the dynamics can be very different for other parameter values.Comment: 41 pages, 2 Tables, 17 Figure

The Search for Extraterrestrial Intelligence (SETI)

A bibliography of reports concerning the Search for Extraterrestrial Intelligence is presented. Cosmic evolution, space communication, and technological advances are discussed along with search strategies and search systems

Frequency reassignment in cellular phone networks

In cellular communications networks, cells use beacon frequencies to ensure the smooth operation of the network, for example in handling call handovers from one cell to another. These frequencies are assigned according to a frequency plan, which is updated from time to time, in response to evolving network requirements. The migration from one frequency plan to a new one proceeds in stages, governed by the network's base station controllers. Existing methods result in periods of reduced network availability or performance during the reassgnment process. The problem posed to the Study Group was to develop a dynamic reassignment algorithm for implementing a new frequency plan so that there is little or no disruption of the network's performance during the transition. This problem was naturally formulated in terms of graph colouring and an effective algorithm was developed based on a straightforward approach of search and random colouring

Full street simplified three player Kuhn poker

We study a simplified version of full street, three player Kuhn poker, in which the weakest card, J, must be checked and/or folded by a player who holds it. The number of nontrivial betting frequencies that must be calculated is thereby reduced from 23 to 11, and all equilibrium solutions can be found analytically. In particular, there are three ranges of values of the pot size, P, for which there are three distinct, coexisting equilibrium solutions. We also study an ordinary differential equation model of repeated play of the game, which we expect to be at least qualitatively accurate when all players both adjust their betting frequencies sufficiently slowly and have sufficiently short memories. We find that none of the equilibrium solutions of the game is asymptotically stable as a solution of the ordinary differential equations. Depending on the pot size, the solution may be periodic, close to periodic or have long chaotic transients. In each case, the rates at which the players accumulate profit closely match those associated with one of the equilibrium solutions of the game

A dam-break driven by a moving source: a simple model for a powder snow avalanche

We study the two-dimensional, irrotational flow of an inviscid, incompressible fluid injected from a line source moving at constant speed along a horizontal boundary, into a second, immiscible, inviscid fluid of lower density. A semi-infinite, horizontal layer sustained by the moving source has previously been studied as a simple model for a powder snow avalanche, Caroll et al. (2012). We show that with fluids of unequal densities, in a frame of reference moving with the source, no steady solution exists, and formulate an initial/boundary value problem that allows us to study the evolution of the flow. After considering the limit of small density difference, we study the fully nonlinear initial/boundary value problem and find that the flow at the head of the layer is effectively a dam-break for the initial conditions that we have used. We study the dynamics of this in detail for small times using the method of matched asymptotic expansions. Finally, we solve the fully nonlinear free boundary problem numerically using an adaptive vortex blob method, after regularising the flow by modifying the initial interface to include a thin layer of the denser fluid that extends to infinity ahead of the source. We find that the disturbance of the interface in the linear theory develops into a dispersive shock in the fully nonlinear initial/boundary value problem and overturns. For sufficiently large Richardson number, overturning can also occur at the head of the layer

Slow travelling wave solutions of the nonlocal Fisher-KPP equation

© 2020 IOP Publishing Ltd & London Mathematical Society. We study travelling wave solutions, u = U(x - ct), of the nonlocal Fisher- KPP equation in one spatial dimension, dimension, (Display equation presented), with D = 1 and c = 1, where = = u is the spatial convolution of the population density, u(x, t), with a continuous, symmetric, strictly positive kernel, =(x), which is decreasing for x > 0 and has a finite derivative as x = 0+, normalized so that = = -= =(x)dx = 1. In addition, we restrict our attention to kernels for which the spatially-uniform steady state u = 1 is stable, so that travelling wave solutions have U = 1 as x - ct → - and U = 0 as x - ct→ for c > 0. We use the formal method of matched asymptotic expansions and numerical methods to solve the travelling wave equation for various kernels, =(x), when c = 1. The most interesting feature of the leading order solution behind the wavefront is a sequence of tall, narrow spikes with O(1) weight, separated by regions where U is exponentially small. The regularity of =(x) at x = 0 is a key factor in determining the number and spacing of the spikes, and the spatial extent of the region where spikes exist

Design of microfluidic networks

Microfluidics is a relatively new and fast growing research area in fluid mechanics. The devices in question are thin wafers containing etched or printed interconnecting channels through which fluids are pumped, which can mix and/or react at various nodes to produce an output product. Microfluidic devices have applications in many manufacturing and chemical detection processes. For example, they can be used to manufacture monodisperse droplets with very well defined properties for pharmaceutical applications; or form the basis for miniaturised ‘lab-on-a-chip’ sensor arrays for detecting biological substances or toxins

Simplified three player Kuhn poker

We study a very small three player poker game (one-third street Kuhn poker), and a simplified version of the game that is interesting because it has three distinct equilibrium solutions. For one-third street Kuhn poker, we are able to find all of the equilibrium solutions analytically. For large enough pot size, P, there is a degree of freedom in the solution that allows one player to transfer profit between the other two players without changing their own profit. This has potentially interesting consequences in repeated play of the game. We also show that in a simplified version of the game with P>5, there is one equilibrium solution if 5P∗. This may be the simplest non-trivial multiplayer poker game with more than one distinct equilibrium solution and provides us with a test case for theories of dynamic strategy adjustment over multiple realisations of the game. We then study a third order system of ordinary differential equations that models the dynamics of three players who try to maximise their expectation by continuously varying their betting frequencies. We find that the dynamics of this system are oscillatory, with two distinct types of solution. We then study a difference equation model, based on repeated play of the game, in which each player continually updates their estimates of the other players' betting frequencies. We find that the dynamics are noisy, but basically oscillatory for short enough estimation periods and slow enough frequency adjustments, but that the dynamics can be very different for other parameter values

Paper Session II-C - The Search for Extra Terrestrial Intelligence

The rationale for SETI is based on three arguments. First, there may be a plurality of inhabited worlds in the universe. Second, we now have sophisticated radiotechnology which may allow us to detect signals transmitted by extraterrestrial civilizations. Third, even if the probabilities of detection are low,, the extraordinary value of an unambiguous positive result justifies spending on SETI some tiny fraction of the gross national product of the planet Earth